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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+27/x^3
-
(x^2-8)/(x-3)
-
x-2+4/(x-2)
-
x^2-9*x+14
-
Expresiones idénticas
- z*sin(z)/(z^ dos - uno)
- z multiplicar por seno de (z) dividir por (z al cuadrado menos 1)
- z multiplicar por seno de (z) dividir por (z en el grado dos menos uno)
- z*sin(z)/(z2-1)
- z*sinz/z2-1
- z*sin(z)/(z²-1)
- z*sin(z)/(z en el grado 2-1)
- zsin(z)/(z^2-1)
- zsin(z)/(z2-1)
- zsinz/z2-1
- zsinz/z^2-1
- z*sin(z) dividir por (z^2-1)
-
Expresiones semejantes
- z*sin(z)/(z^2+1)
-
Expresiones con funciones
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
- sin(3x-29)
Gráfico de la función y = z*sin(z)/(z^2-1)
Solución
Ha introducido
[src]
z*sin(z)
f(z) = --------
2
z - 1
$$f{\left(z \right)} = \frac{z \sin{\left(z \right)}}{z^{2} - 1}$$
f = (z*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:
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 65.9734457253857$$
$$z_{2} = -15.707963267949$$
$$z_{3} = 37.6991118430775$$
$$z_{4} = -69.1150383789755$$
$$z_{5} = -34.5575191894877$$
$$z_{6} = -91.106186954104$$
$$z_{7} = 0$$
$$z_{8} = 6.28318530717959$$
$$z_{9} = -47.1238898038469$$
$$z_{10} = -78.5398163397448$$
$$z_{11} = -28.2743338823081$$
$$z_{12} = 97.3893722612836$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = 40.8407044966673$$
$$z_{15} = 62.8318530717959$$
$$z_{16} = -81.6814089933346$$
$$z_{17} = 43.9822971502571$$
$$z_{18} = -59.6902604182061$$
$$z_{19} = 100.530964914873$$
$$z_{20} = -84.8230016469244$$
$$z_{21} = 69.1150383789755$$
$$z_{22} = 185.353966561798$$
$$z_{23} = -94.2477796076938$$
$$z_{24} = 91.106186954104$$
$$z_{25} = -43.9822971502571$$
$$z_{26} = 78.5398163397448$$
$$z_{27} = 47.1238898038469$$
$$z_{28} = -160.221225333079$$
$$z_{29} = 81.6814089933346$$
$$z_{30} = 163.362817986669$$
$$z_{31} = -72.2566310325652$$
$$z_{32} = -6.28318530717959$$
$$z_{33} = 28.2743338823081$$
$$z_{34} = -100.530964914873$$
$$z_{35} = 257.610597594363$$
$$z_{36} = -65.9734457253857$$
$$z_{37} = 94.2477796076938$$
$$z_{38} = 31.4159265358979$$
$$z_{39} = 50.2654824574367$$
$$z_{40} = -21.9911485751286$$
$$z_{41} = 12.5663706143592$$
$$z_{42} = 15.707963267949$$
$$z_{43} = -75.398223686155$$
$$z_{44} = 892.212313619501$$
$$z_{45} = 72.2566310325652$$
$$z_{46} = 18.8495559215388$$
$$z_{47} = -213.628300444106$$
$$z_{48} = -37.6991118430775$$
$$z_{49} = -50.2654824574367$$
$$z_{50} = -3.14159265358979$$
$$z_{51} = -87.9645943005142$$
$$z_{52} = -25.1327412287183$$
$$z_{53} = -40.8407044966673$$
$$z_{54} = 53.4070751110265$$
$$z_{55} = 9.42477796076938$$
$$z_{56} = 461.8141200777$$
$$z_{57} = -56.5486677646163$$
$$z_{58} = -97.3893722612836$$
$$z_{59} = 2774.02631311979$$
$$z_{60} = -12.5663706143592$$
$$z_{61} = -18.8495559215388$$
$$z_{62} = 84.8230016469244$$
$$z_{63} = 25.1327412287183$$
$$z_{64} = 21.9911485751286$$
$$z_{65} = 59.6902604182061$$
$$z_{66} = -53.4070751110265$$
