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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
-sqrt(-1-x^2)
-
7-x-2*x^2
-
y=x^3+x
-
y=(x^3)/(x^2-4)
-
Expresiones idénticas
- sinx/(x^ dos - cuatro)
- seno de x dividir por (x al cuadrado menos 4)
- seno de x dividir por (x en el grado dos menos cuatro)
- sinx/(x2-4)
- sinx/x2-4
- sinx/(x²-4)
- sinx/(x en el grado 2-4)
- sinx/x^2-4
- sinx dividir por (x^2-4)
-
Expresiones semejantes
- sinx/(x^2+4)
-
Expresiones con funciones
- sinx
- sinx*sinx+cosx
- sinx+sin2x
- sinx:2
- sinx\(x(x-2))
- sinx(π/2)
Gráfico de la función y = sinx/(x^2-4)
Solución
Ha introducido
[src]
sin(x)
f(x) = ------
2
x - 4
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x^{2} - 4}$$
f = sin(x)/(x^2 - 4)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{1} = -2$$
$$x_{2} = 2$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = -18.8495559215388$$
$$x_{2} = -53.4070751110265$$
$$x_{3} = -37.6991118430775$$
$$x_{4} = 128.805298797182$$
$$x_{5} = -15.707963267949$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = -122.522113490002$$
$$x_{8} = -56.5486677646163$$
$$x_{9} = 12.5663706143592$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -31.4159265358979$$
$$x_{12} = 84.8230016469244$$
$$x_{13} = 94.2477796076938$$
$$x_{14} = 21.9911485751286$$
$$x_{15} = 0$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 81.6814089933346$$
$$x_{18} = 40.8407044966673$$
$$x_{19} = -75.398223686155$$
$$x_{20} = -78.5398163397448$$
$$x_{21} = 62.8318530717959$$
$$x_{22} = 100.530964914873$$
$$x_{23} = -21.9911485751286$$
$$x_{24} = 47.1238898038469$$
$$x_{25} = 91.106186954104$$
$$x_{26} = 75.398223686155$$
$$x_{27} = 28.2743338823081$$
$$x_{28} = -292.168116783851$$
$$x_{29} = 34.5575191894877$$
$$x_{30} = 6.28318530717959$$
$$x_{31} = 78.5398163397448$$
$$x_{32} = 72.2566310325652$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = 15.707963267949$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = -47.1238898038469$$
$$x_{37} = 25.1327412287183$$
$$x_{38} = 18.8495559215388$$
$$x_{39} = -94.2477796076938$$
$$x_{40} = -40.8407044966673$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -25.1327412287183$$
$$x_{43} = 53.4070751110265$$
$$x_{44} = -28.2743338823081$$
$$x_{45} = -9.42477796076938$$
$$x_{46} = 87.9645943005142$$
$$x_{47} = -50.2654824574367$$
$$x_{48} = -100.530964914873$$
$$x_{49} = -43.9822971502571$$
$$x_{50} = 50.2654824574367$$
$$x_{51} = -97.3893722612836$$
$$x_{52} = 69.1150383789755$$
$$x_{53} = 59.6902604182061$$
$$x_{54} = 97.3893722612836$$
$$x_{55} = -62.8318530717959$$
$$x_{56} = -72.2566310325652$$
$$x_{57} = 109.955742875643$$
$$x_{58} = -91.106186954104$$
$$x_{59} = -12.5663706143592$$
$$x_{60} = -69.1150383789755$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = 9.42477796076938$$
$$x_{63} = 65.9734457253857$$
$$x_{64} = -65.9734457253857$$
$$x_{65} = -84.8230016469244$$
$$x_{66} = -34.5575191894877$$
$$x_{67} = 131.946891450771$$
$$x_{68} = 43.9822971502571$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = -18.8495559215388$$
$$x_{2} = -53.4070751110265$$
$$x_{3} = -37.6991118430775$$
