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tan(x)/(x^4+x^2+5)
Trazado el gráfico de la función/ tan(x)/(x^4+x^2+5)

Gráfico de la función y = tan(x)/(x^4+x^2+5)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          tan(x)  
f(x) = -----------
        4    2    
       x  + x  + 5
f(x)=tan⁡(x)(x4+x2)+5f{\left(x \right)} = \frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5} 
f = tan(x)/(x^4 + x^2 + 5)
Gráfico de la función
02468-8-6-4-2-1010-1010
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)(x4+x2)+5=0\frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=69.1150383789755x_{1} = 69.1150383789755
x2=65.9734457253857x_{2} = 65.9734457253857
x3=−91.106186954104x_{3} = -91.106186954104
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=12.5663706143592x_{6} = 12.5663706143592
x7=21.9911485751286x_{7} = 21.9911485751286
x8=−69.1150383789755x_{8} = -69.1150383789755
x9=−100.530964914873x_{9} = -100.530964914873
x10=3.14159265358979x_{10} = 3.14159265358979
x11=−3.14159265358979x_{11} = -3.14159265358979
x12=−25.1327412287183x_{12} = -25.1327412287183
x13=−15.707963267949x_{13} = -15.707963267949
x14=−53.4070751110265x_{14} = -53.4070751110265
x15=−72.2566310325652x_{15} = -72.2566310325652
x16=84.8230016469244x_{16} = 84.8230016469244
x17=−81.6814089933346x_{17} = -81.6814089933346
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=18.8495559215388x_{19} = 18.8495559215388
x20=−65.9734457253857x_{20} = -65.9734457253857
x21=94.2477796076938x_{21} = 94.2477796076938
x22=9.42477796076938x_{22} = 9.42477796076938
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=34.5575191894877x_{24} = 34.5575191894877
x25=0x_{25} = 0
x26=97.3893722612836x_{26} = 97.3893722612836
x27=53.4070751110265x_{27} = 53.4070751110265
x28=−62.8318530717959x_{28} = -62.8318530717959
x29=59.6902604182061x_{29} = 59.6902604182061
x30=−28.2743338823081x_{30} = -28.2743338823081
x31=−56.5486677646163x_{31} = -56.5486677646163
x32=91.106186954104x_{32} = 91.106186954104
x33=15.707963267949x_{33} = 15.707963267949
x34=−18.8495559215388x_{34} = -18.8495559215388
x35=6.28318530717959x_{35} = 6.28318530717959
x36=56.5486677646163x_{36} = 56.5486677646163
x37=87.9645943005142x_{37} = 87.9645943005142
x38=31.4159265358979x_{38} = 31.4159265358979
x39=25.1327412287183x_{39} = 25.1327412287183
x40=43.9822971502571x_{40} = 43.9822971502571
x41=−47.1238898038469x_{41} = -47.1238898038469
x42=72.2566310325652x_{42} = 72.2566310325652
x43=−34.5575191894877x_{43} = -34.5575191894877
x44=−97.3893722612836x_{44} = -97.3893722612836
x45=−50.2654824574367x_{45} = -50.2654824574367
x46=100.530964914873x_{46} = 100.530964914873
x47=81.6814089933346x_{47} = 81.6814089933346
x48=−75.398223686155x_{48} = -75.398223686155
x49=40.8407044966673x_{49} = 40.8407044966673
x50=−9.42477796076938x_{50} = -9.42477796076938
x51=78.5398163397448x_{51} = 78.5398163397448
x52=−87.9645943005142x_{52} = -87.9645943005142
x53=37.6991118430775x_{53} = 37.6991118430775
x54=−78.5398163397448x_{54} = -78.5398163397448
x55=−6.28318530717959x_{55} = -6.28318530717959
x56=50.2654824574367x_{56} = 50.2654824574367
x57=−37.6991118430775x_{57} = -37.6991118430775
x58=−43.9822971502571x_{58} = -43.9822971502571
x59=47.1238898038469x_{59} = 47.1238898038469
x60=28.2743338823081x_{60} = 28.2743338823081
x61=62.8318530717959x_{61} = 62.8318530717959
x62=−31.4159265358979x_{62} = -31.4159265358979
x63=−12.5663706143592x_{63} = -12.5663706143592
x64=75.398223686155x_{64} = 75.398223686155
x65=−84.8230016469244x_{65} = -84.8230016469244
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 + 5).
tan⁡(0)(04+02)+5\frac{\tan{\left(0 \right)}}{\left(0^{4} + 0^{2}\right) + 5}
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
(−4x3−2x)tan⁡(x)((x4+x2)+5)2+tan⁡2(x)+1(x4+x2)+5=0\frac{\left(- 4 x^{3} - 2 x\right) \tan{\left(x \right)}}{\left(\left(x^{4} + x^{2}\right) + 5\right)^{2}} + \frac{\tan^{2}{\left(x \right)} + 1}{\left(x^{4} + x^{2}\right) + 5} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=7629.02223278931x_{1} = 7629.02223278931
x2=−4505.35264824098x_{2} = -4505.35264824098
x3=−1548.44044778605x_{3} = -1548.44044778605
x4=1746.60409381025x_{4} = 1746.60409381025
x5=5429.19354372517x_{5} = 5429.19354372517
x6=1137.51393320791x_{6} = 1137.51393320791
x7=−8418.30415728598x_{7} = -8418.30415728598
x8=−1562.14068106912x_{8} = -1562.14068106912
x9=1102.9023573482x_{9} = 1102.9023573482
x10=2021.0746422339x_{10} = 2021.0746422339
x11=−2008.04196293154x_{11} = -2008.04196293154
x12=1128.00599739149x_{12} = 1128.00599739149
x13=4507.30534282228x_{13} = 4507.30534282228
x14=−6283.70892947365x_{14} = -6283.70892947365
Signos de extremos en los puntos:
(7629.022232789311, 8.46560946006915e-16)