$$z_{67} = 512.079602535136$$
$$z_{68} = -31.4159265358979$$
$$z_{69} = -9.42477796076938$$
$$z_{70} = -62.8318530717959$$
$$z_{71} = 56.5486677646163$$
$$z_{72} = 75.398223686155$$
$$z_{73} = 34.5575191894877$$
$$z_{74} = 87.9645943005142$$
o sea hay que resolver la ecuación:
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 65.9734457253857$$
$$z_{2} = -15.707963267949$$
$$z_{3} = 37.6991118430775$$
$$z_{4} = -69.1150383789755$$
$$z_{5} = -34.5575191894877$$
$$z_{6} = -91.106186954104$$
$$z_{7} = 0$$
$$z_{8} = 6.28318530717959$$
$$z_{9} = -47.1238898038469$$
$$z_{10} = -78.5398163397448$$
$$z_{11} = -28.2743338823081$$
$$z_{12} = 97.3893722612836$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = 40.8407044966673$$
$$z_{15} = 62.8318530717959$$
$$z_{16} = -81.6814089933346$$
$$z_{17} = 43.9822971502571$$
$$z_{18} = -59.6902604182061$$
$$z_{19} = 100.530964914873$$
$$z_{20} = -84.8230016469244$$
$$z_{21} = 69.1150383789755$$
$$z_{22} = 185.353966561798$$
$$z_{23} = -94.2477796076938$$
$$z_{24} = 91.106186954104$$
$$z_{25} = -43.9822971502571$$
$$z_{26} = 78.5398163397448$$
$$z_{27} = 47.1238898038469$$
$$z_{28} = -160.221225333079$$
$$z_{29} = 81.6814089933346$$
$$z_{30} = 163.362817986669$$
$$z_{31} = -72.2566310325652$$
$$z_{32} = -6.28318530717959$$
$$z_{33} = 28.2743338823081$$
$$z_{34} = -100.530964914873$$
$$z_{35} = 257.610597594363$$
$$z_{36} = -65.9734457253857$$
$$z_{37} = 94.2477796076938$$
$$z_{38} = 31.4159265358979$$
$$z_{39} = 50.2654824574367$$
$$z_{40} = -21.9911485751286$$
$$z_{41} = 12.5663706143592$$
$$z_{42} = 15.707963267949$$
$$z_{43} = -75.398223686155$$
$$z_{44} = 892.212313619501$$
$$z_{45} = 72.2566310325652$$
$$z_{46} = 18.8495559215388$$
$$z_{47} = -213.628300444106$$
$$z_{48} = -37.6991118430775$$
$$z_{49} = -50.2654824574367$$
$$z_{50} = -3.14159265358979$$
$$z_{51} = -87.9645943005142$$
$$z_{52} = -25.1327412287183$$
$$z_{53} = -40.8407044966673$$
$$z_{54} = 53.4070751110265$$
$$z_{55} = 9.42477796076938$$
$$z_{56} = 461.8141200777$$
$$z_{57} = -56.5486677646163$$
$$z_{58} = -97.3893722612836$$
$$z_{59} = 2774.02631311979$$
$$z_{60} = -12.5663706143592$$
$$z_{61} = -18.8495559215388$$
$$z_{62} = 84.8230016469244$$
$$z_{63} = 25.1327412287183$$
$$z_{64} = 21.9911485751286$$
$$z_{65} = 59.6902604182061$$
$$z_{66} = -53.4070751110265$$
$$z_{67} = 512.079602535136$$
$$z_{68} = -31.4159265358979$$
$$z_{69} = -9.42477796076938$$
$$z_{70} = -62.8318530717959$$
$$z_{71} = 56.5486677646163$$
$$z_{72} = 75.398223686155$$
$$z_{73} = 34.5575191894877$$
$$z_{74} = 87.9645943005142$$
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
$$- \frac{2 z^{2} \sin{\left(z \right)}}{\left(z^{2} - 1\right)^{2}} + \frac{z \cos{\left(z \right)} + \sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 64.38711209614$$
$$z_{2} = -23.5192984918928$$
$$z_{3} = -14.0654715664522$$
$$z_{4} = -54.9596662363795$$
$$z_{5} = 199.48612041292$$
$$z_{6} = -89.5242181426111$$