$$x_{4} = 128.805298797182$$
$$x_{5} = -15.707963267949$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = -122.522113490002$$
$$x_{8} = -56.5486677646163$$
$$x_{9} = 12.5663706143592$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -31.4159265358979$$
$$x_{12} = 84.8230016469244$$
$$x_{13} = 94.2477796076938$$
$$x_{14} = 21.9911485751286$$
$$x_{15} = 0$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 81.6814089933346$$
$$x_{18} = 40.8407044966673$$
$$x_{19} = -75.398223686155$$
$$x_{20} = -78.5398163397448$$
$$x_{21} = 62.8318530717959$$
$$x_{22} = 100.530964914873$$
$$x_{23} = -21.9911485751286$$
$$x_{24} = 47.1238898038469$$
$$x_{25} = 91.106186954104$$
$$x_{26} = 75.398223686155$$
$$x_{27} = 28.2743338823081$$
$$x_{28} = -292.168116783851$$
$$x_{29} = 34.5575191894877$$
$$x_{30} = 6.28318530717959$$
$$x_{31} = 78.5398163397448$$
$$x_{32} = 72.2566310325652$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = 15.707963267949$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = -47.1238898038469$$
$$x_{37} = 25.1327412287183$$
$$x_{38} = 18.8495559215388$$
$$x_{39} = -94.2477796076938$$
$$x_{40} = -40.8407044966673$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -25.1327412287183$$
$$x_{43} = 53.4070751110265$$
$$x_{44} = -28.2743338823081$$
$$x_{45} = -9.42477796076938$$
$$x_{46} = 87.9645943005142$$
$$x_{47} = -50.2654824574367$$
$$x_{48} = -100.530964914873$$
$$x_{49} = -43.9822971502571$$
$$x_{50} = 50.2654824574367$$
$$x_{51} = -97.3893722612836$$
$$x_{52} = 69.1150383789755$$
$$x_{53} = 59.6902604182061$$
$$x_{54} = 97.3893722612836$$
$$x_{55} = -62.8318530717959$$
$$x_{56} = -72.2566310325652$$
$$x_{57} = 109.955742875643$$
$$x_{58} = -91.106186954104$$
$$x_{59} = -12.5663706143592$$
$$x_{60} = -69.1150383789755$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = 9.42477796076938$$
$$x_{63} = 65.9734457253857$$
$$x_{64} = -65.9734457253857$$
$$x_{65} = -84.8230016469244$$
$$x_{66} = -34.5575191894877$$
$$x_{67} = 131.946891450771$$
$$x_{68} = 43.9822971502571$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} - 4\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 45.5090895943713$$
$$x_{2} = -80.0856290117635$$
$$x_{3} = 23.4763396531088$$
$$x_{4} = 4.15222477298826$$
$$x_{5} = 64.3715597764874$$
$$x_{6} = 70.657514030234$$
$$x_{7} = -23.4763396531088$$
$$x_{8} = 80.0856290117635$$
$$x_{9} = -10.8062270456027$$
$$x_{10} = -42.3642209820001$$
$$x_{11} = -58.0850045720611$$
$$x_{12} = 73.8003139321454$$
$$x_{13} = 89.5130400768468$$
$$x_{14} = -26.6281461112737$$
$$x_{15} = -98.9399487990654$$
$$x_{16} = -13.9922722908994$$
$$x_{17} = -83.2281657410082$$
$$x_{18} = 67.51460149576$$
$$x_{19} = -67.51460149576$$
$$x_{20} = 17.1611578256127$$
$$x_{21} = -64.3715597764874$$
$$x_{22} = -48.6535328151386$$
$$x_{23} = -4.15222477298826$$
$$x_{24} = -45.5090895943713$$
$$x_{25} = 61.2283689056001$$
$$x_{26} = 95.7976925376814$$
$$x_{27} = -39.2188237682302$$
$$x_{28} = -290.590437700776$$
$$x_{29} = 42.3642209820001$$
$$x_{30} = -7.57754404149509$$
$$x_{31} = 7.57754404149509$$
$$x_{32} = -20.3212959521274$$
$$x_{33} = -86.3706336762639$$
