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tan(x)/(x-sin(x))
Trazado el gráfico de la función/ tan(x)/(x-sin(x))

Gráfico de la función y = tan(x)/(x-sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         tan(x)  
f(x) = ----------
       x - sin(x)
f(x)=tan⁡(x)x−sin⁡(x)f{\left(x \right)} = \frac{\tan{\left(x \right)}}{x - \sin{\left(x \right)}} 
f = tan(x)/(x - sin(x))
Gráfico de la función
02468-8-6-4-2-1010-25002500
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−sin⁡(x)=0\frac{\tan{\left(x \right)}}{x - \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=31.4159265358979x_{6} = 31.4159265358979
x7=69.1150383789755x_{7} = 69.1150383789755
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=−84.8230016469244x_{10} = -84.8230016469244
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=47.1238898038469x_{12} = 47.1238898038469
x13=−15.707963267949x_{13} = -15.707963267949
x14=−12.5663706143592x_{14} = -12.5663706143592
x15=12.5663706143592x_{15} = 12.5663706143592
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=53.4070751110265x_{17} = 53.4070751110265
x18=72.2566310325652x_{18} = 72.2566310325652
x19=−100.530964914873x_{19} = -100.530964914873
x20=−3.14159265358979x_{20} = -3.14159265358979
x21=34.5575191894877x_{21} = 34.5575191894877
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=6.28318530717959x_{23} = 6.28318530717959
x24=−69.1150383789755x_{24} = -69.1150383789755
x25=65.9734457253857x_{25} = 65.9734457253857
x26=97.3893722612836x_{26} = 97.3893722612836
x27=15.707963267949x_{27} = 15.707963267949
x28=−50.2654824574367x_{28} = -50.2654824574367
x29=−25.1327412287183x_{29} = -25.1327412287183
x30=3.14159265358979x_{30} = 3.14159265358979
x31=−18.8495559215388x_{31} = -18.8495559215388
x32=40.8407044966673x_{32} = 40.8407044966673
x33=18.8495559215388x_{33} = 18.8495559215388
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=37.6991118430775x_{35} = 37.6991118430775
x36=−43.9822971502571x_{36} = -43.9822971502571
x37=−78.5398163397448x_{37} = -78.5398163397448
x38=−6.28318530717959x_{38} = -6.28318530717959
x39=−40.8407044966673x_{39} = -40.8407044966673
x40=43.9822971502571x_{40} = 43.9822971502571
x41=56.5486677646163x_{41} = 56.5486677646163
x42=−65.9734457253857x_{42} = -65.9734457253857
x43=25.1327412287183x_{43} = 25.1327412287183
x44=78.5398163397448x_{44} = 78.5398163397448
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=75.398223686155x_{46} = 75.398223686155
x47=59.6902604182061x_{47} = 59.6902604182061
x48=−34.5575191894877x_{48} = -34.5575191894877
x49=81.6814089933346x_{49} = 81.6814089933346
x50=−47.1238898038469x_{50} = -47.1238898038469
x51=100.530964914873x_{51} = 100.530964914873
x52=−9.42477796076938x_{52} = -9.42477796076938
x53=−75.398223686155x_{53} = -75.398223686155
x54=−72.2566310325652x_{54} = -72.2566310325652
x55=−31.4159265358979x_{55} = -31.4159265358979
x56=28.2743338823081x_{56} = 28.2743338823081
x57=−91.106186954104x_{57} = -91.106186954104
x58=21.9911485751286x_{58} = 21.9911485751286
x59=62.8318530717959x_{59} = 62.8318530717959
x60=9.42477796076938x_{60} = 9.42477796076938
x61=50.2654824574367x_{61} = 50.2654824574367
x62=94.2477796076938x_{62} = 94.2477796076938
x63=91.106186954104x_{63} = 91.106186954104
x64=84.8230016469244x_{64} = 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 - sin(x)).
tan⁡(0)(−1)sin⁡(0)\frac{\tan{\left(0 \right)}}{\left(-1\right) \sin{\left(0 \right)}}
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−sin⁡(x)+(cos⁡(x)−1)tan⁡(x)(x−sin⁡(x))2=0\frac{\tan^{2}{\left(x \right)} + 1}{x - \sin{\left(x \right)}} + \frac{\left(\cos{\left(x \right)} - 1\right) \tan{\left(x \right)}}{\left(x - \sin{\left(x \right)}\right)^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x)/(x - sin(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−sin⁡(x)))y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x \left(x - \sin{\left(x \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−sin⁡(x)))y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x \left(x - \sin{\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:
tan⁡(x)x−sin⁡(x)=−tan⁡(x)−x+sin⁡(x)\frac{\tan{\left(x \right)}}{x - \sin{\left(x \right)}} = - \frac{\tan{\left(x \right)}}{- x + \sin{\left(x \right)}}
- No
tan⁡(x)x−sin⁡(x)=tan⁡(x)−x+sin⁡(x)\frac{\tan{\left(x \right)}}{x - \sin{\left(x \right)}} = \frac{\tan{\left(x \right)}}{- x + \sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tan(x)/(x-sin(x))
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