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<br /> <br /><div class="WordSection1"> <h4><span lang="ES" style="mso-ansi-language: ES; mso-bidi-font-weight: bold;">GEOMETRÍA DE LAS FORMAS</span></h4><div class="MsoNormal"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Sistema de Coordenadas Cartesianas</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenadas Cartesianas</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Para ordenar un espacio determinado se deberá establecer donde se encuentra cada uno de los puntos de ese espacio. Por esto se crearon los ejes de coordenadas donde puntos, ejes y planos son elementos fijos y a los cuales referimos el resto del espacio.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es un sistema formado por dos rectas perpendiculares donde el origen es el punto de intersección entre ellas.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Las proyecciones determinan las distancias al origen y constituyen un par ordenado </span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(x ; y).</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenadas polares</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">En el espacio bidimensional tomamos un eje polar fijando en él un origen, un sentido positivo y una escala.<span style="mso-spacerun: yes;">  </span>Cualquier punto se lo puede determinar por su distancia al origen y el ángulo que se forma con el eje polar. </span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Vectores</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un vector es un segmento orientado, dentro del segmento consideramos a uno de los puntos como origen y al otro como extremo, es decir, hay un orden entre ellos. Un vector se define por tres elementos:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l4 level1 lfo1; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dirección: Dada por la recta que lo contiene.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l4 level1 lfo1; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Sentido: Fija el orden en que hayamos elegido los puntos extremos.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l4 level1 lfo1; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Modulo: longitud del segmento.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Los vectores representan fuerzas, velocidades o aceleraciones; se las llama magnitudes vectoriales.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Los vectores pueden ser:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l20 level1 lfo2; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Iguales: Cuando tiene mismo sentido, dirección y mismo módulo.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l20 level1 lfo2; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Opuestos: Tienen igual módulo y dirección, pero sentido contrario.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Componentes del vector:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Proyecciones del vector sobre los ejes. (x ; y ; z). Es equivalente representar un punto por sus coordenadas o por su vector posición.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Diferencia de vectores: AB = OB – OA = (xb ; yb) – (xa ; ya) = (xb – xa ; yb - ya)</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Sumatoria de vectores: A = (a<sub>x</sub> ; a<sub>y</sub>; a<sub>z</sub>) y B = (b<sub>x</sub> ; b<sub>y</sub> ; b<sub>z</sub>) </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 141.6pt; margin-right: 0cm; margin-top: 0cm; text-align: justify;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">      </span></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">A + B = (a<sub>x</sub> + b<sub>x</sub> ; a<sub>y</sub> + b<sub>y</sub> ; a<sub>z</sub> + b<sub>z</sub>)</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt; text-align: justify; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Multiplicar por un numero real: k. A = (ka<sub>x</sub> ; ka<sub>y</sub> ; ka<sub>z</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuaciones de una recta en un plano</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de la recta que pasa por dos puntos<span style="mso-spacerun: yes;">  </span>(a<sub>X</sub> ; a<sub>Y</sub>) y (b<sub>X</sub> ; b<sub>Y</sub>) la ecuación es:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;"> </span></span><b><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">1</span></b><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">         </span><u><span style="mso-spacerun: yes;"> </span>y – a<sub>Y</sub></u> = <u>b<sub>Y</sub> – a<sub>Y<span style="mso-spacerun: yes;">   </span></sub></u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">             </span>x – a<sub>X</sub><span style="mso-spacerun: yes;">    </span>b<sub>X</sub> – a<sub>X</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si en <b>1</b> hacemos multiplicación cruzada, se obtiene otra equivalente, si la igualamos a ¨t¨ que es un parámetro obtenemos:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><b><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">2 </span></b><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">    </span><u><span style="mso-spacerun: yes;"> </span>y – a<sub>Y</sub></u><span style="mso-spacerun: yes;">   </span>=<span style="mso-spacerun: yes;">   </span><u>x – a<sub>X </sub></u>= t</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">    </span>b<sub>Y</sub> – a<sub>Y</sub><span style="mso-spacerun: yes;">      </span>b<sub>X</sub> – a<sub>X<span style="mso-spacerun: yes;">  </span></sub></span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ecuación simétrica a<sub>X</sub> </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≠</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> b<sub>X</sub> ; a<sub>Y</sub></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≠</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> b<sub>Y</sub>; a<sub>Z</sub> </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≠</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> b<sub>Z</sub></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Y de la ecuación <b>2</b> se obtiene:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><b><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;">3</span></b><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;"><span style="mso-spacerun: yes;">    </span>x – a<sub>X</sub> = t (b<sub>X</sub> – a<sub>X</sub>)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;"><span style="mso-spacerun: yes;">      </span>y – a<sub>Y</sub> = t (b<sub>Y</sub> – a<sub>Y</sub>)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación paramétrica</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si se toman esas diferencias como parte de un vector:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;">(x – a<sub>X</sub>; y – a<sub>Y</sub>) = (t (b<sub>X</sub> – a<sub>X</sub>) ; t (b<sub>Y</sub> ; a<sub>Y</sub>))</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Que es lo mismo que:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;">(x ; y) – (a<sub>X</sub> ; a<sub>Y</sub>) = (t (b<sub>X</sub> ; b<sub>Y</sub>) – (a<sub>X</sub>; a<sub>Y</sub>))</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR;"><span style="mso-spacerun: yes;">                                           </span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">-----<span style="mso-spacerun: yes;">       </span>--------<span style="mso-spacerun: yes;">         </span>---------<span style="mso-spacerun: yes;">      </span>---------</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                              </span>r<span style="mso-spacerun: yes;">            </span>a<span style="mso-spacerun: yes;">                 </span>b<span style="mso-spacerun: yes;">                </span>a</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Entonces queda:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">_<span style="mso-spacerun: yes;">   </span>_<span style="mso-spacerun: yes;">       </span>_<span style="mso-spacerun: yes;">    </span>_<span style="mso-spacerun: yes;">     </span>_<span style="mso-spacerun: yes;">   </span>_<span style="mso-spacerun: yes;">        </span>_<span style="mso-spacerun: yes;">   </span>_</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">r – a = t (b - a) o r = a + t (b - a)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación vectorial</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Espacio tridimensional</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Distancia entre dos puntos A = (a<sub>X</sub>; a<sub>Y</sub> ; a<sub>Z</sub>) y B = (b<sub>X</sub> ; b<sub>Y</sub> ; b<sub>Z</sub>):</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">D (AB)= | </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">√ (b<sub>X</sub> – a<sub>X</sub>)<sup>2</sup> + (b<sub>Y</sub>– a<sub>Y</sub>)<sup>2</sup> + (b<sub>Z</sub> – a<sub>Z</sub>)<sup>2</sup> |</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Números directores de la recta AB a cualquier terna (m,n,p) que cumple con:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">m<span style="mso-spacerun: yes;">   </span></span></u><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">   </span>=<span style="mso-spacerun: yes;">  </span><u><span style="mso-spacerun: yes;">    </span>n<span style="mso-spacerun: yes;">   </span></u><span style="mso-spacerun: yes;">   </span>=<span style="mso-spacerun: yes;">  </span><u><span style="mso-spacerun: yes;">     </span>p<span style="mso-spacerun: yes;">       </span></u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                    </span><span style="mso-spacerun: yes;">                              </span>b<sub>X</sub> – a<sub>X</sub><span style="mso-spacerun: yes;">     </span>b<sub>Y</sub> – a<sub>Y</sub><span style="mso-spacerun: yes;">      </span>b<sub>Z</sub> – a<sub>Z</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Vectores tridimensionales</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Teniendo un vector a = (a<sub>X</sub>; a<sub>Y</sub> ; a<sub>Z</sub>) definimos:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l12 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Igualdad entre vectores (a<sub>X</sub> ; a<sub>Y</sub> ; a<sub>Z</sub>) = (b<sub>X</sub>; b<sub>Y</sub> ; b<sub>Z</sub>)</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                             </span></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">a<sub>X</sub>= b<sub>X</sub> ; a<sub>Y</sub> = b<sub>Y</sub> ; a<sub>Z</sub> = b<sub>Z</sub></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l12 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Suma de vectores a + b = (a<sub>X</sub> + b<sub>X </sub>; a<sub>Y</sub> + b<sub>Y</sub>; a<sub>Z </sub>+ b<sub>Z</sub>)</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l12 level1 lfo4; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Producto de un vector por un escalar k (a<sub>X</sub> ; a<sub>Y</sub> ; a<sub>Z</sub>) = (ka<sub>X</sub> ; ka<sub>Y</sub> ; ka<sub>Z</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Versor: es un vector de modulo uno (i ; j ; k)<span style="mso-spacerun: yes;">   </span>i = (1;0;0)<span style="mso-spacerun: yes;">    </span>j = (0;1;0)<span style="mso-spacerun: yes;">    </span>k = (0;0;1)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ej.: Vector r = (x;y;z) = r = xi + yj + zk (Ecuación canónica)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Modulo del vector |a| = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">√a<sub>X</sub><sup>2 </sup>+ a<sub>Y</sub><sup>2</sup>+ a<sub>Z</sub><sup>2 </sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">El modulo de valor positivo determina la longitud del segmento.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ecuación de la recta en el espacio</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">En el espacio tridimensional la ecuación vectorial de una recta r que pasa por un punto a, representada por su vector posición a = (a<sub>x</sub> ; a<sub>y</sub>; a<sub>z</sub>) y tiene dirección c es:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">r = a + t.c<span style="mso-spacerun: yes;">  </span>(Ecuación vectorial)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">r = (x ; y ; z)<span style="mso-spacerun: yes;">   </span>a = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(a<sub>X</sub> ; a<sub>Y</sub> ; a<sub>Z</sub>)<span style="mso-spacerun: yes;">    </span>c = (b<sub>X</sub> – a<sub>X</sub> ; b<sub>Y</sub>– a<sub>Y</sub> ; b<sub>Z</sub> – a<sub>Z</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Por lo tanto obtenemos la ecuación parametrica:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR; mso-fareast-language: ES;">x = a<sub>X</sub> + t . c<sub>X</sub></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="FR" style="font-family: "Arial","sans-serif"; mso-ansi-language: FR; mso-fareast-language: ES;">y = a<sub>Y</sub> + t . c<sub>Y</sub></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">z = a<sub>Z</sub> + t . c<sub>Z</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Por eliminación del parámetro t se obtiene la ecuación simétrica.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x – a<sub>X</sub></span></u><sub><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">               </span></span></sub><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">y – a<sub>Y</sub></span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">            </span><u>z – a<sub>Z</sub></u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                  </span>c<sub>X</sub><span style="mso-spacerun: yes;">                 </span>c<sub>Y<span style="mso-spacerun: yes;">                          </span></sub>c<sub>Z</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Si llegara a faltar c<sub>X</sub>; c<sub>Y<span style="mso-spacerun: yes;">  </span>o </sub>c<sub>Z </sub>eso indica<sub> </sub>que el plano esta contenido en los ejes restantes.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Dado un vector a = (a<sub>X</sub> ; a<sub>Y</sub> ; a<sub>Z</sub>) sus cosenos directores resultan:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">cos α = <u>a<sub>X</sub></u><span style="mso-spacerun: yes;">    </span>cos β = <u>a<sub>Y</sub></u><sub></sub><span style="mso-spacerun: yes;">    </span>cos γ = <u>a<sub>Z</sub></u></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                   </span><span style="mso-spacerun: yes;">     </span>|a|<span style="mso-spacerun: yes;">                 </span>|a|<span style="mso-spacerun: yes;">                 </span>|a|</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">cos α<sup>2</sup>+ cos β<sup>2</sup> + cos γ<sup>2</sup> = 1</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Paralelismo y perpendicularidad entre rectas:</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Condición de paralelismo</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">: v = k . w<span style="mso-spacerun: yes;">  </span>donde k es un numero real cualquiera, es decir que los vectores asociados tiene componentes proporcionales. </span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Condición de Perpendicularidad:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> exige que los vectores asociados tengan productor escalar 0.<span style="mso-spacerun: yes;">  </span></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">v . w = 0</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Alabeadas:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> Dos rectas son alabeadas cuando no son ni perpendiculares ni paralelas, no tiene ningún punto en común.<u> </u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Producto Escalar</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Sean los vectores a = (a<sub>X</sub> ; a<sub>Y</sub>; a<sub>Z</sub>) y b = (b<sub>X</sub> ; b<sub>Y</sub> ; b<sub>Z</sub>) se define el productor escalar a.b como a . b = |a| . |b| . cos θ donde θ es el ángulo formado por los vectores.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">El resultado del producto escalar es un numero que se interpreta como el producto de la longitud de uno de los vectores por la proyección del otro sobre el. </span></div><div class="MsoBodyText3" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="mso-ansi-language: ES; mso-fareast-language: ES;">El producto escalar cumple las siguientes propiedades:</span></div><ol start="1" style="margin-top: 0cm;" type="1"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l23 level1 lfo5; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Propiedad      conmutativa:<span style="mso-spacerun: yes;">    </span>a . b = b . a</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l23 level1 lfo5; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Propiedad      distributiva:<span style="mso-spacerun: yes;">  </span>a . </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(b + c) = a . b + a . c</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l23 level1 lfo5; tab-stops: list 36.0pt; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">k ( a . b) = (ka.b)</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l23 level1 lfo5; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">a . a = |a|<sup>2</sup></span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l23 level1 lfo5; tab-stops: list 36.0pt; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">a. b = |a| . </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">|b| . cos α</span></li></ol><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ejemplo:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">u = 2i – j – k<span style="mso-spacerun: yes;">              </span>v = 3i + 2j + 8k<span style="mso-spacerun: yes;">    </span><span style="mso-spacerun: yes;">        </span>w = -4y +2j – 2k</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">(u + v) . w</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoHeading7"><span lang="ES" style="mso-bidi-font-size: 12.0pt;">Ecuación del plano</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Sea un plano </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">π que pase por el punto P y sea normal a OP = p. Sea Q un punto genérico de π y OQ = r, siendo r = (x ; y ; z). Como Q pertenece a π, entonces PQ = (r - p) pertenece a π, entonces p<u><span style="mso-spacerun: yes;">  </span>|<span style="mso-spacerun: yes;">  </span></u>(r - p) condición que nos permite escribir<span style="mso-spacerun: yes;">  </span><b>-1-</b>(r - p) . p = 0</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><img height="194" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image002.jpg" width="315" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si a 1 se le hace distributiva se llega a la ecuación vectorial del plano:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">r . u<sub>P</sub>= |p|</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">donde u<sub>P</sub> = <u>p</u>= cos </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">α</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> i + cos β </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">j + cos </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">γ</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> k<span style="mso-spacerun: yes;">            </span>es el versor de la dirección normal p al </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES-AR" style="font-family: "Arial","sans-serif";"><span style="mso-spacerun: yes;">      </span><span style="mso-spacerun: yes;">            </span>|p|<span style="mso-spacerun: yes;">                                                       </span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">plano π y α</span><span lang="ES-AR" style="font-family: "Arial","sans-serif";"> β γ son sus ángulos directores.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES-AR" style="font-family: "Arial","sans-serif";">Entonces<span style="mso-spacerun: yes;">  </span>como </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">r . u<sub>P</sub></span><span lang="ES-AR" style="font-family: "Arial","sans-serif";"> queda expresada por </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">r . u<sub>P</sub></span><span lang="ES-AR" style="font-family: "Arial","sans-serif";"><span style="mso-spacerun: yes;">  </span>= x cos </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">α</span><span lang="ES-AR" style="font-family: "Arial","sans-serif";"> + y cos β</span><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;"> + z cos </span><span lang="ES-AR" style="font-family: "Arial","sans-serif";">γ</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> se llega a la ecuación general del plano</span><span lang="ES-AR" style="font-family: "Arial","sans-serif";"></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">-2-<span style="mso-spacerun: yes;">    </span>A (x<span style="mso-spacerun: yes;">  </span>- x<sub>0</sub>) + B (y – y<sub>0</sub>) + C (z – z<sub>0</sub>) (Ecuación cartesiana del plano)</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si a <b>2 </b>lo resolvemos llagamos a:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ax + By + Cz + D = 0 (Ecuación general del plano)</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ejemplo:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-bidi-font-style: italic; mso-fareast-language: ES;"> Dado el plano </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">π de ecuación:<span style="mso-spacerun: yes;">  </span>x + 2y + 2z -6 = 0. Hacer un grafico aproximado que muestre su posición en el espacio.