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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 =:
-
e^3*x+10*e^2*x
-
x^11
-
(-cos(2*x)-sin(2*x))*exp(x)
-
5/(x^2-16)
-
Expresiones idénticas
- sin(x)^ dos /(x^ dos *cos(x)^ dos)
- seno de (x) al cuadrado dividir por (x al cuadrado multiplicar por coseno de (x) al cuadrado )
- seno de (x) en el grado dos dividir por (x en el grado dos multiplicar por coseno de (x) en el grado dos)
- sin(x)2/(x2*cos(x)2)
- sinx2/x2*cosx2
- sin(x)²/(x²*cos(x)²)
- sin(x) en el grado 2/(x en el grado 2*cos(x) en el grado 2)
- sin(x)^2/(x^2cos(x)^2)
- sin(x)2/(x2cos(x)2)
- sinx2/x2cosx2
- sinx^2/x^2cosx^2
- sin(x)^2 dividir por (x^2*cos(x)^2)
-
Expresiones semejantes
- sinx^2/(x^2*cosx^2)
-
Expresiones con funciones
- Seno sin
- sin(x)/2+(1+x)*exp(-x)
- sin(x)^(2)+2
- sin(3*x)+x^4
- sin(pi*x)/sin(pi*x)
- sin(4*x)*cos(x)+2*sin(3*x)-cos(4*x)*sin(x)
- Coseno cos
- cos(x*sqrt(6))+sin(x*sqrt(6))
- cos(log(x))+sin(log(x))
- cos(3*x-pi/4)*(-2)-1
- cos(3*x)+sin(3*x)
- cos(3*π*x)/x
Gráfico de la función y = sin(x)^2/(x^2*cos(x)^2)
Solución
Ha introducido
[src]
2
sin (x)
f(x) = ----------
2 2
x *cos (x)
$$f{\left(x \right)} = \frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}$$
f = sin(x)^2/((x^2*cos(x)^2))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -75.3982239092641$$
$$x_{2} = 12.5663704030148$$
$$x_{3} = -59.6902604581613$$
$$x_{4} = -47.1238902965186$$
$$x_{5} = 97.389369243544$$
$$x_{6} = 43.9822971695396$$
$$x_{7} = 50.2654824463048$$
$$x_{8} = 78.5398161567348$$
$$x_{9} = 69.1150396761346$$
$$x_{10} = -91.1061874780264$$
$$x_{11} = 65.9734457531786$$
$$x_{12} = -37.6991118774578$$
$$x_{13} = 91.1061859440443$$
$$x_{14} = 9.42477841501709$$
$$x_{15} = -28.2743336713822$$
$$x_{16} = 15.7079634768209$$
$$x_{17} = 81.681405676683$$
$$x_{18} = -100.530964457304$$
$$x_{19} = -47.123885947407$$
$$x_{20} = -97.3893724914597$$
$$x_{21} = -69.1150347483656$$
$$x_{22} = 53.4070756384366$$
$$x_{23} = -40.840703312018$$
$$x_{24} = 18.8495554228473$$
$$x_{25} = 84.8230012060497$$
$$x_{26} = -50.265482255523$$
$$x_{27} = 47.1238887177784$$
$$x_{28} = -69.1150388879671$$
$$x_{29} = -31.4159169560951$$
$$x_{30} = -12.5663738465514$$
$$x_{31} = -78.5398205512158$$
$$x_{32} = 69.1150373343872$$
$$x_{33} = 94.2477796093518$$
$$x_{34} = -56.5486717332894$$
$$x_{35} = 9.42477359245689$$
$$x_{36} = 87.964594336279$$
$$x_{37} = -25.1327416992958$$
$$x_{38} = 53.4070717623554$$
$$x_{39} = 100.530964738197$$
$$x_{40} = 28.2743338650717$$
$$x_{41} = 81.6814092305933$$
$$x_{42} = -21.9911485863883$$
$$x_{43} = 59.6902601279252$$
$$x_{44} = 62.8318577159057$$
$$x_{45} = 25.132740077928$$
$$x_{46} = -65.973445764563$$
$$x_{47} = 47.1238910487187$$
$$x_{48} = 31.4159270424371$$
$$x_{49} = 3.14159344183862$$
$$x_{50} = 84.8230065881377$$
$$x_{51} = 97.3893728214886$$
$$x_{52} = -94.2477794200541$$
$$x_{53} = 72.2566310277122$$
$$x_{54} = -18.8495569812202$$
$$x_{55} = 56.5486675748786$$
$$x_{56} = -84.8230005515572$$
$$x_{57} = 6.28318528372099$$
$$x_{58} = 40.8407040263972$$
$$x_{59} = 62.8318526176435$$
$$x_{60} = -97.3893565888386$$
$$x_{61} = 31.4159229433273$$
$$x_{62} = -53.4070753268242$$
$$x_{63} = 3.14159066667015$$
$$x_{64} = 34.5575189920092$$