$$x_{34} = 83.2281657410082$$
$$x_{35} = 58.0850045720611$$
$$x_{36} = -124.076787974095$$
$$x_{37} = 54.9414368692733$$
$$x_{38} = 48.6535328151386$$
$$x_{39} = -95.7976925376814$$
$$x_{40} = -36.0727582894777$$
$$x_{41} = 10.8062270456027$$
$$x_{42} = 32.9258307018341$$
$$x_{43} = -32.9258307018341$$
$$x_{44} = 98.9399487990654$$
$$x_{45} = -17.1611578256127$$
$$x_{46} = 202.62285496058$$
$$x_{47} = 36.0727582894777$$
$$x_{48} = 92.655391214883$$
$$x_{49} = -73.8003139321454$$
$$x_{50} = -29.777763739304$$
$$x_{51} = -70.657514030234$$
$$x_{52} = -155.495972864649$$
$$x_{53} = 26.6281461112737$$
$$x_{54} = 76.9430150396882$$
$$x_{55} = -76.9430150396882$$
$$x_{56} = -51.7976285839881$$
$$x_{57} = 20.3212959521274$$
$$x_{58} = -61.2283689056001$$
$$x_{59} = 86.3706336762639$$
$$x_{60} = -54.9414368692733$$
$$x_{61} = -92.655391214883$$
$$x_{62} = 51.7976285839881$$
$$x_{63} = 13.9922722908994$$
$$x_{64} = -89.5130400768468$$
$$x_{65} = 39.2188237682302$$
$$x_{66} = 29.777763739304$$
Signos de extremos en los puntos:
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 23.4763396531088$$
$$x_{2} = 4.15222477298826$$
$$x_{3} = 80.0856290117635$$
$$x_{4} = -58.0850045720611$$
$$x_{5} = 73.8003139321454$$
$$x_{6} = -26.6281461112737$$
$$x_{7} = -13.9922722908994$$
$$x_{8} = -83.2281657410082$$
$$x_{9} = 67.51460149576$$
$$x_{10} = 17.1611578256127$$
$$x_{11} = -64.3715597764874$$
$$x_{12} = -45.5090895943713$$
$$x_{13} = 61.2283689056001$$
$$x_{14} = -39.2188237682302$$
$$x_{15} = -290.590437700776$$
$$x_{16} = 42.3642209820001$$
$$x_{17} = -7.57754404149509$$
$$x_{18} = -20.3212959521274$$
$$x_{19} = 54.9414368692733$$
$$x_{20} = 48.6535328151386$$
$$x_{21} = -95.7976925376814$$
$$x_{22} = 10.8062270456027$$
$$x_{23} = -32.9258307018341$$
$$x_{24} = 98.9399487990654$$
$$x_{25} = 36.0727582894777$$
$$x_{26} = 92.655391214883$$
$$x_{27} = -70.657514030234$$
$$x_{28} = -76.9430150396882$$
$$x_{29} = -51.7976285839881$$
$$x_{30} = 86.3706336762639$$
$$x_{31} = -89.5130400768468$$
$$x_{32} = 29.777763739304$$
Puntos máximos de la función:
$$x_{32} = 45.5090895943713$$
$$x_{32} = -80.0856290117635$$
$$x_{32} = 64.3715597764874$$
$$x_{32} = 70.657514030234$$
$$x_{32} = -23.4763396531088$$
$$x_{32} = -10.8062270456027$$
$$x_{32} = -42.3642209820001$$
$$x_{32} = 89.5130400768468$$
$$x_{32} = -98.9399487990654$$
$$x_{32} = -67.51460149576$$
$$x_{32} = -48.6535328151386$$
$$x_{32} = -4.15222477298826$$
$$x_{32} = 95.7976925376814$$
$$x_{32} = 7.57754404149509$$
$$x_{32} = -86.3706336762639$$
$$x_{32} = 83.2281657410082$$
$$x_{32} = 58.0850045720611$$
$$x_{32} = -124.076787974095$$
$$x_{32} = -36.0727582894777$$
$$x_{32} = 32.9258307018341$$
$$x_{32} = -17.1611578256127$$
$$x_{32} = 202.62285496058$$
$$x_{32} = -73.8003139321454$$
$$x_{32} = -29.777763739304$$
$$x_{32} = -155.495972864649$$
$$x_{32} = 26.6281461112737$$
$$x_{32} = 76.9430150396882$$
$$x_{32} = 20.3212959521274$$
$$x_{32} = -61.2283689056001$$
$$x_{32} = -54.9414368692733$$
$$x_{32} = -92.655391214883$$
$$x_{32} = 51.7976285839881$$
$$x_{32} = 13.9922722908994$$
$$x_{32} = 39.2188237682302$$
Decrece en los intervalos
