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y=x*sinx*tg|x|
Trazado el gráfico de la función/ y=x*sinx*tg|x|

Gráfico de la función y = y=x*sinx*tg|x|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = x*sin(x)*tan(|x|)
f(x)=xsin⁡(x)tan⁡(∣x∣)f{\left(x \right)} = x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)} 
f = (x*sin(x))*tan(|x|)
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
xsin⁡(x)tan⁡(∣x∣)=0x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=−59.6902604580197x_{1} = -59.6902604580197
x2=−65.973445764889x_{2} = -65.973445764889
x3=0x_{3} = 0
x4=−9.4247781544401x_{4} = -9.4247781544401
x5=28.2743338651806x_{5} = 28.2743338651806
x6=15.7079634687441x_{6} = 15.7079634687441
x7=3.14159219836402x_{7} = 3.14159219836402
x8=94.2477796093522x_{8} = 94.2477796093522
x9=72.256631027717x_{9} = 72.256631027717
x10=−3.14159308416216x_{10} = -3.14159308416216
x11=−87.9645943585411x_{11} = -87.9645943585411
x12=43.9822971695205x_{12} = 43.9822971695205
x13=−21.9911485864446x_{13} = -21.9911485864446
x14=−37.6991118774422x_{14} = -37.6991118774422
x15=−28.2743336990583x_{15} = -28.2743336990583
x16=−6.2831851284574x_{16} = -6.2831851284574
x17=6.28318528435103x_{17} = 6.28318528435103
x18=−15.7079632967843x_{18} = -15.7079632967843
x19=87.964594336084x_{19} = 87.964594336084
x20=−31.4159267294171x_{20} = -31.4159267294171
x21=−81.6814090385035x_{21} = -81.6814090385035
x22=12.5663704383345x_{22} = 12.5663704383345
x23=50.2654824463364x_{23} = 50.2654824463364
x24=21.9911485852254x_{24} = 21.9911485852254
x25=−43.9822971745285x_{25} = -43.9822971745285
x26=65.9734457530902x_{26} = 65.9734457530902
x27=34.5575190147523x_{27} = 34.5575190147523
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*sin(x))*tan(|x|).
0sin⁡(0)tan⁡(∣0∣)0 \sin{\left(0 \right)} \tan{\left(\left|{0}\right| \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
x(tan⁡2(∣x∣)+1)sin⁡(x)sign⁡(x)+(xcos⁡(x)+sin⁡(x))tan⁡(∣x∣)=0x \left(\tan^{2}{\left(\left|{x}\right| \right)} + 1\right) \sin{\left(x \right)} \operatorname{sign}{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \tan{\left(\left|{x}\right| \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−15.707963267949x_{2} = -15.707963267949
x3=37.6991118430775x_{3} = 37.6991118430775
x4=−69.1150383789755x_{4} = -69.1150383789755
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−91.106186954104x_{6} = -91.106186954104
x7=0x_{7} = 0
x8=6.28318530717959x_{8} = 6.28318530717959
x9=−47.1238898038469x_{9} = -47.1238898038469
x10=−78.5398163397448x_{10} = -78.5398163397448
x11=−28.2743338823081x_{11} = -28.2743338823081
x12=97.3893722612836x_{12} = 97.3893722612836
x13=3.14159265358979x_{13} = 3.14159265358979
x14=40.8407044966673x_{14} = 40.8407044966673
x15=62.8318530717959x_{15} = 62.8318530717959
x16=−81.6814089933346x_{16} = -81.6814089933346
x17=43.9822971502571x_{17} = 43.9822971502571
x18=−84.8230016469244x_{18} = -84.8230016469244
x19=100.530964914873x_{19} = 100.530964914873
x20=69.1150383789755x_{20} = 69.1150383789755
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=91.106186954104x_{22} = 91.106186954104
x23=78.5398163397448x_{23} = 78.5398163397448
x24=47.1238898038469x_{24} = 47.1238898038469
x25=81.6814089933346x_{25} = 81.6814089933346
x26=−72.2566310325652x_{26} = -72.2566310325652
x27=−6.28318530717959x_{27} = -6.28318530717959
x28=28.2743338823081x_{28} = 28.2743338823081
x29=−100.530964914873x_{29} = -100.530964914873
x30=−65.9734457253857x_{30} = -65.9734457253857
x31=94.2477796076938x_{31} = 94.2477796076938
x32=31.4159265358979x_{32} = 31.4159265358979
x33=50.2654824574367x_{33} = 50.2654824574367
x34=−21.9911485751286x_{34} = -21.9911485751286
x35=12.5663706143592x_{35} = 12.5663706143592
x36=15.707963267949x_{36} = 15.707963267949
x37=−75.398223686155x_{37} = -75.398223686155
x38=72.2566310325652x_{38} = 72.2566310325652
x39=18.8495559215388x_{39} = 18.8495559215388
x40=−37.6991118430775x_{40} = -37.6991118430775
x41=−50.2654824574367x_{41} = -50.2654824574367
x42=−3.14159265358979x_{42} = -3.14159265358979
x43=−87.9645943005142x_{43} = -87.9645943005142
x44=−25.1327412287183x_{44} = -25.1327412287183
x45=−40.8407044966673x_{45} = -40.8407044966673
x46=53.4070751110265x_{46} = 53.4070751110265
x47=9.42477796076938x_{47} = 9.42477796076938
x48=−43.9822971502571x_{48} = -43.9822971502571
x49=−56.5486677646163x_{49} = -56.5486677646163
x50=−97.3893722612836x_{50} = -97.3893722612836
x51=−59.6902604182061x_{51} = -59.6902604182061
x52=−12.5663706143592x_{52} = -12.5663706143592
x53=−18.8495559215388x_{53} = -18.8495559215388
x54=84.8230016469244x_{54} = 84.8230016469244
x55=25.1327412287183x_{55} = 25.1327412287183
x56=21.9911485751286x_{56} = 21.9911485751286
x57=59.6902604182061x_{57} = 59.6902604182061
x58=−53.4070751110265x_{58} = -53.4070751110265
x59=−31.4159265358979x_{59} = -31.4159265358979
x60=−9.42477796076938x_{60} = -9.42477796076938
x61=−62.8318530717959x_{61} = -62.8318530717959
x62=56.5486677646163x_{62} = 56.5486677646163
x63=75.398223686155x_{63} = 75.398223686155
x64=34.5575191894877x_{64} = 34.5575191894877
x65=87.9645943005142x_{65} = 87.9645943005142
Signos de extremos en los puntos:
(65.97344572538566, -6.34844983898999e-29)

