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Trazado el gráfico de la función/ tan[x]/(x^4+x^2+x)

Gráfico de la función y = tan[x]/(x^4+x^2+x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          tan(x)  
f(x) = -----------
        4    2    
       x  + x  + x
f(x)=tan⁡(x)x+(x4+x2)f{\left(x \right)} = \frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)} 
f = tan(x)/(x + x^4 + x^2)
Gráfico de la función
02468-8-6-4-2-1010-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−0.682327803828019x_{1} = -0.682327803828019
x2=0x_{2} = 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:
tan⁡(x)x+(x4+x2)=0\frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=43.9822971502571x_{1} = 43.9822971502571
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−43.9822971502571x_{3} = -43.9822971502571
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=−78.5398163397448x_{8} = -78.5398163397448
x9=97.3893722612836x_{9} = 97.3893722612836
x10=9.42477796076938x_{10} = 9.42477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=84.8230016469244x_{12} = 84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=6.28318530717959x_{15} = 6.28318530717959
x16=3.14159265358979x_{16} = 3.14159265358979
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−75.398223686155x_{19} = -75.398223686155
x20=−28.2743338823081x_{20} = -28.2743338823081
x21=−56.5486677646163x_{21} = -56.5486677646163
x22=−65.9734457253857x_{22} = -65.9734457253857
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=−91.106186954104x_{24} = -91.106186954104
x25=50.2654824574367x_{25} = 50.2654824574367
x26=−69.1150383789755x_{26} = -69.1150383789755
x27=−100.530964914873x_{27} = -100.530964914873
x28=56.5486677646163x_{28} = 56.5486677646163
x29=−62.8318530717959x_{29} = -62.8318530717959
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=40.8407044966673x_{31} = 40.8407044966673
x32=100.530964914873x_{32} = 100.530964914873
x33=18.8495559215388x_{33} = 18.8495559215388
x34=62.8318530717959x_{34} = 62.8318530717959
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=94.2477796076938x_{36} = 94.2477796076938
x37=−3.14159265358979x_{37} = -3.14159265358979
x38=21.9911485751286x_{38} = 21.9911485751286
x39=12.5663706143592x_{39} = 12.5663706143592
x40=−84.8230016469244x_{40} = -84.8230016469244
x41=34.5575191894877x_{41} = 34.5575191894877
x42=47.1238898038469x_{42} = 47.1238898038469
x43=−15.707963267949x_{43} = -15.707963267949
x44=53.4070751110265x_{44} = 53.4070751110265
x45=65.9734457253857x_{45} = 65.9734457253857
x46=87.9645943005142x_{46} = 87.9645943005142
x47=91.106186954104x_{47} = 91.106186954104
x48=59.6902604182061x_{48} = 59.6902604182061
x49=69.1150383789755x_{49} = 69.1150383789755
x50=−6.28318530717959x_{50} = -6.28318530717959
x51=75.398223686155x_{51} = 75.398223686155
x52=−37.6991118430775x_{52} = -37.6991118430775
x53=−12.5663706143592x_{53} = -12.5663706143592
x54=−18.8495559215388x_{54} = -18.8495559215388
x55=31.4159265358979x_{55} = 31.4159265358979
x56=−81.6814089933346x_{56} = -81.6814089933346
x57=78.5398163397448x_{57} = 78.5398163397448
x58=15.707963267949x_{58} = 15.707963267949
x59=72.2566310325652x_{59} = 72.2566310325652
x60=37.6991118430775x_{60} = 37.6991118430775
x61=25.1327412287183x_{61} = 25.1327412287183
x62=−47.1238898038469x_{62} = -47.1238898038469
x63=−9.42477796076938x_{63} = -9.42477796076938
x64=−34.5575191894877x_{64} = -34.5575191894877
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 tan(x)/(x^4 + x^2 + x).
