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- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
(x+5)^2-9
-
x^(7/2)-3
-
x^3/
-
x^4-3*x^2+4
-
Expresiones idénticas
- sin(z)/(z^ dos - uno)
- seno de (z) dividir por (z al cuadrado menos 1)
- seno de (z) dividir por (z en el grado dos menos uno)
- sin(z)/(z2-1)
- sinz/z2-1
- sin(z)/(z²-1)
- sin(z)/(z en el grado 2-1)
- sinz/z^2-1
- sin(z) dividir por (z^2-1)
-
Expresiones semejantes
- sin(z)/(z^2+1)
-
Expresiones con funciones
- Seno sin
- sin(x)+x^2-1
- sin(x)/2-1
- sin(x^2-x)
- sin(-sin(x)+3*x)/x
- sin(pi*6*x)
Gráfico de la función y = sin(z)/(z^2-1)
Solución
Ha introducido
[src]
sin(z)
f(z) = ------
2
z - 1
$$f{\left(z \right)} = \frac{\sin{\left(z \right)}}{z^{2} - 1}$$
f = sin(z)/(z^2 - 1)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{1} = -1$$
$$z_{2} = 1$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje Z con f = 0
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 69.1150383789755$$
$$z_{2} = 65.9734457253857$$
$$z_{3} = -91.106186954104$$
$$z_{4} = 248.185819633594$$
$$z_{5} = -59.6902604182061$$
$$z_{6} = -21.9911485751286$$
$$z_{7} = 12.5663706143592$$
$$z_{8} = 21.9911485751286$$
$$z_{9} = -144.51326206513$$
$$z_{10} = -414.690230273853$$
$$z_{11} = -100.530964914873$$
$$z_{12} = -69.1150383789755$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = -3.14159265358979$$
$$z_{15} = -25.1327412287183$$
$$z_{16} = -15.707963267949$$
$$z_{17} = -53.4070751110265$$
$$z_{18} = -72.2566310325652$$
$$z_{19} = 84.8230016469244$$
$$z_{20} = -81.6814089933346$$
$$z_{21} = -94.2477796076938$$
$$z_{22} = 18.8495559215388$$
$$z_{23} = -65.9734457253857$$
$$z_{24} = 94.2477796076938$$
$$z_{25} = 9.42477796076938$$
$$z_{26} = -40.8407044966673$$
$$z_{27} = 34.5575191894877$$
$$z_{28} = 0$$
$$z_{29} = 97.3893722612836$$
$$z_{30} = 53.4070751110265$$
$$z_{31} = -62.8318530717959$$
$$z_{32} = 59.6902604182061$$
$$z_{33} = -28.2743338823081$$
$$z_{34} = -56.5486677646163$$
$$z_{35} = 91.106186954104$$
$$z_{36} = 15.707963267949$$
$$z_{37} = -18.8495559215388$$
$$z_{38} = 6.28318530717959$$
$$z_{39} = 56.5486677646163$$
$$z_{40} = 87.9645943005142$$
$$z_{41} = 31.4159265358979$$
$$z_{42} = 25.1327412287183$$
$$z_{43} = 43.9822971502571$$
$$z_{44} = -47.1238898038469$$
$$z_{45} = 72.2566310325652$$
$$z_{46} = -34.5575191894877$$
$$z_{47} = 47.1238898038469$$
$$z_{48} = -97.3893722612836$$
$$z_{49} = -50.2654824574367$$
$$z_{50} = 100.530964914873$$
$$z_{51} = 81.6814089933346$$
$$z_{52} = -75.398223686155$$
$$z_{53} = 40.8407044966673$$
$$z_{54} = -9.42477796076938$$
$$z_{55} = 78.5398163397448$$
$$z_{56} = -87.9645943005142$$
$$z_{57} = 37.6991118430775$$
$$z_{58} = -78.5398163397448$$
$$z_{59} = -6.28318530717959$$
$$z_{60} = 50.2654824574367$$
$$z_{61} = -37.6991118430775$$
$$z_{62} = -43.9822971502571$$
$$z_{63} = -113.097335529233$$
$$z_{64} = 28.2743338823081$$
$$z_{65} = 62.8318530717959$$
$$z_{66} = -31.4159265358979$$
$$z_{67} = -12.5663706143592$$
