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log(x)^3*sin(x)
Trazado el gráfico de la función/ log(x)^3*sin(x)

Gráfico de la función y = log(x)^3*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          3          
f(x) = log (x)*sin(x)
f(x)=log⁡(x)3sin⁡(x)f{\left(x \right)} = \log{\left(x \right)}^{3} \sin{\left(x \right)} 
f = log(x)^3*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-2020
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)3sin⁡(x)=0\log{\left(x \right)}^{3} \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=−81.6814089933346x_{3} = -81.6814089933346
x4=−69.1150383789755x_{4} = -69.1150383789755
x5=53.4070751110265x_{5} = 53.4070751110265
x6=−53.4070751110265x_{6} = -53.4070751110265
x7=−9.42477796076938x_{7} = -9.42477796076938
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=−40.8407044966673x_{9} = -40.8407044966673
x10=−78.5398163397448x_{10} = -78.5398163397448
x11=−43.9822971502571x_{11} = -43.9822971502571
x12=28.2743338823081x_{12} = 28.2743338823081
x13=62.8318530717959x_{13} = 62.8318530717959
x14=−25.1327412287183x_{14} = -25.1327412287183
x15=−12.5663706143592x_{15} = -12.5663706143592
x16=9.42477796076938x_{16} = 9.42477796076938
x17=50.2654824574367x_{17} = 50.2654824574367
x18=−18.8495559215388x_{18} = -18.8495559215388
x19=−91.106186954104x_{19} = -91.106186954104
x20=−62.8318530717959x_{20} = -62.8318530717959
x21=34.5575191894877x_{21} = 34.5575191894877
x22=−84.8230016469244x_{22} = -84.8230016469244
x23=65.9734457253857x_{23} = 65.9734457253857
x24=3.14159265358979x_{24} = 3.14159265358979
x25=−75.398223686155x_{25} = -75.398223686155
x26=−37.6991118430775x_{26} = -37.6991118430775
x27=−94.2477796076938x_{27} = -94.2477796076938
x28=−65.9734457253857x_{28} = -65.9734457253857
x29=78.5398163397448x_{29} = 78.5398163397448
x30=59.6902604182061x_{30} = 59.6902604182061
x31=100.530964914873x_{31} = 100.530964914873
x32=−72.2566310325652x_{32} = -72.2566310325652
x33=−31.4159265358979x_{33} = -31.4159265358979
x34=21.9911485751286x_{34} = 21.9911485751286
x35=91.106186954104x_{35} = 91.106186954104
x36=−97.3893722612836x_{36} = -97.3893722612836
x37=84.8230016469244x_{37} = 84.8230016469244
x38=−116.238928182822x_{38} = -116.238928182822
x39=−34.5575191894877x_{39} = -34.5575191894877
x40=12.5663706143592x_{40} = 12.5663706143592
x41=−28.2743338823081x_{41} = -28.2743338823081
x42=−3.14159265358979x_{42} = -3.14159265358979
x43=15.707963267949x_{43} = 15.707963267949
x44=18.8495559215388x_{44} = 18.8495559215388
x45=40.8407044966673x_{45} = 40.8407044966673
x46=−47.1238898038469x_{46} = -47.1238898038469
x47=−56.5486677646163x_{47} = -56.5486677646163
x48=81.6814089933346x_{48} = 81.6814089933346
x49=94.2477796076938x_{49} = 94.2477796076938
x50=87.9645943005142x_{50} = 87.9645943005142
x51=1.00008843615999x_{51} = 1.00008843615999
x52=−59.6902604182061x_{52} = -59.6902604182061
x53=47.1238898038469x_{53} = 47.1238898038469
x54=−6.28318530717959x_{54} = -6.28318530717959
x55=−100.530964914873x_{55} = -100.530964914873
x56=6.28318530717959x_{56} = 6.28318530717959
x57=97.3893722612836x_{57} = 97.3893722612836
x58=−15.707963267949x_{58} = -15.707963267949
x59=37.6991118430775x_{59} = 37.6991118430775
x60=−50.2654824574367x_{60} = -50.2654824574367
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=−87.9645943005142x_{65} = -87.9645943005142
x66=72.2566310325652x_{66} = 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)^3*sin(x).
log⁡(0)3sin⁡(0)\log{\left(0 \right)}^{3} \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
log⁡(x)3cos⁡(x)+3log⁡(x)2sin⁡(x)x=0\log{\left(x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=61.2729534415634x_{1} = 61.2729534415634
x2=64.4138302477556x_{2} = 64.4138302477556
x3=58.1321658716346x_{3} = 58.1321658716346
x4=89.5428444653399x_{4} = 89.5428444653399
x5=83.2603534390183x_{5} = 83.2603534390183
x6=20.4688629160207x_{6} = 20.4688629160207
x7=36.1514414910436x_{7} = 36.1514414910436
x8=5.06267917117524x_{8} = 5.06267917117524
x9=45.5703287126706x_{9} = 45.5703287126706
x10=54.9914846404513x_{10} = 54.9914846404513
x11=73.8368718590989x_{11} = 73.8368718590989
x12=8.03139099623953x_{12} = 8.03139099623953
x13=42.4303637527494x_{13} = 42.4303637527494
x14=14.2164990994671x_{14} = 14.2164990994671
x15=98.9667658004856x_{15} = 98.9667658004856
x16=29.8746827273128x_{16} = 29.8746827273128
x17=23.60213003453x_{17} = 23.60213003453
x18=95.8254375766894x_{18} = 95.8254375766894
x19=17.3393298030427x_{19} = 17.3393298030427
x20=48.7105340009097x_{20} = 48.7105340009097
x21=33.0127041187522x_{21} = 33.0127041187522
x22=2.49249324020916x_{22} = 2.49249324020916
x23=39.2907044598106x_{23} = 39.2907044598106
x24=11.1072905657786x_{24} = 11.1072905657786
x25=26.7376687721596x_{25} = 26.7376687721596
x26=86.4015846605975x_{26} = 86.4015846605975
x27=51.8509314096671x_{27} = 51.8509314096671
x28=70.6957994975412x_{28} = 70.6957994975412
x29=76.9779922603215x_{29} = 76.9779922603215
x30=80.1191545145906x_{30} = 80.1191545145906
x31=67.5547826177155x_{31} = 67.5547826177155
x32=92.6841296797232x_{32} = 92.6841296797232
Signos de extremos en los puntos:
(61.272953441563416, -69.6924873395813)