$$z_{7} = -45.531112815032$$
$$z_{8} = 0$$
$$z_{9} = -256.035895465913$$
$$z_{10} = 76.9560219212311$$
$$z_{11} = -67.5294282811446$$
$$z_{12} = -29.8115232181233$$
$$z_{13} = 23.5192984918928$$
$$z_{14} = -64.38711209614$$
$$z_{15} = 36.1005797021386$$
$$z_{16} = -58.1022445549727$$
$$z_{17} = 29.8115232181233$$
$$z_{18} = 86.3822189314905$$
$$z_{19} = 67.5294282811446$$
$$z_{20} = -86.3822189314905$$
$$z_{21} = 4.46996389748549$$
$$z_{22} = 54.9596662363795$$
$$z_{23} = 17.2203623011865$$
$$z_{24} = 48.6741268811873$$
$$z_{25} = 45.531112815032$$
$$z_{26} = -7.7208347779541$$
$$z_{27} = 61.2447215519469$$
$$z_{28} = -39.2443992496797$$
$$z_{29} = -80.0981247364271$$
$$z_{30} = 70.6716800442611$$
$$z_{31} = -95.8081365124445$$
$$z_{32} = 7.7208347779541$$
$$z_{33} = -61.2447215519469$$
$$z_{34} = -10.9025655123245$$
$$z_{35} = 108.375718080332$$
$$z_{36} = 26.6659486334171$$
$$z_{37} = 20.3710658047264$$
$$z_{38} = 98.9500607597783$$
$$z_{39} = 51.8169681067243$$
$$z_{40} = -32.9563331139653$$
$$z_{41} = -83.2401890026084$$
$$z_{42} = -20.3710658047264$$
$$z_{43} = -4.46996389748549$$
$$z_{44} = 42.387887292922$$
$$z_{45} = 73.8138756267966$$
$$z_{46} = 32.9563331139653$$
$$z_{47} = -73.8138756267966$$
$$z_{48} = -51.8169681067243$$
$$z_{49} = -36.1005797021386$$
$$z_{50} = 39.2443992496797$$
$$z_{51} = 83.2401890026084$$
$$z_{52} = 95.8081365124445$$
$$z_{53} = -98.9500607597783$$
$$z_{54} = 10.9025655123245$$
$$z_{55} = -42.387887292922$$
$$z_{56} = -26.6659486334171$$
$$z_{57} = -76.9560219212311$$
$$z_{58} = 58.1022445549727$$
$$z_{59} = -48.6741268811873$$
$$z_{60} = -70.6716800442611$$
$$z_{61} = 14.0654715664522$$
$$z_{62} = 92.6661897639043$$
$$z_{63} = -92.6661897639043$$
$$z_{64} = 80.0981247364271$$
$$z_{65} = 89.5242181426111$$
$$z_{66} = -17.2203623011865$$
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:
$$z_{1} = -23.5192984918928$$
$$z_{2} = -54.9596662363795$$
$$z_{3} = 199.48612041292$$
$$z_{4} = -256.035895465913$$
$$z_{5} = -67.5294282811446$$
$$z_{6} = -29.8115232181233$$
$$z_{7} = 23.5192984918928$$
$$z_{8} = 36.1005797021386$$
$$z_{9} = 29.8115232181233$$
$$z_{10} = 86.3822189314905$$
$$z_{11} = 67.5294282811446$$
$$z_{12} = -86.3822189314905$$
$$z_{13} = 4.46996389748549$$
$$z_{14} = 54.9596662363795$$
$$z_{15} = 17.2203623011865$$
$$z_{16} = 48.6741268811873$$
$$z_{17} = 61.2447215519469$$
$$z_{18} = -80.0981247364271$$
$$z_{19} = -61.2447215519469$$
$$z_{20} = -10.9025655123245$$
$$z_{21} = 98.9500607597783$$
$$z_{22} = -4.46996389748549$$
$$z_{23} = 42.387887292922$$
$$z_{24} = 73.8138756267966$$
$$z_{25} = -73.8138756267966$$
$$z_{26} = -36.1005797021386$$
$$z_{27} = -98.9500607597783$$
$$z_{28} = 10.9025655123245$$
$$z_{29} = -42.387887292922$$
$$z_{30} = -48.6741268811873$$
$$z_{31} = 92.6661897639043$$
$$z_{32} = -92.6661897639043$$
$$z_{33} = 80.0981247364271$$
$$z_{34} = -17.2203623011865$$
Puntos máximos de la función:
$$z_{34} = 64.38711209614$$