$$z_{68} = 75.398223686155$$
$$z_{69} = -84.8230016469244$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 69.1150383789755$$
$$z_{2} = 65.9734457253857$$
$$z_{3} = -91.106186954104$$
$$z_{4} = 248.185819633594$$
$$z_{5} = -59.6902604182061$$
$$z_{6} = -21.9911485751286$$
$$z_{7} = 12.5663706143592$$
$$z_{8} = 21.9911485751286$$
$$z_{9} = -144.51326206513$$
$$z_{10} = -414.690230273853$$
$$z_{11} = -100.530964914873$$
$$z_{12} = -69.1150383789755$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = -3.14159265358979$$
$$z_{15} = -25.1327412287183$$
$$z_{16} = -15.707963267949$$
$$z_{17} = -53.4070751110265$$
$$z_{18} = -72.2566310325652$$
$$z_{19} = 84.8230016469244$$
$$z_{20} = -81.6814089933346$$
$$z_{21} = -94.2477796076938$$
$$z_{22} = 18.8495559215388$$
$$z_{23} = -65.9734457253857$$
$$z_{24} = 94.2477796076938$$
$$z_{25} = 9.42477796076938$$
$$z_{26} = -40.8407044966673$$
$$z_{27} = 34.5575191894877$$
$$z_{28} = 0$$
$$z_{29} = 97.3893722612836$$
$$z_{30} = 53.4070751110265$$
$$z_{31} = -62.8318530717959$$
$$z_{32} = 59.6902604182061$$
$$z_{33} = -28.2743338823081$$
$$z_{34} = -56.5486677646163$$
$$z_{35} = 91.106186954104$$
$$z_{36} = 15.707963267949$$
$$z_{37} = -18.8495559215388$$
$$z_{38} = 6.28318530717959$$
$$z_{39} = 56.5486677646163$$
$$z_{40} = 87.9645943005142$$
$$z_{41} = 31.4159265358979$$
$$z_{42} = 25.1327412287183$$
$$z_{43} = 43.9822971502571$$
$$z_{44} = -47.1238898038469$$
$$z_{45} = 72.2566310325652$$
$$z_{46} = -34.5575191894877$$
$$z_{47} = 47.1238898038469$$
$$z_{48} = -97.3893722612836$$
$$z_{49} = -50.2654824574367$$
$$z_{50} = 100.530964914873$$
$$z_{51} = 81.6814089933346$$
$$z_{52} = -75.398223686155$$
$$z_{53} = 40.8407044966673$$
$$z_{54} = -9.42477796076938$$
$$z_{55} = 78.5398163397448$$
$$z_{56} = -87.9645943005142$$
$$z_{57} = 37.6991118430775$$
$$z_{58} = -78.5398163397448$$
$$z_{59} = -6.28318530717959$$
$$z_{60} = 50.2654824574367$$
$$z_{61} = -37.6991118430775$$
$$z_{62} = -43.9822971502571$$
$$z_{63} = -113.097335529233$$
$$z_{64} = 28.2743338823081$$
$$z_{65} = 62.8318530717959$$
$$z_{66} = -31.4159265358979$$
$$z_{67} = -12.5663706143592$$
$$z_{68} = 75.398223686155$$
$$z_{69} = -84.8230016469244$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d z} f{\left(z \right)} = $$
primera derivada
$$- \frac{2 z \sin{\left(z \right)}}{\left(z^{2} - 1\right)^{2}} + \frac{\cos{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -51.7976718062027$$
$$z_{2} = -58.0850352160434$$
$$z_{3} = -61.2283950657729$$
$$z_{4} = 73.8003288675086$$
$$z_{5} = -42.3643000278463$$
$$z_{6} = -36.0728864084812$$
$$z_{7} = 86.3706429922226$$
$$z_{8} = -83.2281761528687$$
$$z_{9} = 32.9259992567895$$
$$z_{10} = 80.0856406984281$$
$$z_{11} = -4.2502319840436$$
$$z_{12} = 29.7779917432681$$
$$z_{13} = 26.6284652377851$$
$$z_{14} = -73.8003288675086$$
$$z_{15} = -32.9259992567895$$
$$z_{16} = -98.9399549958912$$
$$z_{17} = 42.3643000278463$$
$$z_{18} = 20.3220161353369$$
$$z_{19} = -67.5146210051587$$
$$z_{20} = -64.3715822869017$$
