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