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Otras calculadoras
- 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 =:
-
y=(x+1)^3
-
y=2x
-
2*x^3-3*x
-
3-x^2
- Derivada de:
-
z*sinz
-
Expresiones idénticas
- z*sinz
- z multiplicar por seno de z
- zsinz
-
Expresiones semejantes
- z*sin(z)/(z^2-1)
- (z*(sinz+2))/sinz
-
Expresiones con funciones
- sinz
- sinz/z
Gráfico de la función y = z*sinz
Solución
Gráfico de la función
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:
$$z \sin{\left(z \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 65.9734457253857$$
$$z_{2} = -15.707963267949$$
$$z_{3} = 37.6991118430775$$
$$z_{4} = -69.1150383789755$$
$$z_{5} = -34.5575191894877$$
$$z_{6} = -91.106186954104$$
$$z_{7} = 0$$
$$z_{8} = 6.28318530717959$$
$$z_{9} = -47.1238898038469$$
$$z_{10} = -78.5398163397448$$
$$z_{11} = -28.2743338823081$$
$$z_{12} = 97.3893722612836$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = 40.8407044966673$$
$$z_{15} = 62.8318530717959$$
$$z_{16} = -81.6814089933346$$
$$z_{17} = 43.9822971502571$$
$$z_{18} = -84.8230016469244$$
$$z_{19} = 100.530964914873$$
$$z_{20} = 69.1150383789755$$
$$z_{21} = -94.2477796076938$$
$$z_{22} = 91.106186954104$$
$$z_{23} = 78.5398163397448$$
$$z_{24} = 47.1238898038469$$
$$z_{25} = 81.6814089933346$$
$$z_{26} = -72.2566310325652$$
$$z_{27} = -6.28318530717959$$
$$z_{28} = 28.2743338823081$$
$$z_{29} = -100.530964914873$$
$$z_{30} = -65.9734457253857$$
$$z_{31} = 94.2477796076938$$
$$z_{32} = 31.4159265358979$$
$$z_{33} = 50.2654824574367$$
$$z_{34} = -21.9911485751286$$
$$z_{35} = 12.5663706143592$$
$$z_{36} = 15.707963267949$$
$$z_{37} = -75.398223686155$$
$$z_{38} = 72.2566310325652$$
$$z_{39} = 18.8495559215388$$
$$z_{40} = -37.6991118430775$$
$$z_{41} = -50.2654824574367$$
$$z_{42} = -3.14159265358979$$
$$z_{43} = 697.433569096934$$
$$z_{44} = -87.9645943005142$$
$$z_{45} = -25.1327412287183$$
$$z_{46} = -40.8407044966673$$
$$z_{47} = 53.4070751110265$$
$$z_{48} = 9.42477796076938$$
$$z_{49} = -43.9822971502571$$
$$z_{50} = -56.5486677646163$$
$$z_{51} = -97.3893722612836$$
$$z_{52} = -59.6902604182061$$
$$z_{53} = -12.5663706143592$$
$$z_{54} = -18.8495559215388$$
$$z_{55} = 84.8230016469244$$
$$z_{56} = 25.1327412287183$$
$$z_{57} = 21.9911485751286$$
$$z_{58} = 59.6902604182061$$
$$z_{59} = -53.4070751110265$$
$$z_{60} = -31.4159265358979$$
$$z_{61} = -9.42477796076938$$
$$z_{62} = -62.8318530717959$$
$$z_{63} = 56.5486677646163$$
$$z_{64} = 75.398223686155$$
$$z_{65} = 34.5575191894877$$
$$z_{66} = 87.9645943005142$$
o sea hay que resolver la ecuación:
$$z \sin{\left(z \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = 0$$
$$z_{2} = \pi$$
Solución numérica
$$z_{1} = 65.9734457253857$$
$$z_{2} = -15.707963267949$$
$$z_{3} = 37.6991118430775$$
$$z_{4} = -69.1150383789755$$
$$z_{5} = -34.5575191894877$$
$$z_{6} = -91.106186954104$$
$$z_{7} = 0$$
$$z_{8} = 6.28318530717959$$
$$z_{9} = -47.1238898038469$$
$$z_{10} = -78.5398163397448$$
$$z_{11} = -28.2743338823081$$
$$z_{12} = 97.3893722612836$$
$$z_{13} = 3.14159265358979$$
$$z_{14} = 40.8407044966673$$