$$z_{21} = -39.2189234266452$$
$$z_{22} = 67.5146210051587$$
$$z_{23} = 23.4768059032848$$
$$z_{24} = 64.3715822869017$$
$$z_{25} = -86.3706429922226$$
$$z_{26} = -70.6575310493539$$
$$z_{27} = -89.5130484454873$$
$$z_{28} = -26.6284652377851$$
$$z_{29} = -7.59205618191083$$
$$z_{30} = -95.7976993646524$$
$$z_{31} = 39.2189234266452$$
$$z_{32} = 89.5130484454873$$
$$z_{33} = -17.1623570970183$$
$$z_{34} = 76.9430282181184$$
$$z_{35} = 4.2502319840436$$
$$z_{36} = -212.048072363693$$
$$z_{37} = 10.8111042087213$$
$$z_{38} = -92.6553987604331$$
$$z_{39} = 17.1623570970183$$
$$z_{40} = 83.2281761528687$$
$$z_{41} = -45.5091533451563$$
$$z_{42} = 54.9414730837878$$
$$z_{43} = -76.9430282181184$$
$$z_{44} = 13.9944961126907$$
$$z_{45} = -54.9414730837878$$
$$z_{46} = -48.6535849776189$$
$$z_{47} = 45.5091533451563$$
$$z_{48} = -13.9944961126907$$
$$z_{49} = 70.6575310493539$$
$$z_{50} = 98.9399549958912$$
$$z_{51} = -10.8111042087213$$
$$z_{52} = -20.3220161353369$$
$$z_{53} = 58.0850352160434$$
$$z_{54} = 61.2283950657729$$
$$z_{55} = 51.7976718062027$$
$$z_{56} = 36.0728864084812$$
$$z_{57} = 95.7976993646524$$
$$z_{58} = 472.805464302016$$
$$z_{59} = 92.6553987604331$$
$$z_{60} = -80.0856406984281$$
$$z_{61} = -29.7779917432681$$
$$z_{62} = 48.6535849776189$$
$$z_{63} = 7.59205618191083$$
$$z_{64} = -120.934779700424$$
$$z_{65} = 136.644644187573$$
$$z_{66} = -23.4768059032848$$
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:
$$z_{1} = -51.7976718062027$$
$$z_{2} = -58.0850352160434$$
$$z_{3} = 73.8003288675086$$
$$z_{4} = 86.3706429922226$$
$$z_{5} = -83.2281761528687$$
$$z_{6} = 80.0856406984281$$
$$z_{7} = 29.7779917432681$$
$$z_{8} = -32.9259992567895$$
$$z_{9} = 42.3643000278463$$
$$z_{10} = -64.3715822869017$$
$$z_{11} = -39.2189234266452$$
$$z_{12} = 67.5146210051587$$
$$z_{13} = 23.4768059032848$$
$$z_{14} = -70.6575310493539$$
$$z_{15} = -89.5130484454873$$
$$z_{16} = -26.6284652377851$$
$$z_{17} = -7.59205618191083$$
$$z_{18} = -95.7976993646524$$
$$z_{19} = 4.2502319840436$$
$$z_{20} = 10.8111042087213$$
$$z_{21} = 17.1623570970183$$
$$z_{22} = -45.5091533451563$$
$$z_{23} = 54.9414730837878$$
$$z_{24} = -76.9430282181184$$
$$z_{25} = -13.9944961126907$$
$$z_{26} = 98.9399549958912$$
$$z_{27} = -20.3220161353369$$
$$z_{28} = 61.2283950657729$$
$$z_{29} = 36.0728864084812$$
$$z_{30} = 92.6553987604331$$
$$z_{31} = 48.6535849776189$$
$$z_{32} = -120.934779700424$$
$$z_{33} = 136.644644187573$$
Puntos máximos de la función:
$$z_{33} = -61.2283950657729$$
$$z_{33} = -42.3643000278463$$
$$z_{33} = -36.0728864084812$$
$$z_{33} = 32.9259992567895$$
$$z_{33} = -4.2502319840436$$
$$z_{33} = 26.6284652377851$$
$$z_{33} = -73.8003288675086$$
$$z_{33} = -98.9399549958912$$
$$z_{33} = 20.3220161353369$$
$$z_{33} = -67.5146210051587$$
$$z_{33} = 64.3715822869017$$
$$z_{33} = -86.3706429922226$$
$$z_{33} = 39.2189234266452$$
$$z_{33} = 89.5130484454873$$
$$z_{33} = -17.1623570970183$$
$$z_{33} = 76.9430282181184$$