(-15.707963267948966, -5.8895428941999e-30)

(37.69911184307752, 8.14170409694193e-29)

(-69.11503837897546, 1.3448904736186e-27)

(-34.55751918948773, -1.68111309202325e-28)

(-91.106186954104, -1.39624614979795e-34)

(0, 0)

(6.283185307179586, 3.76930745228793e-31)

(-47.1238898038469, -1.38722161196188e-28)

(-78.53981633974483, -1.8941914820334e-29)

(-28.274333882308138, -3.43478141589738e-29)

(97.3893722612836, -4.58542475390885e-27)

(3.141592653589793, -4.71163431535992e-32)

(40.840704496667314, -1.57001387566644e-28)

(62.83185307179586, 3.76930745228793e-28)

(-81.68140899333463, 1.25601110053315e-27)

(43.982297150257104, 1.29287245613476e-28)

(-84.82300164692441, -3.99087542625273e-27)

(100.53096491487338, 1.54390833245714e-27)

(69.11503837897546, 1.3448904736186e-27)

(-94.2477796076938, 1.10977728956951e-27)

(91.106186954104, -1.39624614979795e-34)

(78.53981633974483, -1.8941914820334e-29)

(47.1238898038469, -1.38722161196188e-28)

(81.68140899333463, 1.25601110053315e-27)