tan⁡(0)04+02\frac{\tan{\left(0 \right)}}{0^{4} + 0^{2}}
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
tan⁡2(x)+1x+(x4+x2)+(−4x3−2x−1)tan⁡(x)(x+(x4+x2))2=0\frac{\tan^{2}{\left(x \right)} + 1}{x + \left(x^{4} + x^{2}\right)} + \frac{\left(- 4 x^{3} - 2 x - 1\right) \tan{\left(x \right)}}{\left(x + \left(x^{4} + x^{2}\right)\right)^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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((tan⁡2(x)+1)tan⁡(x)−(tan⁡2(x)+1)(4x3+2x+1)x(x3+x+1)−(6x2+1−(4x3+2x+1)2x(x3+x+1))tan⁡(x)x(x3+x+1))x(x3+x+1)=0\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(4 x^{3} + 2 x + 1\right)}{x \left(x^{3} + x + 1\right)} - \frac{\left(6 x^{2} + 1 - \frac{\left(4 x^{3} + 2 x + 1\right)^{2}}{x \left(x^{3} + x + 1\right)}\right) \tan{\left(x \right)}}{x \left(x^{3} + x + 1\right)}\right)}{x \left(x^{3} + x + 1\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−37.8037755037868x_{1} = -37.8037755037868
x2=56.6189696063376x_{2} = 56.6189696063376
x3=−94.2901266175658x_{3} = -94.2901266175658
x4=59.7569038287361x_{4} = 59.7569038287361
x5=−47.2080280671997x_{5} = -47.2080280671997
x6=25.2871809599612x_{6} = 25.2871809599612
x7=31.540786181549x_{7} = 31.540786181549
x8=−59.7569042920565x_{8} = -59.7569042920565
x9=97.430358940554x_{9} = 97.430358940554
x10=−81.7302350898235x_{10} = -81.7302350898235
x11=44.0723281206025x_{11} = 44.0723281206025
x12=3.76166887818915x_{12} = 3.76166887818915
x13=−34.6714087410708x_{13} = -34.6714087410708
x14=−66.0338025277351x_{14} = -66.0338025277351
x15=−88.0099512251226x_{15} = -88.0099512251226
x16=12.8522465937936x_{16} = 12.8522465937936
x17=69.1726744432842x_{17} = 69.1726744432842
x18=−22.1661425637521x_{18} = -22.1661425637521
x19=−62.8951985607164x_{19} = -62.8951985607164
x20=−53.4814581946947x_{20} = -53.4814581946947
x21=−28.4124622404836x_{21} = -28.4124622404836
x22=−3.76569172914575x_{22} = -3.76569172914575
x23=−9.78197986317429x_{23} = -9.78197986317429
x24=−31.54079192294x_{24} = -31.54079192294
x25=81.7302349564664x_{25} = 81.7302349564664
x26=94.2901265421304x_{26} = 94.2901265421304
x27=−84.8700294395875x_{27} = -84.8700294395875
x28=−44.0723296666291x_{28} = -44.0723296666291
x29=−12.8524108556204x_{29} = -12.8524108556204
x30=47.2080268885788x_{30} = 47.2080268885788
x31=75.4510914276076x_{31} = 75.4510914276076
x32=19.0510501465303x_{32} = 19.0510501465303
x33=−50.3444452694666x_{33} = -50.3444452694666
x34=66.0338022161497x_{34} = 66.0338022161497
x35=62.8951981826111x_{35} = 62.8951981826111
x36=91.1499873087187x_{36} = 91.1499873087187
x37=−40.9375091321834x_{37} = -40.9375091321834
x38=−78.5905832846552x_{38} = -78.5905832846552
x39=37.8037726761926x_{39} = 37.8037726761926
x40=15.9448296005744x_{40} = 15.9448296005744
x41=88.0099511258307x_{41} = 88.0099511258307
x42=72.3117804442317x_{42} = 72.3117804442317
x43=−15.9449057266939x_{43} = -15.9449057266939
x44=40.9375070645391x_{44} = 40.9375070645391
x45=9.78156791301043x_{45} = 9.78156791301043
x46=78.5905831287815x_{46} = 78.5905831287815
x47=22.1661202398505x_{47} = 22.1661202398505
x48=53.4814574752989x_{48} = 53.4814574752989
x49=−19.0510896342101x_{49} = -19.0510896342101
x50=−25.2871944587931x_{50} = -25.2871944587931
x51=100.570675841292x_{51} = 100.570675841292
x52=50.3444443555414x_{52} = 50.3444443555414
x53=34.6714047727383x_{53} = 34.6714047727383
x54=−6.74793974598142x_{54} = -6.74793974598142
x55=−100.570675899621x_{55} = -100.570675899621
x56=84.8700293248275x_{56} = 84.8700293248275
x57=6.74668135662241x_{57} = 6.74668135662241
x58=−56.6189701802256x_{58} = -56.6189701802256
x59=−75.4510916109501x_{59} = -75.4510916109501