$$z_{15} = 62.8318530717959$$
$$z_{16} = -81.6814089933346$$
$$z_{17} = 43.9822971502571$$
$$z_{18} = -84.8230016469244$$
$$z_{19} = 100.530964914873$$
$$z_{20} = 69.1150383789755$$
$$z_{21} = -94.2477796076938$$
$$z_{22} = 91.106186954104$$
$$z_{23} = 78.5398163397448$$
$$z_{24} = 47.1238898038469$$
$$z_{25} = 81.6814089933346$$
$$z_{26} = -72.2566310325652$$
$$z_{27} = -6.28318530717959$$
$$z_{28} = 28.2743338823081$$
$$z_{29} = -100.530964914873$$
$$z_{30} = -65.9734457253857$$
$$z_{31} = 94.2477796076938$$
$$z_{32} = 31.4159265358979$$
$$z_{33} = 50.2654824574367$$
$$z_{34} = -21.9911485751286$$
$$z_{35} = 12.5663706143592$$
$$z_{36} = 15.707963267949$$
$$z_{37} = -75.398223686155$$
$$z_{38} = 72.2566310325652$$
$$z_{39} = 18.8495559215388$$
$$z_{40} = -37.6991118430775$$
$$z_{41} = -50.2654824574367$$
$$z_{42} = -3.14159265358979$$
$$z_{43} = 697.433569096934$$
$$z_{44} = -87.9645943005142$$
$$z_{45} = -25.1327412287183$$
$$z_{46} = -40.8407044966673$$
$$z_{47} = 53.4070751110265$$
$$z_{48} = 9.42477796076938$$
$$z_{49} = -43.9822971502571$$
$$z_{50} = -56.5486677646163$$
$$z_{51} = -97.3893722612836$$
$$z_{52} = -59.6902604182061$$
$$z_{53} = -12.5663706143592$$
$$z_{54} = -18.8495559215388$$
$$z_{55} = 84.8230016469244$$
$$z_{56} = 25.1327412287183$$
$$z_{57} = 21.9911485751286$$
$$z_{58} = 59.6902604182061$$
$$z_{59} = -53.4070751110265$$
$$z_{60} = -31.4159265358979$$
$$z_{61} = -9.42477796076938$$
$$z_{62} = -62.8318530717959$$
$$z_{63} = 56.5486677646163$$
$$z_{64} = 75.398223686155$$
$$z_{65} = 34.5575191894877$$
$$z_{66} = 87.9645943005142$$
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
$$z \cos{\left(z \right)} + \sin{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 83.2642147040886$$
$$z_{2} = 102.111554139654$$
$$z_{3} = -70.69997803861$$
$$z_{4} = 2.02875783811043$$
$$z_{5} = -83.2642147040886$$
$$z_{6} = 0$$
$$z_{7} = 33.0170010333572$$
$$z_{8} = -14.2074367251912$$
$$z_{9} = -26.7409160147873$$
$$z_{10} = -76.9820093304187$$
$$z_{11} = 98.9702722883957$$
$$z_{12} = -80.1230928148503$$
$$z_{13} = 89.5465575382492$$
$$z_{14} = 4.91318043943488$$
$$z_{15} = -36.1559664195367$$
$$z_{16} = -58.1366632448992$$
$$z_{17} = -54.9960525574964$$
$$z_{18} = -67.5590428388084$$
$$z_{19} = -98.9702722883957$$
$$z_{20} = 20.469167402741$$
$$z_{21} = 39.295350981473$$
$$z_{22} = -42.4350618814099$$
$$z_{23} = 36.1559664195367$$
$$z_{24} = 7.97866571241324$$
$$z_{25} = -29.8785865061074$$
$$z_{26} = -89.5465575382492$$
$$z_{27} = -73.8409691490209$$
$$z_{28} = -45.57503179559$$
$$z_{29} = 23.6042847729804$$
$$z_{30} = -11.085538406497$$
$$z_{31} = -48.7152107175577$$
$$z_{32} = 45.57503179559$$
$$z_{33} = 61.2773745335697$$
$$z_{34} = 76.9820093304187$$
$$z_{35} = -2.02875783811043$$
$$z_{36} = 95.8290108090195$$
$$z_{37} = 42.4350618814099$$
$$z_{38} = 54.9960525574964$$
$$z_{39} = -92.687771772017$$
$$z_{40} = 26.7409160147873$$
$$z_{41} = -4.91318043943488$$
$$z_{42} = -51.855560729152$$
$$z_{43} = -61.2773745335697$$
$$z_{44} = 67.5590428388084$$
$$z_{45} = 70.69997803861$$
$$z_{46} = -39.295350981473$$
$$z_{47} = -20.469167402741$$
