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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x-8/x^4
-
y=x²-2x-8
-
-x^4+x^2
-
x*e^(-x^1)
-
Expresiones idénticas
- sinz/z
- seno de z dividir por z
- sinz dividir por z
-
Expresiones semejantes
- z*sin(z)/(z^2-1)
- sin(z)/(z^2-1)
Gráfico de la función y = sinz/z
Solución
Ha introducido
[src]
sin(z)
f(z) = ------
z
$$f{\left(z \right)} = \frac{\sin{\left(z \right)}}{z}$$
f = sin(z)/z
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$z_{1} = 0$$
$$z_{1} = 0$$
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:
$$\frac{\sin{\left(z \right)}}{z} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = \pi$$
Solución numérica
$$z_{1} = -18.8495559215388$$
$$z_{2} = -53.4070751110265$$
$$z_{3} = -37.6991118430775$$
$$z_{4} = -59.6902604182061$$
$$z_{5} = -15.707963267949$$
$$z_{6} = -113.097335529233$$
$$z_{7} = -56.5486677646163$$
$$z_{8} = 12.5663706143592$$
$$z_{9} = 3.14159265358979$$
$$z_{10} = -31.4159265358979$$
$$z_{11} = 84.8230016469244$$
$$z_{12} = -81.6814089933346$$
$$z_{13} = 94.2477796076938$$
$$z_{14} = 21.9911485751286$$
$$z_{15} = -87.9645943005142$$
$$z_{16} = 81.6814089933346$$
$$z_{17} = 40.8407044966673$$
$$z_{18} = -75.398223686155$$
$$z_{19} = -78.5398163397448$$
$$z_{20} = 62.8318530717959$$
$$z_{21} = 100.530964914873$$
$$z_{22} = -21.9911485751286$$
$$z_{23} = 47.1238898038469$$
$$z_{24} = 91.106186954104$$
$$z_{25} = 75.398223686155$$
$$z_{26} = 28.2743338823081$$
$$z_{27} = 34.5575191894877$$
$$z_{28} = 6.28318530717959$$
$$z_{29} = 78.5398163397448$$
$$z_{30} = 72.2566310325652$$
$$z_{31} = -6.28318530717959$$
$$z_{32} = 15.707963267949$$
$$z_{33} = 31.4159265358979$$
$$z_{34} = -47.1238898038469$$
$$z_{35} = 25.1327412287183$$
$$z_{36} = 18.8495559215388$$
$$z_{37} = 153.9380400259$$
$$z_{38} = -94.2477796076938$$
$$z_{39} = -3.14159265358979$$
$$z_{40} = -40.8407044966673$$
$$z_{41} = 56.5486677646163$$
$$z_{42} = -25.1327412287183$$
$$z_{43} = 53.4070751110265$$
$$z_{44} = -28.2743338823081$$
$$z_{45} = -9.42477796076938$$
$$z_{46} = 87.9645943005142$$
$$z_{47} = -50.2654824574367$$
$$z_{48} = -100.530964914873$$
$$z_{49} = -43.9822971502571$$
$$z_{50} = -223.053078404875$$
$$z_{51} = 50.2654824574367$$
$$z_{52} = -97.3893722612836$$
$$z_{53} = 590.619418874881$$
$$z_{54} = 69.1150383789755$$
$$z_{55} = 59.6902604182061$$
$$z_{56} = 97.3893722612836$$
$$z_{57} = -62.8318530717959$$
$$z_{58} = -72.2566310325652$$
$$z_{59} = -91.106186954104$$
$$z_{60} = -12.5663706143592$$
$$z_{61} = -69.1150383789755$$
$$z_{62} = 37.6991118430775$$
$$z_{63} = 9.42477796076938$$
$$z_{64} = 65.9734457253857$$
$$z_{65} = -65.9734457253857$$
$$z_{66} = -370.707933123596$$
$$z_{67} = -84.8230016469244$$
$$z_{68} = -34.5575191894877$$