(-4505.352648240984, -7.74206184331313e-16)

(-1548.4404477860498, 6.64159954060111e-14)

(1746.6040938102522, -1.31117159594844e-14)

(5429.193543725171, 6.61193383732148e-16)

(1137.5139332079134, 1.572210610648e-13)

(-8418.304157285984, 4.62362366282336e-16)

(-1562.1406810691224, -1.62550516529003e-13)

(1102.9023573481954, 1.39350754388198e-13)

(2021.0746422338989, 9.99321998978588e-14)

(-2008.0419629315381, -3.89261266915056e-14)

(1128.0059973914883, 1.08721375868441e-13)

(4507.305342822278, -2.93155896163763e-15)

(-6283.708929473651, -3.70338369942352e-16)


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
La función no tiene puntos máximos
Crece en todo el eje numérico
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(−2x(2x2+1)(tan⁡2(x)+1)x4+x2+5+(tan⁡2(x)+1)tan⁡(x)−(−4x2(2x2+1)2x4+x2+5+6x2+1)tan⁡(x)x4+x2+5)x4+x2+5=0\frac{2 \left(- \frac{2 x \left(2 x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{4} + x^{2} + 5} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(- \frac{4 x^{2} \left(2 x^{2} + 1\right)^{2}}{x^{4} + x^{2} + 5} + 6 x^{2} + 1\right) \tan{\left(x \right)}}{x^{4} + x^{2} + 5}\right)}{x^{4} + x^{2} + 5} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−91.149987348733x_{1} = -91.149987348733
x2=3.7608338874361x_{2} = 3.7608338874361
x3=62.8951983516334x_{3} = 62.8951983516334
x4=−100.570675868524x_{4} = -100.570675868524
x5=−3.7608338874361x_{5} = -3.7608338874361
x6=47.2080273947305x_{6} = 47.2080273947305
x7=−62.8951983516334x_{7} = -62.8951983516334
x8=22.1661280563682x_{8} = 22.1661280563682
x9=−25.2871859346973x_{9} = -25.2871859346973
x10=19.0510630137868x_{10} = 19.0510630137868
x11=84.8700293777013x_{11} = 84.8700293777013
x12=69.172674560332x_{12} = 69.172674560332
x13=−34.6714063759229x_{13} = -34.6714063759229
x14=44.0723287767864x_{14} = 44.0723287767864
x15=59.7569040345638x_{15} = 59.7569040345638
x16=−72.3117805427902x_{16} = -72.3117805427902
x17=−47.2080273947305x_{17} = -47.2080273947305
x18=−94.2901265771818x_{18} = -94.2901265771818
x19=15.9448518567147x_{19} = 15.9448518567147
x20=40.9375079301731x_{20} = 40.9375079301731
x21=−15.9448518567147x_{21} = -15.9448518567147
x22=31.5407884465079x_{22} = 31.5407884465079
x23=100.570675868524x_{23} = 100.570675868524
x24=−44.0723287767864x_{24} = -44.0723287767864
x25=−28.4124569255978x_{25} = -28.4124569255978
x26=−69.172674560332x_{26} = -69.172674560332
x27=−53.4814577901855x_{27} = -53.4814577901855
x28=−50.3444447520328x_{28} = -50.3444447520328
x29=0x_{29} = 0
x30=−40.9375079301731x_{30} = -40.9375079301731
x31=50.3444447520328x_{31} = 50.3444447520328
x32=75.4510915111814x_{32} = 75.4510915111814
x33=−88.0099511717168x_{33} = -88.0099511717168