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x + 2y + 2z -6 = 0</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x + 2y + 2z = 6</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> + <u>2y</u> + <u>2z</u> = <u>6</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">6<span style="mso-spacerun: yes;">     </span>6<span style="mso-spacerun: yes;">     </span>6<span style="mso-spacerun: yes;">      </span>6</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> + <u>y</u> + <u>z</u> = 1</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                                 </span>6<span style="mso-spacerun: yes;">    </span>3<span style="mso-spacerun: yes;">    </span>3<span style="mso-spacerun: yes;">      </span></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Intersección de planos</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Sea el plano </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ax + By + Cz + D = 0. Sus intersecciones con cada uno de los planos coordenados son las llamadas trazas del plano. Sus ecuación esta dadas por la solución de los sistemas formados por las ecuación del plano y las ecuación de cada plano coordenado.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l19 level1 lfo6; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: ES;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Traza sobre plano (x ; y) = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ax + By + D = 0 ; z = 0</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l19 level1 lfo6; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: ES;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Traza sobre plano (x ; z) = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ax + Cz + D = 0 ; y = 0</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l19 level1 lfo6; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: ES;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Traza sobre plano (y ; z) = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">By + Cz + D = 0 ; x = 0</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La intersección de dos planos, si existe, es una recta cuyas ecuaciones se obtienen eliminando sucesivamente x e y, para obtener en cada caso las funciones lineales que son las ecuaciones de los planos proyectantes de la intersección.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(En el primer plano se despeja x y se introduce su resultado en el otro plano. Luego de obtenido y en función de z se introduce en x para obtener x en función de z. Quedando así la recta x = bla y = bla z = z)</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Paralelismo y perpendicularidad entre planos:</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Dos planos son paralelos si sus vectores normales son paralelos.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">π1 // π2<span style="mso-spacerun: yes;">      </span>si: <u>A<sub>1</sub></u> = <u>B<sub>1</sub></u>= <u>C<sub>1</sub></u></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                                       </span><span style="mso-spacerun: yes;">       </span>A<sub>2</sub><span style="mso-spacerun: yes;">    </span>B<sub>2</sub><span style="mso-spacerun: yes;">     </span>C<sub>2</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Dos planos son perpendiculares si el producto escalar de sus vectores normales es igual a cero.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">π1<u><span style="mso-spacerun: yes;">  </span>|<span style="mso-spacerun: yes;">  </span></u>π2<span style="mso-spacerun: yes;">      </span>si: A<sub>1</sub> . A<sub>2</sub> + B<sub>1.</sub></span><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">B<sub>2 +</sub> C<sub>1 .</sub> C<sub>2 </sub>= 0</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES-AR" style="font-family: "Arial","sans-serif";">Ax + By + Cz + D = 0</span><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;"></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Donde A, B, C son las coordenadas del vector normal.</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Posiciones relativas de rectas y planos</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Paralelismo y perpendicularidad entre rectas y planos</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Condición de paralelismo:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> Se utiliza la recta y la normal al plano (recta). Se usa así la condición v . w = 0 (producto escalar).</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Condición de perpendicularidad: </span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">La recta será perpendicular al plano si es paralela a la normal (plano).<span style="mso-spacerun: yes;">  </span>Se utiliza v = k . w.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Trabajo</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Para fuerzas constantes y que tengan la misma dirección. Se utilizan vectores si se dan. Sino como esta.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">T = f . d</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">T = f . d . cos α</span></div><div class="MsoNormal"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div></div><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span> <div class="WordSection2"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">CURVAS CONICAS</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Las curvas cónicas pueden definirse como lugar geométrico de un conjunto de puntos tales que la distancia de cada punto del conjunto a un cierto punto fijo (foco) esta en relación constante con su distancia a una recta fija (directriz).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La relación de las distancias se denomina excentricidad.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Todas las curvas cónicas nacen del doble cono generado por una recta (generatriz) alrededor de un eje, describiendo una circunferencia (directriz) y manteniéndose siempre pasante por un punto del eje (vértice del cono).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Utilizando planos de corte se obtiene distintas curvas cónicas: elipse, parábola, hipérbola, circunferencia (caso especial de la elipse).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ejemplos en el diseño industrial: elipse (bandeja; chapas de casas) ; circunferencias (ruedas de bicicletas) ; parábola (reja arcada) ; hipérbola…</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Excentricidad:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La excentricidad de una cónica es la relación entre las distancias de un punto P al foco y de P a la directriz. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Parábola: e = 1</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Elipse: 1 > e > 0 (cuanto mas próxima esta la e a 0, mas redondeada es la elipse)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Circunferencia: e = 0</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Hipérbola: e > 1 (cuanto menor sea la e, mas cerradas serán las dos ramas de la hipérbola)</span></div></div><b style="mso-bidi-font-weight: normal;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u></b> <div class="WordSection3"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b style="mso-bidi-font-weight: normal;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Elipse</span></u></b><b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="213" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image004.jpg" width="242" /><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es el conjunto de puntos del plano cuya suma de distancias a los focos es constante. Son curvas cerradas, el plano corta a todas las generatrices.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">C es la distancia entre el foco y el centro. Los dos focos están ubicados en el diámetro mayor. | a | son las coordenadas en x del vértice y | b | las coordenadas en y.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Esta curva es simétrica respecto de ambos ejes. Si se mantiene el valor de a fijo y se varia la distancia focal c, las elipses cambian de forma. A medida que c crece las elipses se van achatando. Se llama excentricidad al cociente entre el valor c y el de a.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                            </span><span style="mso-spacerun: yes;"> </span>e = <u>c</u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                   </span>a<span style="mso-spacerun: yes;">        </span>1 > e > 0</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Deducción de la ecuación:</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Siendo los F<sub>1</sub> = (-c;0) y F<sub>2</sub> = (c;0) y siendo 2a la suma de las distancias PF<sub>1</sub> Y PF<sub>2</sub>, las coordenadas del punto P = (x;y) de la elipse satisfacen la ecuación: </span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">√ (x + c)<sup>2</sup> + y<sup>2</sup> + √ (x - c)<sup>2</sup>+ y<sup>2</sup> = 2a</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Desarrollando esta expresión resulta la ecuación de la elipse:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">x<sup>2</sup></span></u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> + <u>y<sup>2</sup></u><sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">= 1<span style="mso-spacerun: yes;">            </span>donde b<sup>2</sup> = a<sup>2</sup> – c<sup>2</sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de la elipse: </span></u></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(x - h)<sup>2</sup></span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> + <u>(y - k)<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">   </span>centro (h ; k)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="181" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image006.jpg" width="294" /><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">  </span><span style="mso-spacerun: yes;">                                              </span>a<sup>2</sup><span style="mso-spacerun: yes;">            </span>b<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">En ambos casos la suma de las distancias de cualquier punto P de la elipse a cada uno de los focos es igual al diámetro mayor. Pf<sub>1</sub> + Pf<sub>2</sub> = 2 . (diámetro mayor,<span style="mso-spacerun: yes;">  </span>a o b según corresponda).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">El centro de la elipse son las coordenadas (h ; k) siendo h en x y k en y.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Existen dos tipos de elipses:</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Elipse horizontal</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Una de sus características es que a > b.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de los focos:<span style="mso-spacerun: yes;">  </span>c<sup>2</sup> = a<sup>2</sup> – b<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenadas de los focos:<span style="mso-spacerun: yes;">  </span>f<sub>1</sub> = (h – c ; k)<span style="mso-spacerun: yes;">       </span>f<sub>2</sub> = (h + c ; k)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Elipse vertical</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Una de sus características es que a < b</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de los focos: c<sup>2</sup>= b<sup>2</sup> – a<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="132" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image008.jpg" width="135" /><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenada de los focos: f<sub>1</sub> = (h ; k - c)<span style="mso-spacerun: yes;">       </span>f<sub>2</sub> = (h; k + c)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Circunferencia</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es un caso especial de la elipse donde a = b y su excentricidad es 0.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación: (x - h)<sup>2</sup> – (y - k)<sup>2</sup> = r<sup>2</sup></span></div></div><u><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u> <div class="WordSection4"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b style="mso-bidi-font-weight: normal;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Parábola</span></u></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="220" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image010.jpg" width="198" /><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es el conjunto de puntos del plano que equidistan del foco y de una recta (directriz). La parábola es una curva abierta de una sola rama, el plano es paralelo a una generatriz.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La parábola tendrá sus ramas en el eje que no esta al cuadrado. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Las coordenadas de vértice son : (h<sub>; </sub>k).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">P es la distancia del foco a la directriz.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="162" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image012.jpg" width="208" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Deducción de la ecuación:</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Siendo el F = (P/2;0) y la recta directriz x = -P/2, entonces DR = PF; por lo tanto, las coordenadas del punto P = (x;y) de la parábola satisfacen la ecuación:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">x + P/2 = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">√ (P/2 - x)<sup>2</sup> + y<sup>2</sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Desarrollando esta expresión resulta la ecuación de la parábola:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">y<sup>2 </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">= 2p x </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Existen dos tipos:</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si la parábola se desarrolla en el eje x la ecuación es:</span></i></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(y – k)<sup>2</sup>= 2p (x – h)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuaciones de los focos:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">y<sub>F</sub>= k</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">x<sub>F</sub>= h + <u>p</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">              </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de la directriz:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">x = h – <u>p</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">             </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si p es + la parábola es C.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si p es – la parábola es D.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si la parábola se desarrolla en el eje y la ecuación es:</span></i></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(x – h)<sup>2</sup>= 2p (y – k)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuaciones de los focos:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">x<sub>F</sub>= h</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">y<sub>F</sub>= k + <u>p</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">              </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de la directriz:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">y = k – <u>p</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">           </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si p es + la parábola es U</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si p es – la parábola es </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∩</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div></div><u><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u> <div class="WordSection5"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b style="mso-bidi-font-weight: normal;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Hipérbola</span></u></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="218" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image014.jpg" width="201" /><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es el conjunto de puntos del plano cuya diferencia de distancia a los focos es constante. Es una curva abierta de dos ramas, el plano es paralelo a dos generatrices.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La curva es simétrica respecto de ambos ejes y no toca al eje que esta restando.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">El eje que corta la hipérbola es el eje real.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">El centro de la hipérbola son las coordenadas (h ; k) siendo h en x y k en y.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">C es la distancia entre el foco y el centro. A es la coordenada en x del vértice y b la coordenada en y.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Asíntotas:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">y = - <u>b</u>. x<span style="mso-spacerun: yes;">                </span>y =<span style="mso-spacerun: yes;">  </span><u>b</u> . x</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                 </span>a<span style="mso-spacerun: yes;">                           </span>a</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">En ambos casos se verifica que:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">c<sup>2</sup>= a<sup>2</sup> + b<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="150" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image016.jpg" width="241" /><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Deducción de la ecuación:</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Siendo los F<sub>1</sub> = (-c;0) y F<sub>2</sub> = (c;0), debe ser PF<sub>2 </sub>- PF<sub>1</sub> = 2a. Dado el punto P perteneciente a la hipérbola, sus coordenadas satisfacen la ecuación: </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">√ (x + c)<sup>2</sup> + y<sup>2</sup>- √ (x - c)<sup>2</sup> + y<sup>2</sup> = 2a</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Desarrollando esta expresión resulta la ecuación de la hipérbola:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">x<sup>2</sup></span></u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> - <u>y<sup>2</sup></u><sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">= 1<span style="mso-spacerun: yes;">            </span>donde b<sup>2</sup> = c<sup>2</sup> – a<sup>2</sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                  </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Puede ser de dos tipos:</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Hipérbola que corta el eje x</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación general:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(x - h)<sup>2</sup></span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> - <u>(y - k)<sup>2</sup></u> = 1</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">            </span>b<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de los vértices:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">v<sub>1</sub>= (h – a ; k)<span style="mso-spacerun: yes;">       </span>v<sub>2</sub> = (h + a ; k)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenadas de los focos:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">f<sub>1</sub>= (h – c ; k)<span style="mso-spacerun: yes;">       </span>f<sub>2</sub> = (h + c ; k)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Excentricidad: e > 1</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">e = <u>c</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">      </span>a </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Hipérbola que corta el eje y</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación general:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(y - k)<sup>2</sup></span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> - <u>(x - h)<sup>2</sup></u> = 1</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                            </span>b<sup>2</sup><span style="mso-spacerun: yes;">           </span>a<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ecuación de los vértices:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">v<sub>1</sub>= (h ; k - b)<span style="mso-spacerun: yes;">      </span><span style="mso-spacerun: yes;"> </span>v<sub>2</sub> = (h ; k + b)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Coordenadas de los focos:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">f<sub>1</sub>= (h; k - c)<span style="mso-spacerun: yes;">       </span>f<sub>2</sub> = (h ; k + c)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Excentricidad: e > 1</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">e = <u>c</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">      </span>b </span></div><h1><span lang="ES-AR" style="font-size: 12.0pt;">CUADRICAS</span></h1><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Definición: Son superficies en el espacio, generadas por curvas cónicas. Pueden ser regladas o de revolución.</span></div><h2><u><span lang="ES-AR" style="font-size: 12.0pt; font-style: normal; font-weight: normal;">Superficies cilíndricas y de revolución:</span></u></h2><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Se llama superficie cilíndrica, a la superficie formada por el conjunto de todas las rectas que cortan una curva plana y son paralelas a una recta fija que no esta en el plano de la curva. La curva se llama directriz y las rectas generatrices.