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Trazado el gráfico de la función/ tan(x)+1/x

Gráfico de la función y = tan(x)+1/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                1
f(x) = tan(x) + -
                x
f(x)=tan⁡(x)+1xf{\left(x \right)} = \tan{\left(x \right)} + \frac{1}{x} 
f = tan(x) + 1/x
Gráfico de la función
02468-8-6-4-2-1010-200200
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:
tan⁡(x)+1x=0\tan{\left(x \right)} + \frac{1}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−56.5309801938186x_{1} = -56.5309801938186
x2=−37.672573565113x_{2} = -37.672573565113
x3=59.6735041304405x_{3} = 59.6735041304405
x4=72.2427897046973x_{4} = 72.2427897046973
x5=−94.2371684817036x_{5} = -94.2371684817036
x6=34.5285657554621x_{6} = 34.5285657554621
x7=−31.3840740178899x_{7} = -31.3840740178899
x8=−75.3849592185347x_{8} = -75.3849592185347
x9=12.4864543952238x_{9} = 12.4864543952238
x10=62.8159348889734x_{10} = 62.8159348889734
x11=100.521017074687x_{11} = 100.521017074687
x12=94.2371684817036x_{12} = 94.2371684817036
x13=−15.644128370333x_{13} = -15.644128370333
x14=−18.7964043662102x_{14} = -18.7964043662102
x15=6.12125046689807x_{15} = 6.12125046689807
x16=−6.12125046689807x_{16} = -6.12125046689807
x17=−72.2427897046973x_{17} = -72.2427897046973
x18=65.9582857893902x_{18} = 65.9582857893902
x19=−97.3791034786112x_{19} = -97.3791034786112
x20=56.5309801938186x_{20} = 56.5309801938186
x21=37.672573565113x_{21} = 37.672573565113
x22=−53.3883466217256x_{22} = -53.3883466217256
x23=−50.2455828375744x_{23} = -50.2455828375744
x24=40.8162093266346x_{24} = 40.8162093266346
x25=−34.5285657554621x_{25} = -34.5285657554621
x26=78.5270825679419x_{26} = 78.5270825679419
x27=−69.100567727981x_{27} = -69.100567727981
x28=69.100567727981x_{28} = 69.100567727981
x29=31.3840740178899x_{29} = 31.3840740178899
x30=25.0929104121121x_{30} = 25.0929104121121
x31=−65.9582857893902x_{31} = -65.9582857893902
x32=−2.79838604578389x_{32} = -2.79838604578389
x33=15.644128370333x_{33} = 15.644128370333
x34=87.9532251106725x_{34} = 87.9532251106725
x35=−81.6691650818489x_{35} = -81.6691650818489
x36=−43.9595528888955x_{36} = -43.9595528888955
x37=75.3849592185347x_{37} = 75.3849592185347
x38=28.2389365752603x_{38} = 28.2389365752603
x39=−91.0952098694071x_{39} = -91.0952098694071
x40=50.2455828375744x_{40} = 50.2455828375744
x41=81.6691650818489x_{41} = 81.6691650818489
x42=18.7964043662102x_{42} = 18.7964043662102
x43=−84.811211299318x_{43} = -84.811211299318
x44=−59.6735041304405x_{44} = -59.6735041304405
x45=−40.8162093266346x_{45} = -40.8162093266346
x46=91.0952098694071x_{46} = 91.0952098694071
x47=−25.0929104121121x_{47} = -25.0929104121121
x48=−78.5270825679419x_{48} = -78.5270825679419
x49=−62.8159348889734x_{49} = -62.8159348889734
x50=43.9595528888955x_{50} = 43.9595528888955
x51=47.1026627703624x_{51} = 47.1026627703624
x52=97.3791034786112x_{52} = 97.3791034786112
x53=−28.2389365752603x_{53} = -28.2389365752603
x54=9.31786646179107x_{54} = 9.31786646179107
x55=−47.1026627703624x_{55} = -47.1026627703624
x56=84.811211299318x_{56} = 84.811211299318
x57=2.79838604578389x_{57} = 2.79838604578389
x58=−9.31786646179107x_{58} = -9.31786646179107
x59=−100.521017074687x_{59} = -100.521017074687
x60=53.3883466217256x_{60} = 53.3883466217256
x61=21.945612879981x_{61} = 21.945612879981
x62=−12.4864543952238x_{62} = -12.4864543952238
x63=−87.9532251106725x_{63} = -87.9532251106725
x64=−21.945612879981x_{64} = -21.945612879981
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) + 1/x.
