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cos(2*x)^(x^(-2))
Trazado el gráfico de la función/ cos(2*x)^(x^(-2))

Gráfico de la función y = cos(2*x)^(x^(-2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 1 
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                  2
                 x 
f(x) = (cos(2*x))
f(x)=cos⁡1x2(2x)f{\left(x \right)} = \cos^{\frac{1}{x^{2}}}{\left(2 x \right)} 
f = cos(2*x)^(x^(-2))
Gráfico de la función
02468-8-6-4-2-101002
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
cos⁡1x2(2x)=0\cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Solución numérica
x1=0.785398163397448x_{1} = 0.785398163397448
x2=2.35619449019234x_{2} = 2.35619449019234
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en cos(2*x)^(x^(-2)).
cos⁡10(0⋅2)\cos^{\frac{1}{0}}{\left(0 \cdot 2 \right)}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
(−2sin⁡(2x)x2cos⁡(2x)−2log⁡(cos⁡(2x))x3)cos⁡1x2(2x)=0\left(- \frac{2 \sin{\left(2 x \right)}}{x^{2} \cos{\left(2 x \right)}} - \frac{2 \log{\left(\cos{\left(2 x \right)} \right)}}{x^{3}}\right) \cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=87.9645943005142x_{2} = 87.9645943005142
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=40.8407044966673x_{5} = 40.8407044966673
x6=31.4159265358979x_{6} = 31.4159265358979
x7=−37.6991118430775x_{7} = -37.6991118430775
x8=−81.6814089933346x_{8} = -81.6814089933346
x9=−21.9911485751286x_{9} = -21.9911485751286
x10=−15.707963267949x_{10} = -15.707963267949
x11=3.14159265358979x_{11} = 3.14159265358979
x12=12.5663706143592x_{12} = 12.5663706143592
x13=−87.9645943005142x_{13} = -87.9645943005142
x14=−100.530964914873x_{14} = -100.530964914873
x15=−3.14159265358979x_{15} = -3.14159265358979
x16=−78.5398163397448x_{16} = -78.5398163397448
x17=−34.5575191894877x_{17} = -34.5575191894877
x18=53.4070751110265x_{18} = 53.4070751110265
x19=34.5575191894877x_{19} = 34.5575191894877
x20=−94.2477796076938x_{20} = -94.2477796076938
x21=6.28318530717959x_{21} = 6.28318530717959
x22=−91.106186954104x_{22} = -91.106186954104
x23=−69.1150383789755x_{23} = -69.1150383789755
x24=75.398223686155x_{24} = 75.398223686155
x25=97.3893722612836x_{25} = 97.3893722612836
x26=65.9734457253857x_{26} = 65.9734457253857
x27=15.707963267949x_{27} = 15.707963267949
x28=−50.2654824574367x_{28} = -50.2654824574367
x29=−25.1327412287183x_{29} = -25.1327412287183
x30=18.8495559215388x_{30} = 18.8495559215388
x31=−12.5663706143592x_{31} = -12.5663706143592
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−43.9822971502571x_{33} = -43.9822971502571
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=−6.28318530717959x_{35} = -6.28318530717959
x36=43.9822971502571x_{36} = 43.9822971502571
x37=56.5486677646163x_{37} = 56.5486677646163
x38=−65.9734457253857x_{38} = -65.9734457253857
x39=−28.2743338823081x_{39} = -28.2743338823081
x40=78.5398163397448x_{40} = 78.5398163397448
x41=59.6902604182061x_{41} = 59.6902604182061
x42=81.6814089933346x_{42} = 81.6814089933346
x43=−47.1238898038469x_{43} = -47.1238898038469
x44=100.530964914873x_{44} = 100.530964914873
x45=−9.42477796076938x_{45} = -9.42477796076938
x46=−75.398223686155x_{46} = -75.398223686155
x47=−72.2566310325652x_{47} = -72.2566310325652
x48=28.2743338823081x_{48} = 28.2743338823081
x49=−31.4159265358979x_{49} = -31.4159265358979
x50=21.9911485751286x_{50} = 21.9911485751286
x51=62.8318530717959x_{51} = 62.8318530717959
x52=9.42477796076938x_{52} = 9.42477796076938
x53=50.2654824574367x_{53} = 50.2654824574367
x54=94.2477796076938x_{54} = 94.2477796076938
x55=72.2566310325652x_{55} = 72.2566310325652
x56=84.8230016469244x_{56} = 84.8230016469244
Signos de extremos en los puntos:
(-59.69026041820607, 1)

(87.96459430051421, 1)

(-97.3893722612836, 1)

(-56.548667764616276, 1)

(40.840704496667314, 1)

(31.41592653589793, 1)

(-37.69911184307752, 1)

(-81.68140899333463, 1)

(-21.991148575128552, 1)

(-15.707963267948966, 1)

(3.141592653589793, 1)

(12.566370614359172, 1)

(-87.96459430051421, 1)

(-100.53096491487338, 1)

(-3.141592653589793, 1)

(-78.53981633974483, 1)

(-34.55751918948773, 1)

(53.40707511102649, 1)

(34.55751918948773, 1)

(-94.2477796076938, 1)

(6.283185307179586, 1)

(-91.106186954104, 1)

(-69.11503837897546, 1)

(75.39822368615503, 1)

(97.3893722612836, 1)

(65.97344572538566, 1)

(15.707963267948966, 1)

(-50.26548245743669, 1)

(-25.132741228718345, 1)