$$z_{48} = 48.7152107175577$$
$$z_{49} = 51.855560729152$$
$$z_{50} = -86.4053708116885$$
$$z_{51} = 80.1230928148503$$
$$z_{52} = -33.0170010333572$$
$$z_{53} = 29.8785865061074$$
$$z_{54} = -17.3363779239834$$
$$z_{55} = -7.97866571241324$$
$$z_{56} = 14.2074367251912$$
$$z_{57} = 17.3363779239834$$
$$z_{58} = 86.4053708116885$$
$$z_{59} = 11.085538406497$$
$$z_{60} = 58.1366632448992$$
$$z_{61} = -64.4181717218392$$
$$z_{62} = 64.4181717218392$$
$$z_{63} = -95.8290108090195$$
$$z_{64} = 73.8409691490209$$
$$z_{65} = -23.6042847729804$$
$$z_{66} = 92.687771772017$$
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:
$$z_{1} = 0$$
$$z_{2} = 98.9702722883957$$
$$z_{3} = -80.1230928148503$$
$$z_{4} = 4.91318043943488$$
$$z_{5} = -36.1559664195367$$
$$z_{6} = -54.9960525574964$$
$$z_{7} = -67.5590428388084$$
$$z_{8} = -98.9702722883957$$
$$z_{9} = -42.4350618814099$$
$$z_{10} = 36.1559664195367$$
$$z_{11} = -29.8785865061074$$
$$z_{12} = -73.8409691490209$$
$$z_{13} = 23.6042847729804$$
$$z_{14} = -11.085538406497$$
$$z_{15} = -48.7152107175577$$
$$z_{16} = 61.2773745335697$$
$$z_{17} = 42.4350618814099$$
$$z_{18} = 54.9960525574964$$
$$z_{19} = -92.687771772017$$
$$z_{20} = -4.91318043943488$$
$$z_{21} = -61.2773745335697$$
$$z_{22} = 67.5590428388084$$
$$z_{23} = 48.7152107175577$$
$$z_{24} = -86.4053708116885$$
$$z_{25} = 80.1230928148503$$
$$z_{26} = 29.8785865061074$$
$$z_{27} = -17.3363779239834$$
$$z_{28} = 17.3363779239834$$
$$z_{29} = 86.4053708116885$$
$$z_{30} = 11.085538406497$$
$$z_{31} = 73.8409691490209$$
$$z_{32} = -23.6042847729804$$
$$z_{33} = 92.687771772017$$
Puntos máximos de la función:
$$z_{33} = 83.2642147040886$$
$$z_{33} = 102.111554139654$$
$$z_{33} = -70.69997803861$$
$$z_{33} = 2.02875783811043$$
$$z_{33} = -83.2642147040886$$
$$z_{33} = 33.0170010333572$$
$$z_{33} = -14.2074367251912$$
$$z_{33} = -26.7409160147873$$
$$z_{33} = -76.9820093304187$$
$$z_{33} = 89.5465575382492$$
$$z_{33} = -58.1366632448992$$
$$z_{33} = 20.469167402741$$
$$z_{33} = 39.295350981473$$
$$z_{33} = 7.97866571241324$$
$$z_{33} = -89.5465575382492$$
$$z_{33} = -45.57503179559$$
$$z_{33} = 45.57503179559$$
$$z_{33} = 76.9820093304187$$
$$z_{33} = -2.02875783811043$$
$$z_{33} = 95.8290108090195$$
$$z_{33} = 26.7409160147873$$
$$z_{33} = -51.855560729152$$
$$z_{33} = 70.69997803861$$
$$z_{33} = -39.295350981473$$
$$z_{33} = -20.469167402741$$
$$z_{33} = 51.855560729152$$
$$z_{33} = -33.0170010333572$$
$$z_{33} = -7.97866571241324$$
$$z_{33} = 14.2074367251912$$
$$z_{33} = 58.1366632448992$$
$$z_{33} = -64.4181717218392$$
$$z_{33} = 64.4181717218392$$
$$z_{33} = -95.8290108090195$$
Decrece en los intervalos
$$\left[98.9702722883957, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9702722883957\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(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
$$z \cos{\left(z \right)} + \sin{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 83.2642147040886$$
$$z_{2} = 102.111554139654$$
$$z_{3} = -70.69997803861$$
$$z_{4} = 2.02875783811043$$
$$z_{5} = -83.2642147040886$$
$$z_{6} = 0$$