x60=−91.1499873950612x_{60} = -91.1499873950612
x61=−72.31178066136x_{61} = -72.31178066136
x62=−97.4303590067504x_{62} = -97.4303590067504
x63=28.4124536274274x_{63} = 28.4124536274274
x64=−69.1726747023379x_{64} = -69.1726747023379
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=−0.682327803828019x_{1} = -0.682327803828019
x2=0x_{2} = 0

lim⁡x→−0.682327803828019−(2((tan⁡2(x)+1)tan⁡(x)−(tan⁡2(x)+1)(4x3+2x+1)x(x3+x+1)−(6x2+1−(4x3+2x+1)2x(x3+x+1))tan⁡(x)x(x3+x+1))x(x3+x+1))=−2.96208916467994⋅1048\lim_{x \to -0.682327803828019^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(4 x^{3} + 2 x + 1\right)}{x \left(x^{3} + x + 1\right)} - \frac{\left(6 x^{2} + 1 - \frac{\left(4 x^{3} + 2 x + 1\right)^{2}}{x \left(x^{3} + x + 1\right)}\right) \tan{\left(x \right)}}{x \left(x^{3} + x + 1\right)}\right)}{x \left(x^{3} + x + 1\right)}\right) = -2.96208916467994 \cdot 10^{48}
lim⁡x→−0.682327803828019+(2((tan⁡2(x)+1)tan⁡(x)−(tan⁡2(x)+1)(4x3+2x+1)x(x3+x+1)−(6x2+1−(4x3+2x+1)2x(x3+x+1))tan⁡(x)x(x3+x+1))x(x3+x+1))=−2.96208916467994⋅1048\lim_{x \to -0.682327803828019^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(4 x^{3} + 2 x + 1\right)}{x \left(x^{3} + x + 1\right)} - \frac{\left(6 x^{2} + 1 - \frac{\left(4 x^{3} + 2 x + 1\right)^{2}}{x \left(x^{3} + x + 1\right)}\right) \tan{\left(x \right)}}{x \left(x^{3} + x + 1\right)}\right)}{x \left(x^{3} + x + 1\right)}\right) = -2.96208916467994 \cdot 10^{48}
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→0−(2((tan⁡2(x)+1)tan⁡(x)−(tan⁡2(x)+1)(4x3+2x+1)x(x3+x+1)−(6x2+1−(4x3+2x+1)2x(x3+x+1))tan⁡(x)x(x3+x+1))x(x3+x+1))=83\lim_{x \to 0^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(4 x^{3} + 2 x + 1\right)}{x \left(x^{3} + x + 1\right)} - \frac{\left(6 x^{2} + 1 - \frac{\left(4 x^{3} + 2 x + 1\right)^{2}}{x \left(x^{3} + x + 1\right)}\right) \tan{\left(x \right)}}{x \left(x^{3} + x + 1\right)}\right)}{x \left(x^{3} + x + 1\right)}\right) = \frac{8}{3}
lim⁡x→0+(2((tan⁡2(x)+1)tan⁡(x)−(tan⁡2(x)+1)(4x3+2x+1)x(x3+x+1)−(6x2+1−(4x3+2x+1)2x(x3+x+1))tan⁡(x)x(x3+x+1))x(x3+x+1))=83\lim_{x \to 0^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(4 x^{3} + 2 x + 1\right)}{x \left(x^{3} + x + 1\right)} - \frac{\left(6 x^{2} + 1 - \frac{\left(4 x^{3} + 2 x + 1\right)^{2}}{x \left(x^{3} + x + 1\right)}\right) \tan{\left(x \right)}}{x \left(x^{3} + x + 1\right)}\right)}{x \left(x^{3} + x + 1\right)}\right) = \frac{8}{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
[100.570675841292,∞)\left[100.570675841292, \infty\right)
Convexa en los intervalos
(−∞,−100.570675899621]\left(-\infty, -100.570675899621\right]
Asíntotas verticales
Hay:
x1=−0.682327803828019x_{1} = -0.682327803828019
x2=0x_{2} = 0
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→−∞(tan⁡(x)x+(x4+x2))y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)x+(x4+x2))y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x)/(x^4 + x^2 + 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→−∞(tan⁡(x)x(x+(x4+x2)))y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x \left(x + \left(x^{4} + x^{2}\right)\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)x(x+(x4+x2)))y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x \left(x + \left(x^{4} + x^{2}\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:
tan⁡(x)x+(x4+x2)=−tan⁡(x)x4+x2−x\frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)} = - \frac{\tan{\left(x \right)}}{x^{4} + x^{2} - x}
- No
tan⁡(x)x+(x4+x2)=tan⁡(x)x4+x2−x\frac{\tan{\left(x \right)}}{x + \left(x^{4} + x^{2}\right)} = \frac{\tan{\left(x \right)}}{x^{4} + x^{2} - x}
- No
es decir, función
no es
par ni impar
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