$$z_{34} = -14.0654715664522$$
$$z_{34} = -89.5242181426111$$
$$z_{34} = -45.531112815032$$
$$z_{34} = 0$$
$$z_{34} = 76.9560219212311$$
$$z_{34} = -64.38711209614$$
$$z_{34} = -58.1022445549727$$
$$z_{34} = 45.531112815032$$
$$z_{34} = -7.7208347779541$$
$$z_{34} = -39.2443992496797$$
$$z_{34} = 70.6716800442611$$
$$z_{34} = -95.8081365124445$$
$$z_{34} = 7.7208347779541$$
$$z_{34} = 108.375718080332$$
$$z_{34} = 26.6659486334171$$
$$z_{34} = 20.3710658047264$$
$$z_{34} = 51.8169681067243$$
$$z_{34} = -32.9563331139653$$
$$z_{34} = -83.2401890026084$$
$$z_{34} = -20.3710658047264$$
$$z_{34} = 32.9563331139653$$
$$z_{34} = -51.8169681067243$$
$$z_{34} = 39.2443992496797$$
$$z_{34} = 83.2401890026084$$
$$z_{34} = 95.8081365124445$$
$$z_{34} = -26.6659486334171$$
$$z_{34} = -76.9560219212311$$
$$z_{34} = 58.1022445549727$$
$$z_{34} = -70.6716800442611$$
$$z_{34} = 14.0654715664522$$
$$z_{34} = 89.5242181426111$$
Decrece en los intervalos
$$\left[199.48612041292, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -256.035895465913\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(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
$$- \frac{2 z^{2} \sin{\left(z \right)}}{\left(z^{2} - 1\right)^{2}} + \frac{z \cos{\left(z \right)} + \sin{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 64.38711209614$$
$$z_{2} = -23.5192984918928$$
$$z_{3} = -14.0654715664522$$
$$z_{4} = -54.9596662363795$$
$$z_{5} = 199.48612041292$$
$$z_{6} = -89.5242181426111$$
$$z_{7} = -45.531112815032$$
$$z_{8} = 0$$
$$z_{9} = -256.035895465913$$
$$z_{10} = 76.9560219212311$$
$$z_{11} = -67.5294282811446$$
$$z_{12} = -29.8115232181233$$
$$z_{13} = 23.5192984918928$$
$$z_{14} = -64.38711209614$$
$$z_{15} = 36.1005797021386$$
$$z_{16} = -58.1022445549727$$
$$z_{17} = 29.8115232181233$$
$$z_{18} = 86.3822189314905$$
$$z_{19} = 67.5294282811446$$
$$z_{20} = -86.3822189314905$$
$$z_{21} = 4.46996389748549$$
$$z_{22} = 54.9596662363795$$
$$z_{23} = 17.2203623011865$$
$$z_{24} = 48.6741268811873$$
$$z_{25} = 45.531112815032$$
$$z_{26} = -7.7208347779541$$
$$z_{27} = 61.2447215519469$$
$$z_{28} = -39.2443992496797$$
$$z_{29} = -80.0981247364271$$
$$z_{30} = 70.6716800442611$$
$$z_{31} = -95.8081365124445$$
$$z_{32} = 7.7208347779541$$
$$z_{33} = -61.2447215519469$$
$$z_{34} = -10.9025655123245$$
$$z_{35} = 108.375718080332$$
$$z_{36} = 26.6659486334171$$
$$z_{37} = 20.3710658047264$$
$$z_{38} = 98.9500607597783$$
$$z_{39} = 51.8169681067243$$
$$z_{40} = -32.9563331139653$$
$$z_{41} = -83.2401890026084$$
$$z_{42} = -20.3710658047264$$
$$z_{43} = -4.46996389748549$$
$$z_{44} = 42.387887292922$$
$$z_{45} = 73.8138756267966$$
$$z_{46} = 32.9563331139653$$
$$z_{47} = -73.8138756267966$$
$$z_{48} = -51.8169681067243$$
$$z_{49} = -36.1005797021386$$
$$z_{50} = 39.2443992496797$$
$$z_{51} = 83.2401890026084$$
$$z_{52} = 95.8081365124445$$
$$z_{53} = -98.9500607597783$$
$$z_{54} = 10.9025655123245$$
$$z_{55} = -42.387887292922$$