(-72.25663103256524, -2.93139900017185e-27)

(-6.283185307179586, 3.76930745228793e-31)

(28.274333882308138, -3.43478141589738e-29)

(-100.53096491487338, 1.54390833245714e-27)

(-65.97344572538566, -6.34844983898999e-29)

(94.2477796076938, 1.10977728956951e-27)

(31.41592653589793, 4.71163431535992e-29)

(50.26548245743669, 1.92988541557142e-28)

(-21.991148575128552, -1.61609057016845e-29)

(12.566370614359172, 3.01544596183035e-30)

(15.707963267948966, -5.8895428941999e-30)

(-75.39822368615503, 6.51336327755355e-28)

(72.25663103256524, -2.93139900017185e-27)

(18.84955592153876, 1.01771301211774e-29)

(-37.69911184307752, 8.14170409694193e-29)

(-50.26548245743669, 1.92988541557142e-28)

(-3.141592653589793, -4.71163431535992e-32)

(-87.96459430051421, 1.03429796490781e-27)

(-25.132741228718345, 2.41235676946428e-29)

(-40.840704496667314, -1.57001387566644e-28)

(53.40707511102649, -1.15535214562331e-28)

(9.42477796076938, -1.27214126514718e-30)

(-43.982297150257104, 1.29287245613476e-28)

(-56.548667764616276, 2.7478251327179e-28)

(-97.3893722612836, -4.58542475390885e-27)

(-59.69026041820607, -8.97021321364436e-29)

(-12.566370614359172, 3.01544596183035e-30)

(-18.84955592153876, 1.01771301211774e-29)

(84.82300164692441, -3.99087542625273e-27)

(25.132741228718345, 2.41235676946428e-29)

(21.991148575128552, -1.61609057016845e-29)

(59.69026041820607, -8.97021321364436e-29)

(-53.40707511102649, -1.15535214562331e-28)

(-31.41592653589793, 4.71163431535992e-29)

(-9.42477796076938, -1.27214126514718e-30)

(-62.83185307179586, 3.76930745228793e-28)

(56.548667764616276, 2.7478251327179e-28)

(75.39822368615503, 6.51336327755355e-28)

(34.55751918948773, -1.68111309202325e-28)

(87.96459430051421, 1.03429796490781e-27)