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Se llama superficie reglada a la superficie que cumple con la condición de que por cada una de sus puntos pasa al menos una recta, llamada generatriz rectilínea, que tiene en común con la superficie un segmento que contiene dicho punto. Ejemplo: superficies cilíndricas y cónicas.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Se llama superficie de revolución a la superficie que se obtiene rotando una curva plana en torno a un eje. Se dice que la curva genera la superficie.</span></div><div class="MsoNormal"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><img height="236" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image018.jpg" width="493" /></span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Teniendo una curva generatriz (C) definida sobre el plano (y , z) y siendo z el eje de revolución, un punto cualquiera P<sub>O</sub> describirá una circunferencia de centro M y de radio MP<sub>O</sub> = MP (siendo P un punto de la superficie de revolución).</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Teniendo la ecuación de la curva F(y,z) = 0 entonces tenemos que </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">y = √ x<sup>2</sup> + y<sup>2</sup></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La variable de revolución queda igual y la otra variable presente la reemplazo por la ecuación, entonces se incluye la variable no presente hasta ahora.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Ejemplo:</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup> + 4z<sup>2</sup>= 16<span style="mso-spacerun: yes;">  </span>gira en el eje x</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Entonces:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">z = √ z<sup>2</sup> + y<sup>2</sup></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Reemplazando esta ecuación en la original tengo:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup> + 4(√ z<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = 16 </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup> + 4y<sup>2</sup> + 4z<sup>2</sup> = 16 </span></div><div class="MsoNormal"><br /></div></div><b style="mso-bidi-font-weight: normal;"><u><span lang="ES-AR" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES-AR; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u></b> <div class="WordSection6"> <h3><u><span lang="ES-AR" style="font-size: 12.0pt; mso-bidi-font-weight: normal;">Superficie esférica</span></u></h3><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La superficie esférica es generada por la rotación de una circunferencia alrededor de uno de sus diámetros. Como lugar geométrico, la esfera es el conjunto de puntos que equidistan de su centro (C= (x;y;z)).</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Para cualquiera de estos puntos se deberá satisfacer la ecuación:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">(x – x<sub>O</sub>)<sup>2</sup> + (y – y<sub>O</sub>)<sup>2 </sup>+ (z – z<sub>O</sub>)<sup>2</sup> = r<sup>2</sup></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Si se desarrolla esta ecuación: </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2 </sup>+y<sup>2 </sup>+z<sup>2</sup>+ Dx + Ey + Fz + G = 0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Cuando el centro coincide con el origen de coordenadas: C = (0,0,0) entonces x<sub>O</sub>= y<sub>O</sub>= z<sub>O</sub> = 0 y en consecuencia la ecuación es:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2 </sup>+y<sup>2 </sup>+z<sup>2</sup> = r<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><img align="left" height="286" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image020.jpg" width="292" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una circunferencia: (x – x<sub>O</sub>)<sup>2</sup> + (y – y<sub>O</sub>)<sup>2 </sup>= r<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una circunferencia: (x – x<sub>O</sub>)<sup>2</sup> + (z – z<sub>O</sub>)<sup>2</sup>= r<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una circunferencia: (y – y<sub>O</sub>)<sup>2 </sup>+ (z – z<sub>O</sub>)<sup>2</sup>= r<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(<u>+</u>x;0;0)<span style="mso-spacerun: yes;">       </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(0;<u>+</u>y;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0<span style="mso-spacerun: yes;">      </span>(0;0;<u>+</u>z)</span></div></div><u><span lang="ES-AR" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES-AR; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u> <div class="WordSection7"> <h2><u><span lang="ES-AR" style="font-size: 12.0pt; font-style: normal; font-weight: normal;">Ecuaciones de las cuádricas</span></u></h2><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Se trata de polinomios de segundo grado y su diferencia con las cónicas es que estas son funciones de dos variables (x;y) mientras que las cuádricas son funciones de tres variables (x;y;z).</span></div><h4><span lang="ES-AR"> </span></h4><h4><span lang="ES-AR">Elipsoides</span></h4><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Siendo a, b, c positivos; la longitud de los semiejes<span style="mso-spacerun: yes;">  </span>del elipsoide en la dirección de los ejes x,y,z respectivamente se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                          </span><span style="mso-spacerun: yes;">                 </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Los elipsoides pueden ser considerados como generados por una elipse variable que se traslada paralela al plano (x,y) [Reglada]</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><img align="left" height="240" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image022.jpg" width="243" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una elipse: <u>x<sup>2</sup></u>+ <u>y<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">              </span><span style="mso-spacerun: yes;">         </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una elipse: <u><span style="mso-spacerun: yes;"> </span>x<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                        </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una elipse: <u>y<sup>2</sup></u>+ <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                        </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(x;0;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(0;y;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0<span style="mso-spacerun: yes;">      </span>(0;0;z)</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Para que exista intersección la distancia entre elipse y elipse debe ser mayor o igual que cero.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Como caso particular los elipsoides pueden ser de revolución. Si lo fueran alrededor del eje y, los semiejes a y c serian iguales, ya que la traza con el plano (xz) seria una circunferencia.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La ecuación:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                           </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">           </span>a = c</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Si tuviera los tres semiejes iguales (a=b=c), la ecuación seria:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                           </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">    </span><span style="mso-spacerun: yes;">       </span>a = c = b</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Osea:<span style="mso-spacerun: yes;">     </span>x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup>= a<sup>2</sup><span style="mso-spacerun: yes;">    </span>entonces seria una esfera de r = a</span></div><div class="MsoNormal"><br /></div></div><b style="mso-bidi-font-weight: normal;"><span lang="ES-AR" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES-AR; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection8"> <h4><span lang="ES-AR">Hiperboloides</span></h4><div class="MsoNormal"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Hiperboloide de una hoja:</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Siendo a, b, c positivos se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> - <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">   </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><img align="left" height="264" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image024.jpg" width="209" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es un elipse (las dos variables son positivas): <u>x<sup>2</sup></u> + <u>y<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                                          </span>a<sup>2</sup><span style="mso-spacerun: yes;">   </span>b<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una hipérbola (variable de distinto signo): <u>x<sup>2</sup></u> - <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                                        </span>a<sup>2</sup><span style="mso-spacerun: yes;">   </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x = 0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una hipérbolas (variables de distinto signos): <u>y<sup>2</sup></u> - <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                                             </span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">b<sup>2</sup><span style="mso-spacerun: yes;">   </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(x;0;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(0;y;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: No hay intersección.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">El hiperboloide de una hoja es una superficie reglada pero cuando a = b, el hiperboloide será de revolución alrededor del eje z y su ecuación será:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> - <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                          </span><span style="mso-spacerun: yes;">                  </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">   </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span>a = b</span></div><div class="MsoNormal"><br /></div></div><u><span style="font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></u> <div class="WordSection9"> <div class="MsoNormal"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Hiperboloide de dos hojas:</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> Siendo a, b, c positivos se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">- x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> - <u>y<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">       </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">No existe esta traza.<u> - x<sup>2</sup></u> - <u>y<sup>2</sup></u> = 1 (algo negativo menos algo no puede dar +)</span></div><div class="MsoNormal"><img align="left" height="257" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image026.jpg" width="213" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                   </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">        </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una hipérbola: - <u>x<sup>2</sup></u>+ <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                           </span><span style="mso-spacerun: yes;">    </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una hipérbolas: - <u>y<sup>2</sup></u>+ <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                 </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: No existe</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: No existe</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0<span style="mso-spacerun: yes;">      </span>(0;0;z)</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">En el caso en que a = b el hiperboloide es de revolución alrededor del eje z</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">- x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> - <u>y<sup>2</sup></u> + <u>z<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                    </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span>a = b</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son elipses: <u>- x<sup>2</sup></u>- <u>y<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                      </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">           </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son hipérbolas: - <u>x<sup>2</sup></u>+ <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son hipérbolas: - <u>y<sup>2</sup></u>+ <u>z<sup>2</sup></u> = 1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                            </span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: No existe</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: No existe</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0<span style="mso-spacerun: yes;">      </span>(0;0;z)</span></div><div class="MsoNormal"><br /></div></div><b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection10"> <h4><span lang="ES" style="mso-ansi-language: ES;">Paraboloides</span></h4><div class="MsoNormal"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Paraboloide elíptico:</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Siendo a, b, c positivos se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> = <u>c.z</u><span style="mso-spacerun: yes;">   </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                             </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">          </span>siendo c>0</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><img align="left" height="276" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image028.jpg" width="236" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es un punto (origen): <u>x<sup>2</sup></u>+ <u>y<sup>2</sup></u><span style="mso-spacerun: yes;">  </span>= 0<span style="mso-spacerun: yes;">            </span></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                  </span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">a<sup>2</sup><span style="mso-spacerun: yes;">     </span>b<sup>2</sup><span style="mso-spacerun: yes;">  </span></span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una parábola:<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> = c.z </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                             </span>a<sup>2</sup><span style="mso-spacerun: yes;">             </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una parábola:<span style="mso-spacerun: yes;">  </span><u>y<sup>2</sup></u> = c.z </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                             </span>b<sup>2</sup><span style="mso-spacerun: yes;">            </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(x;0;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0<span style="mso-spacerun: yes;">      </span>(0;y;0)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0<span style="mso-spacerun: yes;">      </span>(0;0;z)</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">En el caso en que a = b, el paraboloide será de revolución alrededor del eje z:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> = <u>z</u><span style="mso-spacerun: yes;">   </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                              </span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<span style="mso-spacerun: yes;">       </span>a = b</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Paraboloide hiperbólico:</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Siendo a, b, c positivos se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">- x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> = c.z<span style="mso-spacerun: yes;">   </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                              </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">             </span>siendo c>0</span></div><div class="MsoNormal"><img align="left" height="293" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image030.jpg" width="253" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son dos rectas (asintotas): -<u>x<sup>2</sup></u> + <u>y<sup>2</sup></u> = 0<span style="mso-spacerun: yes;">   </span>(se distribuye)</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                             </span><span style="mso-spacerun: yes;">                </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una parábola de eje z:<span style="mso-spacerun: yes;">  </span>-<u>x<sup>2</sup></u> = c.z </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">             </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es una parábola de eje z: <span style="mso-spacerun: yes;"> </span><u>y<sup>2</sup></u> = c.z </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                           </span>b<sup>2</sup><span style="mso-spacerun: yes;">            </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">El paraboloide hiperbólico no puede ser de revolución, ya que ninguna de sus secciones planas es elíptica, pero es una superficie reglada, pues por cada uno de sus puntos para un generatriz que es asíntota del sistema de hipérbolas.</span></div><div class="MsoNormal" style="tab-stops: list 36.0pt;"><br /></div><div class="MsoNormal" style="tab-stops: list 36.0pt;"><b><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Cono cuádrico</span></b><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">: Siendo a, b, c positivos se define:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x<sup>2</sup></span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"> + <u>y<sup>2</sup></u> = <u>z<sup>2</sup></u><span style="mso-spacerun: yes;">   </span>(ecuación canónica)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                              </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">       </span></span></div><div class="MsoNormal"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><img height="185" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image032.jpg" width="152" /></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Es un punto (centro 0,0,0)<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> + <u>y<sup>2</sup></u>= 0 </span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                            </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">      </span><span style="mso-spacerun: yes;">     </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (xz) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son dos rectas:<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> = <u>z<sup>2</sup></u><span style="mso-spacerun: yes;">  </span>(de distribuye y entonces (<u>x</u> + <u>z</u>). (<u>x</u> - <u>z</u>)<span style="mso-spacerun: yes;">  </span>= 0) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                           </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">                                            </span>a<span style="mso-spacerun: yes;">    </span>c<span style="mso-spacerun: yes;">     </span>a<span style="mso-spacerun: yes;">   </span>c</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                         </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Son dos rectas:<span style="mso-spacerun: yes;">  </span><u>y<sup>2</sup></u> = <u>z<sup>2</sup></u><span style="mso-spacerun: yes;">  </span>(de distribuye y entonces (<u>y</u> + <u>z</u>). (<u>y</u> - <u>z</u>)<span style="mso-spacerun: yes;">  </span>= 0) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                           </span>b<sup>2</sup><span style="mso-spacerun: yes;">    </span>c<sup>2</sup><span style="mso-spacerun: yes;">                                            </span>b<span style="mso-spacerun: yes;">    </span>c<span style="mso-spacerun: yes;">     </span>b<span style="mso-spacerun: yes;">   </span>c</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Intersección eje z: x = 0<span style="mso-spacerun: yes;">        </span>y = 0</span></div><div class="MsoNormal"><br /></div><h4><span lang="ES-AR">Cilindros cuádricos</span></h4><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Se llama superficie cilíndrica, a la superficie formada por el conjunto de todas las rectas que cortan una curva plana y son paralelas a una recta fija que no esta en el plano de la curva. La curva se llama directriz y las rectas generatrices.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La superficie engendrada por una elipse, hipérbola o parábola que se mueve paralelamente a si misma, manteniendo centro o vértice sobre una recta perpendicular a su plano es un cilindro.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Si son de generatrices paralelas al eje z, perpendiculares al plano (x;y) tiene ecuaciones para cualquier valor de z.