$$\left[98.9399487990654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -290.590437700776\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} - 4\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 45.5090895943713$$
$$x_{2} = -80.0856290117635$$
$$x_{3} = 23.4763396531088$$
$$x_{4} = 4.15222477298826$$
$$x_{5} = 64.3715597764874$$
$$x_{6} = 70.657514030234$$
$$x_{7} = -23.4763396531088$$
$$x_{8} = 80.0856290117635$$
$$x_{9} = -10.8062270456027$$
$$x_{10} = -42.3642209820001$$
$$x_{11} = -58.0850045720611$$
$$x_{12} = 73.8003139321454$$
$$x_{13} = 89.5130400768468$$
$$x_{14} = -26.6281461112737$$
$$x_{15} = -98.9399487990654$$
$$x_{16} = -13.9922722908994$$
$$x_{17} = -83.2281657410082$$
$$x_{18} = 67.51460149576$$
$$x_{19} = -67.51460149576$$
$$x_{20} = 17.1611578256127$$
$$x_{21} = -64.3715597764874$$
$$x_{22} = -48.6535328151386$$
$$x_{23} = -4.15222477298826$$
$$x_{24} = -45.5090895943713$$
$$x_{25} = 61.2283689056001$$
$$x_{26} = 95.7976925376814$$
$$x_{27} = -39.2188237682302$$
$$x_{28} = -290.590437700776$$
$$x_{29} = 42.3642209820001$$
$$x_{30} = -7.57754404149509$$
$$x_{31} = 7.57754404149509$$
$$x_{32} = -20.3212959521274$$
$$x_{33} = -86.3706336762639$$
$$x_{34} = 83.2281657410082$$
$$x_{35} = 58.0850045720611$$
$$x_{36} = -124.076787974095$$
$$x_{37} = 54.9414368692733$$
$$x_{38} = 48.6535328151386$$
$$x_{39} = -95.7976925376814$$
$$x_{40} = -36.0727582894777$$
$$x_{41} = 10.8062270456027$$
$$x_{42} = 32.9258307018341$$
$$x_{43} = -32.9258307018341$$
$$x_{44} = 98.9399487990654$$
$$x_{45} = -17.1611578256127$$
$$x_{46} = 202.62285496058$$
$$x_{47} = 36.0727582894777$$
$$x_{48} = 92.655391214883$$
$$x_{49} = -73.8003139321454$$
$$x_{50} = -29.777763739304$$
$$x_{51} = -70.657514030234$$
$$x_{52} = -155.495972864649$$
$$x_{53} = 26.6281461112737$$
$$x_{54} = 76.9430150396882$$
$$x_{55} = -76.9430150396882$$
$$x_{56} = -51.7976285839881$$
$$x_{57} = 20.3212959521274$$
$$x_{58} = -61.2283689056001$$
$$x_{59} = 86.3706336762639$$
$$x_{60} = -54.9414368692733$$
$$x_{61} = -92.655391214883$$
$$x_{62} = 51.7976285839881$$
$$x_{63} = 13.9922722908994$$
$$x_{64} = -89.5130400768468$$
$$x_{65} = 39.2188237682302$$
$$x_{66} = 29.777763739304$$
Signos de extremos en los puntos:
(45.50908959437133, 0.000483306558706209)
(-80.08562901176347, 0.000155964659973988)
(23.476339653108777, -0.00182099790998716)
(4.152224772988261, -0.0639807988925772)
(64.37155977648736, 0.000241446798078732)
(70.65751403023398, 0.000200381305938913)
(-23.476339653108777, 0.00182099790998716)
(80.08562901176347, -0.000155964659973988)
(-10.806227045602704, 0.0087087677703681)
(-42.36422098200009, 0.000557808305842402)
(-58.08500457206114, -0.000296571379601852)
(73.80031393214543, -0.000183672018953665)
(89.51304007684683, 0.000124834826472013)
(-26.62814611127368, -0.00141429382702696)
(-98.9399487990654, 0.000102175165394175)
(-13.992272290899354, -0.00515956885743496)
(-83.22816574100821, -0.000144405824378095)
(67.51460149576005, -0.000219480051529099)
(-67.51460149576005, 0.000219480051529099)
(17.16115782561268, -0.00341850130927523)
(-64.37155977648736, -0.000241446798078732)