$$z_{33} = -212.048072363693$$
$$z_{33} = -92.6553987604331$$
$$z_{33} = 83.2281761528687$$
$$z_{33} = 13.9944961126907$$
$$z_{33} = -54.9414730837878$$
$$z_{33} = -48.6535849776189$$
$$z_{33} = 45.5091533451563$$
$$z_{33} = 70.6575310493539$$
$$z_{33} = -10.8111042087213$$
$$z_{33} = 58.0850352160434$$
$$z_{33} = 51.7976718062027$$
$$z_{33} = 95.7976993646524$$
$$z_{33} = 472.805464302016$$
$$z_{33} = -80.0856406984281$$
$$z_{33} = -29.7779917432681$$
$$z_{33} = 7.59205618191083$$
$$z_{33} = -23.4768059032848$$
Decrece en los intervalos
$$\left[136.644644187573, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -120.934779700424\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d z} f{\left(z \right)} = $$
primera derivada
$$- \frac{2 z \sin{\left(z \right)}}{\left(z^{2} - 1\right)^{2}} + \frac{\cos{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -51.7976718062027$$
$$z_{2} = -58.0850352160434$$
$$z_{3} = -61.2283950657729$$
$$z_{4} = 73.8003288675086$$
$$z_{5} = -42.3643000278463$$
$$z_{6} = -36.0728864084812$$
$$z_{7} = 86.3706429922226$$
$$z_{8} = -83.2281761528687$$
$$z_{9} = 32.9259992567895$$
$$z_{10} = 80.0856406984281$$
$$z_{11} = -4.2502319840436$$
$$z_{12} = 29.7779917432681$$
$$z_{13} = 26.6284652377851$$
$$z_{14} = -73.8003288675086$$
$$z_{15} = -32.9259992567895$$
$$z_{16} = -98.9399549958912$$
$$z_{17} = 42.3643000278463$$
$$z_{18} = 20.3220161353369$$
$$z_{19} = -67.5146210051587$$
$$z_{20} = -64.3715822869017$$
$$z_{21} = -39.2189234266452$$
$$z_{22} = 67.5146210051587$$
$$z_{23} = 23.4768059032848$$
$$z_{24} = 64.3715822869017$$
$$z_{25} = -86.3706429922226$$
$$z_{26} = -70.6575310493539$$
$$z_{27} = -89.5130484454873$$
$$z_{28} = -26.6284652377851$$
$$z_{29} = -7.59205618191083$$
$$z_{30} = -95.7976993646524$$
$$z_{31} = 39.2189234266452$$
$$z_{32} = 89.5130484454873$$
$$z_{33} = -17.1623570970183$$
$$z_{34} = 76.9430282181184$$
$$z_{35} = 4.2502319840436$$
$$z_{36} = -212.048072363693$$
$$z_{37} = 10.8111042087213$$
$$z_{38} = -92.6553987604331$$
$$z_{39} = 17.1623570970183$$
$$z_{40} = 83.2281761528687$$
$$z_{41} = -45.5091533451563$$
$$z_{42} = 54.9414730837878$$
$$z_{43} = -76.9430282181184$$
$$z_{44} = 13.9944961126907$$
$$z_{45} = -54.9414730837878$$
$$z_{46} = -48.6535849776189$$
$$z_{47} = 45.5091533451563$$
$$z_{48} = -13.9944961126907$$
$$z_{49} = 70.6575310493539$$
$$z_{50} = 98.9399549958912$$
$$z_{51} = -10.8111042087213$$
$$z_{52} = -20.3220161353369$$
$$z_{53} = 58.0850352160434$$
$$z_{54} = 61.2283950657729$$
$$z_{55} = 51.7976718062027$$
$$z_{56} = 36.0728864084812$$
$$z_{57} = 95.7976993646524$$
$$z_{58} = 472.805464302016$$
$$z_{59} = 92.6553987604331$$
$$z_{60} = -80.0856406984281$$
$$z_{61} = -29.7779917432681$$
$$z_{62} = 48.6535849776189$$
$$z_{63} = 7.59205618191083$$
$$z_{64} = -120.934779700424$$
$$z_{65} = 136.644644187573$$
$$z_{66} = -23.4768059032848$$
Signos de extremos en los puntos:
(-51.79767180620269, -0.000372578407377583)
(-58.08503521604338, -0.000296307594083531)