$$z_{7} = 33.0170010333572$$
$$z_{8} = -14.2074367251912$$
$$z_{9} = -26.7409160147873$$
$$z_{10} = -76.9820093304187$$
$$z_{11} = 98.9702722883957$$
$$z_{12} = -80.1230928148503$$
$$z_{13} = 89.5465575382492$$
$$z_{14} = 4.91318043943488$$
$$z_{15} = -36.1559664195367$$
$$z_{16} = -58.1366632448992$$
$$z_{17} = -54.9960525574964$$
$$z_{18} = -67.5590428388084$$
$$z_{19} = -98.9702722883957$$
$$z_{20} = 20.469167402741$$
$$z_{21} = 39.295350981473$$
$$z_{22} = -42.4350618814099$$
$$z_{23} = 36.1559664195367$$
$$z_{24} = 7.97866571241324$$
$$z_{25} = -29.8785865061074$$
$$z_{26} = -89.5465575382492$$
$$z_{27} = -73.8409691490209$$
$$z_{28} = -45.57503179559$$
$$z_{29} = 23.6042847729804$$
$$z_{30} = -11.085538406497$$
$$z_{31} = -48.7152107175577$$
$$z_{32} = 45.57503179559$$
$$z_{33} = 61.2773745335697$$
$$z_{34} = 76.9820093304187$$
$$z_{35} = -2.02875783811043$$
$$z_{36} = 95.8290108090195$$
$$z_{37} = 42.4350618814099$$
$$z_{38} = 54.9960525574964$$
$$z_{39} = -92.687771772017$$
$$z_{40} = 26.7409160147873$$
$$z_{41} = -4.91318043943488$$
$$z_{42} = -51.855560729152$$
$$z_{43} = -61.2773745335697$$
$$z_{44} = 67.5590428388084$$
$$z_{45} = 70.69997803861$$
$$z_{46} = -39.295350981473$$
$$z_{47} = -20.469167402741$$
$$z_{48} = 48.7152107175577$$
$$z_{49} = 51.855560729152$$
$$z_{50} = -86.4053708116885$$
$$z_{51} = 80.1230928148503$$
$$z_{52} = -33.0170010333572$$
$$z_{53} = 29.8785865061074$$
$$z_{54} = -17.3363779239834$$
$$z_{55} = -7.97866571241324$$
$$z_{56} = 14.2074367251912$$
$$z_{57} = 17.3363779239834$$
$$z_{58} = 86.4053708116885$$
$$z_{59} = 11.085538406497$$
$$z_{60} = 58.1366632448992$$
$$z_{61} = -64.4181717218392$$
$$z_{62} = 64.4181717218392$$
$$z_{63} = -95.8290108090195$$
$$z_{64} = 73.8409691490209$$
$$z_{65} = -23.6042847729804$$
$$z_{66} = 92.687771772017$$
Signos de extremos en los puntos:
(83.26421470408864, 83.2582103729533)
(102.11155413965392, 102.106657886316)
(-70.69997803861, 70.6929069615931)
(2.028757838110434, 1.81970574115965)
(-83.26421470408864, 83.2582103729533)
(0, 0)
(33.017001033357246, 33.0018677308454)
(-14.207436725191188, 14.1723741137743)
(-26.74091601478731, 26.7222376646974)
(-76.98200933041872, 76.9755151282637)
(98.9702722883957, -98.9652206531187)
(-80.12309281485025, -80.1168531456592)
(89.54655753824919, 89.5409743728852)
(4.913180439434884, -4.81446988971227)
(-36.15596641953672, -36.1421453722421)
(-58.13666324489916, 58.1280647280857)
(-54.99605255749639, -54.9869632496976)
(-67.5590428388084, -67.5516431209725)
(-98.9702722883957, -98.9652206531187)
(20.46916740274095, 20.4447840582523)
(39.295350981472986, 39.2826330068918)
(-42.43506188140989, -42.4232840772591)
(36.15596641953672, -36.1421453722421)
(7.978665712413241, 7.91672737158778)
(-29.878586506107393, -29.8618661591868)
(-89.54655753824919, 89.5409743728852)
(-73.8409691490209, -73.8341987715416)
(-45.57503179559002, 45.5640648360268)
(23.604284772980407, -23.5831306496334)
(-11.085538406497022, -11.04070801593)
(-48.715210717557724, -48.7049502253679)
(45.57503179559002, 45.5640648360268)
(61.277374533569656, -61.2692165444766)