(-48.653532815138554, 0.000422802805149809)
(-4.152224772988261, 0.0639807988925772)
(-45.50908959437133, -0.000483306558706209)
(61.22836890560009, -0.000266886242151348)
(95.7976925376814, 0.000108989467051541)
(-39.21882376823015, -0.000650990789484618)
(-290.5904377007758, -1.18426160073303e-5)
(42.36422098200009, -0.000557808305842402)
(-7.577544041495088, -0.0180091484531043)
(7.577544041495088, 0.0180091484531043)
(-20.32129595212744, -0.00243326959458963)
(-86.37063367626389, 0.000134086233502956)
(83.22816574100821, 0.000144405824378095)
(58.08500457206114, 0.000296571379601852)
(-124.07678797409463, 6.49643845271334e-5)
(54.94143686927333, -0.000331503052571738)
(48.653532815138554, -0.000422802805149809)
(-95.7976925376814, -0.000108989467051541)
(-36.072758289477676, 0.0007696756958536)
(10.806227045602704, -0.0087087677703681)
(32.92583070183405, 0.000924115453595151)
(-32.92583070183405, -0.000924115453595151)
(98.9399487990654, -0.000102175165394175)
(-17.16115782561268, 0.00341850130927523)
(202.62285496058013, 2.43581496647271e-5)
(36.072758289477676, -0.0007696756958536)
(92.65539121488297, -0.000116509077378612)
(-73.80031393214543, 0.000183672018953665)
(-29.777763739304003, 0.00113029856328114)
(-70.65751403023398, -0.000200381305938913)
(-155.49597286464936, 4.1361628405071e-5)
(26.62814611127368, 0.00141429382702696)
(76.94301503968819, 0.000168969479119145)
(-76.94301503968819, -0.000168969479119145)
(-51.79762858398812, -0.000372995628973321)
(20.32129595212744, 0.00243326959458963)
(-61.22836890560009, 0.000266886242151348)
(86.37063367626389, -0.000134086233502956)
(-54.94143686927333, 0.000331503052571738)
(-92.65539121488297, 0.000116509077378612)
(51.79762858398812, 0.000372995628973321)
(13.992272290899354, 0.00515956885743496)
(-89.51304007684683, -0.000124834826472013)
(39.21882376823015, 0.000650990789484618)
(29.777763739304003, -0.00113029856328114)
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 23.4763396531088$$
$$x_{2} = 4.15222477298826$$
$$x_{3} = 80.0856290117635$$
$$x_{4} = -58.0850045720611$$
$$x_{5} = 73.8003139321454$$
$$x_{6} = -26.6281461112737$$
$$x_{7} = -13.9922722908994$$
$$x_{8} = -83.2281657410082$$
$$x_{9} = 67.51460149576$$
$$x_{10} = 17.1611578256127$$
$$x_{11} = -64.3715597764874$$
$$x_{12} = -45.5090895943713$$
$$x_{13} = 61.2283689056001$$
$$x_{14} = -39.2188237682302$$
$$x_{15} = -290.590437700776$$
$$x_{16} = 42.3642209820001$$
$$x_{17} = -7.57754404149509$$
$$x_{18} = -20.3212959521274$$
$$x_{19} = 54.9414368692733$$
$$x_{20} = 48.6535328151386$$
$$x_{21} = -95.7976925376814$$
$$x_{22} = 10.8062270456027$$
$$x_{23} = -32.9258307018341$$
$$x_{24} = 98.9399487990654$$
$$x_{25} = 36.0727582894777$$
$$x_{26} = 92.655391214883$$
$$x_{27} = -70.657514030234$$
$$x_{28} = -76.9430150396882$$
$$x_{29} = -51.7976285839881$$
$$x_{30} = 86.3706336762639$$
$$x_{31} = -89.5130400768468$$
$$x_{32} = 29.777763739304$$
Puntos máximos de la función:
$$x_{32} = 45.5090895943713$$
$$x_{32} = -80.0856290117635$$
$$x_{32} = 64.3715597764874$$
$$x_{32} = 70.657514030234$$
$$x_{32} = -23.4763396531088$$