(64.41383024775558, 72.2637650284379)

(58.13216587163458, 67.0525603965331)

(89.54284446533988, 90.8019251907425)

(83.26035343901827, 86.463675148813)

(20.468862916020687, 27.4812868432645)

(36.15144149104357, -46.1677094492915)

(5.062679171175242, -4.00738236246644)

(45.57032871267055, 55.7021656471767)

(54.99148464045134, -64.3392173341535)

(73.83687185909886, -79.6065697917808)

(8.031390996239526, 8.9006328028635)

(42.4303637527494, -52.634958189165)

(14.216499099467082, 18.6437213828936)

(98.96676580048565, -97.0031586507996)

(29.87468272731276, -39.1833294541818)

(23.602130034530013, -31.5690573732181)

(95.82543757668938, 94.9743773579989)

(17.33932980304268, -23.1791662900454)

(48.71053400090973, -58.6703593940542)

(33.012704118752204, 42.7464677559037)

(2.4924932402091593, 0.460459077966658)

(39.29070445981061, 49.4600967590969)

(11.107290565778559, -13.8687767133005)

(26.737668772159612, 35.4632686286516)

(86.4015846605975, -88.6545457182069)

(51.85093140966714, 61.5471381349689)

(70.69579949754117, 77.217113333416)

(76.97799226032154, 81.9422464140317)

(80.11915451459063, -84.227058638229)

(67.5547826177155, -74.7706823820439)

(92.68412967972316, -92.9078942326061)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=61.2729534415634x_{1} = 61.2729534415634
x2=36.1514414910436x_{2} = 36.1514414910436
x3=5.06267917117524x_{3} = 5.06267917117524
x4=54.9914846404513x_{4} = 54.9914846404513
x5=73.8368718590989x_{5} = 73.8368718590989
x6=42.4303637527494x_{6} = 42.4303637527494
x7=98.9667658004856x_{7} = 98.9667658004856
x8=29.8746827273128x_{8} = 29.8746827273128
x9=23.60213003453x_{9} = 23.60213003453
x10=17.3393298030427x_{10} = 17.3393298030427
x11=48.7105340009097x_{11} = 48.7105340009097
x12=11.1072905657786x_{12} = 11.1072905657786
x13=86.4015846605975x_{13} = 86.4015846605975
x14=80.1191545145906x_{14} = 80.1191545145906
x15=67.5547826177155x_{15} = 67.5547826177155
x16=92.6841296797232x_{16} = 92.6841296797232
Puntos máximos de la función:
x16=64.4138302477556x_{16} = 64.4138302477556
x16=58.1321658716346x_{16} = 58.1321658716346
x16=89.5428444653399x_{16} = 89.5428444653399
x16=83.2603534390183x_{16} = 83.2603534390183
x16=20.4688629160207x_{16} = 20.4688629160207
x16=45.5703287126706x_{16} = 45.5703287126706
x16=8.03139099623953x_{16} = 8.03139099623953
x16=14.2164990994671x_{16} = 14.2164990994671
x16=95.8254375766894x_{16} = 95.8254375766894
x16=33.0127041187522x_{16} = 33.0127041187522
x16=2.49249324020916x_{16} = 2.49249324020916
x16=39.2907044598106x_{16} = 39.2907044598106
x16=26.7376687721596x_{16} = 26.7376687721596
x16=51.8509314096671x_{16} = 51.8509314096671
x16=70.6957994975412x_{16} = 70.6957994975412
x16=76.9779922603215x_{16} = 76.9779922603215
Decrece en los intervalos
[98.9667658004856,∞)\left[98.9667658004856, \infty\right)
Crece en los intervalos
(−∞,5.06267917117524]\left(-\infty, 5.06267917117524\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)}^{3} \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)3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)}^{3} \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)^3*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)3sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{3} \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)3sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3} \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)3sin⁡(x)=−log⁡(−x)3sin⁡(x)\log{\left(x \right)}^{3} \sin{\left(x \right)} = - \log{\left(- x \right)}^{3} \sin{\left(x \right)}
- No
log⁡(x)3sin⁡(x)=log⁡(−x)3sin⁡(x)\log{\left(x \right)}^{3} \sin{\left(x \right)} = \log{\left(- x \right)}^{3} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = log(x)^3*sin(x)
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