(76.98200933041872, 76.9755151282637)
(-2.028757838110434, 1.81970574115965)
(95.82901080901948, 95.8237936084657)
(42.43506188140989, -42.4232840772591)
(54.99605255749639, -54.9869632496976)
(-92.687771772017, -92.6823777880592)
(26.74091601478731, 26.7222376646974)
(-4.913180439434884, -4.81446988971227)
(-51.85556072915197, 51.8459212502015)
(-61.277374533569656, -61.2692165444766)
(67.5590428388084, -67.5516431209725)
(70.69997803861, 70.6929069615931)
(-39.295350981472986, 39.2826330068918)
(-20.46916740274095, 20.4447840582523)
(48.715210717557724, -48.7049502253679)
(51.85556072915197, 51.8459212502015)
(-86.40537081168854, -86.3995847156108)
(80.12309281485025, -80.1168531456592)
(-33.017001033357246, 33.0018677308454)
(29.878586506107393, -29.8618661591868)
(-17.33637792398336, -17.3076086078585)
(-7.978665712413241, 7.91672737158778)
(14.207436725191188, 14.1723741137743)
(17.33637792398336, -17.3076086078585)
(86.40537081168854, -86.3995847156108)
(11.085538406497022, -11.04070801593)
(58.13666324489916, 58.1280647280857)
(-64.41817172183916, 64.4104113393753)
(64.41817172183916, 64.4104113393753)
(-95.82901080901948, 95.8237936084657)
(73.8409691490209, -73.8341987715416)
(-23.604284772980407, -23.5831306496334)
(92.687771772017, -92.6823777880592)
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:
$$z_{1} = 0$$
$$z_{2} = 98.9702722883957$$
$$z_{3} = -80.1230928148503$$
$$z_{4} = 4.91318043943488$$
$$z_{5} = -36.1559664195367$$
$$z_{6} = -54.9960525574964$$
$$z_{7} = -67.5590428388084$$
$$z_{8} = -98.9702722883957$$
$$z_{9} = -42.4350618814099$$
$$z_{10} = 36.1559664195367$$
$$z_{11} = -29.8785865061074$$
$$z_{12} = -73.8409691490209$$
$$z_{13} = 23.6042847729804$$
$$z_{14} = -11.085538406497$$
$$z_{15} = -48.7152107175577$$
$$z_{16} = 61.2773745335697$$
$$z_{17} = 42.4350618814099$$
$$z_{18} = 54.9960525574964$$
$$z_{19} = -92.687771772017$$
$$z_{20} = -4.91318043943488$$
$$z_{21} = -61.2773745335697$$
$$z_{22} = 67.5590428388084$$
$$z_{23} = 48.7152107175577$$
$$z_{24} = -86.4053708116885$$
$$z_{25} = 80.1230928148503$$
$$z_{26} = 29.8785865061074$$
$$z_{27} = -17.3363779239834$$
$$z_{28} = 17.3363779239834$$
$$z_{29} = 86.4053708116885$$
$$z_{30} = 11.085538406497$$
$$z_{31} = 73.8409691490209$$
$$z_{32} = -23.6042847729804$$
$$z_{33} = 92.687771772017$$
Puntos máximos de la función:
$$z_{33} = 83.2642147040886$$
$$z_{33} = 102.111554139654$$
$$z_{33} = -70.69997803861$$
$$z_{33} = 2.02875783811043$$
$$z_{33} = -83.2642147040886$$
$$z_{33} = 33.0170010333572$$
$$z_{33} = -14.2074367251912$$
$$z_{33} = -26.7409160147873$$
$$z_{33} = -76.9820093304187$$
$$z_{33} = 89.5465575382492$$
$$z_{33} = -58.1366632448992$$
$$z_{33} = 20.469167402741$$
$$z_{33} = 39.295350981473$$
$$z_{33} = 7.97866571241324$$
$$z_{33} = -89.5465575382492$$
$$z_{33} = -45.57503179559$$
$$z_{33} = 45.57503179559$$
$$z_{33} = 76.9820093304187$$
$$z_{33} = -2.02875783811043$$
$$z_{33} = 95.8290108090195$$
$$z_{33} = 26.7409160147873$$
$$z_{33} = -51.855560729152$$
$$z_{33} = 70.69997803861$$
$$z_{33} = -39.295350981473$$
$$z_{33} = -20.469167402741$$
$$z_{33} = 51.855560729152$$
