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(x^3-x)*tan(x)
Trazado el gráfico de la función/ (x^3-x)*tan(x)

Gráfico de la función y = (x^3-x)*tan(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       / 3    \       
f(x) = \x  - x/*tan(x)
f(x)=(x3−x)tan⁡(x)f{\left(x \right)} = \left(x^{3} - x\right) \tan{\left(x \right)} 
f = (x^3 - x)*tan(x)
Gráfico de la función
02468-8-6-4-2-1010-2500025000
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:
(x3−x)tan⁡(x)=0\left(x^{3} - x\right) \tan{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−1x_{1} = -1
x2=0x_{2} = 0
x3=1x_{3} = 1
x4=πx_{4} = \pi
Solución numérica
x1=−75.398223686155x_{1} = -75.398223686155
x2=47.1238898038469x_{2} = 47.1238898038469
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=9.42477796076938x_{4} = 9.42477796076938
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−97.3893722612836x_{6} = -97.3893722612836
x7=−62.8318530717959x_{7} = -62.8318530717959
x8=87.9645943005142x_{8} = 87.9645943005142
x9=−87.9645943005142x_{9} = -87.9645943005142
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=6.28318530717959x_{11} = 6.28318530717959
x12=59.6902604182061x_{12} = 59.6902604182061
x13=−47.1238898038469x_{13} = -47.1238898038469
x14=−40.8407044966673x_{14} = -40.8407044966673
x15=100.530964914873x_{15} = 100.530964914873
x16=62.8318530717959x_{16} = 62.8318530717959
x17=3.14159265358979x_{17} = 3.14159265358979
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−69.1150383789755x_{19} = -69.1150383789755
x20=97.3893722612836x_{20} = 97.3893722612836
x21=12.5663706143592x_{21} = 12.5663706143592
x22=94.2477796076938x_{22} = 94.2477796076938
x23=31.4159265358979x_{23} = 31.4159265358979
x24=25.1327412287183x_{24} = 25.1327412287183
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=−94.2477796076938x_{26} = -94.2477796076938
x27=−59.6902604182061x_{27} = -59.6902604182061
x28=−56.5486677646163x_{28} = -56.5486677646163
x29=81.6814089933346x_{29} = 81.6814089933346
x30=43.9822971502571x_{30} = 43.9822971502571
x31=−91.106186954104x_{31} = -91.106186954104
x32=15.707963267949x_{32} = 15.707963267949
x33=34.5575191894877x_{33} = 34.5575191894877
x34=21.9911485751286x_{34} = 21.9911485751286
x35=40.8407044966673x_{35} = 40.8407044966673
x36=69.1150383789755x_{36} = 69.1150383789755
x37=65.9734457253857x_{37} = 65.9734457253857
x38=−72.2566310325652x_{38} = -72.2566310325652
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=91.106186954104x_{40} = 91.106186954104
x41=53.4070751110265x_{41} = 53.4070751110265
x42=−28.2743338823081x_{42} = -28.2743338823081
x43=56.5486677646163x_{43} = 56.5486677646163
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=−18.8495559215388x_{45} = -18.8495559215388
x46=−100.530964914873x_{46} = -100.530964914873
x47=−53.4070751110265x_{47} = -53.4070751110265
x48=−15.707963267949x_{48} = -15.707963267949
x49=84.8230016469244x_{49} = 84.8230016469244
x50=72.2566310325652x_{50} = 72.2566310325652
x51=18.8495559215388x_{51} = 18.8495559215388
x52=0x_{52} = 0
x53=−43.9822971502571x_{53} = -43.9822971502571
x54=−84.8230016469244x_{54} = -84.8230016469244
x55=−78.5398163397448x_{55} = -78.5398163397448
x56=−12.5663706143592x_{56} = -12.5663706143592
x57=75.398223686155x_{57} = 75.398223686155
x58=−6.28318530717959x_{58} = -6.28318530717959
x59=78.5398163397448x_{59} = 78.5398163397448
x60=−50.2654824574367x_{60} = -50.2654824574367
x61=−81.6814089933346x_{61} = -81.6814089933346
x62=50.2654824574367x_{62} = 50.2654824574367
x63=−9.42477796076938x_{63} = -9.42477796076938
x64=37.6991118430775x_{64} = 37.6991118430775
x65=−25.1327412287183x_{65} = -25.1327412287183
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 (x^3 - x)*tan(x).
(03−0)tan⁡(0)\left(0^{3} - 0\right) \tan{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
(3x2−1)tan⁡(x)+(x3−x)(tan⁡2(x)+1)=0\left(3 x^{2} - 1\right) \tan{\left(x \right)} + \left(x^{3} - x\right) \left(\tan^{2}{\left(x \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−0.74501822309569x_{2} = -0.74501822309569
x3=0.74501822309569x_{3} = 0.74501822309569
x4=1.16326146403121⋅10−14x_{4} = 1.16326146403121 \cdot 10^{-14}
x5=−0.745018223095691x_{5} = -0.745018223095691
Signos de extremos en los puntos:
(0, 0)

(-0.7450182230956899, -0.305748437940331)

(0.7450182230956899, -0.305748437940331)

(1.1632614640312125e-14, -1.35317723370004e-28)

(-0.745018223095691, -0.305748437940331)