$$x_{65} = -75.3982096453948$$
$$x_{66} = -15.7079632965878$$
$$x_{67} = 40.8407088324578$$
$$x_{68} = -12.5663700636969$$
$$x_{69} = -43.9822971743658$$
$$x_{70} = -9.42477815198073$$
$$x_{71} = -78.5398158689808$$
$$x_{72} = 21.9911485852142$$
$$x_{73} = -18.8495546439997$$
$$x_{74} = 91.1061882995801$$
$$x_{75} = 59.6902606477915$$
$$x_{76} = -84.8230028788551$$
$$x_{77} = -34.5575186847907$$
$$x_{78} = -81.6814090387721$$
$$x_{79} = -25.1327370334014$$
$$x_{80} = -6.28318506808401$$
$$x_{81} = -31.415926743419$$
$$x_{82} = 12.5663723824066$$
$$x_{83} = -9.42476997567688$$
$$x_{84} = 25.1327424029566$$
$$x_{85} = 34.5575103669975$$
$$x_{86} = -62.8318542597133$$
$$x_{87} = 75.3982242304841$$
$$x_{88} = -62.8318519368231$$
$$x_{89} = -91.1061835114386$$
$$x_{90} = -87.9645943579888$$
$$x_{91} = -3.1415929974916$$
$$x_{92} = -56.5486672790266$$
$$x_{93} = 18.8495598353757$$
$$x_{94} = 75.3982205160105$$
$$x_{95} = -72.2566308380522$$
$$x_{96} = -34.557522891703$$
$$x_{97} = -40.8407056349543$$
$$x_{98} = 37.6991120642897$$
$$x_{99} = -53.4070631301154$$
o sea hay que resolver la ecuación:
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -75.3982239092641$$
$$x_{2} = 12.5663704030148$$
$$x_{3} = -59.6902604581613$$
$$x_{4} = -47.1238902965186$$
$$x_{5} = 97.389369243544$$
$$x_{6} = 43.9822971695396$$
$$x_{7} = 50.2654824463048$$
$$x_{8} = 78.5398161567348$$
$$x_{9} = 69.1150396761346$$
$$x_{10} = -91.1061874780264$$
$$x_{11} = 65.9734457531786$$
$$x_{12} = -37.6991118774578$$
$$x_{13} = 91.1061859440443$$
$$x_{14} = 9.42477841501709$$
$$x_{15} = -28.2743336713822$$
$$x_{16} = 15.7079634768209$$
$$x_{17} = 81.681405676683$$
$$x_{18} = -100.530964457304$$
$$x_{19} = -47.123885947407$$
$$x_{20} = -97.3893724914597$$
$$x_{21} = -69.1150347483656$$
$$x_{22} = 53.4070756384366$$
$$x_{23} = -40.840703312018$$
$$x_{24} = 18.8495554228473$$
$$x_{25} = 84.8230012060497$$
$$x_{26} = -50.265482255523$$
$$x_{27} = 47.1238887177784$$
$$x_{28} = -69.1150388879671$$
$$x_{29} = -31.4159169560951$$
$$x_{30} = -12.5663738465514$$
$$x_{31} = -78.5398205512158$$
$$x_{32} = 69.1150373343872$$
$$x_{33} = 94.2477796093518$$
$$x_{34} = -56.5486717332894$$
$$x_{35} = 9.42477359245689$$
$$x_{36} = 87.964594336279$$
$$x_{37} = -25.1327416992958$$
$$x_{38} = 53.4070717623554$$
$$x_{39} = 100.530964738197$$
$$x_{40} = 28.2743338650717$$
$$x_{41} = 81.6814092305933$$
$$x_{42} = -21.9911485863883$$
$$x_{43} = 59.6902601279252$$
$$x_{44} = 62.8318577159057$$
$$x_{45} = 25.132740077928$$
$$x_{46} = -65.973445764563$$
$$x_{47} = 47.1238910487187$$
$$x_{48} = 31.4159270424371$$
$$x_{49} = 3.14159344183862$$
$$x_{50} = 84.8230065881377$$
$$x_{51} = 97.3893728214886$$
$$x_{52} = -94.2477794200541$$
$$x_{53} = 72.2566310277122$$
$$x_{54} = -18.8495569812202$$
$$x_{55} = 56.5486675748786$$
$$x_{56} = -84.8230005515572$$
$$x_{57} = 6.28318528372099$$
$$x_{58} = 40.8407040263972$$
$$x_{59} = 62.8318526176435$$
$$x_{60} = -97.3893565888386$$
$$x_{61} = 31.4159229433273$$
$$x_{62} = -53.4070753268242$$
$$x_{63} = 3.14159066667015$$
$$x_{64} = 34.5575189920092$$
$$x_{65} = -75.3982096453948$$
$$x_{66} = -15.7079632965878$$
$$x_{67} = 40.8407088324578$$
$$x_{68} = -12.5663700636969$$
$$x_{69} = -43.9822971743658$$
$$x_{70} = -9.42477815198073$$