$$z_{69} = 43.9822971502571$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(z \right)}}{z} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Z:
Solución analítica
$$z_{1} = \pi$$
Solución numérica
$$z_{1} = -18.8495559215388$$
$$z_{2} = -53.4070751110265$$
$$z_{3} = -37.6991118430775$$
$$z_{4} = -59.6902604182061$$
$$z_{5} = -15.707963267949$$
$$z_{6} = -113.097335529233$$
$$z_{7} = -56.5486677646163$$
$$z_{8} = 12.5663706143592$$
$$z_{9} = 3.14159265358979$$
$$z_{10} = -31.4159265358979$$
$$z_{11} = 84.8230016469244$$
$$z_{12} = -81.6814089933346$$
$$z_{13} = 94.2477796076938$$
$$z_{14} = 21.9911485751286$$
$$z_{15} = -87.9645943005142$$
$$z_{16} = 81.6814089933346$$
$$z_{17} = 40.8407044966673$$
$$z_{18} = -75.398223686155$$
$$z_{19} = -78.5398163397448$$
$$z_{20} = 62.8318530717959$$
$$z_{21} = 100.530964914873$$
$$z_{22} = -21.9911485751286$$
$$z_{23} = 47.1238898038469$$
$$z_{24} = 91.106186954104$$
$$z_{25} = 75.398223686155$$
$$z_{26} = 28.2743338823081$$
$$z_{27} = 34.5575191894877$$
$$z_{28} = 6.28318530717959$$
$$z_{29} = 78.5398163397448$$
$$z_{30} = 72.2566310325652$$
$$z_{31} = -6.28318530717959$$
$$z_{32} = 15.707963267949$$
$$z_{33} = 31.4159265358979$$
$$z_{34} = -47.1238898038469$$
$$z_{35} = 25.1327412287183$$
$$z_{36} = 18.8495559215388$$
$$z_{37} = 153.9380400259$$
$$z_{38} = -94.2477796076938$$
$$z_{39} = -3.14159265358979$$
$$z_{40} = -40.8407044966673$$
$$z_{41} = 56.5486677646163$$
$$z_{42} = -25.1327412287183$$
$$z_{43} = 53.4070751110265$$
$$z_{44} = -28.2743338823081$$
$$z_{45} = -9.42477796076938$$
$$z_{46} = 87.9645943005142$$
$$z_{47} = -50.2654824574367$$
$$z_{48} = -100.530964914873$$
$$z_{49} = -43.9822971502571$$
$$z_{50} = -223.053078404875$$
$$z_{51} = 50.2654824574367$$
$$z_{52} = -97.3893722612836$$
$$z_{53} = 590.619418874881$$
$$z_{54} = 69.1150383789755$$
$$z_{55} = 59.6902604182061$$
$$z_{56} = 97.3893722612836$$
$$z_{57} = -62.8318530717959$$
$$z_{58} = -72.2566310325652$$
$$z_{59} = -91.106186954104$$
$$z_{60} = -12.5663706143592$$
$$z_{61} = -69.1150383789755$$
$$z_{62} = 37.6991118430775$$
$$z_{63} = 9.42477796076938$$
$$z_{64} = 65.9734457253857$$
$$z_{65} = -65.9734457253857$$
$$z_{66} = -370.707933123596$$
$$z_{67} = -84.8230016469244$$
$$z_{68} = -34.5575191894877$$
$$z_{69} = 43.9822971502571$$
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
$$\frac{\cos{\left(z \right)}}{z} - \frac{\sin{\left(z \right)}}{z^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -86.3822220347287$$
$$z_{2} = -4.49340945790906$$
$$z_{3} = -42.3879135681319$$
$$z_{4} = -32.9563890398225$$
$$z_{5} = -67.5294347771441$$
$$z_{6} = -20.3713029592876$$
$$z_{7} = 23.519452498689$$
$$z_{8} = 61.2447302603744$$
$$z_{9} = 7.72525183693771$$
$$z_{10} = -92.6661922776228$$
$$z_{11} = 14.0661939128315$$
$$z_{12} = -45.5311340139913$$
$$z_{13} = 48.6741442319544$$
$$z_{14} = 70.6716857116195$$
$$z_{15} = 64.3871195905574$$