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:
x1=−0.74501822309569x_{1} = -0.74501822309569
x2=0.74501822309569x_{2} = 0.74501822309569
x3=−0.745018223095691x_{3} = -0.745018223095691
Puntos máximos de la función:
x3=0x_{3} = 0
x3=1.16326146403121⋅10−14x_{3} = 1.16326146403121 \cdot 10^{-14}
Decrece en los intervalos
[−0.74501822309569,0]∪[0.74501822309569,∞)\left[-0.74501822309569, 0\right] \cup \left[0.74501822309569, \infty\right)
Crece en los intervalos
(−∞,−0.745018223095691]\left(-\infty, -0.745018223095691\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(x(x2−1)(tan⁡2(x)+1)tan⁡(x)+3xtan⁡(x)+(3x2−1)(tan⁡2(x)+1))=02 \left(x \left(x^{2} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 x \tan{\left(x \right)} + \left(3 x^{2} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−81.6446937555336x_{1} = -81.6446937555336
x2=−72.2151308721608x_{2} = -72.2151308721608
x3=18.6914180658702x_{3} = 18.6914180658702
x4=−59.6400337867216x_{4} = -59.6400337867216
x5=−40.7673505273091x_{5} = -40.7673505273091
x6=−43.914169748432x_{6} = -43.914169748432
x7=−84.7876453360242x_{7} = -84.7876453360242
x8=37.6196641250491x_{8} = 37.6196641250491
x9=2.27639684529328x_{9} = 2.27639684529328
x10=−25.013809217733x_{10} = -25.013809217733
x11=34.4708756385137x_{11} = 34.4708756385137
x12=−78.5016335791697x_{12} = -78.5016335791697
x13=−18.6914180658702x_{13} = -18.6914180658702
x14=91.073267590934x_{14} = 91.073267590934
x15=100.50113024604x_{15} = 100.50113024604
x16=9.11399178681792x_{16} = 9.11399178681792
x17=21.8553751585661x_{17} = 21.8553751585661
x18=−94.2159569713689x_{18} = -94.2159569713689
x19=53.3509485725081x_{19} = 53.3509485725081
x20=97.3585756490797x_{20} = 97.3585756490797
x21=−37.6196641250491x_{21} = -37.6196641250491
x22=−91.073267590934x_{22} = -91.073267590934
x23=59.6400337867216x_{23} = 59.6400337867216
x24=25.013809217733x_{24} = 25.013809217733
x25=−12.3309587290799x_{25} = -12.3309587290799
x26=84.7876453360242x_{26} = 84.7876453360242
x27=−5.82819309906884x_{27} = -5.82819309906884
x28=78.5016335791697x_{28} = 78.5016335791697
x29=−56.4956547093161x_{29} = -56.4956547093161
x30=40.7673505273091x_{30} = 40.7673505273091
x31=47.0602944247052x_{31} = 47.0602944247052
x32=75.3584512533474x_{32} = 75.3584512533474
x33=50.2058542581421x_{33} = 50.2058542581421
x34=15.5187142563086x_{34} = 15.5187142563086
x35=−97.3585756490797x_{35} = -97.3585756490797
x36=−53.3509485725081x_{36} = -53.3509485725081
x37=−31.3206571098888x_{37} = -31.3206571098888
x38=81.6446937555336x_{38} = 81.6446937555336
x39=−47.0602944247052x_{39} = -47.0602944247052
x40=−75.3584512533474x_{40} = -75.3584512533474
x41=−28.1685365194601x_{41} = -28.1685365194601
x42=−100.50113024604x_{42} = -100.50113024604
x43=69.0716536441652x_{43} = 69.0716536441652
x44=56.4956547093161x_{44} = 56.4956547093161
x45=−34.4708756385137x_{45} = -34.4708756385137
x46=94.2159569713689x_{46} = 94.2159569713689
x47=0.45901672478924x_{47} = 0.45901672478924
x48=72.2151308721608x_{48} = 72.2151308721608
x49=28.1685365194601x_{49} = 28.1685365194601
x50=−65.9279972073077x_{50} = -65.9279972073077
x51=−9.11399178681792x_{51} = -9.11399178681792
x52=−87.9304999411323x_{52} = -87.9304999411323
x53=−21.8553751585661x_{53} = -21.8553751585661
x54=−2.27639684529328x_{54} = -2.27639684529328
x55=−69.0716536441652x_{55} = -69.0716536441652
x56=−62.7841347390722x_{56} = -62.7841347390722
x57=−15.5187142563086x_{57} = -15.5187142563086
x58=12.3309587290799x_{58} = 12.3309587290799
x59=62.7841347390722x_{59} = 62.7841347390722
x60=43.914169748432x_{60} = 43.914169748432
x61=87.9304999411323x_{61} = 87.9304999411323
x62=31.3206571098888x_{62} = 31.3206571098888
x63=5.82819309906884x_{63} = 5.82819309906884
x64=65.9279972073077x_{64} = 65.9279972073077
x65=−50.2058542581421x_{65} = -50.2058542581421

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
[100.50113024604,∞)\left[100.50113024604, \infty\right)
Convexa en los intervalos
[−2.27639684529328,0.45901672478924]\left[-2.27639684529328, 0.45901672478924\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞((x3−x)tan⁡(x))y = \lim_{x \to -\infty}\left(\left(x^{3} - x\right) \tan{\left(x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((x3−x)tan⁡(x))y = \lim_{x \to \infty}\left(\left(x^{3} - x\right) \tan{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x^3 - x)*tan(x), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞((x3−x)tan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(x^{3} - x\right) \tan{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((x3−x)tan⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\left(x^{3} - x\right) \tan{\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:
(x3−x)tan⁡(x)=−(−x3+x)tan⁡(x)\left(x^{3} - x\right) \tan{\left(x \right)} = - \left(- x^{3} + x\right) \tan{\left(x \right)}
- No
(x3−x)tan⁡(x)=(−x3+x)tan⁡(x)\left(x^{3} - x\right) \tan{\left(x \right)} = \left(- x^{3} + x\right) \tan{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (x^3-x)*tan(x)
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