$$x_{32} = -10.8062270456027$$
$$x_{32} = -42.3642209820001$$
$$x_{32} = 89.5130400768468$$
$$x_{32} = -98.9399487990654$$
$$x_{32} = -67.51460149576$$
$$x_{32} = -48.6535328151386$$
$$x_{32} = -4.15222477298826$$
$$x_{32} = 95.7976925376814$$
$$x_{32} = 7.57754404149509$$
$$x_{32} = -86.3706336762639$$
$$x_{32} = 83.2281657410082$$
$$x_{32} = 58.0850045720611$$
$$x_{32} = -124.076787974095$$
$$x_{32} = -36.0727582894777$$
$$x_{32} = 32.9258307018341$$
$$x_{32} = -17.1611578256127$$
$$x_{32} = 202.62285496058$$
$$x_{32} = -73.8003139321454$$
$$x_{32} = -29.777763739304$$
$$x_{32} = -155.495972864649$$
$$x_{32} = 26.6281461112737$$
$$x_{32} = 76.9430150396882$$
$$x_{32} = 20.3212959521274$$
$$x_{32} = -61.2283689056001$$
$$x_{32} = -54.9414368692733$$
$$x_{32} = -92.655391214883$$
$$x_{32} = 51.7976285839881$$
$$x_{32} = 13.9922722908994$$
$$x_{32} = 39.2188237682302$$
Decrece en los intervalos
$$\left[98.9399487990654, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -290.590437700776\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -34.4409194318359$$
$$x_{2} = -69.0570585223772$$
$$x_{3} = -81.6323744858714$$
$$x_{4} = 5.40521123343275$$
$$x_{5} = -103.633943397032$$
$$x_{6} = 91.0622362167619$$
$$x_{7} = -72.2011806345949$$
$$x_{8} = 65.9126941265009$$
$$x_{9} = -65.9126941265009$$
$$x_{10} = 21.80589274806$$
$$x_{11} = -100.491142008147$$
$$x_{12} = -12.2287224949723$$
$$x_{13} = 119.346994135494$$
$$x_{14} = -37.5923546185444$$
$$x_{15} = -59.6230841163499$$
$$x_{16} = -87.91907042905$$
$$x_{17} = 62.768050805063$$
$$x_{18} = 28.13129974445$$
$$x_{19} = 87.91907042905$$
$$x_{20} = 97.348262428604$$
$$x_{21} = 24.9713492935255$$
$$x_{22} = -31.2874672441028$$
$$x_{23} = 31.2874672441028$$
$$x_{24} = 56.4777396209399$$
$$x_{25} = 84.7757877051701$$
$$x_{26} = -28.13129974445$$
$$x_{27} = -56.4777396209399$$
$$x_{28} = -47.0386736392297$$
$$x_{29} = 15.4437856317982$$
$$x_{30} = -43.890940829429$$
$$x_{31} = 50.1856303934487$$
$$x_{32} = -84.7757877051701$$
$$x_{33} = -97.348262428604$$
$$x_{34} = 72.2011806345949$$
$$x_{35} = -109.919338493527$$
$$x_{36} = -5.40521123343275$$
$$x_{37} = 43.890940829429$$
$$x_{38} = -53.3319498938786$$
$$x_{39} = -40.7422494725529$$
$$x_{40} = 37.5923546185444$$
$$x_{41} = 0$$
$$x_{42} = -15.4437856317982$$
$$x_{43} = 94.2052968101638$$
$$x_{44} = -75.3450909469212$$
$$x_{45} = -62.768050805063$$
$$x_{46} = -91.0622362167619$$
$$x_{47} = -24.9713492935255$$
$$x_{48} = 78.4888150282294$$
$$x_{49} = -78.4888150282294$$
$$x_{50} = -21.80589274806$$
$$x_{51} = 81.6323744858714$$
$$x_{52} = 18.6319434077619$$
$$x_{53} = -8.95019260991225$$
$$x_{54} = 75.3450909469212$$
$$x_{55} = 69.0570585223772$$
$$x_{56} = -50.1856303934487$$
$$x_{57} = 34.4409194318359$$
$$x_{58} = 59.6230841163499$$
$$x_{59} = 47.0386736392297$$
$$x_{60} = -94.2052968101638$$
$$x_{61} = 40.7422494725529$$
$$x_{62} = 53.3319498938786$$
$$x_{63} = 100.491142008147$$
$$x_{64} = 166.480379738649$$
$$x_{65} = -18.6319434077619$$
$$x_{66} = 12.2287224949723$$
$$x_{67} = 8.95019260991225$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$\lim_{x \to -2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 2$$