</span></div><div class="MsoNormal"><img align="left" height="233" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image034.jpg" width="308" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Cilindro elíptico</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">: <u>x<sup>2</sup></u> + <u>y<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                          </span>a<sup>2</sup><span style="mso-spacerun: yes;">     </span>b<sup>2</sup><span style="mso-spacerun: yes;">  </span><span style="mso-spacerun: yes;">         </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es una elipse:<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> + <u>y<sup>2 </sup></u>= 1</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                        </span>a<sup>2</sup><span style="mso-spacerun: yes;">     </span>b<sup>2</sup><span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xz) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Son dos rectas:<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> =<span style="mso-spacerun: yes;">  </span>1 </span></div><div class="MsoNormal"><img align="left" height="192" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image036.jpg" width="171" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                           </span>a<sup>2</sup><span style="mso-spacerun: yes;">           </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación x=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Son dos rectas:<span style="mso-spacerun: yes;">  </span><u>y<sup>2</sup></u> =<span style="mso-spacerun: yes;">  </span>1 </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                           </span>b<sup>2</sup><span style="mso-spacerun: yes;">           </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje y: x = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje z: No existe</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><img align="left" height="199" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image038.jpg" width="264" /><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Cilindro parabólico:</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">  </span>x<sup>2</sup>= 2pz<span style="mso-spacerun: yes;">   </span>o<span style="mso-spacerun: yes;">    </span>y<sup>2</sup> = 2px</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación z=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es el eje y (apoyan todas las parábolas)</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xz) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación y=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es una parábola </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación x=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es el eje y.<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje x: x = 0</span></div><div class="MsoNormal"><img align="left" height="242" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image040.jpg" width="320" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje y: y = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje z: z = 0</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Cilindro hiperbólico</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">:<span style="mso-spacerun: yes;">   </span><u>x<sup>2</sup></u>- <u>y<sup>2</sup></u> = 1<span style="mso-spacerun: yes;">   </span></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                                  </span>a<sup>2</sup><span style="mso-spacerun: yes;">    </span>b<sup>2</sup><span style="mso-spacerun: yes;">           </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xy)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación z=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es una hipérbola:<span style="mso-spacerun: yes;">  </span><u>x<sup>2</sup></u> - <u>y<sup>2 </sup></u>= 1</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                              </span>a<sup>2</sup><span style="mso-spacerun: yes;">   </span>b<sup>2</sup><span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (xz) </span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación y=0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Es un punto. <u>x<sup>2</sup></u> = 1</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;"><span style="mso-spacerun: yes;">                     </span>a<sup>2</sup><span style="mso-spacerun: yes;">     </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">La traza sobre el plano (yz)</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Plano de ecuación x=0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">No existe</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección con los ejes coordenados</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje x: y = 0<span style="mso-spacerun: yes;">        </span>z = 0</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje y: No existe</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 10.0pt;">Intersección eje z: No existe</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">HELICE Y HELICOIDE</span></b></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Hélice cilíndrica circular:</span></u></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La hélice circular, es la curva trazada por un punto P, que se mueve con movimiento circular uniforme.<span style="mso-spacerun: yes;">  </span>Este consiste en una rotación alrededor<span style="mso-spacerun: yes;">  </span>del eje del cilindro y una traslación en la dirección z.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Un punto sobre la circunferencia (centro O y radio R) gira alrededor del centro con una velocidad angular </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">ω = <u>σ</u> = cte</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                                 </span>t</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">A la vez, el centro se desplaza por el eje z a una velocidad de traslación constante.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Si R es el radio del cilindro sostén entonces:</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x = R. cos α</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">y = R. sen α</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">z = k . ωt</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">P es el paso de la hélice y es la distancia entre dos intersecciones consecutivas de la hélice.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">P = k . 2π</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Helicoide recto</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Un helicoide recto es el lugar geométrico de las rectas paralelas al plano de la base de una hélice circular, cortan con un eje.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Si la hélice circular es:</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x = R. cos α</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">y = R. sen α</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">z = k . ωt</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Una recta paralela al plano es una generatriz del helicoide si corta al eje z y a la hélice.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La ecuación del helicoide es:</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Z = k arc tg <u>y</u><span style="mso-spacerun: yes;">                     </span>donde <u>y</u> = tg ωt<span style="mso-spacerun: yes;">                 </span>z = k . ωt</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">   </span><span style="mso-spacerun: yes;">                </span>x<span style="mso-spacerun: yes;">                                </span>x</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal"><br /></div></div><b><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection11"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">GRAFOS </span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición de grafo</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se llama grafo a una terna G = (V , A , </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">φ</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">) donde V y A son conjuntos finitos y </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">φ es una aplicación que hace corresponder a cada elemento de A un par de elementos de V.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Los elementos de V son los vértices de G, y los elementos de A son las aristas de G, y φ es la aplicación de incidencia que asocia a cada arista sus dos vértices.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">La representación grafica de un grafo se efectúa asociando a cada vértice un punto del plano de dibujo y a cada arista una línea que une los puntos asociados con los vértices.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">El grado de un vértice es el numero de aristas que en el inciden. Un vértice se dice aislado si su grado es nulo y pendiente si su grado es 1.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dos o mas aristas se llaman múltiples si tienen por extremos los mismos vértices.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un lazo es una arista cuyos dos extremos coinciden en un vértice. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un vértice y una arista son incidentes si el vértice es extremo de la arista.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dos vértices son adyacentes si son extremos de la misma arista.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dos aristas son adyacentes si tienen un vértice en común.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dos grafos P y P´ son isomorfos si tienen la misma cantidad de vértices y aristas y se mantiene las relaciones de adyacencia e incidencia entre ambos.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Conceptos no orientados</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l3 level1 lfo9; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Arista: Existe una arista entre dos vértices x e y distintos del grafo si existe un arco que va de x a y o de y a x.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l3 level1 lfo9; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cadena: Sucesión de aristas adyacentes.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l3 level1 lfo9; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ciclo: Cadena finita en la que el vértice inicial coincide con el final.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Tipos de grafos</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo vacío</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es todo grafo que no posee aristas, aunque pueda contener vértices.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo sencillo</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es todo grafo que no tiene ni lazos ni aristas múltiples.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo k – regular</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es en el que todos los vértices tiene igual grado k.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo completo de n vértices</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es todo grafo sencillo en el que todo par de vértices determina una arista. Todos sus vértices tienen grado n –1 y el numero de aristas es n <u>n-1</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                                                                                                        </span><span style="mso-spacerun: yes;">  </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="166" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image042.jpg" width="642" /></span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div></div><i><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></i> <div class="WordSection12"> <div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo complemento de G (C<sub>G</sub>)</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es el que tiene los mismo vértices que G y cuyas aristas no pertenecen a G.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Subgrafo</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es cuando los vértices y las aristas están incluidos en los vértices y las aristas de G. Pueden tomarse respecto de un vértice (se anula el vértice y todas las aristas que en el inciden) o bien respecto de una arista (se anula la arista).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="169" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image044.jpg" width="642" /></span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo euleriano</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es cuando todas sus aristas pueden recorrerse en un solo trazo sin pasar dos veces por alguna de ellas y volver al punto inicial. Para que un grafo sea euleriano, solo puede tener como máximo dos vértices a los que concurra un número impar de aristas. En todos los demás vértices debe incidir un número par de aristas. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i style="mso-bidi-font-style: normal;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo euleriano restringido:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es cuando se pueden recorrer todos los vértices sin repetir ninguno y no vuelvo al vértice inicial.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="193" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image046.jpg" width="642" /></span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo hamiltoniano</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es cuando existe un recorrido que pasa por todos los vértices una sola vez, sin necesidad de recorrer todas las aristas.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo p – coloreado</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Son grafos de V vértices y p subconjuntos de pares no ordenados de elementos de V, determinados por otras tantas aplicaciones </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">φ.</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> Un grafo p – coloreado posee aristas de p clases distintas que se colorean en forma diferente.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="169" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image048.jpg" width="642" /></span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafo rotulado</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: Es el que tiene sus vértices individualizados por números o letras.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Especificación y representación de grafos.</span></u></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level1 lfo7; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Enumerando sus vértices y sus aristas, agregando un listado de las relaciones entre esas partes.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level1 lfo7; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Mediante matrices. Las matrices son arreglos rectangulares de números cuya dimensión esta dada por el numero de filas multiplicado por el numero de columnas. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 72.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level2 lfo7; tab-stops: list 72.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Courier New"; mso-ansi-language: ES; mso-fareast-font-family: "Courier New";"><span style="mso-list: Ignore;">o<span style="font: 7.0pt "Times New Roman";">   </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La matriz de incidencia tiene n filas y k columnas, donde cada fila corresponde a un vértice y cada columna a una arista. En el lugar de cruce se escribe un 1 si el vértice y la arista son incidentes y un 0 si no lo son.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 72.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level2 lfo7; tab-stops: list 72.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Courier New"; mso-ansi-language: ES; mso-fareast-font-family: "Courier New";"><span style="mso-list: Ignore;">o<span style="font: 7.0pt "Times New Roman";">   </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La matriz de adyacencia de vértices es cuadrada y tiene n filas por n columnas. En el lugar de cruce se escribe un 1 si el vértice si los vértices son adyacentes o un 0 si no lo son.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 72.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level2 lfo7; tab-stops: list 72.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Courier New"; mso-ansi-language: ES; mso-fareast-font-family: "Courier New";"><span style="mso-list: Ignore;">o<span style="font: 7.0pt "Times New Roman";">   </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La matriz de adyacencia de aristas es cuadrada y tiene k filas por k columnas. En el cruce se escribe un 1 si las aristas son adyacentes y un 0 si no lo son.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level1 lfo7; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Mediante rejillas: En ellas los vértices y las aristas se representan en una grilla ortogonal, colocando un punto en las intersecciones cuando se cumple la relación de incidencia (o de adyacencia).</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level1 lfo7; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Diagrama de punto entrelazados: es en los que los vértices y/o las aristas se representan mediante dos columnas de puntos enfrentados, uniendo los puntos cuando se cumple la relación.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l24 level1 lfo7; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Representación poligonal: es en la que los vértices se ubican formando un polígono regular, y las aristas aparecen como lados. En algunos casos, se trata de convertir esta representación en otra con mínimos cruces.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafos dirigidos o dígrafos</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un dígrafo es un grafo cuyas aristas tienen un sentido u orientación determinado. En este caso, las aristas se denominen arcos. El arco (ab) esta dado por un par ordenado: a es el vértice inicial y b es el vértice final.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un grafo se dice fuertemente conexo si entre dos vértices cualquiera existe un camino de cualquier longitud que va de uno a otro.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un grafo es conexo si entre dos vértices cualquiera existe una cadena.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es no conexo cuando tiene partes separadas.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Una componente fuertemente conexa es una parte del dígrafo que sea fuertemente conexa.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="166" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image050.jpg" width="642" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Conceptos Orientados</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Vértices: Puntos que representan los elementos del conjunto V.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Arcos: Líneas orientadas que unen pares de vértices y representan los elementos del conjunto A.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Extremos inicial y extremos final de un arco: Vértice del que parte un arco y vértice al que llega.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Camino: Sucesión de arcos adyacentes tales que el extremo final de uno coincide con el extremo inicial del siguiente.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Longitud: Numero de arcos del camino.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Circuito: Camino en el que el vértice inicial coincide con el final.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l15 level1 lfo8; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Lazo: Circuito de longitud 1.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafos Planos</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un grafo es plano si existe un grafo isomorfo que puede dibujarse en el plano de modo que las aristas solo se crucen en los vértices.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un grafo es no plano cuando no existe un isomorfo que pueda dibujarse sin que sus aristas se crucen.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="211" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image052.jpg" width="178" /></span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Teorema de Kuratowski</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La condición necesaria para que un grafo sea plano es que no admita subgrafos ni del tipo K<sub>3,3</sub> ni del tipo K<sub>5.<span style="mso-spacerun: yes;">  </span></sub>Siendo estos los grafos no planos que definen toda una familia de subgrafos no planos. Solo vértices de grado mayor o igual que 4 son candidatos para el K<sub>5</sub>, y solo vértices de grado mayor o igual que 3 son candidatos para el K<sub>3,3</sub>.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="169" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image054.jpg" width="642" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafos poligonales</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Llamaremos grafo poligonal a un grafo plano conexo que es reunión de ciclos, tal que existe un ciclo mínimo y otro máximo. En todo grafo poligonal se cuenta no solamente el numero de vértices V y el de aristas A, si no también el de caras C, incluida la cara del infinito.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si se consideran los 5 poliedros regulares (tetraedro, octaedro, icosaedro, cubo y dodecaedro) se puede comprobar que, entre el numero C de caras y el numero V de vértices y el numero A de aristas, vale la formula de Euler: C + V = A + 2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La formula de Euler es también valida en cualquier grafo poligonal. Un grafo poligonal es regular si el grado de cada vértice es el mismo. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un grafo es completamente regular si es regular y además cada cara tiene el mismo número de aristas limitantes. Los grafos completamente regulares no triviales son los asociados con los 5 poliedros regulares.