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Trazado el gráfico de la función/ tan(x)*sqrt(x)

Gráfico de la función y = tan(x)*sqrt(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=−75.398223686155x_{1} = -75.398223686155
x2=47.1238898038469x_{2} = 47.1238898038469
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=9.42477796076938x_{4} = 9.42477796076938
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−97.3893722612836x_{6} = -97.3893722612836
x7=−62.8318530717959x_{7} = -62.8318530717959
x8=87.9645943005142x_{8} = 87.9645943005142
x9=−87.9645943005142x_{9} = -87.9645943005142
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=6.28318530717959x_{11} = 6.28318530717959
x12=59.6902604182061x_{12} = 59.6902604182061
x13=−47.1238898038469x_{13} = -47.1238898038469
x14=−40.8407044966673x_{14} = -40.8407044966673
x15=100.530964914873x_{15} = 100.530964914873
x16=62.8318530717959x_{16} = 62.8318530717959
x17=3.14159265358979x_{17} = 3.14159265358979
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−69.1150383789755x_{19} = -69.1150383789755
x20=97.3893722612836x_{20} = 97.3893722612836
x21=12.5663706143592x_{21} = 12.5663706143592
x22=94.2477796076938x_{22} = 94.2477796076938
x23=31.4159265358979x_{23} = 31.4159265358979
x24=25.1327412287183x_{24} = 25.1327412287183
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=−94.2477796076938x_{26} = -94.2477796076938
x27=−59.6902604182061x_{27} = -59.6902604182061
x28=−56.5486677646163x_{28} = -56.5486677646163
x29=81.6814089933346x_{29} = 81.6814089933346
x30=43.9822971502571x_{30} = 43.9822971502571
x31=−91.106186954104x_{31} = -91.106186954104
x32=15.707963267949x_{32} = 15.707963267949
x33=34.5575191894877x_{33} = 34.5575191894877
x34=21.9911485751286x_{34} = 21.9911485751286
x35=40.8407044966673x_{35} = 40.8407044966673
x36=69.1150383789755x_{36} = 69.1150383789755
x37=65.9734457253857x_{37} = 65.9734457253857
x38=−72.2566310325652x_{38} = -72.2566310325652
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=91.106186954104x_{40} = 91.106186954104
x41=53.4070751110265x_{41} = 53.4070751110265
x42=−28.2743338823081x_{42} = -28.2743338823081
x43=56.5486677646163x_{43} = 56.5486677646163
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=−18.8495559215388x_{45} = -18.8495559215388
x46=−100.530964914873x_{46} = -100.530964914873
x47=−53.4070751110265x_{47} = -53.4070751110265
x48=−15.707963267949x_{48} = -15.707963267949
x49=84.8230016469244x_{49} = 84.8230016469244
x50=72.2566310325652x_{50} = 72.2566310325652
x51=18.8495559215388x_{51} = 18.8495559215388
x52=0x_{52} = 0
x53=−43.9822971502571x_{53} = -43.9822971502571
x54=−84.8230016469244x_{54} = -84.8230016469244
x55=−78.5398163397448x_{55} = -78.5398163397448
x56=−12.5663706143592x_{56} = -12.5663706143592
x57=75.398223686155x_{57} = 75.398223686155
x58=−6.28318530717959x_{58} = -6.28318530717959
x59=78.5398163397448x_{59} = 78.5398163397448
x60=−50.2654824574367x_{60} = -50.2654824574367
x61=−81.6814089933346x_{61} = -81.6814089933346
x62=50.2654824574367x_{62} = 50.2654824574367
x63=−9.42477796076938x_{63} = -9.42477796076938
x64=37.6991118430775x_{64} = 37.6991118430775
x65=−25.1327412287183x_{65} = -25.1327412287183
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)*sqrt(x).
0tan⁡(0)\sqrt{0} \tan{\left(0 \right)}
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
x(tan⁡2(x)+1)+tan⁡(x)2x=0\sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{\tan{\left(x \right)}}{2 \sqrt{x}} = 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
