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log(x)^2*sin(x)
Trazado el gráfico de la función/ log(x)^2*sin(x)

Gráfico de la función y = log(x)^2*sin(x)

v

Gráfico:

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Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2          
f(x) = log (x)*sin(x)
f(x)=log⁡(x)2sin⁡(x)f{\left(x \right)} = \log{\left(x \right)}^{2} \sin{\left(x \right)} 
f = log(x)^2*sin(x)
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:
log⁡(x)2sin⁡(x)=0\log{\left(x \right)}^{2} \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
x2=πx_{2} = \pi
Solución numérica
x1=31.4159265358979x_{1} = 31.4159265358979
x2=69.1150383789755x_{2} = 69.1150383789755
x3=53.4070751110265x_{3} = 53.4070751110265
x4=−6.28318530717959x_{4} = -6.28318530717959
x5=−3.14159265358979x_{5} = -3.14159265358979
x6=−69.1150383789755x_{6} = -69.1150383789755
x7=−9.42477796076938x_{7} = -9.42477796076938
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−53.4070751110265x_{9} = -53.4070751110265
x10=−94.2477796076938x_{10} = -94.2477796076938
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=28.2743338823081x_{12} = 28.2743338823081
x13=−72.2566310325652x_{13} = -72.2566310325652
x14=62.8318530717959x_{14} = 62.8318530717959
x15=9.42477796076938x_{15} = 9.42477796076938
x16=50.2654824574367x_{16} = 50.2654824574367
x17=−116.238928182822x_{17} = -116.238928182822
x18=−21.9911485751286x_{18} = -21.9911485751286
x19=−91.106186954104x_{19} = -91.106186954104
x20=−62.8318530717959x_{20} = -62.8318530717959
x21=34.5575191894877x_{21} = 34.5575191894877
x22=−47.1238898038469x_{22} = -47.1238898038469
x23=65.9734457253857x_{23} = 65.9734457253857
x24=−34.5575191894877x_{24} = -34.5575191894877
x25=−40.8407044966673x_{25} = -40.8407044966673
x26=3.14159265358979x_{26} = 3.14159265358979
x27=78.5398163397448x_{27} = 78.5398163397448
x28=59.6902604182061x_{28} = 59.6902604182061
x29=100.530964914873x_{29} = 100.530964914873
x30=−31.4159265358979x_{30} = -31.4159265358979
x31=21.9911485751286x_{31} = 21.9911485751286
x32=91.106186954104x_{32} = 91.106186954104
x33=−81.6814089933346x_{33} = -81.6814089933346
x34=−97.3893722612836x_{34} = -97.3893722612836
x35=84.8230016469244x_{35} = 84.8230016469244
x36=−87.9645943005142x_{36} = -87.9645943005142
x37=12.5663706143592x_{37} = 12.5663706143592
x38=−28.2743338823081x_{38} = -28.2743338823081
x39=−59.6902604182061x_{39} = -59.6902604182061
x40=−56.5486677646163x_{40} = -56.5486677646163
x41=15.707963267949x_{41} = 15.707963267949
x42=18.8495559215388x_{42} = 18.8495559215388
x43=40.8407044966673x_{43} = 40.8407044966673
x44=−43.9822971502571x_{44} = -43.9822971502571
x45=−84.8230016469244x_{45} = -84.8230016469244
x46=−100.530964914873x_{46} = -100.530964914873
x47=81.6814089933346x_{47} = 81.6814089933346
x48=−75.398223686155x_{48} = -75.398223686155
x49=−50.2654824574367x_{49} = -50.2654824574367
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−78.5398163397448x_{51} = -78.5398163397448
x52=87.9645943005142x_{52} = 87.9645943005142
x53=47.1238898038469x_{53} = 47.1238898038469
x54=−18.8495559215388x_{54} = -18.8495559215388
x55=−12.5663706143592x_{55} = -12.5663706143592
x56=−65.9734457253857x_{56} = -65.9734457253857
x57=6.28318530717959x_{57} = 6.28318530717959
x58=97.3893722612836x_{58} = 97.3893722612836
x59=37.6991118430775x_{59} = 37.6991118430775
x60=−15.707963267949x_{60} = -15.707963267949
x61=43.9822971502571x_{61} = 43.9822971502571
x62=56.5486677646163x_{62} = 56.5486677646163
x63=25.1327412287183x_{63} = 25.1327412287183
x64=75.398223686155x_{64} = 75.398223686155
x65=72.2566310325652x_{65} = 72.2566310325652
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 log(x)^2*sin(x).
log⁡(0)2sin⁡(0)\log{\left(0 \right)}^{2} \sin{\left(0 \right)}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−log⁡(x)2sin⁡(x)+4log⁡(x)cos⁡(x)x−2(log⁡(x)−1)sin⁡(x)x2=0- \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{4 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{2 \left(\log{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=59.7066394917018x_{1} = 59.7066394917018
x2=15.7992708977347x_{2} = 15.7992708977347
x3=1.65706451750942x_{3} = 1.65706451750942
x4=9.60585403951963x_{4} = 9.60585403951963
x5=84.8336186997604x_{5} = 84.8336186997604
x6=53.4258900441622x_{6} = 53.4258900441622
x7=69.1286964375608x_{7} = 69.1286964375608
x8=6.59122807282926x_{8} = 6.59122807282926
x9=91.1159155945839x_{9} = 91.1159155945839
x10=12.689444473502x_{10} = 12.689444473502
x11=94.2571139671403x_{11} = 94.2571139671403
x12=37.7282990086415x_{12} = 37.7282990086415
x13=22.0496746158449x_{13} = 22.0496746158449
x14=3.79739269085901x_{14} = 3.79739269085901
x15=78.5514844556215x_{15} = 78.5514844556215
x16=97.3983409430341x_{16} = 97.3983409430341
x17=44.0063067584764x_{17} = 44.0063067584764
x18=31.4527729390447x_{18} = 31.4527729390447
x19=34.5901303273917x_{19} = 34.5901303273917
x20=87.9747491999982x_{20} = 87.9747491999982
x21=50.2857805952779x_{21} = 50.2857805952779
x22=28.3165366149909x_{22} = 28.3165366149909
x23=18.9212415222179x_{23} = 18.9212415222179
x24=25.181905221086x_{24} = 25.181905221086
x25=81.6925286637411x_{25} = 81.6925286637411
x26=47.145901016022x_{26} = 47.145901016022
x27=75.4104924106057x_{27} = 75.4104924106057
x28=40.867072151817x_{28} = 40.867072151817
x29=72.2695600625994x_{29} = 72.2695600625994
x30=65.9879124364584x_{30} = 65.9879124364584
x31=100.53959359966x_{31} = 100.53959359966
x32=62.8472213411914x_{32} = 62.8472213411914
x33=56.5661872052363x_{33} = 56.5661872052363

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
[97.3983409430341,∞)\left[97.3983409430341, \infty\right)
Convexa en los intervalos
(−∞,3.79739269085901]\left(-\infty, 3.79739269085901\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)}^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(log⁡(x)2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)}^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(x)^2*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)2sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(x)2sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
log⁡(x)2sin⁡(x)=−log⁡(−x)2sin⁡(x)\log{\left(x \right)}^{2} \sin{\left(x \right)} = - \log{\left(- x \right)}^{2} \sin{\left(x \right)}
- No
log⁡(x)2sin⁡(x)=log⁡(−x)2sin⁡(x)\log{\left(x \right)}^{2} \sin{\left(x \right)} = \log{\left(- x \right)}^{2} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = log(x)^2*sin(x)
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