$$z_{16} = 36.1006222443756$$
$$z_{17} = 95.8081387868617$$
$$z_{18} = -29.811598790893$$
$$z_{19} = -76.9560263103312$$
$$z_{20} = -10.9041216594289$$
$$z_{21} = 98.9500628243319$$
$$z_{22} = 76.9560263103312$$
$$z_{23} = 45.5311340139913$$
$$z_{24} = 39.2444323611642$$
$$z_{25} = -98.9500628243319$$
$$z_{26} = -89.5242209304172$$
$$z_{27} = -61.2447302603744$$
$$z_{28} = 17.2207552719308$$
$$z_{29} = 92.6661922776228$$
$$z_{30} = 4.49340945790906$$
$$z_{31} = 54.9596782878889$$
$$z_{32} = -394.267341680887$$
$$z_{33} = -48.6741442319544$$
$$z_{34} = 67.5294347771441$$
$$z_{35} = -7.72525183693771$$
$$z_{36} = -17.2207552719308$$
$$z_{37} = -4355.81798462425$$
$$z_{38} = 86.3822220347287$$
$$z_{39} = 32.9563890398225$$
$$z_{40} = -26.6660542588127$$
$$z_{41} = 26.6660542588127$$
$$z_{42} = 80.0981286289451$$
$$z_{43} = 108.375719651675$$
$$z_{44} = -95.8081387868617$$
$$z_{45} = 20.3713029592876$$
$$z_{46} = -83.2401924707234$$
$$z_{47} = 10.9041216594289$$
$$z_{48} = 83.2401924707234$$
$$z_{49} = 89.5242209304172$$
$$z_{50} = 29.811598790893$$
$$z_{51} = 58.1022547544956$$
$$z_{52} = -54.9596782878889$$
$$z_{53} = -64.3871195905574$$
$$z_{54} = -39.2444323611642$$
$$z_{55} = -14.0661939128315$$
$$z_{56} = -70.6716857116195$$
$$z_{57} = -73.8138806006806$$
$$z_{58} = 73.8138806006806$$
$$z_{59} = -36.1006222443756$$
$$z_{60} = -58.1022547544956$$
$$z_{61} = 42.3879135681319$$
$$z_{62} = -51.8169824872797$$
$$z_{63} = -23.519452498689$$
$$z_{64} = 51.8169824872797$$
$$z_{65} = -80.0981286289451$$
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:
$$z_{1} = -86.3822220347287$$
$$z_{2} = -4.49340945790906$$
$$z_{3} = -42.3879135681319$$
$$z_{4} = -67.5294347771441$$
$$z_{5} = 23.519452498689$$
$$z_{6} = 61.2447302603744$$
$$z_{7} = -92.6661922776228$$
$$z_{8} = 48.6741442319544$$
$$z_{9} = 36.1006222443756$$
$$z_{10} = -29.811598790893$$
$$z_{11} = -10.9041216594289$$
$$z_{12} = 98.9500628243319$$
$$z_{13} = -98.9500628243319$$
$$z_{14} = -61.2447302603744$$
$$z_{15} = 17.2207552719308$$
$$z_{16} = 92.6661922776228$$
$$z_{17} = 4.49340945790906$$
$$z_{18} = 54.9596782878889$$
$$z_{19} = -394.267341680887$$
$$z_{20} = -48.6741442319544$$
$$z_{21} = 67.5294347771441$$
$$z_{22} = -17.2207552719308$$
$$z_{23} = 86.3822220347287$$
$$z_{24} = 80.0981286289451$$
$$z_{25} = 10.9041216594289$$
$$z_{26} = 29.811598790893$$
$$z_{27} = -54.9596782878889$$
$$z_{28} = -73.8138806006806$$
$$z_{29} = 73.8138806006806$$
$$z_{30} = -36.1006222443756$$
$$z_{31} = 42.3879135681319$$
$$z_{32} = -23.519452498689$$
$$z_{33} = -80.0981286289451$$
Puntos máximos de la función:
$$z_{33} = -32.9563890398225$$
$$z_{33} = -20.3713029592876$$
$$z_{33} = 7.72525183693771$$
$$z_{33} = 14.0661939128315$$
$$z_{33} = -45.5311340139913$$
$$z_{33} = 70.6716857116195$$
$$z_{33} = 64.3871195905574$$