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="169" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image056.jpg" width="207" /></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="326" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image058.jpg" width="562" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Grafos duales</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Sea G un grafo plano y conexo, si se construye un grafo G´</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 42.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l8 level1 lfo10; tab-stops: list 42.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">A cada cara de G le corresponderá un vértice de G´</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 42.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l8 level1 lfo10; tab-stops: list 42.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">A cada vértice de G le corresponderá una cara de G´</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 42.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l8 level1 lfo10; tab-stops: list 42.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">A cada arista de G le corresponderá una arista de G´</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 42.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l8 level1 lfo10; tab-stops: list 42.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dos vértices de G´ están unidos por una arista si las caras correspondientes de G tienen una arista en común. Entonces G´ es el grafo dual de G</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dado un grafo plano y conexo G, si se construye su grafo dual G´ y luego el dual G´´ de G´ , G Y G ´´ son isomorfos.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="159" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image060.jpg" width="527" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Mosaicos regulares</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es un tipo especial de recubrimiento del plano. Los diferentes tipos de mosaicos se obtienen siguiendo un principio general de repetición de un modulo en dos direcciones, con condiciones restrictivas de acoplamiento y regularidad.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">No puede haber huecos ni superposiciones. La arista de un modulo tiene que ser coincidente con la arista del otro. La sumatoria de los ángulos que concurren de los vértices tiene que ser 360°. Cada ángulo interno mide:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">(n - 2</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">) . 180°</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">n</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">En cada vértice tendremos entonces el siguiente numero de polígonos:</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                                 </span><u>360</u>°<span style="mso-spacerun: yes;">          </span>=<span style="mso-spacerun: yes;">  </span><u>2 n</u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                           </span><u>(n - 2</u>). 180°<span style="mso-spacerun: yes;">        </span>n -2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                    </span><span style="mso-spacerun: yes;">                           </span>n</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Como este numero debe ser entero, para n > 2, n tiene que ser igual a 3 , 4 o 6. Esto significa que plano puede recubrirse totalmente con mosaicos triangulares, cuadrados y hexagonales.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="309" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image062.jpg" width="599" /></span></div></div><b><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection13"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">SIMETRÍA</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La definición geométrica habla de la correspondencia exacta en la disposición regular de las partes, puntos de un cuerpo o figura con relación a un centro, un eje o un plano.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se dice que una configuración espacial es simétrica respecto de un plano E si puede superponerse sobre si misma por reflexión en dicho plano.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Simetría traslatoria, rotatoria y afín</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Simetría: Es un conjunto de las transformaciones del plano que dejan invariantes las figuras. La simetría de una figura cualquiera del espacio que descripta por un subgrupo de dicho grupo. Estos subgrupos son:</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Traslación:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> Queda definida por un vector que indica en que dirección se traslada y cuanto se traslada.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Axial o reflexiva:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> Dos puntos P y P´ son simétricos respecto a una recta ¨r¨ llamada eje de simetría. Si se encuentran sobre un mismo plano perpendicular a dicha recta y equidistan de la misma.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Giro centro ¨o¨ y ángulo de giro ¨</span></u><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">α</span></u><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">¨:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> Es la transformación que a todo punto P le hace corresponder otro punto P´, tal que OP = OP´ y el angulo POP´ = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">α</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="302" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image064.jpg" width="642" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Rotaciones = antihorario positivo</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Simetría axial = espejo</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Simetría central equivale a una rotación 180</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: JA;">º</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: JA;">La simetría axial (composición) es conmutativa.</span></div></div><b><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection14"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">NUMERO DE ORO</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">El numero de oro corresponde matemáticamente a la división de un segmento en media y extrema razón. En efecto sea el segmento AB que se quiere dividir mediante un punto C en dos partes de manera que: </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">AB</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> = <u>AC</u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">AC<span style="mso-spacerun: yes;">    </span>CB</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Llamando AC = a;<span style="mso-spacerun: yes;">  </span>CB = b se tiene la relación 1+ <u>b</u> = <u>a </u></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                                               </span>a<span style="mso-spacerun: yes;">    </span>b</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Esta igualdad se puede escribir, indicando con x = <u>a</u> como:</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                                           </span><span style="mso-spacerun: yes;">                      </span>b<span style="mso-spacerun: yes;">   </span></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">1+<u>1</u>= x </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">    </span>x</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">De donde resulta:</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x<sup>2</sup>= 1+x<span style="mso-spacerun: yes;">          </span></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Esta ecuación de segundo grado en x, tiene como solución positiva el siguiente valor que no es mas que el numero de oro Ф.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x = <u>1 + √5</u> = 1,618...</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">          </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ф = 1,618...</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Se puede obtener también el número de oro, como el cociente de las longitudes de una diagonal y un lado de un pentágono regular.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Sección Áurea</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Un rectángulo es áureo, si sus lados están en la relación 1:</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ф (número de oro).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Obtención de un rectángulo áureo:</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">A partir de un cuadrado de lados = b; marcamos el punto medio de un lado. Unimos ese punto con un vértice opuesto. Tomamos la medida de ese segmento y la trasladamos hacia el lado donde marcamos el punto medio.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">El rectángulo áureo tiene base a y altura b y se cumple:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">a</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> = Ф</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                                    </span>b</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="187" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image066.jpg" width="642" /></span></b></div></div><b><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span></b> <div class="WordSection15"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">DERIVADAS </span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición de derivada</span></u><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La derivada de un función f (x) en un punto x<sub>o</sub>, indicada por f´(x), es un numero real que mide la pendiente de la recta tangente a la curva que representa la función.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Funciones crecientes y decrecientes</span></u></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l5 level1 lfo11; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">F(x) es creciente en un intervalo (a , b) si f (x<sub>1</sub>) < f (x<sub>2</sub>) y su derivada resulta f´(x) > 0</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l5 level1 lfo11; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">F(x) es decreciente en un intervalo (a , b) si f (x<sub>1</sub>) > f (x<sub>2</sub>) y su derivada resulta f´(x)<0</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Concavidad de las funciones</span></u></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l2 level1 lfo12; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si los puntos de f (x) están por encima de la recta tangente (f´´(x)), entonces la concavidad será hacia arriba. En caso contrario, si se encuentran por debajo, la concavidad será hacia abajo.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l2 level1 lfo12; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se llama punto de inflexión, al punto en que la curva cambia de concavidad en un sentido u otro. En este punto, f´´(x) = 0</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Máximos y mínimos de un función</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Para determinar los máximos y mínimos, se utiliza el criterio de la 2da derivada (condición suficiente):</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l11 level1 lfo13; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si f´´(x<sub>o</sub>)>0, hay un mínimo local en x<sub>o</sub></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l11 level1 lfo13; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si f´´(x<sub>o</sub>)<0, hay un máximo local en x<sub>o</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Derivadas</span></b></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Aplicaciones físicas: (velocidades)</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Desde una plataforma ubicada a 20 m de altura se arroja un proyectil verticalmente y hacia arriba, con una velocidad inicial de 50 m/seg.<span style="mso-spacerun: yes;">  </span>Si la ecuación horaria del mismo es </span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">s(t) = 20 + 50t + 5t<sup>2</sup> Calcular. </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">La velocidad del proyectil en el instante t = 2.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Si se deriva la ecuación se tiene la velocidad instantánea.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">S´(t) = 50 - 10t</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">S´(2) = 50 - 10 . 2 = 50 – 20 = 30 m/seg</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Aplicaciones geométricas: (optimizacion)</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Si se desea cercar una superficie rectangular con una pared en una de sus costados y se dispone de 100 m de alambre para cerco. Determinar el área máxima que se puede abarcar con esa cantidad de alambre:</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Sea x el ancho en metros de la superficie a cercar e y su longitud, de modo que y = 100 – 2x</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">El area total esta dada por:</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A(x) = xy = x (100 – 2x)</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A(x) = 100 x – 2x<sup>2</sup></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">La funcion A(x) es derivable</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A´(x) = 100 – 4x</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A´´(x) = -4 </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Entonces para buscar el valor que maximiza la funcion A´(x) = 0</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">100 – 4x = 0</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;"> </span>x = 25</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo2: Hallar dos números positivos cuya suma sea 20 y además la suma de sus cuadrados sea mínima.</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Dos números tales que su suma sea 20</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">X + y = 20</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Y la suma de sus cuadrados sea mínima</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">X<sup>2</sup> + y<sup>2</sup> = mínimo </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Despejo en la primera ecuación: y = 20 – x</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Reemplazo en la 2da: x<sup>2</sup> + (20-x)<sup>2</sup></span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Desarrollo: x<sup>2</sup> + 20<sup>2</sup> – 2.20 + x<sup>2</sup></span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Llegue a esto que tiene que ser el mínimo: 2x<sup>2</sup>– 40x +400 = mínimo</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Derivo: 4x – 40</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Igualo a cero para buscar puntos críticos: 4x – 40 = 0 </span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">X = 40/4 = 10</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Reemplazo en las ecuaciones y obtengo que números me tiene que dar.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Integrales</span></b></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Aplicaciones físicas: (momentos y baricentro)</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo:</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Aplicaciones geométricas: (cálculo de áreas; volumen de revolución)</span></div><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Calcular el área limitada por las curvas:</span></div><ol start="1" style="margin-top: 0cm;" type="1"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-list: l14 level1 lfo22; tab-stops: list 36.0pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Y = 2x ; y = 0 x =      2</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-list: l14 level1 lfo22; tab-stops: list 36.0pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">y = x<sup>3</sup>     – 6x<sup>2</sup> + 8x ; y=0</span></li></ol><div class="MsoNormal"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo 2 = Calcular el volumen de revolución.</span></div><ol start="1" style="margin-top: 0cm;" type="1"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-list: l1 level1 lfo23; tab-stops: list 36.0pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">y = 2x<sup>2</sup>     desde x = 0 hasta x = 5</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-list: l1 level1 lfo23; tab-stops: list 36.0pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">x<sup>2</sup> + y<sup>2</sup>     = 4 desde x = -r<span style="mso-spacerun: yes;">  </span>hasta x = r</span></li></ol><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">INTEGRALES</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Ley de Barrow: Sea F (x) una primitiva de f (x):</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">a</span></sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∫<sup>b</sup> f(x) dx = F(b) – F(a)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Dadas dos funciones, con dos intervalos distintos, en los cuales en el primer intervalo f(x) es el techo y g(x) el piso; en el segundo intervalo f(x) es el piso y g(x) el techo.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">a</span></sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∫<sup>b</sup> [f(x) – g(x)] dx + <sub>b</sub>∫<sup>c</sup>[g(x) – f(x)] dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Trabajo de una fuerza</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Dada una fuerza constante, el trabajo será T = F. d (siendo d la distancia a la que se ha desplazado el objeto) Sin embargo siendo F una fuerza no constante, se utilizara la integral definida para resolver el trabajo en distintos intervalos; obteniendo el trabajo total al sumar dichos intervalos. <span style="mso-spacerun: yes;">              </span></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">T = <sub>a</sub>∫<sup>b</sup> f(x) dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Un ejemplo del trabajo efectuado por una fuerza variable es el estiramiento o compresión de un resorte, de acuerdo con la ley de Hooke la fuerza F requerida es proporcional a la longitud del resorte:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">F (x) =<span style="mso-spacerun: yes;">  </span>k. x</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Donde k es la constante del resorte. Si el resorte se estira desde un punto a hasta un punto b:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">T = k <sub>a</sub>∫<sup>b</sup> x dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momentos y centro de gravedad</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momentos y centro de gravedad de un sistema de puntos materiales sobre una recta:</span></u></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Se define momento de primer orden (momento estático) M<sup>(1)</sup> respecto del origen O, a la suma de los productos de las abscisas (x<sub>1</sub>, x<sub>2</sub>,x<sub>N</sub>) por las masas correspondientes.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sup> (1) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.xi /<span style="mso-spacerun: yes;">  </span>M<sup>(1) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.yi</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Se define momento de segundo orden (momento de inercia) M<sup>(2)</sup> respecto de un origen O, a la suma de los productos de las masas por los cuadrados de las distancias al origen.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sup> (2) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.xi<sup>2</sup> /<span style="mso-spacerun: yes;">  </span>M<sup>(2) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.yi<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">El centro de gravedad es donde se concentra la masa total del sistema, su momento estático seria igual al momento estático del sistema.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">El valor x<sub>G </sub>de la posición de un punto sobre la recta, que tomado como nuevo origen, anula el momento estático, se llama abscisa del centro de gravedad. El punto G es el centro de gravedad del sistema de masas mi distribuidas en los punto de abscisa xi.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x<sub>G</sub><sup> </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.xi</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">      </span>y<sub>G</sub><sup> </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.yi</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                              </span>∑ mi<span style="mso-spacerun: yes;">               </span>∑ mi</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">G = (x<sub>G ;</sub> y<sub>G</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momentos y centro de gravedad de un sistema de puntos materiales sobre un plano.