(-61.22839506577286, 0.000266672614366939)
(73.80032886750858, -0.000183570831307369)
(-42.36430002784626, 0.000556875375921231)
(-36.072886408481175, 0.000767899860645288)
(86.37064299222264, -0.000134032303380199)
(-83.22817615286866, -0.000144343274284351)
(32.925999256789495, 0.00092155585086097)
(80.0856406984281, -0.000155891696428242)
(-4.250231984043597, 0.0524535903376383)
(29.777991743268093, -0.0011264701660197)
(26.62846523778512, 0.00140830162078119)
(-73.80032886750858, 0.000183570831307369)
(-32.925999256789495, -0.00092155585086097)
(-98.93995499589117, 0.000102143849300632)
(42.36430002784626, -0.000556875375921231)
(20.322016135336863, 0.0024155503091525)
(-67.51462100515869, 0.000219335568761136)
(-64.37158228690171, -0.000241271950626197)
(-39.2189234266452, -0.000649720249572517)
(67.51462100515869, -0.000219335568761136)
(23.476805903284752, -0.00181106789272178)
(64.37158228690171, 0.000241271950626197)
(-86.37064299222264, 0.000134032303380199)
(-70.65753104935389, -0.000200260872068306)
(-89.5130484454873, -0.000124788081105762)
(-26.62846523778512, -0.00140830162078119)
(-7.592056181910829, -0.0170534046093743)
(-95.79769936465237, -0.000108953834823294)
(39.2189234266452, 0.000649720249572517)
(89.5130484454873, 0.000124788081105762)
(-17.162357097018344, 0.00338356230302691)
(76.94302821811836, 0.000168883841534665)
(4.250231984043597, -0.0524535903376383)
(-212.0480723636933, 2.22393292118406e-5)
(10.811104208721284, -0.00848320511338927)
(-92.65539876043312, 0.000116468359027335)
(17.162357097018344, -0.00338356230302691)
(83.22817615286866, 0.000144343274284351)
(-45.509153345156335, -0.000482606141533831)
(54.941473083787834, -0.00033117347900486)
(-76.94302821811836, -0.000168883841534665)
(13.994496112690735, 0.00508011541536346)
(-54.941473083787834, 0.00033117347900486)
(-48.653584977618934, 0.000422266745161591)
(45.509153345156335, 0.000482606141533831)
(-13.994496112690735, -0.00508011541536346)
(70.65753104935389, 0.000200260872068306)
(98.93995499589117, -0.000102143849300632)
(-10.811104208721284, 0.00848320511338927)
(-20.322016135336863, -0.0024155503091525)
(58.08503521604338, 0.000296307594083531)
(61.22839506577286, -0.000266672614366939)
(51.79767180620269, 0.000372578407377583)
(36.072886408481175, -0.000767899860645288)
(95.79769936465237, 0.000108953834823294)
(472.8054643020164, 4.47335209914827e-6)
(92.65539876043312, -0.000116468359027335)
(-80.0856406984281, 0.000155891696428242)
(-29.777991743268093, 0.0011264701660197)
(48.653584977618934, -0.000422266745161591)
(7.592056181910829, 0.0170534046093743)
(-120.93477970042365, -6.83703606024318e-5)
(136.6446441875733, -5.35539505172292e-5)
(-23.476805903284752, 0.00181106789272178)
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:
$$z_{1} = -51.7976718062027$$
$$z_{2} = -58.0850352160434$$
$$z_{3} = 73.8003288675086$$
$$z_{4} = 86.3706429922226$$
$$z_{5} = -83.2281761528687$$
$$z_{6} = 80.0856406984281$$
$$z_{7} = 29.7779917432681$$