$$z_{33} = 95.8081387868617$$
$$z_{33} = -76.9560263103312$$
$$z_{33} = 76.9560263103312$$
$$z_{33} = 45.5311340139913$$
$$z_{33} = 39.2444323611642$$
$$z_{33} = -89.5242209304172$$
$$z_{33} = -7.72525183693771$$
$$z_{33} = -4355.81798462425$$
$$z_{33} = 32.9563890398225$$
$$z_{33} = -26.6660542588127$$
$$z_{33} = 26.6660542588127$$
$$z_{33} = 108.375719651675$$
$$z_{33} = -95.8081387868617$$
$$z_{33} = 20.3713029592876$$
$$z_{33} = -83.2401924707234$$
$$z_{33} = 83.2401924707234$$
$$z_{33} = 89.5242209304172$$
$$z_{33} = 58.1022547544956$$
$$z_{33} = -64.3871195905574$$
$$z_{33} = -39.2444323611642$$
$$z_{33} = -14.0661939128315$$
$$z_{33} = -70.6716857116195$$
$$z_{33} = -58.1022547544956$$
$$z_{33} = -51.8169824872797$$
$$z_{33} = 51.8169824872797$$
Decrece en los intervalos
$$\left[98.9500628243319, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -394.267341680887\right]$$
$$\frac{d}{d z} f{\left(z \right)} = 0$$
(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
$$\frac{\cos{\left(z \right)}}{z} - \frac{\sin{\left(z \right)}}{z^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = -86.3822220347287$$
$$z_{2} = -4.49340945790906$$
$$z_{3} = -42.3879135681319$$
$$z_{4} = -32.9563890398225$$
$$z_{5} = -67.5294347771441$$
$$z_{6} = -20.3713029592876$$
$$z_{7} = 23.519452498689$$
$$z_{8} = 61.2447302603744$$
$$z_{9} = 7.72525183693771$$
$$z_{10} = -92.6661922776228$$
$$z_{11} = 14.0661939128315$$
$$z_{12} = -45.5311340139913$$
$$z_{13} = 48.6741442319544$$
$$z_{14} = 70.6716857116195$$
$$z_{15} = 64.3871195905574$$
$$z_{16} = 36.1006222443756$$
$$z_{17} = 95.8081387868617$$
$$z_{18} = -29.811598790893$$
$$z_{19} = -76.9560263103312$$
$$z_{20} = -10.9041216594289$$
$$z_{21} = 98.9500628243319$$
$$z_{22} = 76.9560263103312$$
$$z_{23} = 45.5311340139913$$
$$z_{24} = 39.2444323611642$$
$$z_{25} = -98.9500628243319$$
$$z_{26} = -89.5242209304172$$
$$z_{27} = -61.2447302603744$$
$$z_{28} = 17.2207552719308$$
$$z_{29} = 92.6661922776228$$
$$z_{30} = 4.49340945790906$$
$$z_{31} = 54.9596782878889$$
$$z_{32} = -394.267341680887$$
$$z_{33} = -48.6741442319544$$
$$z_{34} = 67.5294347771441$$
$$z_{35} = -7.72525183693771$$
$$z_{36} = -17.2207552719308$$
$$z_{37} = -4355.81798462425$$
$$z_{38} = 86.3822220347287$$
$$z_{39} = 32.9563890398225$$
$$z_{40} = -26.6660542588127$$
$$z_{41} = 26.6660542588127$$
$$z_{42} = 80.0981286289451$$
$$z_{43} = 108.375719651675$$
$$z_{44} = -95.8081387868617$$
$$z_{45} = 20.3713029592876$$
$$z_{46} = -83.2401924707234$$
$$z_{47} = 10.9041216594289$$
$$z_{48} = 83.2401924707234$$
$$z_{49} = 89.5242209304172$$
$$z_{50} = 29.811598790893$$
$$z_{51} = 58.1022547544956$$
$$z_{52} = -54.9596782878889$$
$$z_{53} = -64.3871195905574$$
$$z_{54} = -39.2444323611642$$
$$z_{55} = -14.0661939128315$$
$$z_{56} = -70.6716857116195$$
$$z_{57} = -73.8138806006806$$