</span></u></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Se define momento de primer orden M<sub>x</sub><sup> (1) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">respecto del eje de las abscisas a:</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">M<sub>x</sub><sup> (1) </sup></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">= </span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∑ mi.yi<span style="mso-spacerun: yes;">   </span>/<span style="mso-spacerun: yes;">   </span>M<sub>y</sub><sup>(1) </sup></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">= </span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∑ mi.xi</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Los momentos de inercia son:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sub>x</sub><sup> (2) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ yi<sup>2</sup>.mi<span style="mso-spacerun: yes;">   </span>/<span style="mso-spacerun: yes;">   </span>M<sub>y</sub><sup>(2) </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ xi<sup>2</sup>.mi</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momento polar: </span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sub>O</sub><sup>(2)</sup> = M<sub>x</sub><sup> (2)</sup>+ M<sub>y</sub><sup>(2)</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Para hallar el punto G = (x<sub>G ;</sub> y<sub>G</sub>):</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                             </span>x<sub>G</sub><sup></sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.xi</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">  </span>= <u>M</u><sub>y</sub><sup> (1)</sup><span style="mso-spacerun: yes;">         </span>y<sub>G</sub><sup> </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ mi.yi</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> = <u>M</u><sub>x</sub><sup>(1)</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                       </span>∑ mi<span style="mso-spacerun: yes;">        </span>M<span style="mso-spacerun: yes;">                     </span>∑ mi<span style="mso-spacerun: yes;">        </span>M</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momentos y centro de gravedad de superficies y volúmenes.</span></u></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Centro de gravedad de una superficie (lamina homogénea):</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Teniendo una figura plana delimitada por la curva AB (y = f (x)) y el eje de las abscisas, entonces el momento estático es:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">M<sub>x</sub><sup> (1) </sup></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">= k </span><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">a</span></sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∫<sup>b</sup> xy dx<span style="mso-spacerun: yes;">   </span>M<sub>y</sub><sup>(1) </sup></span><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">= k </span><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">a</span></sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">∫<sup>b</sup> y<sup>2</sup> dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Teniendo en cuenta que las masa total M<sub>T</sub> de la figura es M<sub>T</sub> = k . A , siendo k la densidad y A el área de la figura plana:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">A =</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> </span><sub><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">a</span></sub><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∫<sup>b</sup> y dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Las coordenadas del centro de gravedad son:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x<sub>G</sub><sup> </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">1 </span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><sub>a</sub>∫<sup>b</sup> xy dx<span style="mso-spacerun: yes;">     </span>y<sub>G</sub><sup> </sup></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">1 </span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><sub>a</sub>∫<sup>b</sup> y<sup>2</sup> dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                     </span>A<span style="mso-spacerun: yes;">                          </span>2A</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Si la figura plana tiene un eje de simetría, el centro de gravedad deberá estar sobre ese eje y si posee dos, se hallara en la intersección de ambos.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Centro de gravedad de un sólido de revolución.</span></u></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Solamente se puede obtener el centro de gravedad de un sólido de revolución, mediante una integral definida, ya que por razones de simetría el centro de gravedad estará ubicado sobre el eje en que ha girado el arco de curva plana que forma el sólido.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">En el sólido de revolución la única coordenada del centro de gravedad que hay que obtener es la eje de revolución, debido a que las otras coordenadas serán cero.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">x<sub>G</sub><sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">=</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> <u>π <sub>a</sub>∫<sup>b</sup>xy<sup>2</sup> dx</u></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">       </span>V</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Donde V es el volumen del cuerpo de revolución:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">V<sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">=</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> π <sub>a</sub>∫<sup>b</sup> y<sup>2</sup>dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Deducción del volumen de revolución:</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-bidi-font-style: italic; mso-fareast-language: ES;">Se debe trasladar a un plano 2d y dividir el intervalo </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">[a;b] en n partes.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">a = x<sub>0</sub> < x<sub>1</sub> < x<sub>3</sub>< x<sub>i</sub> = b</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">x <sub>i-1</sub> < c<sub>i</sub> < x<sub>i</sub>donde c<sub>i</sub> = x = ancho de cada parte</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Si el volumen se obtiene por generar la curva alrededor del eje x.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Vt = ∑ π. f(c<sub>i</sub>)<sup>2 </sup>∆x<sub>i</sub><span style="mso-bidi-font-style: italic;"></span></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">V= lim </span></i><sub><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∆xi → 0<span style="mso-spacerun: yes;">    </span></span></sub><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ π. f(c<sub>i</sub>)<sup>2 </sup>∆x<sub>i</sub><span style="mso-bidi-font-style: italic;"></span></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><sup><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                       </span><span style="mso-spacerun: yes;">                                        </span>n → ∞</span></sup></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">V<sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">=</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> π <sub>a</sub>∫<sup>b</sup> f(x)<sup>2</sup>dx<span style="mso-spacerun: yes;">    </span>F(x) = y</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">V<sup> </sup></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">=</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> π <sub>a</sub>∫<sup>b</sup> y<sup>2</sup>dx</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Momento de Inercia de Placas Planas</span></u></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Se consideran placas planas a aquellos cuerpos donde una de las dimensiones, el espesor, es despreciable respecto de las otras dos. Para calcular sus momentos de inercia se utiliza el Teorema de Steiner. Este teorema se utiliza cuando el momento de inercia no tiene su centro en el eje en cuestión, y se encuentra separado a una distancia d.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sub>xx</sub><sup>(2)</sup> = M<sup>(2)</sup> + A.d<sup>2</sup></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Momento de Inercia de una superficie según su ubicación en el plano.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l9 level1 lfo14; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cuerpo Rectangular apoyado sobre el eje:</span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">M<sub>XX</sub><sup>(2) </sup>= <u>b . h<sup>3</sup></u><span style="mso-spacerun: yes;">    </span>M<sub>yy</sub><sup>(2) </sup>= <u>h . b<sup>3</sup></u></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">                                                               </span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">3<span style="mso-spacerun: yes;">                      </span>3</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l9 level1 lfo14; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cuerpo Rectangular, el eje pasa por el centro:</span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">M<sub>XX´</sub><sup>(2) </sup>= <u>b . h<sup>3</sup></u><span style="mso-spacerun: yes;">    </span>M<sub>yy´</sub><sup>(2) </sup>= <u>h . b<sup>3</sup></u></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">                                                              </span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">12<span style="mso-spacerun: yes;">                      </span>12</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l9 level1 lfo14; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">    </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cuerpo Rectangular, el eje separado del cuerpo:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">M<sub>xx</sub><sup>(2)</sup> = M<sup>(2)</sup> + A.d<sup>2</sup></span></div></div><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span> <div class="WordSection16"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">PROBABILIDAD</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Espacio muestral:</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es el conjunto de los resultados diferentes derivados de un experimento aleatorio. Cada uno de los resultados que pertenece a un espacio muestral será un suceso.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l17 level1 lfo24; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Suceso seguro P = 1</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l17 level1 lfo24; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Suceso imposible P = 0</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l17 level1 lfo24; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ocurrencia incierta P = n</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l17 level1 lfo24; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Punto de indiferencia P = 0,5</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición clásica o canónica (Laplace)</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La probabilidad se define como el número de casos favorables a un suceso divididos por el número de casos posibles. Es aplicable cuando los números son finitos y cuando los casos posibles son equiprobables.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La probabilidad de que ocurra un suceso A es:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">P<sub>(A)</sub>= <u>c</u> (casos favorables)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                   </span>n (numero total de casos)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ejemplo: El dado, la probabilidad de sacar dos seis es igual a 1/6.1/6 = 1/36</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición basada en la frecuencia (Von Misses)</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Los sucesos deben ser indefinidamente repetibles bajo las mismas condiciones. Si en n pruebas, el suceso A se presento una cierta cantidad de veces Y<sub>(A)</sub> entonces:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">f<sub>(A)</sub>= <u>Y<sub>(A)</sub></u><span style="mso-spacerun: yes;">  </span>(es la frecuencia absoluta de aparición del suceso a)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                 </span>n<span style="mso-spacerun: yes;">     </span>(numero de casos)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición axiomática (Kolmogroff)</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se establece un conjunto de axiomas, estos se basan en la teoría de conjuntos y se visualizan mediante el diagrama de Venn.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Los sucesos se pueden clasificar en:</span></i></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l13 level1 lfo15; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Incompatibles (excluyentes o mutuamente      excluyentes): A </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∩ B = Θ (si un suceso      se esta dando, el otro no puede ocurrir).</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"></span></li></ul><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Ejemplo: Se tira un dado tres veces. Cual es la probabilidad de que salgan todos ases?.<span style="mso-spacerun: yes;">  </span>Es un suceso independientes porque lo que ocurren en la primera tirada no influye en las demás tiradas. </span></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Entonces: P<sub>(3A)</sub>= P<sub>(A1)</sub> . </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">P<sub>(A2)</sub> </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">. P</span><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(A3) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">= 1/6 . 1/6 . 1/6 = 1/216</span></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><br /></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l13 level1 lfo15; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Compatibles (no      excluyentes): </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∩      B ≠ Θ (si un suceso se esta dando el otro puede ocurrir también).</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> Se clasifican en:</span></li><ul style="margin-top: 0cm;" type="circle"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l13 level2 lfo15; tab-stops: list 72.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Compatibles       condicionados: La ocurrencia de uno de los sucesos condiciona la       ocurrencia del otro suceso.</span></li></ul></ul><div class="MsoNormal" style="margin-left: 54.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Se extraen y sin reposición 2 naipes de una baraja española de 40. Calcular la probabilidad de que sean dos cartas de oro. Al sacar la primera carta de oro, modifico la posibilidad de que salga otra de oro en la segunda vez.</span></div><div class="MsoNormal" style="margin-left: 54.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Entonces: P<sub>(O1 </sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∩</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> O2)</span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> = P<sub>(O1)</sub> . P<sub>(O2)</sub> = 10/40 . 9/40</span></div><div class="MsoNormal" style="margin-left: 54.0pt; mso-layout-grid-align: none; text-autospace: none;"><br /></div><ul style="margin-top: 0cm;" type="disc"><ul style="margin-top: 0cm;" type="circle"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l13 level2 lfo15; tab-stops: list 72.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Compatibles       independientes: Cada suceso posee su propio espacio muestral. La       ocurrencia de uno no modifica la ocurrencia del otro.</span></li></ul></ul><div class="MsoNormal" style="margin-left: 54.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Un alumno decide contestar las respuestas de un parcial eligiendo al azar, cual es la probabilidad de que acierte las 5? Responde las preguntas son sucesos independientes entonces la intersección es el productor de las probabilidades.</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">  </span>P<sub>(V1 ∩ V2 ∩ V3 ∩ V4 ∩ V5)</sub> = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(V1)</sub> . </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">P<sub>(</sub></span><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">V2</span></sub><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">) </span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">. P<sub>(V3) </sub></span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">. P<sub>(</sub></span><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">V4</span></sub><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">) </span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">. P<sub>(V5)</sub></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoHeading8"><span lang="ES" style="mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt;">Axiomas de la probabilidad</span></div><div style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Axioma 1</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">: La probabilidad de que ocurra un suceso A es un numero real no negativo:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(A)</sub> </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≥ 0</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Axioma 2:</span></i><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> La probabilidad de un suceso cierto es igual a la unidad:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(E)</sub> </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">= 1</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><i><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Axioma 3:</span></i><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> La suma de dos sucesos excluyentes es igual a la suma de las probabilidades de cada uno de ellos. Si A </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∩ B = Θ entonces P<sub>(A U B)</sub> = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(A) </sub>+ P<sub>(B)</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Propiedades y reglas de cálculo de la probabilidad</span></u></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">La probabilidad es un numero real comprendido      entre 0 y 1 teniendo P<sub>(E)</sub> </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">= 1 y a su vez<span style="mso-spacerun: yes;">       </span>E = A U Ā entonces P<sub>(A U Ā)</sub> = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(A)</sub>     + P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Ā</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">= 1 </span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Dados dos sucesos      compatibles, la probabilidad de que ocurra uno de ellos es igual a la suma      de las probabilidades individuales menos la probabilidad de que ambos se      presenten simultáneamente P<sub>(A U B)</sub> = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(A)</sub>     + P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">- P<sub>(A∩ B)</sub></span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">La probabilidad de que      el evento A vaya a ocurrir, sabiendo que el evento B ha sucedido, se      denomina como probabilidad condicional del suceso A, respecto del suceso B      entonces P<sub>(A/B)</sub> = <u>P<sub>(A∩ B)</sub></u></span></li></ul><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                   </span>P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">)</span></sub></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none; text-indent: 17.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Ejemplo: Un equipo electrónico consta de dos circuitos A y B. La probabilidad de que </span></div><div class="MsoNormal" style="margin-left: 18.6pt; mso-layout-grid-align: none; text-autospace: none; text-indent: 17.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">A falle es de 0,2. Que falle solamente B es de 0,15. Y que fallen ambos a la vez 0,15. </span></div><div class="MsoNormal" style="margin-left: 36.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Calcular que falle A sabiendo que B fallo.</span></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none; text-indent: 17.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P(falle a / sabiendo que b fallo) = <u>P</u></span><u><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(A∩ B)</span></sub></u><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></sub></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                        </span><span style="mso-spacerun: yes;">                     </span>P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">)</span></sub></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none; text-indent: 35.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">=<sub> </sub>P</span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(A∩ B) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">+</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> P</span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(B∩ |A|) </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">= 0,15 + 0,15 = 0,30</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none; text-indent: 17.