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  • Gráfico de la función y =:
  • 6*x-x^3 6*x-x^3
  • -(4-x^2)^(1/2) -(4-x^2)^(1/2)
  • 3-(x+2)/(x^2+2*x) 3-(x+2)/(x^2+2*x)
  • 4/(3+2x-x^2) 4/(3+2x-x^2)

  • Expresiones idénticas

  • x^ dos *sinx/lnx
  • x al cuadrado multiplicar por seno de x dividir por lnx
  • x en el grado dos multiplicar por seno de x dividir por lnx
  • x2*sinx/lnx
  • x²*sinx/lnx
  • x en el grado 2*sinx/lnx
  • x^2sinx/lnx
  • x2sinx/lnx
  • x^2*sinx dividir por lnx
Trazado el gráfico de la función/ x^2*sinx/lnx

Gráfico de la función y = x^2*sinx/lnx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        2       
       x *sin(x)
f(x) = ---------
         log(x)
f(x)=x2sin⁡(x)log⁡(x)f{\left(x \right)} = \frac{x^{2} \sin{\left(x \right)}}{\log{\left(x \right)}} 
f = (x^2*sin(x))/log(x)
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=1x_{1} = 1
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:
x2sin⁡(x)log⁡(x)=0\frac{x^{2} \sin{\left(x \right)}}{\log{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=12.5663706143592x_{1} = 12.5663706143592
x2=−9.42477796076938x_{2} = -9.42477796076938
x3=78.5398163397448x_{3} = 78.5398163397448
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−84.8230016469244x_{5} = -84.8230016469244
x6=−122.522113490002x_{6} = -122.522113490002
x7=−47.1238898038469x_{7} = -47.1238898038469
x8=50.2654824574367x_{8} = 50.2654824574367
x9=81.6814089933346x_{9} = 81.6814089933346
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=−50.2654824574367x_{11} = -50.2654824574367
x12=3.14159265358979x_{12} = 3.14159265358979
x13=−91.106186954104x_{13} = -91.106186954104
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=18.8495559215388x_{15} = 18.8495559215388
x16=−28.2743338823081x_{16} = -28.2743338823081
x17=56.5486677646163x_{17} = 56.5486677646163
x18=−62.8318530717959x_{18} = -62.8318530717959
x19=75.398223686155x_{19} = 75.398223686155
x20=15.707963267949x_{20} = 15.707963267949
x21=21.9911485751286x_{21} = 21.9911485751286
x22=6.28318530717959x_{22} = 6.28318530717959
x23=69.1150383789755x_{23} = 69.1150383789755
x24=62.8318530717959x_{24} = 62.8318530717959
x25=−75.398223686155x_{25} = -75.398223686155
x26=9.42477796076938x_{26} = 9.42477796076938
x27=−40.8407044966673x_{27} = -40.8407044966673
x28=59.6902604182061x_{28} = 59.6902604182061
x29=−65.9734457253857x_{29} = -65.9734457253857
x30=−31.4159265358979x_{30} = -31.4159265358979
x31=28.2743338823081x_{31} = 28.2743338823081
x32=31.4159265358979x_{32} = 31.4159265358979
x33=94.2477796076938x_{33} = 94.2477796076938
x34=−72.2566310325652x_{34} = -72.2566310325652
x35=−43.9822971502571x_{35} = -43.9822971502571
x36=53.4070751110265x_{36} = 53.4070751110265
x37=37.6991118430775x_{37} = 37.6991118430775
x38=−87.9645943005142x_{38} = -87.9645943005142
x39=−34.5575191894877x_{39} = -34.5575191894877
x40=−69.1150383789755x_{40} = -69.1150383789755
x41=−56.5486677646163x_{41} = -56.5486677646163
x42=−81.6814089933346x_{42} = -81.6814089933346
x43=−78.5398163397448x_{43} = -78.5398163397448
x44=−53.4070751110265x_{44} = -53.4070751110265