$$z_{58} = 73.8138806006806$$
$$z_{59} = -36.1006222443756$$
$$z_{60} = -58.1022547544956$$
$$z_{61} = 42.3879135681319$$
$$z_{62} = -51.8169824872797$$
$$z_{63} = -23.519452498689$$
$$z_{64} = 51.8169824872797$$
$$z_{65} = -80.0981286289451$$
Signos de extremos en los puntos:
(-86.38222203472871, -0.0115756804584678)
(-4.493409457909064, -0.217233628211222)
(-42.38791356813192, -0.0235850682290164)
(-32.956389039822476, 0.0303291711863103)
(-67.52943477714412, -0.0148067339465492)
(-20.37130295928756, 0.0490296240140742)
(23.519452498689006, -0.0424796169776126)
(61.2447302603744, -0.0163257593209978)
(7.725251836937707, 0.128374553525899)
(-92.66619227762284, -0.0107907938495342)
(14.066193912831473, 0.0709134594504622)
(-45.53113401399128, 0.0219576982284824)
(48.674144231954386, -0.0205404540417537)
(70.6716857116195, 0.0141485220648664)
(64.38711959055742, 0.0155291838074613)
(36.10062224437561, -0.0276897323011492)
(95.8081387868617, 0.0104369581345658)
(-29.81159879089296, -0.0335251350213988)
(-76.95602631033118, 0.0129933369870427)
(-10.904121659428899, -0.0913252028230577)
(98.95006282433188, -0.010105591736504)
(76.95602631033118, 0.0129933369870427)
(45.53113401399128, 0.0219576982284824)
(39.24443236116419, 0.0254730530928808)
(-98.95006282433188, -0.010105591736504)
(-89.52422093041719, 0.0111694646341736)
(-61.2447302603744, -0.0163257593209978)
(17.22075527193077, -0.0579718023461539)
(92.66619227762284, -0.0107907938495342)
(4.493409457909064, -0.217233628211222)
(54.959678287888934, -0.0181921463218031)
(-394.26734168088706, -0.00253634191261283)
(-48.674144231954386, -0.0205404540417537)
(67.52943477714412, -0.0148067339465492)
(-7.725251836937707, 0.128374553525899)
(-17.22075527193077, -0.0579718023461539)
(-4355.817984624248, 0.000229577998248987)
(86.38222203472871, -0.0115756804584678)
(32.956389039822476, 0.0303291711863103)
(-26.666054258812675, 0.0374745199939312)
(26.666054258812675, 0.0374745199939312)
(80.09812862894512, -0.012483713321779)
(108.37571965167469, 0.00922676625078197)
(-95.8081387868617, 0.0104369581345658)
(20.37130295928756, 0.0490296240140742)
(-83.2401924707234, 0.0120125604820527)
(10.904121659428899, -0.0913252028230577)
(83.2401924707234, 0.0120125604820527)
(89.52422093041719, 0.0111694646341736)
(29.81159879089296, -0.0335251350213988)
(58.10225475449559, 0.0172084874716279)
(-54.959678287888934, -0.0181921463218031)
(-64.38711959055742, 0.0155291838074613)
(-39.24443236116419, 0.0254730530928808)
(-14.066193912831473, 0.0709134594504622)
(-70.6716857116195, 0.0141485220648664)
(-73.81388060068065, -0.01354634434514)
(73.81388060068065, -0.01354634434514)
(-36.10062224437561, -0.0276897323011492)
(-58.10225475449559, 0.0172084874716279)
(42.38791356813192, -0.0235850682290164)
(-51.81698248727967, 0.019295099487588)