$$z_{33} = -33.0170010333572$$
$$z_{33} = -7.97866571241324$$
$$z_{33} = 14.2074367251912$$
$$z_{33} = 58.1366632448992$$
$$z_{33} = -64.4181717218392$$
$$z_{33} = 64.4181717218392$$
$$z_{33} = -95.8290108090195$$
Decrece en los intervalos
$$\left[98.9702722883957, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9702722883957\right]$$
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
$$- z \sin{\left(z \right)} + 2 \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -44.0276918992479$$
$$z_{2} = -28.3447768697864$$
$$z_{3} = 87.9873209346887$$
$$z_{4} = 44.0276918992479$$
$$z_{5} = -1.0768739863118$$
$$z_{6} = -59.7237354324305$$
$$z_{7} = -15.8336114149477$$
$$z_{8} = -40.8895777660408$$
$$z_{9} = -72.2842925036825$$
$$z_{10} = -50.3052188363296$$
$$z_{11} = -100.550852725424$$
$$z_{12} = 94.2689923093066$$
$$z_{13} = 28.3447768697864$$
$$z_{14} = -53.4444796697636$$
$$z_{15} = -97.4099011706723$$
$$z_{16} = -47.1662676027767$$
$$z_{17} = -75.4247339745236$$
$$z_{18} = -94.2689923093066$$
$$z_{19} = -87.9873209346887$$
$$z_{20} = 81.7058821480364$$
$$z_{21} = -9.62956034329743$$
$$z_{22} = 6.57833373272234$$
$$z_{23} = 72.2842925036825$$
$$z_{24} = 59.7237354324305$$
$$z_{25} = 37.7520396346102$$
$$z_{26} = 91.1281305511393$$
$$z_{27} = -31.479374920314$$
$$z_{28} = 9.62956034329743$$
$$z_{29} = 97.4099011706723$$
$$z_{30} = -84.8465692433091$$
$$z_{31} = 40.8895777660408$$
$$z_{32} = 3.6435971674254$$
$$z_{33} = 1.0768739863118$$
$$z_{34} = 69.1439554764926$$
$$z_{35} = -6.57833373272234$$
$$z_{36} = 50.3052188363296$$
$$z_{37} = 15.8336114149477$$
$$z_{38} = -91.1281305511393$$
$$z_{39} = -22.0814757672807$$
$$z_{40} = 84.8465692433091$$
$$z_{41} = 18.954681766529$$
$$z_{42} = -56.5839987378634$$
$$z_{43} = -66.0037377708277$$
$$z_{44} = 47.1662676027767$$
$$z_{45} = 53.4444796697636$$
$$z_{46} = -37.7520396346102$$
$$z_{47} = -12.7222987717666$$
$$z_{48} = -25.2119030642106$$
$$z_{49} = 34.6152330552306$$
$$z_{50} = 31.479374920314$$
$$z_{51} = -3.6435971674254$$
$$z_{52} = 75.4247339745236$$
$$z_{53} = -34.6152330552306$$
$$z_{54} = -128.820822990274$$
$$z_{55} = 25.2119030642106$$
$$z_{56} = 100.550852725424$$
$$z_{57} = 62.863657228703$$
$$z_{58} = -78.5652673845995$$
$$z_{59} = 56.5839987378634$$
$$z_{60} = 78.5652673845995$$
$$z_{61} = -18.954681766529$$
$$z_{62} = -62.863657228703$$
$$z_{63} = 22.0814757672807$$
$$z_{64} = -81.7058821480364$$
$$z_{65} = -69.1439554764926$$
$$z_{66} = 12.7222987717666$$
$$z_{67} = 66.0037377708277$$
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
$$\left[97.4099011706723, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.550852725424\right]$$
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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
$$- z \sin{\left(z \right)} + 2 \cos{\left(z \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -44.0276918992479$$
$$z_{2} = -28.3447768697864$$
$$z_{3} = 87.9873209346887$$
$$z_{4} = 44.0276918992479$$
$$z_{5} = -1.0768739863118$$
$$z_{6} = -59.7237354324305$$
$$z_{7} = -15.8336114149477$$
$$z_{8} = -40.8895777660408$$