x45=47.1238898038469x_{45} = 47.1238898038469
x46=91.106186954104x_{46} = 91.106186954104
x47=−12.5663706143592x_{47} = -12.5663706143592
x48=43.9822971502571x_{48} = 43.9822971502571
x49=−37.6991118430775x_{49} = -37.6991118430775
x50=97.3893722612836x_{50} = 97.3893722612836
x51=−100.530964914873x_{51} = -100.530964914873
x52=0x_{52} = 0
x53=−6.28318530717959x_{53} = -6.28318530717959
x54=100.530964914873x_{54} = 100.530964914873
x55=34.5575191894877x_{55} = 34.5575191894877
x56=65.9734457253857x_{56} = 65.9734457253857
x57=−25.1327412287183x_{57} = -25.1327412287183
x58=−18.8495559215388x_{58} = -18.8495559215388
x59=−15.707963267949x_{59} = -15.707963267949
x60=40.8407044966673x_{60} = 40.8407044966673
x61=25.1327412287183x_{61} = 25.1327412287183
x62=−97.3893722612836x_{62} = -97.3893722612836
x63=84.8230016469244x_{63} = 84.8230016469244
x64=72.2566310325652x_{64} = 72.2566310325652
x65=−94.2477796076938x_{65} = -94.2477796076938
x66=87.9645943005142x_{66} = 87.9645943005142
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 (x^2*sin(x))/log(x).
02sin⁡(0)log⁡(0)\frac{0^{2} \sin{\left(0 \right)}}{\log{\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
−xsin⁡(x)log⁡(x)2+x2cos⁡(x)+2xsin⁡(x)log⁡(x)=0- \frac{x \sin{\left(x \right)}}{\log{\left(x \right)}^{2}} + \frac{x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}}{\log{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=55.0096818013075x_{1} = 55.0096818013075
x2=8.0408451025572x_{2} = 8.0408451025572
x3=11.1369534645168x_{3} = 11.1369534645168
x4=83.2735037685111x_{4} = 83.2735037685111
x5=80.1327208973489x_{5} = 80.1327208973489
x6=89.5552363688572x_{6} = 89.5552363688572
x7=17.3734327909215x_{7} = 17.3734327909215
x8=33.0385580651894x_{8} = 33.0385580651894
x9=14.2506102502786x_{9} = 14.2506102502786
x10=70.7107929100546x_{10} = 70.7107929100546
x11=92.6961749920025x_{11} = 92.6961749920025
x12=45.591200805792x_{12} = 45.591200805792
x13=73.8513567169975x_{13} = 73.8513567169975
x14=42.4523055827707x_{14} = 42.4523055827707
x15=86.4143442421817x_{15} = 86.4143442421817
x16=26.7668070704804x_{16} = 26.7668070704804
x17=0x_{17} = 0
x18=67.5703222786305x_{18} = 67.5703222786305
x19=51.8699417249451x_{19} = 51.8699417249451
x20=98.9781743228469x_{20} = 98.9781743228469
x21=20.5015780565376x_{21} = 20.5015780565376
x22=61.2897163928766x_{22} = 61.2897163928766
x23=4.98209830649058x_{23} = 4.98209830649058
x24=23.6330727100421x_{24} = 23.6330727100421
x25=64.4299581202857x_{25} = 64.4299581202857
x26=95.8371556201252x_{26} = 95.8371556201252
x27=48.7304326581782x_{27} = 48.7304326581782
x28=36.1758617604665x_{28} = 36.1758617604665
x29=76.9920025298171x_{29} = 76.9920025298171
x30=39.3138246956213x_{30} = 39.3138246956213
x31=29.9021113575113x_{31} = 29.9021113575113
x32=58.1496164151911x_{32} = 58.1496164151911
Signos de extremos en los puntos:
(55.009681801307465, -754.716715969124)

(8.040845102557205, 30.4766683938832)

(11.136953464516813, -50.9462771643042)

(83.27350376851109, 1567.77458241356)

(80.13272089734888, -1464.44938886271)

(89.55523636885715, 1783.94185367132)