2x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1x−tan⁡(x)4x32=02 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x}} - \frac{\tan{\left(x \right)}}{4 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2.97267489465031x_{1} = 2.97267489465031
x2=9.37139905890989x_{2} = 9.37139905890989
x3=56.5398243251235x_{3} = 56.5398243251235
x4=−87.9589097974476x_{4} = -87.9589097974476
x5=100.52599105631x_{5} = 100.52599105631
x6=87.9589097974476x_{6} = 87.9589097974476
x7=−53.397711277025x_{7} = -53.397711277025
x8=−59.6818825684128x_{8} = -59.6818825684128
x9=91.1006984944297x_{9} = 91.1006984944297
x10=50.2555331401204x_{10} = 50.2555331401204
x11=78.5334495829027x_{11} = 78.5334495829027
x12=−50.2555331401204x_{12} = -50.2555331401204
x13=−40.8284578305388x_{13} = -40.8284578305388
x14=6.2024870457813x_{14} = 6.2024870457813
x15=−84.8171065755666x_{15} = -84.8171065755666
x16=−9.37139905890989x_{16} = -9.37139905890989
x17=−78.5334495829027x_{17} = -78.5334495829027
x18=15.6760621125375x_{18} = 15.6760621125375
x19=−81.6752871523214x_{19} = -81.6752871523214
x20=−56.5398243251235x_{20} = -56.5398243251235
x21=37.6858438726898x_{21} = 37.6858438726898
x22=−72.2497105343831x_{22} = -72.2497105343831
x23=65.965865975172x_{23} = 65.965865975172
x24=−47.1132768850395x_{24} = -47.1132768850395
x25=−2.97267489465031x_{25} = -2.97267489465031
x26=−18.8229895430946x_{26} = -18.8229895430946
x27=43.9709257551348x_{27} = 43.9709257551348
x28=40.8284578305388x_{28} = 40.8284578305388
x29=−12.5264445146365x_{29} = -12.5264445146365
x30=21.9683866353213x_{30} = 21.9683866353213
x31=−25.112829773268x_{31} = -25.112829773268
x32=−28.2566380026491x_{32} = -28.2566380026491
x33=−65.965865975172x_{33} = -65.965865975172
x34=12.5264445146365x_{34} = 12.5264445146365
x35=−91.1006984944297x_{35} = -91.1006984944297
x36=−6.2024870457813x_{36} = -6.2024870457813
x37=53.397711277025x_{37} = 53.397711277025
x38=−69.1078032428841x_{38} = -69.1078032428841
x39=−97.3842379376276x_{39} = -97.3842379376276
x40=59.6818825684128x_{40} = 59.6818825684128
x41=−43.9709257551348x_{41} = -43.9709257551348
x42=−15.6760621125375x_{42} = -15.6760621125375
x43=18.8229895430946x_{43} = 18.8229895430946
x44=25.112829773268x_{44} = 25.112829773268
x45=−94.2424741193764x_{45} = -94.2424741193764
x46=75.3915915982233x_{46} = 75.3915915982233
x47=47.1132768850395x_{47} = 47.1132768850395
x48=−62.8238942325121x_{48} = -62.8238942325121
x49=−75.3915915982233x_{49} = -75.3915915982233
x50=94.2424741193764x_{50} = 94.2424741193764
x51=62.8238942325121x_{51} = 62.8238942325121
x52=72.2497105343831x_{52} = 72.2497105343831
x53=−100.52599105631x_{53} = -100.52599105631
x54=28.2566380026491x_{54} = 28.2566380026491
x55=−37.6858438726898x_{55} = -37.6858438726898
x56=−31.4000022978176x_{56} = -31.4000022978176
x57=97.3842379376276x_{57} = 97.3842379376276
x58=81.6752871523214x_{58} = 81.6752871523214
x59=−21.9683866353213x_{59} = -21.9683866353213
x60=84.8171065755666x_{60} = 84.8171065755666
x61=31.4000022978176x_{61} = 31.4000022978176
x62=34.5430439901451x_{62} = 34.5430439901451
x63=−34.5430439901451x_{63} = -34.5430439901451
x64=69.1078032428841x_{64} = 69.1078032428841

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
[100.52599105631,∞)\left[100.52599105631, \infty\right)
Convexa en los intervalos
(−∞,2.97267489465031]\left(-\infty, 2.97267489465031\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→−∞(xtan⁡(x))y = \lim_{x \to -\infty}\left(\sqrt{x} \tan{\left(x \right)}\right)
True

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

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