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=37.6991118430775x_{1} = 37.6991118430775
x2=−69.1150383789755x_{2} = -69.1150383789755
x3=0x_{3} = 0
x4=6.28318530717959x_{4} = 6.28318530717959
x5=62.8318530717959x_{5} = 62.8318530717959
x6=−81.6814089933346x_{6} = -81.6814089933346
x7=43.9822971502571x_{7} = 43.9822971502571
x8=100.530964914873x_{8} = 100.530964914873
x9=69.1150383789755x_{9} = 69.1150383789755
x10=−94.2477796076938x_{10} = -94.2477796076938
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−6.28318530717959x_{12} = -6.28318530717959
x13=−100.530964914873x_{13} = -100.530964914873
x14=94.2477796076938x_{14} = 94.2477796076938
x15=31.4159265358979x_{15} = 31.4159265358979
x16=50.2654824574367x_{16} = 50.2654824574367
x17=12.5663706143592x_{17} = 12.5663706143592
x18=−75.398223686155x_{18} = -75.398223686155
x19=18.8495559215388x_{19} = 18.8495559215388
x20=−37.6991118430775x_{20} = -37.6991118430775
x21=−50.2654824574367x_{21} = -50.2654824574367
x22=−87.9645943005142x_{22} = -87.9645943005142
x23=−25.1327412287183x_{23} = -25.1327412287183
x24=−43.9822971502571x_{24} = -43.9822971502571
x25=−56.5486677646163x_{25} = -56.5486677646163
x26=−12.5663706143592x_{26} = -12.5663706143592
x27=−18.8495559215388x_{27} = -18.8495559215388
x28=25.1327412287183x_{28} = 25.1327412287183
x29=−31.4159265358979x_{29} = -31.4159265358979
x30=−62.8318530717959x_{30} = -62.8318530717959
x31=56.5486677646163x_{31} = 56.5486677646163
x32=75.398223686155x_{32} = 75.398223686155
x33=87.9645943005142x_{33} = 87.9645943005142
Puntos máximos de la función:
x33=65.9734457253857x_{33} = 65.9734457253857
x33=−15.707963267949x_{33} = -15.707963267949
x33=−34.5575191894877x_{33} = -34.5575191894877
x33=−91.106186954104x_{33} = -91.106186954104
x33=−47.1238898038469x_{33} = -47.1238898038469
x33=−78.5398163397448x_{33} = -78.5398163397448
x33=−28.2743338823081x_{33} = -28.2743338823081
x33=97.3893722612836x_{33} = 97.3893722612836
x33=3.14159265358979x_{33} = 3.14159265358979
x33=40.8407044966673x_{33} = 40.8407044966673
x33=−84.8230016469244x_{33} = -84.8230016469244
x33=91.106186954104x_{33} = 91.106186954104
x33=78.5398163397448x_{33} = 78.5398163397448
x33=47.1238898038469x_{33} = 47.1238898038469
x33=−72.2566310325652x_{33} = -72.2566310325652
x33=28.2743338823081x_{33} = 28.2743338823081
x33=−65.9734457253857x_{33} = -65.9734457253857
x33=−21.9911485751286x_{33} = -21.9911485751286
x33=15.707963267949x_{33} = 15.707963267949
x33=72.2566310325652x_{33} = 72.2566310325652
x33=−3.14159265358979x_{33} = -3.14159265358979
x33=−40.8407044966673x_{33} = -40.8407044966673
x33=53.4070751110265x_{33} = 53.4070751110265
x33=9.42477796076938x_{33} = 9.42477796076938
x33=−97.3893722612836x_{33} = -97.3893722612836
x33=−59.6902604182061x_{33} = -59.6902604182061
x33=84.8230016469244x_{33} = 84.8230016469244
x33=21.9911485751286x_{33} = 21.9911485751286
x33=59.6902604182061x_{33} = 59.6902604182061
x33=−53.4070751110265x_{33} = -53.4070751110265
x33=−9.42477796076938x_{33} = -9.42477796076938
x33=34.5575191894877x_{33} = 34.5575191894877
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
(−∞,−100.530964914873]\left(-\infty, -100.530964914873\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
2x(tan⁡(∣x∣)sign⁡2(x)+δ(x))(tan⁡2(∣x∣)+1)sin⁡(x)−(xsin⁡(x)−2cos⁡(x))tan⁡(∣x∣)+2(xcos⁡(x)+sin⁡(x))(tan⁡2(∣x∣)+1)sign⁡(x)=02 x \left(\tan{\left(\left|{x}\right| \right)} \operatorname{sign}^{2}{\left(x \right)} + \delta\left(x\right)\right) \left(\tan^{2}{\left(\left|{x}\right| \right)} + 1\right) \sin{\left(x \right)} - \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \tan{\left(\left|{x}\right| \right)} + 2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(\left|{x}\right| \right)} + 1\right) \operatorname{sign}{\left(x \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 x->+oo y x->-oo
True

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

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xsin⁡(x)tan⁡(∣x∣))y = \lim_{x \to \infty}\left(x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*sin(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→−∞(sin⁡(x)tan⁡(∣x∣))y = x \lim_{x \to -\infty}\left(\sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡(x)tan⁡(∣x∣))y = x \lim_{x \to \infty}\left(\sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)}\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:
xsin⁡(x)tan⁡(∣x∣)=xsin⁡(x)tan⁡(∣x∣)x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)} = x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)}
- Sí
xsin⁡(x)tan⁡(∣x∣)=−xsin⁡(x)tan⁡(∣x∣)x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)} = - x \sin{\left(x \right)} \tan{\left(\left|{x}\right| \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = y=x*sinx*tg|x|
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