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P(falle a / sabiendo que b fallo) = <u>P</u></span><u><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(A∩ B)</span></sub></u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> = <u>0,15 </u>= 0,5</span></div><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                             </span>P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">)</span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">      </span>0,30</span></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Dados dos sucesos independientes, la      probabilidad de que ocurran ambos es igual al producto de las      probabilidades de ocurrencia de cada uno de ellos por separado. Se dice      que dos sucesos A y B son independientes si se cumple: </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">P<sub>(A/B)</sub> = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">A</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">)      </span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">entonces P<sub>(</sub></span><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">A</span></sub><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">)</span></sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> = </span><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">P<sub>(A∩ B) </sub></span></u><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"></span></u></li></ul><div class="MsoNormal" style="margin-left: 18.0pt; mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                  </span>P<sub>(</sub></span><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">B</span></sub><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">)</span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoHeading8"><span lang="ES" style="mso-bidi-font-family: Arial; mso-bidi-font-size: 12.0pt;">Teorema de Bayes</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si en lugar de un suceso A, se tiene un conjunto de n sucesos A mutuamente excluyentes, que pueden producir el suceso B. Si la realización de un suceso B depende necesariamente de que ocurra uno de los acontecimientos excluyentes (A<sub>1</sub>; A<sub>2</sub>; A<sub>n</sub>) y se sabe que B se ha producido la probabilidad de que el suceso A, se haya producido juntamente con B esta dada por la expresión:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">P<sub>(A1/B)</sub> =<span style="mso-spacerun: yes;">  </span><u><span style="mso-spacerun: yes;">                    </span></u></span><u><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">P</span></u><u><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">(B/A1</span></sub></u><u><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">)</span></sub></u><u><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> . P<sub>(</sub></span></u><u><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">A1</span></sub></u><u><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">)</span></sub></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;">                                                           </span>P</span><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US; mso-fareast-language: ES;">(B/A1</span></sub><sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;">)</span></sub><span style="font-family: "Arial","sans-serif"; mso-ansi-language: EN-US;"> . </span><span lang="ES-AR" style="font-family: "Arial","sans-serif";">P<sub>(</sub></span><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">A1</span></sub><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">) + </span></sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">P</span><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">(B/A2</span></sub><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">)</span></sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";"> . P<sub>(</sub></span><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">A2</span></sub><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">) + </span></sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">P</span><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">(B/An</span></sub><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">)</span></sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";"> . P<sub>(</sub></span><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif"; mso-fareast-language: ES;">An</span></sub><sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";">)</span></sub><span lang="ES-AR" style="font-family: "Arial","sans-serif";"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Ejemplo: Tres cajas A1, A2, A3, contienen bolillas de las cuales x son rojas, el suceso B es igual a sacar una roja; que probabilidad hay que sea de la caja A1.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div></div><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span> <div class="WordSection17"> <div style="margin-bottom: .0001pt; margin: 0cm;"><b><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">ESTADÍSTICA</span></b></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La estadística es un método de descripción numérica de conjuntos numerosos, es decir, un método de descripción cuantitativa que utiliza el dato numérico como soporte. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Variable cierta: se llama al atributo (por ejemplo la altura).</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Frecuencias absolutas: Es el numero de veces que se repite el valor de la variable (cantidad de individuos para un valor determinado) Yi.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l22 level1 lfo16; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Frecuencias relativas: indica el porcentual o proporción de repetición de la variable. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                         </span>fi = <u>Yi</u><span style="mso-spacerun: yes;">  </span></span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                               </span>N<span style="mso-spacerun: yes;">  </span>(numero total de población)</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ fi = 1 porque N = ∑</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> Yi</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Medidas de posición</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Representan los valores características del conjunto, es decir la moda, la media, la mediana y los fractiles. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Modo o moda:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es el valor mas frecuente de la variable, el valor dominante. Un conjunto observado M<sub>O</sub>puede ser unimodal (moda única), cabe suponer que la serie de frecuencias corresponde a una población homogénea. Si la grafica presenta mas de un máximo, se llama multimodal o plurimodal y corresponde a una población heterogénea. La serie también pude ser amodal (carente de moda o todos los valores tienen la misma frecuencia).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">En el histograma el valor de la moda se halla buscando la intersección de las líneas punteadas en el intervalo modal de máxima frecuencia.</span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">M<sub>O</sub> = i +<u><span style="mso-spacerun: yes;">   </span>A<span style="mso-spacerun: yes;">    </span></u><span style="mso-spacerun: yes;"> </span>. ancho (intervalo)</span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">A + B</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">i es el valor menor del intervalo. / A es la altura a la izquierda<span style="mso-spacerun: yes;">  </span>/ B es la altura a la derecha</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="136" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image068.jpg" width="250" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Media aritmética:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es la suma de todos los valores observados dividido por el total de observaciones.</span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">X = </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">∑ <u>(xi.yi)</u> = ∑ (xi.fi)</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">    </span>N</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><img align="left" height="1" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image070.jpg" width="326" /><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Mediana (M<sub>E</sub>):</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es el valor de la variable cierta que divide la serie de frecuencias en dos partes iguales de individuos observados, ordenados por el valor creciente del atributo. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La mediana se determina gráficamente en forma aproximada como la abcisa que corresponde a la coordenada de: <u>N</u>en el grafico de frecuencias<span style="mso-spacerun: yes;">                 </span>2</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">acumuladas. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">                                            </span></span></div><div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm; text-align: center;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Fractiles: son los valores que representan una fracción del conjunto observado. Se usan cuando los anteriores valores característicos no representan el conjunto observado (o es muy disperso). Su calculo es muy parecido al de la mediana. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l21 level1 lfo17; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Quartiles: dividen en 4 partes al conjunto observado.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l21 level1 lfo17; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Deciles: dividen en 10 partes.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l21 level1 lfo17; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Percentiles en 10 partes.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cálculos de fractiles: gráficamente los cuartiles se determinan mediante el polígono de frecuencias acumuladas. Así Q<sub>1</sub> y Q<sub>3 </sub>serán las abscisas correspondientes a las ordenadas en <u>N</u> y 3<u>N</u> respectivamente. Q<sub>2</sub> es igual a <u>N</u>y Q<sub>4 </sub>es igual a N</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 35.4pt; margin-right: 0cm; margin-top: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-spacerun: yes;">             </span>4<span style="mso-spacerun: yes;">       </span>4<span style="mso-spacerun: yes;">                                        </span><span style="mso-spacerun: yes;">           </span>2 </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Para calcular estas medidas se acumulan las frecuencias, sumando a cada frecuencia absoluta todas las anteriores o todas las posteriores, con lo que se obtienen las frecuencias acumuladas. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l10 level1 lfo18; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Frecuencia acumulada izquierda absoluta F<sub>AI</sub>como el numero de individuos observados que poseen valores menores o iguales que un determinado valor de la variable. </span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l10 level1 lfo18; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Frecuencia acumulada derecha absoluta G<sub>AI</sub> como el numero de individuos observados que poseen valores mayores o iguales que un determinado valor de la variable. </span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si se divide cada frecuencia acumulada por el numero total de individuos observados N, se obtiene las frecuencias acumuladas relativas, F<sub>I</sub> y G<sub>I</sub></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Parámetros de dispersión o desvío:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> son las diferencia entre un valor cualquiera que puede tomar la variable y el valor medio. Estos parámetro indican la distancia a la que se encuentran los correspondientes valores respecto a la referencia. La suma de todas las desviaciones con respecto a la media es igual a 0.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">e<sub>I</sub>= x<sub>i</sub> - X</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Varianza:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es el promedio de los desvíos elevados al cuadrado, multiplicados por fi. La varianza representa la variabilidad que tienen los datos entre si, ósea, el área de dispersión de los datos tomando como centro el promedio.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">σ<sup>2</sup>= ∑ (x<sub>i</sub><sup>2</sup>.f<sub>i</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dispersión o desvío típico:</span></i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"> es la raíz cuadrada de la varianza. Representa la variabilidad o la distancia de los datos en promedio respecto de la media.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ = ∑ √ (x<sub>i</sub><sup>2</sup>.f<sub>i</sub>)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si el valor de </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">σ es grande, esta alejado de la mediana.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Si el valor de </span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">σ es pequeño, esta cerca de la mediana.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Por esto se introduce el concepto de coeficiente de variación: es la relación que existe entre el desvió estándar y la media aritmética, multiplicada por 100.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">C<sub>V</sub>= </span><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">σ</span></u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;"> . 100</span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">X</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><i><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Representación grafica</span></i></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se efectúa mediante un histograma, que esta formado por rectángulos cuya base es el intervalo de clase y cuya altura es igual a la frecuencia.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">También existe la representación poligonal, que en el histograma une los puntos medios de la base superior, se<span style="mso-spacerun: yes;">  </span>llama polígono de frecuencias y se indica con línea punteada.</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="191" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image072.jpg" width="335" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Además esta el grafico de frecuencias acumuladas izquierda F<sub>AI</sub> en termino de x.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="177" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image073.jpg" width="298" /></span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Ejemplo:</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">De una población de 500 personas se registran sus pesos obteniéndose los siguientes valores</span></div><div class="MsoNormal"><br /></div><div align="center"> <table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480;"> <tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">  <td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 158.4pt;" valign="top" width="211">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Peso en Kg.</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">50 – 60</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">60 – 70 </span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">70 – 80 </span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">80 – 90 </span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 57.8pt;" valign="top" width="77">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">90 – 100 </span></div></td> </tr><tr style="mso-yfti-irow: 1; mso-yfti-lastrow: yes;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 158.4pt;" valign="top" width="211">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Nro de personas</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">30</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">150</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">180</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">90</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 57.8pt;" valign="top" width="77">  <div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">50</span></div></td> </tr></tbody></table></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Averiguar moda, mediana, valor medio, desviación estándar, varianza.</span></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Graficar Histograma y frecuencia acumulada.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; mso-yfti-tbllook: 480; width: 631px;"> <tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">  <td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Intervalo</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Cantidad de datos yi</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Valor Medio ci</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Frecuencia Relativa</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">fi = yi/N</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Media</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">X= ci .fi</span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><br /></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">(ci-X)<sup>2</sup></span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Varianza</span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">σ<sup>2</sup> = </span></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">(ci-X)<sup>2 </sup>. fi</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">Frec. Acumul</span></div></td> </tr><tr style="mso-yfti-irow: 1;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">50–60</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">30</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">55</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,06</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">3,3</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">384,16</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">23,0496</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">30</span></div></td> </tr><tr style="mso-yfti-irow: 2;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">60–70</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">150</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">65</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,3</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">19,5</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">92,16</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">27,648</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">180</span></div></td> </tr><tr style="mso-yfti-irow: 3;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">70–80</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">180</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">75</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,36</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">27</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,16</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,0576</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">360</span></div></td> </tr><tr style="mso-yfti-irow: 4;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">80–90</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">80</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">85</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,18</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">15,3</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">108,16</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">19,4688</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">450</span></div></td> </tr><tr style="mso-yfti-irow: 5;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">90–100</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">50</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">95</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">0,1</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">9,5</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">416,16</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">41,61</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">500</span></div></td> </tr><tr style="mso-yfti-irow: 6; mso-yfti-lastrow: yes;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.85pt;" valign="top" width="76">  <div align="center" class="MsoNormal" style="text-align: center;"><br /></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 58.85pt;" valign="top" width="78">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">N = 500</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 43.5pt;" valign="top" width="58">  <div align="center" class="MsoNormal" style="text-align: center;"><br /></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 70.2pt;" valign="top" width="94">  <div align="center" class="MsoNormal" style="text-align: center;"><br /></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 55.0pt;" valign="top" width="73">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">X= 74,6</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 63.0pt;" valign="top" width="84">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><br /></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 72.0pt;" valign="top" width="96">  <div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial;">σ<sup>2</sup>=111,84</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 54.0pt;" valign="top" width="72">  <div align="center" class="MsoNormal" style="text-align: center;"><br /></div></td> </tr></tbody></table><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">σ= √111,84</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">σ= 10,57</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Moda = valor inicial intervalo +<u><span style="mso-spacerun: yes;">    </span>a<span style="mso-spacerun: yes;">      </span></u>. (ancho del intervalo)</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                                                  </span>a + b</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><img align="left" height="1" hspace="12" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image075.jpg" width="438" /><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Moda = 70 + (<u><span style="mso-spacerun: yes;">     </span>30<span style="mso-spacerun: yes;">    </span></u>. 