$$x_{71} = -78.5398158689808$$
$$x_{72} = 21.9911485852142$$
$$x_{73} = -18.8495546439997$$
$$x_{74} = 91.1061882995801$$
$$x_{75} = 59.6902606477915$$
$$x_{76} = -84.8230028788551$$
$$x_{77} = -34.5575186847907$$
$$x_{78} = -81.6814090387721$$
$$x_{79} = -25.1327370334014$$
$$x_{80} = -6.28318506808401$$
$$x_{81} = -31.415926743419$$
$$x_{82} = 12.5663723824066$$
$$x_{83} = -9.42476997567688$$
$$x_{84} = 25.1327424029566$$
$$x_{85} = 34.5575103669975$$
$$x_{86} = -62.8318542597133$$
$$x_{87} = 75.3982242304841$$
$$x_{88} = -62.8318519368231$$
$$x_{89} = -91.1061835114386$$
$$x_{90} = -87.9645943579888$$
$$x_{91} = -3.1415929974916$$
$$x_{92} = -56.5486672790266$$
$$x_{93} = 18.8495598353757$$
$$x_{94} = 75.3982205160105$$
$$x_{95} = -72.2566308380522$$
$$x_{96} = -34.557522891703$$
$$x_{97} = -40.8407056349543$$
$$x_{98} = 37.6991120642897$$
$$x_{99} = -53.4070631301154$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$2 \frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 2 x \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{x^{4} \cos^{4}{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = 97.3893722612836$$
$$x_{27} = 15.707963267949$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = 3.14159265358979$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 18.8495559215388$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = -78.5398163397448$$
$$x_{38} = -6.28318530717959$$
$$x_{39} = -40.8407044966673$$
$$x_{40} = 43.9822971502571$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 78.5398163397448$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = 75.398223686155$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -34.5575191894877$$
$$x_{49} = 81.6814089933346$$
$$x_{50} = -47.1238898038469$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -9.42477796076938$$
$$x_{53} = -75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -31.4159265358979$$
$$x_{56} = 28.2743338823081$$
$$x_{57} = -91.106186954104$$
$$x_{58} = 21.9911485751286$$
$$x_{59} = 62.8318530717959$$
$$x_{60} = 9.42477796076938$$
$$x_{61} = 50.2654824574367$$
$$x_{62} = 94.2477796076938$$
$$x_{63} = 91.106186954104$$
$$x_{64} = 84.8230016469244$$
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:
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = 97.3893722612836$$
$$x_{27} = 15.707963267949$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = 3.14159265358979$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 18.8495559215388$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = -78.5398163397448$$
$$x_{38} = -6.28318530717959$$
$$x_{39} = -40.8407044966673$$
$$x_{40} = 43.9822971502571$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 78.5398163397448$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = 75.398223686155$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -34.5575191894877$$
$$x_{49} = 81.6814089933346$$
$$x_{50} = -47.1238898038469$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -9.42477796076938$$
$$x_{53} = -75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -31.4159265358979$$
$$x_{56} = 28.2743338823081$$
$$x_{57} = -91.106186954104$$
$$x_{58} = 21.9911485751286$$
$$x_{59} = 62.8318530717959$$
$$x_{60} = 9.42477796076938$$