$$z_{8} = -32.9259992567895$$
$$z_{9} = 42.3643000278463$$
$$z_{10} = -64.3715822869017$$
$$z_{11} = -39.2189234266452$$
$$z_{12} = 67.5146210051587$$
$$z_{13} = 23.4768059032848$$
$$z_{14} = -70.6575310493539$$
$$z_{15} = -89.5130484454873$$
$$z_{16} = -26.6284652377851$$
$$z_{17} = -7.59205618191083$$
$$z_{18} = -95.7976993646524$$
$$z_{19} = 4.2502319840436$$
$$z_{20} = 10.8111042087213$$
$$z_{21} = 17.1623570970183$$
$$z_{22} = -45.5091533451563$$
$$z_{23} = 54.9414730837878$$
$$z_{24} = -76.9430282181184$$
$$z_{25} = -13.9944961126907$$
$$z_{26} = 98.9399549958912$$
$$z_{27} = -20.3220161353369$$
$$z_{28} = 61.2283950657729$$
$$z_{29} = 36.0728864084812$$
$$z_{30} = 92.6553987604331$$
$$z_{31} = 48.6535849776189$$
$$z_{32} = -120.934779700424$$
$$z_{33} = 136.644644187573$$
Puntos máximos de la función:
$$z_{33} = -61.2283950657729$$
$$z_{33} = -42.3643000278463$$
$$z_{33} = -36.0728864084812$$
$$z_{33} = 32.9259992567895$$
$$z_{33} = -4.2502319840436$$
$$z_{33} = 26.6284652377851$$
$$z_{33} = -73.8003288675086$$
$$z_{33} = -98.9399549958912$$
$$z_{33} = 20.3220161353369$$
$$z_{33} = -67.5146210051587$$
$$z_{33} = 64.3715822869017$$
$$z_{33} = -86.3706429922226$$
$$z_{33} = 39.2189234266452$$
$$z_{33} = 89.5130484454873$$
$$z_{33} = -17.1623570970183$$
$$z_{33} = 76.9430282181184$$
$$z_{33} = -212.048072363693$$
$$z_{33} = -92.6553987604331$$
$$z_{33} = 83.2281761528687$$
$$z_{33} = 13.9944961126907$$
$$z_{33} = -54.9414730837878$$
$$z_{33} = -48.6535849776189$$
$$z_{33} = 45.5091533451563$$
$$z_{33} = 70.6575310493539$$
$$z_{33} = -10.8111042087213$$
$$z_{33} = 58.0850352160434$$
$$z_{33} = 51.7976718062027$$
$$z_{33} = 95.7976993646524$$
$$z_{33} = 472.805464302016$$
$$z_{33} = -80.0856406984281$$
$$z_{33} = -29.7779917432681$$
$$z_{33} = 7.59205618191083$$
$$z_{33} = -23.4768059032848$$
Decrece en los intervalos
$$\left[136.644644187573, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -120.934779700424\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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:
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = $$
segunda derivada
$$\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -47.03878961526$$
$$z_{2} = -113.061952083617$$
$$z_{3} = 34.4412164059948$$
$$z_{4} = -81.632396588364$$
$$z_{5} = -40.742428305889$$
$$z_{6} = 81.632396588364$$
$$z_{7} = 15.4472262173652$$
$$z_{8} = -31.2878642896602$$
$$z_{9} = 78.4888398981535$$
$$z_{10} = -87.9190881163862$$
$$z_{11} = 56.4778065045334$$
$$z_{12} = -12.2358820049333$$
$$z_{13} = 21.8070821937415$$
$$z_{14} = -59.6231409396343$$
$$z_{15} = -24.9721361872599$$
$$z_{16} = 69.0570950600626$$
$$z_{17} = 84.7758074362733$$
$$z_{18} = 50.1857258235068$$
$$z_{19} = -18.6338695613689$$
$$z_{20} = 12.2358820049333$$
$$z_{21} = -62.7680994904373$$
$$z_{22} = -94.2053111846222$$
$$z_{23} = 65.91273615789$$
$$z_{24} = -53.3320293640533$$
$$z_{25} = -100.491153848292$$
$$z_{26} = 24.9721361872599$$
$$z_{27} = 18.6338695613689$$