(-23.519452498689006, -0.0424796169776126)
(51.81698248727967, 0.019295099487588)
(-80.09812862894512, -0.012483713321779)
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:
$$z_{1} = -86.3822220347287$$
$$z_{2} = -4.49340945790906$$
$$z_{3} = -42.3879135681319$$
$$z_{4} = -67.5294347771441$$
$$z_{5} = 23.519452498689$$
$$z_{6} = 61.2447302603744$$
$$z_{7} = -92.6661922776228$$
$$z_{8} = 48.6741442319544$$
$$z_{9} = 36.1006222443756$$
$$z_{10} = -29.811598790893$$
$$z_{11} = -10.9041216594289$$
$$z_{12} = 98.9500628243319$$
$$z_{13} = -98.9500628243319$$
$$z_{14} = -61.2447302603744$$
$$z_{15} = 17.2207552719308$$
$$z_{16} = 92.6661922776228$$
$$z_{17} = 4.49340945790906$$
$$z_{18} = 54.9596782878889$$
$$z_{19} = -394.267341680887$$
$$z_{20} = -48.6741442319544$$
$$z_{21} = 67.5294347771441$$
$$z_{22} = -17.2207552719308$$
$$z_{23} = 86.3822220347287$$
$$z_{24} = 80.0981286289451$$
$$z_{25} = 10.9041216594289$$
$$z_{26} = 29.811598790893$$
$$z_{27} = -54.9596782878889$$
$$z_{28} = -73.8138806006806$$
$$z_{29} = 73.8138806006806$$
$$z_{30} = -36.1006222443756$$
$$z_{31} = 42.3879135681319$$
$$z_{32} = -23.519452498689$$
$$z_{33} = -80.0981286289451$$
Puntos máximos de la función:
$$z_{33} = -32.9563890398225$$
$$z_{33} = -20.3713029592876$$
$$z_{33} = 7.72525183693771$$
$$z_{33} = 14.0661939128315$$
$$z_{33} = -45.5311340139913$$
$$z_{33} = 70.6716857116195$$
$$z_{33} = 64.3871195905574$$
$$z_{33} = 95.8081387868617$$
$$z_{33} = -76.9560263103312$$
$$z_{33} = 76.9560263103312$$
$$z_{33} = 45.5311340139913$$
$$z_{33} = 39.2444323611642$$
$$z_{33} = -89.5242209304172$$
$$z_{33} = -7.72525183693771$$
$$z_{33} = -4355.81798462425$$
$$z_{33} = 32.9563890398225$$
$$z_{33} = -26.6660542588127$$
$$z_{33} = 26.6660542588127$$
$$z_{33} = 108.375719651675$$
$$z_{33} = -95.8081387868617$$
$$z_{33} = 20.3713029592876$$
$$z_{33} = -83.2401924707234$$
$$z_{33} = 83.2401924707234$$
$$z_{33} = 89.5242209304172$$
$$z_{33} = 58.1022547544956$$
$$z_{33} = -64.3871195905574$$
$$z_{33} = -39.2444323611642$$
$$z_{33} = -14.0661939128315$$
$$z_{33} = -70.6716857116195$$
$$z_{33} = -58.1022547544956$$
$$z_{33} = -51.8169824872797$$
$$z_{33} = 51.8169824872797$$
Decrece en los intervalos
$$\left[98.9500628243319, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -394.267341680887\right]$$
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
$$\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 131.931731514843$$
$$z_{2} = -81.6569138240367$$
$$z_{3} = -1288.05143523817$$
$$z_{4} = -25.052825280993$$
$$z_{5} = 84.7994143922025$$
$$z_{6} = 25.052825280993$$
$$z_{7} = -342.42775856009$$
$$z_{8} = 50.2256516491831$$
$$z_{9} = 2.0815759778181$$
$$z_{10} = 56.5132704621986$$
$$z_{11} = -9.20584014293667$$
$$z_{12} = -56.5132704621986$$
$$z_{13} = -53.3695918204908$$
$$z_{14} = -5.94036999057271$$