$$x_{61} = 50.2654824574367$$
$$x_{62} = 94.2477796076938$$
$$x_{63} = 91.106186954104$$
$$x_{64} = 84.8230016469244$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$2 \frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 2 x \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{x^{4} \cos^{4}{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = 97.3893722612836$$
$$x_{27} = 15.707963267949$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = 3.14159265358979$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 18.8495559215388$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = -78.5398163397448$$
$$x_{38} = -6.28318530717959$$
$$x_{39} = -40.8407044966673$$
$$x_{40} = 43.9822971502571$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 78.5398163397448$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = 75.398223686155$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -34.5575191894877$$
$$x_{49} = 81.6814089933346$$
$$x_{50} = -47.1238898038469$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -9.42477796076938$$
$$x_{53} = -75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -31.4159265358979$$
$$x_{56} = 28.2743338823081$$
$$x_{57} = -91.106186954104$$
$$x_{58} = 21.9911485751286$$
$$x_{59} = 62.8318530717959$$
$$x_{60} = 9.42477796076938$$
$$x_{61} = 50.2654824574367$$
$$x_{62} = 94.2477796076938$$
$$x_{63} = 91.106186954104$$
$$x_{64} = 84.8230016469244$$
Signos de extremos en los puntos:
(-59.69026041820607, 4.21786179228739e-34)
(-62.83185307179586, 1.51957436358475e-33)
(-97.3893722612836, 4.96414972110828e-33)
(87.96459430051421, 1.51957436358475e-33)
(-56.548667764616276, 1.51957436358475e-33)
(31.41592653589793, 1.51957436358475e-33)
(69.11503837897546, 4.07351440822617e-33)
(-37.69911184307752, 1.51957436358475e-33)
(-81.68140899333463, 2.30474995774498e-33)
(-84.82300164692441, 6.539239761642e-33)
(-21.991148575128552, 1.51957436358475e-33)
(47.1238898038469, 1.32563038769384e-33)
(-15.707963267948966, 1.51957436358475e-33)
(-12.566370614359172, 1.51957436358475e-33)
(12.566370614359172, 1.51957436358475e-33)
(-87.96459430051421, 1.51957436358475e-33)
(53.40707511102649, 7.58434347792408e-34)
(72.25663103256524, 7.77037197267108e-33)
(-100.53096491487338, 1.51957436358475e-33)
(-3.141592653589793, 1.51957436358475e-33)
(34.55751918948773, 4.07351440822617e-33)
(-94.2477796076938, 1.32563038769384e-33)
(6.283185307179586, 1.51957436358475e-33)
(-69.11503837897546, 4.07351440822617e-33)
(65.97344572538566, 2.21085464398688e-34)
(97.3893722612836, 4.96414972110828e-33)
(15.707963267948966, 1.51957436358475e-33)
(-50.26548245743669, 1.51957436358475e-33)
(-25.132741228718345, 1.51957436358475e-33)
(3.141592653589793, 1.51957436358475e-33)
(-18.84955592153876, 1.51957436358475e-33)
(40.840704496667314, 2.30474995774498e-33)
(18.84955592153876, 1.51957436358475e-33)
(-53.40707511102649, 7.58434347792408e-34)
(37.69911184307752, 1.51957436358475e-33)
(-43.982297150257104, 1.51957436358475e-33)
(-78.53981633974483, 3.90979723557589e-35)
(-6.283185307179586, 1.51957436358475e-33)
(-40.840704496667314, 2.30474995774498e-33)
(43.982297150257104, 1.51957436358475e-33)
(56.548667764616276, 1.51957436358475e-33)
(-65.97344572538566, 2.21085464398688e-34)