$$z_{56} = -26.6659486334171$$
$$z_{57} = -76.9560219212311$$
$$z_{58} = 58.1022445549727$$
$$z_{59} = -48.6741268811873$$
$$z_{60} = -70.6716800442611$$
$$z_{61} = 14.0654715664522$$
$$z_{62} = 92.6661897639043$$
$$z_{63} = -92.6661897639043$$
$$z_{64} = 80.0981247364271$$
$$z_{65} = 89.5242181426111$$
$$z_{66} = -17.2203623011865$$
Signos de extremos en los puntos:
(64.38711209614003, 0.0155329305638574)
(-23.51929849189281, -0.0425565503632636)
(-14.065471566452166, 0.071273704843965)
(-54.95966623637955, -0.0181981710779705)
(199.48612041292, -0.00501294306442466)
(-89.52421814261109, 0.011170858450225)
(-45.53111281503197, 0.0219682951476771)
(0, 0)
(-256.0358954659132, -0.00390573214421328)
(76.95602192123113, 0.0129955313516866)
(-67.52942828114465, -0.0148099815899088)
(-29.81152321812332, -0.0335629000697628)
(23.51929849189281, -0.0425565503632636)
(-64.38711209614003, 0.0155329305638574)
(36.100579702138575, -0.0277109952385317)
(-58.10224455497274, 0.0172135864769991)
(29.81152321812332, -0.0335629000697628)
(86.38221893149047, -0.0115772319735735)
(67.52942828114465, -0.0148099815899088)
(-86.38221893149047, -0.0115772319735735)
(4.469963897485487, -0.228615608199587)
(54.95966623637955, -0.0181981710779705)
(17.220362301186473, -0.0581679527920454)
(48.67412688118727, -0.0205491275919096)
(45.53111281503197, 0.0219682951476771)
(-7.720834777954096, 0.130563549425582)
(61.244721551946924, -0.0163301129538142)
(-39.24439924967973, 0.0254896034489584)
(-80.09812473642714, -0.0124856594290107)
(70.67168004426111, 0.0141513554601957)
(-95.80813651244455, 0.010438095281174)
(7.720834777954096, 0.130563549425582)
(-61.244721551946924, -0.0163301129538142)
(-10.902565512324477, -0.0920999140067109)
(108.37571808033167, 0.00922755188884025)
(26.665948633417063, 0.0375272953860803)
(20.371065804726353, 0.0491480573048362)
(98.95006075977834, -0.0101066239605399)
(51.816968106724275, 0.0193022884246921)
(-32.9563331139653, 0.0303571212035259)
(-83.24018900260842, 0.0120142944143532)
(-20.371065804726353, 0.0491480573048362)
(-4.469963897485487, -0.228615608199587)
(42.387887292922, -0.0235982021713917)
(73.8138756267966, -0.0135488310588046)
(32.9563331139653, 0.0303571212035259)
(-73.8138756267966, -0.0135488310588046)
(-51.816968106724275, 0.0193022884246921)
(-36.100579702138575, -0.0277109952385317)
(39.24439924967973, 0.0254896034489584)
(83.24018900260842, 0.0120142944143532)
(95.80813651244455, 0.010438095281174)
(-98.95006075977834, -0.0101066239605399)
(10.902565512324477, -0.0920999140067109)
(-42.387887292922, -0.0235982021713917)
(-26.665948633417063, 0.0375272953860803)
(-76.95602192123113, 0.0129955313516866)
(58.10224455497274, 0.0172135864769991)
(-48.67412688118727, -0.0205491275919096)
(-70.67168004426111, 0.0141513554601957)
(14.065471566452166, 0.071273704843965)
(92.66618976390433, -0.0107920506355808)
(-92.66618976390433, -0.0107920506355808)
(80.09812473642714, -0.0124856594290107)