$$z_{15} = 5.94036999057271$$
$$z_{16} = -94.2265525745684$$
$$z_{17} = -18.7426455847748$$
$$z_{18} = 81.6569138240367$$
$$z_{19} = -12.404445021902$$
$$z_{20} = 12.404445021902$$
$$z_{21} = -65.9431119046552$$
$$z_{22} = -21.8996964794928$$
$$z_{23} = 78.5143405319308$$
$$z_{24} = 47.0813974121542$$
$$z_{25} = -100.511065295271$$
$$z_{26} = 28.2033610039524$$
$$z_{27} = -62.8000005565198$$
$$z_{28} = -2.0815759778181$$
$$z_{29} = 62.8000005565198$$
$$z_{30} = -34.499514921367$$
$$z_{31} = 15.5792364103872$$
$$z_{32} = -91.0842274914688$$
$$z_{33} = -15.5792364103872$$
$$z_{34} = 65.9431119046552$$
$$z_{35} = 18.7426455847748$$
$$z_{36} = -69.0860849466452$$
$$z_{37} = -40.7916552312719$$
$$z_{38} = 59.6567290035279$$
$$z_{39} = 75.3716854092873$$
$$z_{40} = -87.9418500396598$$
$$z_{41} = 37.6459603230864$$
$$z_{42} = 21.8996964794928$$
$$z_{43} = 69.0860849466452$$
$$z_{44} = 87.9418500396598$$
$$z_{45} = 53.3695918204908$$
$$z_{46} = 34.499514921367$$
$$z_{47} = -37.6459603230864$$
$$z_{48} = -78.5143405319308$$
$$z_{49} = -59.6567290035279$$
$$z_{50} = 9.20584014293667$$
$$z_{51} = 97.368830362901$$
$$z_{52} = -47.0813974121542$$
$$z_{53} = -1790.70669566846$$
$$z_{54} = -72.2289377620154$$
$$z_{55} = -84.7994143922025$$
$$z_{56} = 40.7916552312719$$
$$z_{57} = 31.3520917265645$$
$$z_{58} = 72.2289377620154$$
$$z_{59} = 91.0842274914688$$
$$z_{60} = -28.2033610039524$$
$$z_{61} = -97.368830362901$$
$$z_{62} = -75.3716854092873$$
$$z_{63} = -43.9367614714198$$
$$z_{64} = 94.2265525745684$$
$$z_{65} = -50.2256516491831$$
$$z_{66} = 43.9367614714198$$
$$z_{67} = 100.511065295271$$
$$z_{68} = -31.3520917265645$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = 0$$
$$\lim_{z \to 0^-}\left(\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z}\right) = - \frac{1}{3}$$
$$\lim_{z \to 0^+}\left(\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z}\right) = - \frac{1}{3}$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[97.368830362901, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -1790.70669566846\right]$$
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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
$$\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$z_{1} = 131.931731514843$$
$$z_{2} = -81.6569138240367$$
$$z_{3} = -1288.05143523817$$
$$z_{4} = -25.052825280993$$
$$z_{5} = 84.7994143922025$$
$$z_{6} = 25.052825280993$$
$$z_{7} = -342.42775856009$$
$$z_{8} = 50.2256516491831$$
$$z_{9} = 2.0815759778181$$
$$z_{10} = 56.5132704621986$$
$$z_{11} = -9.20584014293667$$
$$z_{12} = -56.5132704621986$$
$$z_{13} = -53.3695918204908$$
$$z_{14} = -5.94036999057271$$
$$z_{15} = 5.94036999057271$$
$$z_{16} = -94.2265525745684$$
$$z_{17} = -18.7426455847748$$
$$z_{18} = 81.6569138240367$$
$$z_{19} = -12.404445021902$$
$$z_{20} = 12.404445021902$$
$$z_{21} = -65.9431119046552$$