$$z_{9} = -72.2842925036825$$
$$z_{10} = -50.3052188363296$$
$$z_{11} = -100.550852725424$$
$$z_{12} = 94.2689923093066$$
$$z_{13} = 28.3447768697864$$
$$z_{14} = -53.4444796697636$$
$$z_{15} = -97.4099011706723$$
$$z_{16} = -47.1662676027767$$
$$z_{17} = -75.4247339745236$$
$$z_{18} = -94.2689923093066$$
$$z_{19} = -87.9873209346887$$
$$z_{20} = 81.7058821480364$$
$$z_{21} = -9.62956034329743$$
$$z_{22} = 6.57833373272234$$
$$z_{23} = 72.2842925036825$$
$$z_{24} = 59.7237354324305$$
$$z_{25} = 37.7520396346102$$
$$z_{26} = 91.1281305511393$$
$$z_{27} = -31.479374920314$$
$$z_{28} = 9.62956034329743$$
$$z_{29} = 97.4099011706723$$
$$z_{30} = -84.8465692433091$$
$$z_{31} = 40.8895777660408$$
$$z_{32} = 3.6435971674254$$
$$z_{33} = 1.0768739863118$$
$$z_{34} = 69.1439554764926$$
$$z_{35} = -6.57833373272234$$
$$z_{36} = 50.3052188363296$$
$$z_{37} = 15.8336114149477$$
$$z_{38} = -91.1281305511393$$
$$z_{39} = -22.0814757672807$$
$$z_{40} = 84.8465692433091$$
$$z_{41} = 18.954681766529$$
$$z_{42} = -56.5839987378634$$
$$z_{43} = -66.0037377708277$$
$$z_{44} = 47.1662676027767$$
$$z_{45} = 53.4444796697636$$
$$z_{46} = -37.7520396346102$$
$$z_{47} = -12.7222987717666$$
$$z_{48} = -25.2119030642106$$
$$z_{49} = 34.6152330552306$$
$$z_{50} = 31.479374920314$$
$$z_{51} = -3.6435971674254$$
$$z_{52} = 75.4247339745236$$
$$z_{53} = -34.6152330552306$$
$$z_{54} = -128.820822990274$$
$$z_{55} = 25.2119030642106$$
$$z_{56} = 100.550852725424$$
$$z_{57} = 62.863657228703$$
$$z_{58} = -78.5652673845995$$
$$z_{59} = 56.5839987378634$$
$$z_{60} = 78.5652673845995$$
$$z_{61} = -18.954681766529$$
$$z_{62} = -62.863657228703$$
$$z_{63} = 22.0814757672807$$
$$z_{64} = -81.7058821480364$$
$$z_{65} = -69.1439554764926$$
$$z_{66} = 12.7222987717666$$
$$z_{67} = 66.0037377708277$$
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
$$\left[97.4099011706723, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.550852725424\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con z->+oo y z->-oo
$$\lim_{z \to -\infty}\left(z \sin{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to \infty}\left(z \sin{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to -\infty}\left(z \sin{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{z \to \infty}\left(z \sin{\left(z \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función z*sin(z), dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty} \sin{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to \infty} \sin{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to -\infty} \sin{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -1, 1\right\rangle z$$
$$\lim_{z \to \infty} \sin{\left(z \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -1, 1\right\rangle z$$
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:
$$z \sin{\left(z \right)} = z \sin{\left(z \right)}$$
- Sí
$$z \sin{\left(z \right)} = - z \sin{\left(z \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$z \sin{\left(z \right)} = z \sin{\left(z \right)}$$
- Sí
$$z \sin{\left(z \right)} = - z \sin{\left(z \right)}$$
- No
es decir, función
es
par
Gráfico