- es el punto de flexión
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[166.480379738649, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -109.919338493527\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -34.4409194318359$$
$$x_{2} = -69.0570585223772$$
$$x_{3} = -81.6323744858714$$
$$x_{4} = 5.40521123343275$$
$$x_{5} = -103.633943397032$$
$$x_{6} = 91.0622362167619$$
$$x_{7} = -72.2011806345949$$
$$x_{8} = 65.9126941265009$$
$$x_{9} = -65.9126941265009$$
$$x_{10} = 21.80589274806$$
$$x_{11} = -100.491142008147$$
$$x_{12} = -12.2287224949723$$
$$x_{13} = 119.346994135494$$
$$x_{14} = -37.5923546185444$$
$$x_{15} = -59.6230841163499$$
$$x_{16} = -87.91907042905$$
$$x_{17} = 62.768050805063$$
$$x_{18} = 28.13129974445$$
$$x_{19} = 87.91907042905$$
$$x_{20} = 97.348262428604$$
$$x_{21} = 24.9713492935255$$
$$x_{22} = -31.2874672441028$$
$$x_{23} = 31.2874672441028$$
$$x_{24} = 56.4777396209399$$
$$x_{25} = 84.7757877051701$$
$$x_{26} = -28.13129974445$$
$$x_{27} = -56.4777396209399$$
$$x_{28} = -47.0386736392297$$
$$x_{29} = 15.4437856317982$$
$$x_{30} = -43.890940829429$$
$$x_{31} = 50.1856303934487$$
$$x_{32} = -84.7757877051701$$
$$x_{33} = -97.348262428604$$
$$x_{34} = 72.2011806345949$$
$$x_{35} = -109.919338493527$$
$$x_{36} = -5.40521123343275$$
$$x_{37} = 43.890940829429$$
$$x_{38} = -53.3319498938786$$
$$x_{39} = -40.7422494725529$$
$$x_{40} = 37.5923546185444$$
$$x_{41} = 0$$
$$x_{42} = -15.4437856317982$$
$$x_{43} = 94.2052968101638$$
$$x_{44} = -75.3450909469212$$
$$x_{45} = -62.768050805063$$
$$x_{46} = -91.0622362167619$$
$$x_{47} = -24.9713492935255$$
$$x_{48} = 78.4888150282294$$
$$x_{49} = -78.4888150282294$$
$$x_{50} = -21.80589274806$$
$$x_{51} = 81.6323744858714$$
$$x_{52} = 18.6319434077619$$
$$x_{53} = -8.95019260991225$$
$$x_{54} = 75.3450909469212$$
$$x_{55} = 69.0570585223772$$
$$x_{56} = -50.1856303934487$$
$$x_{57} = 34.4409194318359$$
$$x_{58} = 59.6230841163499$$
$$x_{59} = 47.0386736392297$$
$$x_{60} = -94.2052968101638$$
$$x_{61} = 40.7422494725529$$
$$x_{62} = 53.3319498938786$$
$$x_{63} = 100.491142008147$$
$$x_{64} = 166.480379738649$$
$$x_{65} = -18.6319434077619$$
$$x_{66} = 12.2287224949723$$
$$x_{67} = 8.95019260991225$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$\lim_{x \to -2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 2$$
- es el punto de flexión
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[166.480379738649, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -109.919338493527\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{1} = -2$$
$$x_{2} = 2$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/(x^2 - 4), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = - \frac{\sin{\left(x \right)}}{x^{2} - 4}$$
- No
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = \frac{\sin{\left(x \right)}}{x^{2} - 4}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = - \frac{\sin{\left(x \right)}}{x^{2} - 4}$$
- No
$$\frac{\sin{\left(x \right)}}{x^{2} - 4} = \frac{\sin{\left(x \right)}}{x^{2} - 4}$$
- No
es decir, función
no es
par ni impar