(17.37343279092147, -105.250641258395)

(33.03855806518939, 311.658493320776)

(14.250610250278603, 75.9464608595787)

(70.71079291005462, 1173.73341477123)

(92.69617499200248, -1896.74936078458)

(45.591200805792035, 543.770598259678)

(73.85135671699746, -1267.40876716107)

(42.45230558277072, -480.393533673302)

(86.41434424218173, -1674.27796616356)

(26.766807070480397, 217.521467901829)

(0, 0)

(67.57032227863053, -1083.31660170947)

(51.86994172494514, 680.968344185969)

(98.97817432284694, -2131.73133617601)

(20.501578056537618, 138.695138529106)

(61.28971639287658, -912.351955850313)

(4.982098306490585, -14.8980018502629)

(23.633072710042107, -176.153036626547)

(64.42995812028568, 996.181314907557)

(95.83715562012517, 2012.68434240149)

(48.73043265817815, -610.641423341998)

(36.175861760466546, -364.289624183622)

(76.99200252981709, 1364.32094558695)

(39.31382469562128, 420.551421998622)

(29.902111357511284, -262.714455916325)

(58.149616415191076, 831.854505112042)


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=55.0096818013075x_{1} = 55.0096818013075
x2=11.1369534645168x_{2} = 11.1369534645168
x3=80.1327208973489x_{3} = 80.1327208973489
x4=17.3734327909215x_{4} = 17.3734327909215
x5=92.6961749920025x_{5} = 92.6961749920025
x6=73.8513567169975x_{6} = 73.8513567169975
x7=42.4523055827707x_{7} = 42.4523055827707
x8=86.4143442421817x_{8} = 86.4143442421817
x9=67.5703222786305x_{9} = 67.5703222786305
x10=98.9781743228469x_{10} = 98.9781743228469
x11=61.2897163928766x_{11} = 61.2897163928766
x12=4.98209830649058x_{12} = 4.98209830649058
x13=23.6330727100421x_{13} = 23.6330727100421
x14=48.7304326581782x_{14} = 48.7304326581782
x15=36.1758617604665x_{15} = 36.1758617604665
x16=29.9021113575113x_{16} = 29.9021113575113
Puntos máximos de la función:
x16=8.0408451025572x_{16} = 8.0408451025572
x16=83.2735037685111x_{16} = 83.2735037685111
x16=89.5552363688572x_{16} = 89.5552363688572
x16=33.0385580651894x_{16} = 33.0385580651894
x16=14.2506102502786x_{16} = 14.2506102502786
x16=70.7107929100546x_{16} = 70.7107929100546
x16=45.591200805792x_{16} = 45.591200805792
x16=26.7668070704804x_{16} = 26.7668070704804
x16=51.8699417249451x_{16} = 51.8699417249451
x16=20.5015780565376x_{16} = 20.5015780565376
x16=64.4299581202857x_{16} = 64.4299581202857
x16=95.8371556201252x_{16} = 95.8371556201252
x16=76.9920025298171x_{16} = 76.9920025298171
x16=39.3138246956213x_{16} = 39.3138246956213
x16=58.1496164151911x_{16} = 58.1496164151911
Decrece en los intervalos
[98.9781743228469,∞)\left[98.9781743228469, \infty\right)
Crece en los intervalos
(−∞,4.98209830649058]\left(-\infty, 4.98209830649058\right]
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
−x2sin⁡(x)+4xcos⁡(x)+(1+2log⁡(x))sin⁡(x)log⁡(x)−2(xcos⁡(x)+2sin⁡(x))log⁡(x)+2sin⁡(x)log⁡(x)=0\frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{\log{\left(x \right)}} + 2 \sin{\left(x \right)}}{\log{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9.73825804999703x_{1} = 9.73825804999703
x2=37.7902199005762x_{2} = 37.7902199005762
x3=94.2855263214674x_{3} = 94.2855263214674
x4=1.7369604085904x_{4} = 1.7369604085904
x5=81.7247781203759x_{5} = 81.7247781203759
x6=62.8877406029215x_{6} = 62.8877406029215
x7=78.5848649651498x_{7} = 78.5848649651498
x8=91.145197141779x_{8} = 91.145197141779
x9=66.0267645568129x_{9} = 66.0267645568129
x10=34.6564454970524x_{10} = 34.6564454970524
x11=22.1418726336934x_{11} = 22.1418726336934
x12=88.0049560896319x_{12} = 88.0049560896319
x13=59.7489785224634x_{13} = 59.7489785224634
x14=84.8648127427499x_{14} = 84.8648127427499
x15=47.197557221622x_{15} = 47.197557221622
x16=12.8138952257971x_{16} = 12.8138952257971
x17=31.5241455319092x_{17} = 31.5241455319092
x18=50.3347372722883x_{18} = 50.3347372722883
x19=0x_{19} = 0
x20=44.0609816570793x_{20} = 44.0609816570793
x21=15.9120042788467x_{21} = 15.9120042788467
x22=56.6105205920834x_{22} = 56.6105205920834
x23=6.70565140095935x_{23} = 6.70565140095935
x24=72.3054648371234x_{24} = 72.3054648371234
x25=3.75655871416371x_{25} = 3.75655871416371
x26=19.0229634824438x_{26} = 19.0229634824438
x27=69.166015596746x_{27} = 69.166015596746
x28=53.4724187059505x_{28} = 53.4724187059505
x29=97.4259352641823x_{29} = 97.4259352641823
x30=40.9251439862157x_{30} = 40.9251439862157
x31=28.3937790363139x_{31} = 28.3937790363139
x32=75.4450880938076x_{32} = 75.4450880938076
x33=25.2660160084775x_{33} = 25.2660160084775
x34=100.566416632366x_{34} = 100.566416632366
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=1x_{1} = 1