$$z_{28} = 91.0622521332484$$
$$z_{29} = 37.5925825893023$$
$$z_{30} = 5.5229319930101$$
$$z_{31} = 0$$
$$z_{32} = -5.5229319930101$$
$$z_{33} = 75.3451190666404$$
$$z_{34} = 100.491153848292$$
$$z_{35} = 119.347001200998$$
$$z_{36} = -75.3451190666404$$
$$z_{37} = 94.2053111846222$$
$$z_{38} = 43.8910837148979$$
$$z_{39} = 31.2878642896602$$
$$z_{40} = 40.742428305889$$
$$z_{41} = -50.1857258235068$$
$$z_{42} = 8.9698877316131$$
$$z_{43} = -15.4472262173652$$
$$z_{44} = 87.9190881163862$$
$$z_{45} = -8.9698877316131$$
$$z_{46} = 109.919347538894$$
$$z_{47} = -34.4412164059948$$
$$z_{48} = -56.4778065045334$$
$$z_{49} = -103.633954191491$$
$$z_{50} = -65.91273615789$$
$$z_{51} = -91.0622521332484$$
$$z_{52} = -69.0570950600626$$
$$z_{53} = -78.4888398981535$$
$$z_{54} = -28.1318477063748$$
$$z_{55} = 53.3320293640533$$
$$z_{56} = 47.03878961526$$
$$z_{57} = -84.7758074362733$$
$$z_{58} = -37.5925825893023$$
$$z_{59} = 241.886097146371$$
$$z_{60} = 28.1318477063748$$
$$z_{61} = 72.2012125964132$$
$$z_{62} = 62.7680994904373$$
$$z_{63} = -21.8070821937415$$
$$z_{64} = -43.8910837148979$$
$$z_{65} = -72.2012125964132$$
$$z_{66} = -97.3482754540447$$
$$z_{67} = 59.6231409396343$$
$$z_{68} = 97.3482754540447$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$\lim_{z \to -1^-}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to -1^+}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = \infty$$
- los límites no son iguales, signo
$$z_{1} = -1$$
- es el punto de flexión
$$\lim_{z \to 1^-}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to 1^+}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = \infty$$
- los límites no son iguales, signo
$$z_{2} = 1$$
- 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
$$\left[241.886097146371, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -103.633954191491\right]$$
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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:
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = $$
segunda derivada
$$\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -47.03878961526$$
$$z_{2} = -113.061952083617$$
$$z_{3} = 34.4412164059948$$
$$z_{4} = -81.632396588364$$
$$z_{5} = -40.742428305889$$
$$z_{6} = 81.632396588364$$
$$z_{7} = 15.4472262173652$$
$$z_{8} = -31.2878642896602$$
$$z_{9} = 78.4888398981535$$
$$z_{10} = -87.9190881163862$$
$$z_{11} = 56.4778065045334$$
$$z_{12} = -12.2358820049333$$
$$z_{13} = 21.8070821937415$$
$$z_{14} = -59.6231409396343$$
$$z_{15} = -24.9721361872599$$
$$z_{16} = 69.0570950600626$$
$$z_{17} = 84.7758074362733$$
$$z_{18} = 50.1857258235068$$
$$z_{19} = -18.6338695613689$$
$$z_{20} = 12.2358820049333$$
$$z_{21} = -62.7680994904373$$
$$z_{22} = -94.2053111846222$$
$$z_{23} = 65.91273615789$$
$$z_{24} = -53.3320293640533$$
$$z_{25} = -100.491153848292$$
$$z_{26} = 24.9721361872599$$
$$z_{27} = 18.6338695613689$$
$$z_{28} = 91.0622521332484$$
$$z_{29} = 37.5925825893023$$
$$z_{30} = 5.5229319930101$$
