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Trazado el gráfico de la función/ tg(x)/x^2

Gráfico de la función y = tg(x)/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       tan(x)
f(x) = ------
          2  
         x
f(x)=tan⁡(x)x2f{\left(x \right)} = \frac{\tan{\left(x \right)}}{x^{2}} 
f = tan(x)/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=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)x2=0\frac{\tan{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=59.6902604182061x_{1} = 59.6902604182061
x2=−100.530964914873x_{2} = -100.530964914873
x3=−15.707963267949x_{3} = -15.707963267949
x4=−25.1327412287183x_{4} = -25.1327412287183
x5=84.8230016469244x_{5} = 84.8230016469244
x6=−75.398223686155x_{6} = -75.398223686155
x7=97.3893722612836x_{7} = 97.3893722612836
x8=−50.2654824574367x_{8} = -50.2654824574367
x9=81.6814089933346x_{9} = 81.6814089933346
x10=−72.2566310325652x_{10} = -72.2566310325652
x11=91.106186954104x_{11} = 91.106186954104
x12=50.2654824574367x_{12} = 50.2654824574367
x13=−43.9822971502571x_{13} = -43.9822971502571
x14=−37.6991118430775x_{14} = -37.6991118430775
x15=25.1327412287183x_{15} = 25.1327412287183
x16=−65.9734457253857x_{16} = -65.9734457253857
x17=−53.4070751110265x_{17} = -53.4070751110265
x18=−18.8495559215388x_{18} = -18.8495559215388
x19=−59.6902604182061x_{19} = -59.6902604182061
x20=15.707963267949x_{20} = 15.707963267949
x21=9.42477796076938x_{21} = 9.42477796076938
x22=18.8495559215388x_{22} = 18.8495559215388
x23=−56.5486677646163x_{23} = -56.5486677646163
x24=−6.28318530717959x_{24} = -6.28318530717959
x25=−62.8318530717959x_{25} = -62.8318530717959
x26=12.5663706143592x_{26} = 12.5663706143592
x27=56.5486677646163x_{27} = 56.5486677646163
x28=40.8407044966673x_{28} = 40.8407044966673
x29=3.14159265358979x_{29} = 3.14159265358979
x30=−21.9911485751286x_{30} = -21.9911485751286
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=6.28318530717959x_{32} = 6.28318530717959
x33=69.1150383789755x_{33} = 69.1150383789755
x34=72.2566310325652x_{34} = 72.2566310325652
x35=−78.5398163397448x_{35} = -78.5398163397448
x36=37.6991118430775x_{36} = 37.6991118430775
x37=21.9911485751286x_{37} = 21.9911485751286
x38=47.1238898038469x_{38} = 47.1238898038469
x39=34.5575191894877x_{39} = 34.5575191894877
x40=−97.3893722612836x_{40} = -97.3893722612836
x41=−31.4159265358979x_{41} = -31.4159265358979
x42=100.530964914873x_{42} = 100.530964914873
x43=−47.1238898038469x_{43} = -47.1238898038469
x44=28.2743338823081x_{44} = 28.2743338823081
x45=94.2477796076938x_{45} = 94.2477796076938
x46=−40.8407044966673x_{46} = -40.8407044966673
x47=−12.5663706143592x_{47} = -12.5663706143592
x48=−34.5575191894877x_{48} = -34.5575191894877
x49=−28.2743338823081x_{49} = -28.2743338823081
x50=78.5398163397448x_{50} = 78.5398163397448
x51=−94.2477796076938x_{51} = -94.2477796076938
x52=−91.106186954104x_{52} = -91.106186954104
x53=43.9822971502571x_{53} = 43.9822971502571
x54=75.398223686155x_{54} = 75.398223686155
x55=62.8318530717959x_{55} = 62.8318530717959
x56=−3.14159265358979x_{56} = -3.14159265358979
x57=87.9645943005142x_{57} = 87.9645943005142
x58=53.4070751110265x_{58} = 53.4070751110265
x59=−81.6814089933346x_{59} = -81.6814089933346
x60=−87.9645943005142x_{60} = -87.9645943005142
x61=65.9734457253857x_{61} = 65.9734457253857
x62=−69.1150383789755x_{62} = -69.1150383789755
x63=31.4159265358979x_{63} = 31.4159265358979
x64=−9.42477796076938x_{64} = -9.42477796076938
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^2.