(25.132741228718345, 1.51957436358475e-33)
(78.53981633974483, 3.90979723557589e-35)
(-28.274333882308138, 1.51957436358475e-33)
(75.39822368615503, 1.51957436358475e-33)
(59.69026041820607, 4.21786179228739e-34)
(-34.55751918948773, 4.07351440822617e-33)
(81.68140899333463, 2.30474995774498e-33)
(-47.1238898038469, 1.32563038769384e-33)
(100.53096491487338, 1.51957436358475e-33)
(-9.42477796076938, 1.51957436358475e-33)
(-75.39822368615503, 1.51957436358475e-33)
(-72.25663103256524, 7.77037197267108e-33)
(-31.41592653589793, 1.51957436358475e-33)
(28.274333882308138, 1.51957436358475e-33)
(-91.106186954104, 1.84636847666697e-40)
(21.991148575128552, 1.51957436358475e-33)
(62.83185307179586, 1.51957436358475e-33)
(9.42477796076938, 1.51957436358475e-33)
(50.26548245743669, 1.51957436358475e-33)
(94.2477796076938, 1.32563038769384e-33)
(91.106186954104, 1.84636847666697e-40)
(84.82300164692441, 6.539239761642e-33)
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:
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = 97.3893722612836$$
$$x_{27} = 15.707963267949$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = -25.1327412287183$$
$$x_{30} = 3.14159265358979$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 18.8495559215388$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = -78.5398163397448$$
$$x_{38} = -6.28318530717959$$
$$x_{39} = -40.8407044966673$$
$$x_{40} = 43.9822971502571$$
$$x_{41} = 56.5486677646163$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 78.5398163397448$$
$$x_{45} = -28.2743338823081$$
$$x_{46} = 75.398223686155$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -34.5575191894877$$
$$x_{49} = 81.6814089933346$$
$$x_{50} = -47.1238898038469$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -9.42477796076938$$
$$x_{53} = -75.398223686155$$
$$x_{54} = -72.2566310325652$$
$$x_{55} = -31.4159265358979$$
$$x_{56} = 28.2743338823081$$
$$x_{57} = -91.106186954104$$
$$x_{58} = 21.9911485751286$$
$$x_{59} = 62.8318530717959$$
$$x_{60} = 9.42477796076938$$
$$x_{61} = 50.2654824574367$$
$$x_{62} = 94.2477796076938$$
$$x_{63} = 91.106186954104$$
$$x_{64} = 84.8230016469244$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[100.530964914873, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -100.530964914873\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
$$x_{3} = 4.71238898038469$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)^2/((x^2*cos(x)^2)), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin^{2}{\left(x \right)}}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin^{2}{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin^{2}{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{x^{2} \cos^{2}{\left(x \right)}} \sin^{2}{\left(x \right)}}{x}\right)$$
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = \frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}$$
- Sí
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = - \frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = \frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}$$
- Sí
$$\frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}} = - \frac{\sin^{2}{\left(x \right)}}{x^{2} \cos^{2}{\left(x \right)}}$$
- No
es decir, función
es
par
Gráfico