$$z_{31} = 0$$
$$z_{32} = -5.5229319930101$$
$$z_{33} = 75.3451190666404$$
$$z_{34} = 100.491153848292$$
$$z_{35} = 119.347001200998$$
$$z_{36} = -75.3451190666404$$
$$z_{37} = 94.2053111846222$$
$$z_{38} = 43.8910837148979$$
$$z_{39} = 31.2878642896602$$
$$z_{40} = 40.742428305889$$
$$z_{41} = -50.1857258235068$$
$$z_{42} = 8.9698877316131$$
$$z_{43} = -15.4472262173652$$
$$z_{44} = 87.9190881163862$$
$$z_{45} = -8.9698877316131$$
$$z_{46} = 109.919347538894$$
$$z_{47} = -34.4412164059948$$
$$z_{48} = -56.4778065045334$$
$$z_{49} = -103.633954191491$$
$$z_{50} = -65.91273615789$$
$$z_{51} = -91.0622521332484$$
$$z_{52} = -69.0570950600626$$
$$z_{53} = -78.4888398981535$$
$$z_{54} = -28.1318477063748$$
$$z_{55} = 53.3320293640533$$
$$z_{56} = 47.03878961526$$
$$z_{57} = -84.7758074362733$$
$$z_{58} = -37.5925825893023$$
$$z_{59} = 241.886097146371$$
$$z_{60} = 28.1318477063748$$
$$z_{61} = 72.2012125964132$$
$$z_{62} = 62.7680994904373$$
$$z_{63} = -21.8070821937415$$
$$z_{64} = -43.8910837148979$$
$$z_{65} = -72.2012125964132$$
$$z_{66} = -97.3482754540447$$
$$z_{67} = 59.6231409396343$$
$$z_{68} = 97.3482754540447$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$\lim_{z \to -1^-}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to -1^+}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = \infty$$
- los límites no son iguales, signo
$$z_{1} = -1$$
- es el punto de flexión
$$\lim_{z \to 1^-}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to 1^+}\left(\frac{- \frac{4 z \cos{\left(z \right)}}{z^{2} - 1} - \sin{\left(z \right)} + \frac{2 \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1}}{z^{2} - 1}\right) = \infty$$
- los límites no son iguales, signo
$$z_{2} = 1$$
- 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
$$\left[241.886097146371, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -103.633954191491\right]$$
Asíntotas verticales
Hay:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{1} = -1$$
$$z_{2} = 1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con z->+oo y z->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
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(z)/(z^2 - 1), dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z \left(z^{2} - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z \left(z^{2} - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z \left(z^{2} - 1\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z \left(z^{2} - 1\right)}\right) = 0$$
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(-z) и f = -f(-z).
Pues, comprobamos:
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = - \frac{\sin{\left(z \right)}}{z^{2} - 1}$$
- No
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = \frac{\sin{\left(z \right)}}{z^{2} - 1}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = - \frac{\sin{\left(z \right)}}{z^{2} - 1}$$
- No
$$\frac{\sin{\left(z \right)}}{z^{2} - 1} = \frac{\sin{\left(z \right)}}{z^{2} - 1}$$
- No
es decir, función
no es
par ni impar