$$z_{22} = -21.8996964794928$$
$$z_{23} = 78.5143405319308$$
$$z_{24} = 47.0813974121542$$
$$z_{25} = -100.511065295271$$
$$z_{26} = 28.2033610039524$$
$$z_{27} = -62.8000005565198$$
$$z_{28} = -2.0815759778181$$
$$z_{29} = 62.8000005565198$$
$$z_{30} = -34.499514921367$$
$$z_{31} = 15.5792364103872$$
$$z_{32} = -91.0842274914688$$
$$z_{33} = -15.5792364103872$$
$$z_{34} = 65.9431119046552$$
$$z_{35} = 18.7426455847748$$
$$z_{36} = -69.0860849466452$$
$$z_{37} = -40.7916552312719$$
$$z_{38} = 59.6567290035279$$
$$z_{39} = 75.3716854092873$$
$$z_{40} = -87.9418500396598$$
$$z_{41} = 37.6459603230864$$
$$z_{42} = 21.8996964794928$$
$$z_{43} = 69.0860849466452$$
$$z_{44} = 87.9418500396598$$
$$z_{45} = 53.3695918204908$$
$$z_{46} = 34.499514921367$$
$$z_{47} = -37.6459603230864$$
$$z_{48} = -78.5143405319308$$
$$z_{49} = -59.6567290035279$$
$$z_{50} = 9.20584014293667$$
$$z_{51} = 97.368830362901$$
$$z_{52} = -47.0813974121542$$
$$z_{53} = -1790.70669566846$$
$$z_{54} = -72.2289377620154$$
$$z_{55} = -84.7994143922025$$
$$z_{56} = 40.7916552312719$$
$$z_{57} = 31.3520917265645$$
$$z_{58} = 72.2289377620154$$
$$z_{59} = 91.0842274914688$$
$$z_{60} = -28.2033610039524$$
$$z_{61} = -97.368830362901$$
$$z_{62} = -75.3716854092873$$
$$z_{63} = -43.9367614714198$$
$$z_{64} = 94.2265525745684$$
$$z_{65} = -50.2256516491831$$
$$z_{66} = 43.9367614714198$$
$$z_{67} = 100.511065295271$$
$$z_{68} = -31.3520917265645$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = 0$$
$$\lim_{z \to 0^-}\left(\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z}\right) = - \frac{1}{3}$$
$$\lim_{z \to 0^+}\left(\frac{- \sin{\left(z \right)} - \frac{2 \cos{\left(z \right)}}{z} + \frac{2 \sin{\left(z \right)}}{z^{2}}}{z}\right) = - \frac{1}{3}$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[97.368830362901, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -1790.70669566846\right]$$
Asíntotas verticales
Hay:
$$z_{1} = 0$$
$$z_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con z->+oo y z->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z}\right) = 0$$
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, dividida por z con z->+oo y z ->-oo
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{z \to -\infty}\left(\frac{\sin{\left(z \right)}}{z^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{z \to \infty}\left(\frac{\sin{\left(z \right)}}{z^{2}}\right) = 0$$
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:
$$\frac{\sin{\left(z \right)}}{z} = \frac{\sin{\left(z \right)}}{z}$$
- No
$$\frac{\sin{\left(z \right)}}{z} = - \frac{\sin{\left(z \right)}}{z}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(z \right)}}{z} = \frac{\sin{\left(z \right)}}{z}$$
- No
$$\frac{\sin{\left(z \right)}}{z} = - \frac{\sin{\left(z \right)}}{z}$$
- No
es decir, función
no es
par ni impar