tan⁡(0)02\frac{\tan{\left(0 \right)}}{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)+1x2−2tan⁡(x)x3=0\frac{\tan^{2}{\left(x \right)} + 1}{x^{2}} - \frac{2 \tan{\left(x \right)}}{x^{3}} = 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)−2(tan⁡2(x)+1)x+3tan⁡(x)x2)x2=0\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{3 \tan{\left(x \right)}}{x^{2}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=18.9538314073642x_{1} = 18.9538314073642
x2=−59.7237073696423x_{2} = -59.7237073696423
x3=6.56206743448403x_{3} = 6.56206743448403
x4=3.57990203616361x_{4} = 3.57990203616361
x5=−53.4444405432462x_{5} = -53.4444405432462
x6=66.0037169665853x_{6} = 66.0037169665853
x7=37.751929127791x_{7} = 37.751929127791
x8=94.268985157597x_{8} = 94.268985157597
x9=−6.56206743448403x_{9} = -6.56206743448403
x10=72.2842766569239x_{10} = 72.2842766569239
x11=9.62366096208515x_{11} = 9.62366096208515
x12=75.4247200230927x_{12} = 75.4247200230927
x13=31.4791850671807x_{13} = 31.4791850671807
x14=−72.2842766569239x_{14} = -72.2842766569239
x15=−25.2115361587402x_{15} = -25.2115361587402
x16=91.1281226349785x_{16} = 91.1281226349785
x17=−81.7058711694478x_{17} = -81.7058711694478
x18=78.5652550380417x_{18} = 78.5652550380417
x19=97.4098946880995x_{19} = 97.4098946880995
x20=−62.863633155918x_{20} = -62.863633155918
x21=40.8894906794103x_{21} = 40.8894906794103
x22=−28.3445175898144x_{22} = -28.3445175898144
x23=25.2115361587402x_{23} = 25.2115361587402
x24=−22.0809329417135x_{24} = -22.0809329417135
x25=28.3445175898144x_{25} = 28.3445175898144
x26=−40.8894906794103x_{26} = -40.8894906794103
x27=−50.3051719455312x_{27} = -50.3051719455312
x28=−97.4098946880995x_{28} = -97.4098946880995
x29=34.6150899463703x_{29} = 34.6150899463703
x30=−44.027622064428x_{30} = -44.027622064428
x31=44.027622064428x_{31} = 44.027622064428
x32=12.7196005943114x_{32} = 12.7196005943114
x33=−34.6150899463703x_{33} = -34.6150899463703
x34=69.1439373752036x_{34} = 69.1439373752036
x35=15.8321741729874x_{35} = 15.8321741729874
x36=−3.57990203616361x_{36} = -3.57990203616361
x37=−69.1439373752036x_{37} = -69.1439373752036
x38=−91.1281226349785x_{38} = -91.1281226349785
x39=−31.4791850671807x_{39} = -31.4791850671807
x40=−12.7196005943114x_{40} = -12.7196005943114
x41=−15.8321741729874x_{41} = -15.8321741729874
x42=56.5839657532757x_{42} = 56.5839657532757
x43=100.550846831073x_{43} = 100.550846831073
x44=50.3051719455312x_{44} = 50.3051719455312
x45=−66.0037169665853x_{45} = -66.0037169665853
x46=−94.268985157597x_{46} = -94.268985157597
x47=−100.550846831073x_{47} = -100.550846831073
x48=−18.9538314073642x_{48} = -18.9538314073642
x49=−37.751929127791x_{49} = -37.751929127791
x50=81.7058711694478x_{50} = 81.7058711694478
x51=−9.62366096208515x_{51} = -9.62366096208515
x52=59.7237073696423x_{52} = 59.7237073696423
x53=−56.5839657532757x_{53} = -56.5839657532757
x54=53.4444405432462x_{54} = 53.4444405432462
x55=−87.987312141176x_{55} = -87.987312141176
x56=−84.8465594379176x_{56} = -84.8465594379176
x57=62.863633155918x_{57} = 62.863633155918
x58=22.0809329417135x_{58} = 22.0809329417135
x59=−78.5652550380417x_{59} = -78.5652550380417
x60=−75.4247200230927x_{60} = -75.4247200230927
x61=84.8465594379176x_{61} = 84.8465594379176
x62=87.987312141176x_{62} = 87.987312141176
x63=47.1662107531913x_{63} = 47.1662107531913
x64=−47.1662107531913x_{64} = -47.1662107531913
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)−2(tan⁡2(x)+1)x+3tan⁡(x)x2)x2)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{3 \tan{\left(x \right)}}{x^{2}}\right)}{x^{2}}\right) = -\infty
lim⁡x→0+(2((tan⁡2(x)+1)tan⁡(x)−2(tan⁡2(x)+1)x+3tan⁡(x)x2)x2)=∞\lim_{x \to 0^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{3 \tan{\left(x \right)}}{x^{2}}\right)}{x^{2}}\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
[100.550846831073,∞)\left[100.550846831073, \infty\right)
Convexa en los intervalos
(−∞,−100.550846831073]\left(-\infty, -100.550846831073\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)x2)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x^{2}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)x2)y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x^{2}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x)/x^2, 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)xx2)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x x^{2}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)xx2)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x x^{2}}\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)x2=−tan⁡(x)x2\frac{\tan{\left(x \right)}}{x^{2}} = - \frac{\tan{\left(x \right)}}{x^{2}}
- No
tan⁡(x)x2=tan⁡(x)x2\frac{\tan{\left(x \right)}}{x^{2}} = \frac{\tan{\left(x \right)}}{x^{2}}
- No
es decir, función
no es
par ni impar
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