10)</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">                       </span>30 + 90</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Moda = 72,5</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;"><img height="290" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image077.jpg" width="642" /></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">180</span></u><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">= <u>70</u></span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">10<span style="mso-spacerun: yes;">     </span><span style="mso-spacerun: yes;">  </span>x</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">x = 3,8</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Me = 70 + 3,8 = 73,8</span></div></div><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-AR;"><br clear="all" style="mso-break-type: section-break; page-break-before: always;" /></span> <div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Variable Aleatoria</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Cuando se lleva a cabo una experiencia aleatoria, se obtiene una cantidad que lleva asociada una probabilidad de ocurrencia; dicha cantidad recibe el nombre de variable aleatoria. Esta nos permite pasar de los resultados experimentales a una función numérica. El valor numérico depende del resultado del experimento.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">La variable aleatoria queda definida por una tabla del siguiente tipo:</span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><img height="73" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image079.jpg" width="610" /></span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Es variable porque son posibles diferentes valores numéricos. Es aleatoria porque el valor obtenido depende de cual de los posibles resultados aparezca<span style="mso-spacerun: yes;">  </span>y porque involucra la probabilidad de los resultados del espacio muestral.</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Las variables aleatorias llevan asociadas una probabilidad de ocurrencia y se calculan antes de la experiencia (a priori).</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Las variables aleatorias pueden ser discretas o continuas:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l18 level1 lfo19; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Discreta: Si el numero de valores que puede tener es contable (finito o infinito). Se utiliza, por ejemplo, para expresar cantidades de unidades defectuosas o le cantidad de vehículos que arriban o parten.</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l18 level1 lfo19; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Continua: Si se puede tomar cualquier valor numérico en un intervalo o conjuntos de intervalos. Se utiliza, por ejemplo, para expresar escalas de medición (tiempo, peso, distancia, temperatura, etc.)</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><u><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Distribuciones probabilísticas discretas</span></u></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Definición: es una realización mutuamente excluyente de todos los resultados numéricos posibles para esa variable aleatoria, de modo tal que la probabilidad de ocurrencia se relaciona en particular con cada caso. Se define: </span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">P(x = r) o P (r)<span style="mso-spacerun: yes;">  </span>r = valores que puede tomar x = variable aleatoria</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Se debe cumplir que la suma de las probabilidades puntuales de todos los valores sea igual a 1.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Se define como función de distribución<span style="mso-spacerun: yes;">  </span>o función de probabilidad acumulada a la probabilidad de que la variable aleatoria x sea </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">mayor o igual ; o menor o igual a un valor de r.</span></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l16 level1 lfo20; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Izquierda: </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P(x </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">     r) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ [P(r<sub>i</sub>)]</span></li><li class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-list: l16 level1 lfo20; tab-stops: list 36.0pt; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">Derecha: P(x </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≥</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> r) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ [P(r<sub>i</sub>)]</span></li></ul><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES; mso-fareast-language: ES;">Para las variables aleatorias se definen dos parámetros:</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l7 level1 lfo21; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Esperanza matemática: es el valor promedio de una variable aleatoria después de infinitas observaciones. </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">E(x) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ r<sub>i</sub>. P(r<sub>i</sub>)</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l7 level1 lfo21; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Varianza: es el valor esperado al cuadrado de las desviaciones de la variable aleatoria respecto de su esperanza matemática. Indica cuan dispersa puede ser la distribución:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">V(x)= </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ [ (r<sub>i</sub>- μ)<sup>2</sup> . P(r<sub>i</sub>) ]</span></div><div style="margin-bottom: .0001pt; margin: 0cm;"><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;"><span style="mso-tab-count: 1;">            </span>Si se introducen desvíos resulta</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">V(x)= </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ [ e<sub>i</sub><sup>2</sup>. P(r<sub>i</sub>) ]</span></div><div style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-list: l7 level1 lfo21; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span lang="ES" style="font-family: Symbol; mso-ansi-language: ES; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">         </span></span></span><span lang="ES" style="font-family: "Arial","sans-serif"; mso-ansi-language: ES;">Dispersión: es el valor que mide la dispersión de los valores que toma la variable respecto de la esperanza matemática.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">D(x) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">√σ² = σ</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Ejemplo: Siendo la variable aleatoria x la suma de las caras superiores que resulta al arrojar dos dados, se desea saber cual es la probabilidad de que cada valor que toma la variable y su esperanza matemática.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480;"> <tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">  <td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.7pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">R1</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">2</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">3</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">4</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">5</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">6</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">7</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">8</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">9</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">10</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">11</span></div></td>  <td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">12</span></div></td> </tr><tr style="mso-yfti-irow: 1; mso-yfti-lastrow: yes;">  <td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.7pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">P(ri)</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/36</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/18</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/12</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/9</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">5/36</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/6</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">5/36</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/9</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/12</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/18</span></div></td>  <td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 40.75pt;" valign="top" width="54">  <div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">1/36</span></div></td> </tr></tbody></table><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">E(x) = 2.1/36 + 3.1/18 …....... = 7</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">V(x)= </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">∑ [ (r<sub>i</sub> - 7)<sup>2</sup>. P(r<sub>i</sub>) ]</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">D(x) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">√σ² = σ = 2,4152</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Distribuciones probabilísticas continuas</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Esta caracterizada por una función que recibe el nombre de función de densidad de probabilidad. Esta función proporciona un medio para determinar la probabilidad de ocurrencia en un intervalo variable (a </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> x </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> b)</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">, ya que cada valor que toma la variable aleatoria no se le puede asociar una probabilidad de ocurrencia, porque esta probabilidad vale cero.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Propiedades:</span></div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l6 level1 lfo25; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: "MS Shell Dlg","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-fareast-font-family: "MS Shell Dlg"; mso-fareast-language: ES;"><span style="mso-list: Ignore;">a)<span style="font: 7.0pt "Times New Roman";">     </span></span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Especificar el dominio de la variable: -∞ </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> x </span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> ∞</span></div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l6 level1 lfo25; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: "MS Shell Dlg","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-fareast-font-family: "MS Shell Dlg"; mso-fareast-language: ES;"><span style="mso-list: Ignore;">b)<span style="font: 7.0pt "Times New Roman";">     </span></span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Para todo valor de x que toma la variable, Φ(x) es positiva.</span></div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l6 level1 lfo25; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: "MS Shell Dlg","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-fareast-font-family: "MS Shell Dlg"; mso-fareast-language: ES;"><span style="mso-list: Ignore;">c)<span style="font: 7.0pt "Times New Roman";">      </span></span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Las funciones de distribución de probabilidad acumulada izquierda y derecha de la </span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none; text-indent: 35.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">normal son:<sub><span style="mso-spacerun: yes;">                                                                      </span>–<u>(a-</u>μ<u>)2</u></sub></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none; text-indent: 35.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Izquierda: </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P(x </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≤</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> a) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Φ(a) = <u><span style="mso-spacerun: yes;">    </span>1<span style="mso-spacerun: yes;">      </span></u>. -<sub>∞ </sub>∫<sup>a</sup> e<span style="mso-spacerun: yes;">   </span><sup>2</sup></span><sup><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span></sup><sup><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">2<span style="mso-spacerun: yes;">   </span></span></sup><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">dx</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                 </span><span style="mso-spacerun: yes;">                                      </span></span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> √2</span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">π</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                                                                     </span>–<u>(a-</u>μ<u> )2</u></span></sub></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none; text-indent: 35.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Derecha: </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;">P(x </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">≥</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial;"> a) = </span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Φ(a) = <u><span style="mso-spacerun: yes;">    </span>1<span style="mso-spacerun: yes;">      </span></u>.<span style="mso-spacerun: yes;">  </span><sub>a</sub>∫<sup>∞</sup>e<span style="mso-spacerun: yes;">   </span><sup>2</sup></span><sup><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span></sup><sup><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">2<span style="mso-spacerun: yes;">   </span></span></sup><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">dx</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                </span><span style="mso-tab-count: 1;">    </span><span style="mso-spacerun: yes;">           </span><span style="mso-spacerun: yes;">   </span></span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> √2</span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">π</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 36.0pt; margin-right: 0cm; margin-top: 0cm; mso-layout-grid-align: none; mso-list: l6 level1 lfo25; tab-stops: list 36.0pt; text-autospace: none; text-indent: -18.0pt;"><span lang="ES" style="font-family: "MS Shell Dlg","sans-serif"; font-size: 12.0pt; mso-ansi-language: ES; mso-fareast-font-family: "MS Shell Dlg"; mso-fareast-language: ES;"><span style="mso-list: Ignore;">d)<span style="font: 7.0pt "Times New Roman";">     </span></span></span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">El área encerrada bajo la curva que describe dicha función, en todo el dominio es igual a 1.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none; text-indent: 35.4pt;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">-<sub>∞ </sub>∫<sup>∞</sup> Φ(x) dx</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><u><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Distribución normal</span></u></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Las duraciones son variables aleatorias que se comportan según la normal. La apariencia graficas de la distribución es una curva simétrica con forma de campana (Campana de Gauss) que se extiende sin límite tanto en la dirección positiva como la negativa.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Se define la función de densidad de la probabilidad normal como:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                         </span>–<u>(</u></span></sub><u><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">x</span></sub></u><u><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">- </span></sub></u><sub><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">μ</span></sub><u><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">)2</span></sub></u><sub><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></sub></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Φ</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">(</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">x</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">)=<u><span style="mso-spacerun: yes;">    </span>1<span style="mso-spacerun: yes;">      </span></u>.</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">e</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">   </span><sup>2</sup></span><sup><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span></sup><sup><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;">2</span></sup><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                                                                   </span></span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> √2</span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">π</span><span style="font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Donde μ es la esperanza matemática de la variable y la </span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"> dispersión.</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><img height="128" src="file:///C:\Users\Brian\AppData\Local\Temp\msohtmlclip1\01\clip_image081.jpg" width="328" /></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">El cálculo de estas funciones de distribución puede resolverse mediante el uso de integrales o mediante el uso de tablas.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">El valor de Φ(a) estará dado por el área bajo la campana de Gauss hasta el valor a.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Como las integrales no se pueden evaluar directamente, se utilizan las tablas, donde a esta distribución se le encuentra otra distribución estándar, a partir de una variable reducida (Z) que permite calcular estas funciones para cualquier distribución normal: </span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Z = <u>x – μ </u><span style="mso-spacerun: yes;">                    </span>variable estándar = centrada en la campana eje y</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"><span style="mso-spacerun: yes;">                        </span></span><span lang="EL" style="font-size: 12.0pt; mso-ansi-language: EL; mso-bidi-font-family: Arial; mso-fareast-language: ES;">σ</span><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;"></span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Esta variable reducida Z es normal con el eje de simetría en cero y puntos de inflexión en 1 y -1. Los valores de Z pueden encontrarse en tablas apropiadas y reducen el problema a una suma o resta de áreas de acuerdo con lo que se pida.</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Por la propiedad que tiene la distribución normal estándar de distribuirse simétricamente alrededor de su media (que vale cero) se demuestra que:</span></div><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; text-align: center; text-autospace: none;"><span lang="ES" style="font-size: 12.0pt; mso-ansi-language: ES; mso-bidi-font-family: Arial; mso-fareast-language: ES;">Φ(-z) = 1 – Φ (z)</span></div><div class="MsoNormal" style="mso-layout-grid-align: none; text-autospace: none;"><br /></div><div class="MsoNormal"><span style="font-size: 12.0pt; mso-bidi-font-family: Arial;">Ejemplo: Los pesos de un grupo de 600 personas se distribuye normalmente con valor medio 67 y desvío strandar 5. Calcular cual es la probabilidad de que pesen entre 60 y 80.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="FR" style="font-size: 12.0pt; mso-ansi-language: FR; mso-bidi-font-family: Arial;">Z1= <u>60 – 67</u> = -1,4<span style="mso-spacerun: yes;">                </span>Z2 = <u>80 – 67</u>= 2,6</span></div><div class="MsoNormal"><span lang="FR" style="font-size: 12.0pt; mso-ansi-language: FR; mso-bidi-font-family: Arial;"><span style="mso-spacerun: yes;">          </span><span style="mso-spacerun: yes;">  </span>5<span style="mso-spacerun: yes;">                                            </span>5</span></div><br /><div class="MsoNormal"><b><span lang="ES" style="font-family: "Arial","sans-serif"; font-size: 11.0pt; mso-ansi-language: ES; mso-bidi-font-size: 10.0pt; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"><br /></span></b></div><br /><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="FR" style="font-size: 12.0pt; mso-ansi-language: FR; mso-bidi-font-family: Arial;">P (60 < x < 80)= P (-1,4 < x < 2,6) = P(0 < x < 2,6) - P (0 < x < 1,4)</span></div><div class="MsoNormal"><span lang="FR" style="font-size: 12.0pt; mso-ansi-language: FR; mso-bidi-font-family: Arial;"><span style="mso-tab-count: 5;">                                                            </span><span style="mso-spacerun: yes;">  </span>= 0,9953 - (1 – 0,9192)<span style="mso-spacerun: yes;">  </span></span></div><div class="MsoNormal"><span lang="FR" style="font-size: 12.0pt; mso-ansi-language: FR; mso-bidi-font-family: Arial;"><span style="mso-tab-count: 5;">                                                            </span><span style="mso-spacerun: yes;">  </span>= 0,9953 - 0,0808 </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm;"><br /></div>