(18.84955592153876, 1)

(-12.566370614359172, 1)

(37.69911184307752, 1)

(-43.982297150257104, 1)

(-53.40707511102649, 1)

(-6.283185307179586, 1)

(43.982297150257104, 1)

(56.548667764616276, 1)

(-65.97344572538566, 1)

(-28.274333882308138, 1)

(78.53981633974483, 1)

(59.69026041820607, 1)

(81.68140899333463, 1)

(-47.1238898038469, 1)

(100.53096491487338, 1)

(-9.42477796076938, 1)

(-75.39822368615503, 1)

(-72.25663103256524, 1)

(28.274333882308138, 1)

(-31.41592653589793, 1)

(21.991148575128552, 1)

(62.83185307179586, 1)

(9.42477796076938, 1)

(50.26548245743669, 1)

(94.2477796076938, 1)

(72.25663103256524, 1)

(84.82300164692441, 1)


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x56=−59.6902604182061x_{56} = -59.6902604182061
x56=87.9645943005142x_{56} = 87.9645943005142
x56=−97.3893722612836x_{56} = -97.3893722612836
x56=−56.5486677646163x_{56} = -56.5486677646163
x56=40.8407044966673x_{56} = 40.8407044966673
x56=31.4159265358979x_{56} = 31.4159265358979
x56=−37.6991118430775x_{56} = -37.6991118430775
x56=−81.6814089933346x_{56} = -81.6814089933346
x56=−21.9911485751286x_{56} = -21.9911485751286
x56=−15.707963267949x_{56} = -15.707963267949
x56=3.14159265358979x_{56} = 3.14159265358979
x56=12.5663706143592x_{56} = 12.5663706143592
x56=−87.9645943005142x_{56} = -87.9645943005142
x56=−100.530964914873x_{56} = -100.530964914873
x56=−3.14159265358979x_{56} = -3.14159265358979
x56=−78.5398163397448x_{56} = -78.5398163397448
x56=−34.5575191894877x_{56} = -34.5575191894877
x56=53.4070751110265x_{56} = 53.4070751110265
x56=34.5575191894877x_{56} = 34.5575191894877
x56=−94.2477796076938x_{56} = -94.2477796076938
x56=6.28318530717959x_{56} = 6.28318530717959
x56=−91.106186954104x_{56} = -91.106186954104
x56=−69.1150383789755x_{56} = -69.1150383789755
x56=75.398223686155x_{56} = 75.398223686155
x56=97.3893722612836x_{56} = 97.3893722612836
x56=65.9734457253857x_{56} = 65.9734457253857
x56=15.707963267949x_{56} = 15.707963267949
x56=−50.2654824574367x_{56} = -50.2654824574367
x56=−25.1327412287183x_{56} = -25.1327412287183
x56=18.8495559215388x_{56} = 18.8495559215388
x56=−12.5663706143592x_{56} = -12.5663706143592
x56=37.6991118430775x_{56} = 37.6991118430775
x56=−43.9822971502571x_{56} = -43.9822971502571
x56=−53.4070751110265x_{56} = -53.4070751110265
x56=−6.28318530717959x_{56} = -6.28318530717959
x56=43.9822971502571x_{56} = 43.9822971502571
x56=56.5486677646163x_{56} = 56.5486677646163
x56=−65.9734457253857x_{56} = -65.9734457253857
x56=−28.2743338823081x_{56} = -28.2743338823081
x56=78.5398163397448x_{56} = 78.5398163397448
x56=59.6902604182061x_{56} = 59.6902604182061
x56=81.6814089933346x_{56} = 81.6814089933346
x56=−47.1238898038469x_{56} = -47.1238898038469
x56=100.530964914873x_{56} = 100.530964914873
x56=−9.42477796076938x_{56} = -9.42477796076938
x56=−75.398223686155x_{56} = -75.398223686155
x56=−72.2566310325652x_{56} = -72.2566310325652
x56=28.2743338823081x_{56} = 28.2743338823081
x56=−31.4159265358979x_{56} = -31.4159265358979
x56=21.9911485751286x_{56} = 21.9911485751286
x56=62.8318530717959x_{56} = 62.8318530717959
x56=9.42477796076938x_{56} = 9.42477796076938
x56=50.2654824574367x_{56} = 50.2654824574367
x56=94.2477796076938x_{56} = 94.2477796076938
x56=72.2566310325652x_{56} = 72.2566310325652
x56=84.8230016469244x_{56} = 84.8230016469244
Decrece en los intervalos
(−∞,−100.530964914873]\left(-\infty, -100.530964914873\right]
Crece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞cos⁡1x2(2x)=1\lim_{x \to -\infty} \cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞cos⁡1x2(2x)=1\lim_{x \to \infty} \cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(2*x)^(x^(-2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡1x2(2x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{\frac{1}{x^{2}}}{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡1x2(2x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{\frac{1}{x^{2}}}{\left(2 x \right)}}{x}\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(-x) и f = -f(-x).
Pues, comprobamos:
cos⁡1x2(2x)=cos⁡1x2(2x)\cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = \cos^{\frac{1}{x^{2}}}{\left(2 x \right)}
- Sí
cos⁡1x2(2x)=−cos⁡1x2(2x)\cos^{\frac{1}{x^{2}}}{\left(2 x \right)} = - \cos^{\frac{1}{x^{2}}}{\left(2 x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = cos(2*x)^(x^(-2))
    © Sr Examen – Калькулятор