lim⁡x→1−(−x2sin⁡(x)+4xcos⁡(x)+(1+2log⁡(x))sin⁡(x)log⁡(x)−2(xcos⁡(x)+2sin⁡(x))log⁡(x)+2sin⁡(x)log⁡(x))=−∞\lim_{x \to 1^-}\left(\frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{\log{\left(x \right)}} + 2 \sin{\left(x \right)}}{\log{\left(x \right)}}\right) = -\infty
lim⁡x→1+(−x2sin⁡(x)+4xcos⁡(x)+(1+2log⁡(x))sin⁡(x)log⁡(x)−2(xcos⁡(x)+2sin⁡(x))log⁡(x)+2sin⁡(x)log⁡(x))=∞\lim_{x \to 1^+}\left(\frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{\log{\left(x \right)}} + 2 \sin{\left(x \right)}}{\log{\left(x \right)}}\right) = \infty
- los límites no son iguales, signo
x1=1x_{1} = 1
- 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
[97.4259352641823,∞)\left[97.4259352641823, \infty\right)
Convexa en los intervalos
(−∞,3.75655871416371]\left(-\infty, 3.75655871416371\right]
Asíntotas verticales
Hay:
x1=1x_{1} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x2sin⁡(x)log⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{x^{2} \sin{\left(x \right)}}{\log{\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→∞(x2sin⁡(x)log⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{x^{2} \sin{\left(x \right)}}{\log{\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 (x^2*sin(x))/log(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xsin⁡(x)log⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)}}{\log{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞(xsin⁡(x)log⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)}}{\log{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
x2sin⁡(x)log⁡(x)=−x2sin⁡(x)log⁡(−x)\frac{x^{2} \sin{\left(x \right)}}{\log{\left(x \right)}} = - \frac{x^{2} \sin{\left(x \right)}}{\log{\left(- x \right)}}
- No
x2sin⁡(x)log⁡(x)=x2sin⁡(x)log⁡(−x)\frac{x^{2} \sin{\left(x \right)}}{\log{\left(x \right)}} = \frac{x^{2} \sin{\left(x \right)}}{\log{\left(- x \right)}}
- No
es decir, función
no es
par ni impar
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