x34=53.4814577901855x_{34} = 53.4814577901855
x35=−78.590583200109x_{35} = -78.590583200109
x36=37.8037738409624x_{36} = 37.8037738409624
x37=25.2871859346973x_{37} = 25.2871859346973
x38=6.74671474060891x_{38} = 6.74671474060891
x39=91.149987348733x_{39} = 91.149987348733
x40=−6.74671474060891x_{40} = -6.74671474060891
x41=97.4303589713879x_{41} = 97.4303589713879
x42=9.7816361116334x_{42} = 9.7816361116334
x43=94.2901265771818x_{43} = 94.2901265771818
x44=−75.4510915111814x_{44} = -75.4510915111814
x45=−12.852286570046x_{45} = -12.852286570046
x46=28.4124569255978x_{46} = 28.4124569255978
x47=−9.7816361116334x_{47} = -9.7816361116334
x48=−84.8700293777013x_{48} = -84.8700293777013
x49=−81.7302350177074x_{49} = -81.7302350177074
x50=56.6189698595128x_{50} = 56.6189698595128
x51=81.7302350177074x_{51} = 81.7302350177074
x52=−66.0338023562199x_{52} = -66.0338023562199
x53=72.3117805427902x_{53} = 72.3117805427902
x54=66.0338023562199x_{54} = 66.0338023562199
x55=−59.7569040345638x_{55} = -59.7569040345638
x56=12.852286570046x_{56} = 12.852286570046
x57=−22.1661280563682x_{57} = -22.1661280563682
x58=88.0099511717168x_{58} = 88.0099511717168
x59=78.590583200109x_{59} = 78.590583200109
x60=−56.6189698595128x_{60} = -56.6189698595128
x61=34.6714063759229x_{61} = 34.6714063759229
x62=−19.0510630137868x_{62} = -19.0510630137868
x63=−31.5407884465079x_{63} = -31.5407884465079
x64=−37.8037738409624x_{64} = -37.8037738409624
x65=−97.4303589713879x_{65} = -97.4303589713879

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
[100.570675868524,∞)\left[100.570675868524, \infty\right)
Convexa en los intervalos
(−∞,−100.570675868524]\left(-\infty, -100.570675868524\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→−∞(tan⁡(x)(x4+x2)+5)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)(x4+x2)+5)y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x)/(x^4 + x^2 + 5), 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((x4+x2)+5))y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x \left(\left(x^{4} + x^{2}\right) + 5\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)x((x4+x2)+5))y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x \left(\left(x^{4} + x^{2}\right) + 5\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)(x4+x2)+5=−tan⁡(x)(x4+x2)+5\frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5} = - \frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5}
- No
tan⁡(x)(x4+x2)+5=tan⁡(x)(x4+x2)+5\frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5} = \frac{\tan{\left(x \right)}}{\left(x^{4} + x^{2}\right) + 5}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tan(x)/(x^4+x^2+5)
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