(89.52421814261109, 0.011170858450225)
(-17.220362301186473, -0.0581679527920454)
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:
$$z_{1} = -23.5192984918928$$
$$z_{2} = -54.9596662363795$$
$$z_{3} = 199.48612041292$$
$$z_{4} = -256.035895465913$$
$$z_{5} = -67.5294282811446$$
$$z_{6} = -29.8115232181233$$
$$z_{7} = 23.5192984918928$$
$$z_{8} = 36.1005797021386$$
$$z_{9} = 29.8115232181233$$
$$z_{10} = 86.3822189314905$$
$$z_{11} = 67.5294282811446$$
$$z_{12} = -86.3822189314905$$
$$z_{13} = 4.46996389748549$$
$$z_{14} = 54.9596662363795$$
$$z_{15} = 17.2203623011865$$
$$z_{16} = 48.6741268811873$$
$$z_{17} = 61.2447215519469$$
$$z_{18} = -80.0981247364271$$
$$z_{19} = -61.2447215519469$$
$$z_{20} = -10.9025655123245$$
$$z_{21} = 98.9500607597783$$
$$z_{22} = -4.46996389748549$$
$$z_{23} = 42.387887292922$$
$$z_{24} = 73.8138756267966$$
$$z_{25} = -73.8138756267966$$
$$z_{26} = -36.1005797021386$$
$$z_{27} = -98.9500607597783$$
$$z_{28} = 10.9025655123245$$
$$z_{29} = -42.387887292922$$
$$z_{30} = -48.6741268811873$$
$$z_{31} = 92.6661897639043$$
$$z_{32} = -92.6661897639043$$
$$z_{33} = 80.0981247364271$$
$$z_{34} = -17.2203623011865$$
Puntos máximos de la función:
$$z_{34} = 64.38711209614$$
$$z_{34} = -14.0654715664522$$
$$z_{34} = -89.5242181426111$$
$$z_{34} = -45.531112815032$$
$$z_{34} = 0$$
$$z_{34} = 76.9560219212311$$
$$z_{34} = -64.38711209614$$
$$z_{34} = -58.1022445549727$$
$$z_{34} = 45.531112815032$$
$$z_{34} = -7.7208347779541$$
$$z_{34} = -39.2443992496797$$
$$z_{34} = 70.6716800442611$$
$$z_{34} = -95.8081365124445$$
$$z_{34} = 7.7208347779541$$
$$z_{34} = 108.375718080332$$
$$z_{34} = 26.6659486334171$$
$$z_{34} = 20.3710658047264$$
$$z_{34} = 51.8169681067243$$
$$z_{34} = -32.9563331139653$$
$$z_{34} = -83.2401890026084$$
$$z_{34} = -20.3710658047264$$
$$z_{34} = 32.9563331139653$$
$$z_{34} = -51.8169681067243$$
$$z_{34} = 39.2443992496797$$
$$z_{34} = 83.2401890026084$$
$$z_{34} = 95.8081365124445$$
$$z_{34} = -26.6659486334171$$
$$z_{34} = -76.9560219212311$$
$$z_{34} = 58.1022445549727$$
$$z_{34} = -70.6716800442611$$
$$z_{34} = 14.0654715664522$$
$$z_{34} = 89.5242181426111$$
Decrece en los intervalos
$$\left[199.48612041292, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -256.035895465913\right]$$
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
$$\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -37.6458850314086$$
$$z_{2} = 100.511061353637$$
$$z_{3} = -75.3716760575087$$
$$z_{4} = 34.499417014021$$
$$z_{5} = 69.0860728006109$$
$$z_{6} = 31.3519611339483$$
$$z_{7} = 59.6567101316809$$
$$z_{8} = -9.2003236611918$$
$$z_{9} = -81.6569064709099$$
$$z_{10} = 18.7420275079985$$
$$z_{11} = 94.2265477901003$$
$$z_{12} = 9.2003236611918$$
$$z_{13} = -28.2031813468519$$
$$z_{14} = 72.2289271346397$$
$$z_{15} = -18.7420275079985$$
$$z_{16} = -31.3519611339483$$
$$z_{17} = -65.9430979361068$$
$$z_{18} = 56.5132482585039$$
$$z_{19} = 12.4022644580604$$