tan⁡(0)+10\tan{\left(0 \right)} + \frac{1}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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)+1−1x2=0\tan^{2}{\left(x \right)} + 1 - \frac{1}{x^{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)+1x3)=02 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{1}{x^{3}}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−6.27914616840886x_{1} = -6.27914616840886
x2=12.5658666236671x_{2} = 12.5658666236671
x3=65.9734422428741x_{3} = 65.9734422428741
x4=−65.9734422428741x_{4} = -65.9734422428741
x5=18.8494066053354x_{5} = 18.8494066053354
x6=87.9645928313298x_{6} = 87.9645928313298
x7=−59.6902557161297x_{7} = -59.6902557161297
x8=−69.1150353500932x_{8} = -69.1150353500932
x9=56.5486622345162x_{9} = 56.5486622345162
x10=−28.2742896413126x_{10} = -28.2742896413126
x11=−9.42358300730682x_{11} = -9.42358300730682
x12=6.27914616840886x_{12} = 6.27914616840886
x13=−78.5398142756465x_{13} = -78.5398142756465
x14=−37.6990931789674x_{14} = -37.6990931789674
x15=91.1061856317236x_{15} = 91.1061856317236
x16=−87.9645928313298x_{16} = -87.9645928313298
x17=72.2566283818265x_{17} = 72.2566283818265
x18=50.2654745835232x_{18} = 50.2654745835232
x19=37.6990931789674x_{19} = 37.6990931789674
x20=−31.4158942842642x_{20} = -31.4158942842642
x21=−97.3893711786903x_{21} = -97.3893711786903
x22=−12.5658666236671x_{22} = -12.5658666236671
x23=−56.5486622345162x_{23} = -56.5486622345162
x24=47.1238802478309x_{24} = 47.1238802478309
x25=−40.8406898168452x_{25} = -40.8406898168452
x26=97.3893711786903x_{26} = 97.3893711786903
x27=−62.8318490403533x_{27} = -62.8318490403533
x28=28.2742896413126x_{28} = 28.2742896413126
x29=−53.4070685464944x_{29} = -53.4070685464944
x30=69.1150353500932x_{30} = 69.1150353500932
x31=−81.6814071583587x_{31} = -81.6814071583587
x32=34.5574949583816x_{32} = 34.5574949583816
x33=100.530963930635x_{33} = 100.530963930635
x34=−25.1326782369668x_{34} = -25.1326782369668
x35=−47.1238802478309x_{35} = -47.1238802478309
x36=25.1326782369668x_{36} = 25.1326782369668
x37=9.42358300730682x_{37} = 9.42358300730682
x38=15.7077052429814x_{38} = 15.7077052429814
x39=94.2477784131925x_{39} = 94.2477784131925
x40=−84.8230000083766x_{40} = -84.8230000083766
x41=−75.3982213531445x_{41} = -75.3982213531445
x42=−43.982285396773x_{42} = -43.982285396773
x43=−34.5574949583816x_{43} = -34.5574949583816
x44=21.9910545461263x_{44} = 21.9910545461263
x45=84.8230000083766x_{45} = 84.8230000083766
x46=31.4158942842642x_{46} = 31.4158942842642
x47=−50.2654745835232x_{47} = -50.2654745835232
x48=59.6902557161297x_{48} = 59.6902557161297
x49=−15.7077052429814x_{49} = -15.7077052429814
x50=43.982285396773x_{50} = 43.982285396773
x51=62.8318490403533x_{51} = 62.8318490403533
x52=75.3982213531445x_{52} = 75.3982213531445
x53=53.4070685464944x_{53} = 53.4070685464944
x54=−18.8494066053354x_{54} = -18.8494066053354
x55=−94.2477784131925x_{55} = -94.2477784131925
x56=40.8406898168452x_{56} = 40.8406898168452
x57=−91.1061856317236x_{57} = -91.1061856317236
x58=−3.10834412267136x_{58} = -3.10834412267136
x59=−72.2566283818265x_{59} = -72.2566283818265
x60=−100.530963930635x_{60} = -100.530963930635
x61=−21.9910545461263x_{61} = -21.9910545461263
x62=3.10834412267136x_{62} = 3.10834412267136
x63=81.6814071583587x_{63} = 81.6814071583587
x64=78.5398142756465x_{64} = 78.5398142756465
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=0x_{1} = 0

lim⁡x→0−(2((tan⁡2(x)+1)tan⁡(x)+1x3))=−∞\lim_{x \to 0^-}\left(2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{1}{x^{3}}\right)\right) = -\infty
lim⁡x→0+(2((tan⁡2(x)+1)tan⁡(x)+1x3))=∞\lim_{x \to 0^+}\left(2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{1}{x^{3}}\right)\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[100.530963930635,∞)\left[100.530963930635, \infty\right)
Convexa en los intervalos
(−∞,−100.530963930635]\left(-\infty, -100.530963930635\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
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(tan⁡(x)+1x)y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \frac{1}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)+1x)y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \frac{1}{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x) + 1/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)+1xx)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \frac{1}{x}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)+1xx)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \frac{1}{x}}{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:
tan⁡(x)+1x=−tan⁡(x)−1x\tan{\left(x \right)} + \frac{1}{x} = - \tan{\left(x \right)} - \frac{1}{x}
- No
tan⁡(x)+1x=tan⁡(x)+1x\tan{\left(x \right)} + \frac{1}{x} = \tan{\left(x \right)} + \frac{1}{x}
- No
es decir, función
no es
par ni impar
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