$$z_{20} = -91.0842221942899$$
$$z_{21} = 973.891668990563$$
$$z_{22} = -56.5132482585039$$
$$z_{23} = -78.5143322594471$$
$$z_{24} = -87.9418441537954$$
$$z_{25} = -50.2256200032935$$
$$z_{26} = -47.0813589805145$$
$$z_{27} = 40.7915960867494$$
$$z_{28} = 81.6569064709099$$
$$z_{29} = -62.7999843815971$$
$$z_{30} = -40.7915960867494$$
$$z_{31} = 47.0813589805145$$
$$z_{32} = -34.499417014021$$
$$z_{33} = 5.91739496841265$$
$$z_{34} = 398.977254104831$$
$$z_{35} = -53.3695654514671$$
$$z_{36} = 50.2256200032935$$
$$z_{37} = 62.7999843815971$$
$$z_{38} = 53.3695654514671$$
$$z_{39} = -25.052568442955$$
$$z_{40} = -100.511061353637$$
$$z_{41} = 21.8993108150839$$
$$z_{42} = 87.9418441537954$$
$$z_{43} = 25.052568442955$$
$$z_{44} = 43.9367141641847$$
$$z_{45} = -15.5781517879098$$
$$z_{46} = 28.2031813468519$$
$$z_{47} = -5.91739496841265$$
$$z_{48} = -72.2289271346397$$
$$z_{49} = -69.0860728006109$$
$$z_{50} = 65.9430979361068$$
$$z_{51} = 37.6458850314086$$
$$z_{52} = 97.3688260270429$$
$$z_{53} = -97.3688260270429$$
$$z_{54} = -21.8993108150839$$
$$z_{55} = -12.4022644580604$$
$$z_{56} = 84.7994078270634$$
$$z_{57} = 75.3716760575087$$
$$z_{58} = 15.5781517879098$$
$$z_{59} = -94.2265477901003$$
$$z_{60} = -84.7994078270634$$
$$z_{61} = 78.5143322594471$$
$$z_{62} = -59.6567101316809$$
$$z_{63} = -43.9367141641847$$
$$z_{64} = 91.0842221942899$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$\lim_{z \to -1^-}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = \infty$$
$$\lim_{z \to -1^+}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = -\infty$$
- los límites no son iguales, signo
$$z_{1} = -1$$
- es el punto de flexión
$$\lim_{z \to 1^-}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to 1^+}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = \infty$$
- 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
$$\left[398.977254104831, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.511061353637\right]$$
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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
$$\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -37.6458850314086$$
$$z_{2} = 100.511061353637$$
$$z_{3} = -75.3716760575087$$
$$z_{4} = 34.499417014021$$
$$z_{5} = 69.0860728006109$$
$$z_{6} = 31.3519611339483$$
$$z_{7} = 59.6567101316809$$
$$z_{8} = -9.2003236611918$$
$$z_{9} = -81.6569064709099$$
$$z_{10} = 18.7420275079985$$
$$z_{11} = 94.2265477901003$$
$$z_{12} = 9.2003236611918$$
$$z_{13} = -28.2031813468519$$
$$z_{14} = 72.2289271346397$$
$$z_{15} = -18.7420275079985$$
$$z_{16} = -31.3519611339483$$
$$z_{17} = -65.9430979361068$$
$$z_{18} = 56.5132482585039$$
$$z_{19} = 12.4022644580604$$
$$z_{20} = -91.0842221942899$$
$$z_{21} = 973.891668990563$$
$$z_{22} = -56.5132482585039$$
$$z_{23} = -78.5143322594471$$
$$z_{24} = -87.9418441537954$$
$$z_{25} = -50.2256200032935$$
$$z_{26} = -47.0813589805145$$
$$z_{27} = 40.7915960867494$$
$$z_{28} = 81.6569064709099$$
$$z_{29} = -62.7999843815971$$
$$z_{30} = -40.7915960867494$$
$$z_{31} = 47.0813589805145$$
$$z_{32} = -34.499417014021$$
$$z_{33} = 5.91739496841265$$
$$z_{34} = 398.977254104831$$
$$z_{35} = -53.3695654514671$$
$$z_{36} = 50.2256200032935$$
$$z_{37} = 62.7999843815971$$
$$z_{38} = 53.3695654514671$$
$$z_{39} = -25.052568442955$$
$$z_{40} = -100.511061353637$$
$$z_{41} = 21.8993108150839$$
$$z_{42} = 87.9418441537954$$
$$z_{43} = 25.052568442955$$
$$z_{44} = 43.9367141641847$$
$$z_{45} = -15.5781517879098$$
$$z_{46} = 28.2031813468519$$
$$z_{47} = -5.91739496841265$$
$$z_{48} = -72.2289271346397$$
$$z_{49} = -69.0860728006109$$
$$z_{50} = 65.9430979361068$$
$$z_{51} = 37.6458850314086$$
$$z_{52} = 97.3688260270429$$
$$z_{53} = -97.3688260270429$$
$$z_{54} = -21.8993108150839$$
$$z_{55} = -12.4022644580604$$
$$z_{56} = 84.7994078270634$$
$$z_{57} = 75.3716760575087$$
$$z_{58} = 15.5781517879098$$
$$z_{59} = -94.2265477901003$$
$$z_{60} = -84.7994078270634$$
$$z_{61} = 78.5143322594471$$
$$z_{62} = -59.6567101316809$$
$$z_{63} = -43.9367141641847$$
$$z_{64} = 91.0842221942899$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$\lim_{z \to -1^-}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = \infty$$
$$\lim_{z \to -1^+}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = -\infty$$
- los límites no son iguales, signo
$$z_{1} = -1$$
- es el punto de flexión
$$\lim_{z \to 1^-}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = -\infty$$
$$\lim_{z \to 1^+}\left(\frac{- z \sin{\left(z \right)} - \frac{4 z \left(z \cos{\left(z \right)} + \sin{\left(z \right)}\right)}{z^{2} - 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} - 1} - 1\right) \sin{\left(z \right)}}{z^{2} - 1} + 2 \cos{\left(z \right)}}{z^{2} - 1}\right) = \infty$$
- 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
$$\left[398.977254104831, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.511061353637\right]$$
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
$$\lim_{z \to -\infty}\left(\frac{z \sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{z \sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{z \to -\infty}\left(\frac{z \sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{z \sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
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 (z*sin(z))/(z^2 - 1), dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2} - 1}\right) = 0$$
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:
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = \frac{z \sin{\left(z \right)}}{z^{2} - 1}$$
- Sí
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = - \frac{z \sin{\left(z \right)}}{z^{2} - 1}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = \frac{z \sin{\left(z \right)}}{z^{2} - 1}$$
- Sí
$$\frac{z \sin{\left(z \right)}}{z^{2} - 1} = - \frac{z \sin{\left(z \right)}}{z^{2} - 1}$$
- No
es decir, función
es
par
Gráfico