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Trazado el gráfico de la función/ ln((x^2)-4)/logx(x^2-9)(x-5)sinx

Gráfico de la función y = ln((x^2)-4)/logx(x^2-9)(x-5)sinx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2    \                        
       log\x  - 4/ / 2    \               
f(x) = -----------*\x  - 9/*(x - 5)*sin(x)
          log(x)
f(x)=log⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x)f{\left(x \right)} = \frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \sin{\left(x \right)} 
f = (((log(x^2 - 4)/log(x))*(x^2 - 9))*(x - 5))*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-10001000
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:
log⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x)=0\frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−3x_{1} = -3
x2=0x_{2} = 0
x3=3x_{3} = 3
x4=5x_{4} = 5
x5=−5x_{5} = - \sqrt{5}
x6=5x_{6} = \sqrt{5}
x7=πx_{7} = \pi
Solución numérica
x1=−34.5575191894877x_{1} = -34.5575191894877
x2=62.8318530717959x_{2} = 62.8318530717959
x3=3x_{3} = 3
x4=91.106186954104x_{4} = 91.106186954104
x5=−3.14159265358979x_{5} = -3.14159265358979
x6=72.2566310325652x_{6} = 72.2566310325652
x7=40.8407044966673x_{7} = 40.8407044966673
x8=−69.1150383789755x_{8} = -69.1150383789755
x9=34.5575191894877x_{9} = 34.5575191894877
x10=−12.5663706143592x_{10} = -12.5663706143592
x11=−62.8318530717959x_{11} = -62.8318530717959
x12=50.2654824574367x_{12} = 50.2654824574367
x13=21.9911485751286x_{13} = 21.9911485751286
x14=−37.6991118430775x_{14} = -37.6991118430775
x15=−47.1238898038469x_{15} = -47.1238898038469
x16=−75.398223686155x_{16} = -75.398223686155
x17=59.6902604182061x_{17} = 59.6902604182061
x18=84.8230016469244x_{18} = 84.8230016469244
x19=−128.805298797182x_{19} = -128.805298797182
x20=−84.8230016469244x_{20} = -84.8230016469244
x21=31.4159265358979x_{21} = 31.4159265358979
x22=9.42477796076938x_{22} = 9.42477796076938
x23=−94.2477796076938x_{23} = -94.2477796076938
x24=−43.9822971502571x_{24} = -43.9822971502571
x25=97.3893722612836x_{25} = 97.3893722612836
x26=−91.106186954104x_{26} = -91.106186954104
x27=−6.28318530717959x_{27} = -6.28318530717959
x28=15.707963267949x_{28} = 15.707963267949
x29=28.2743338823081x_{29} = 28.2743338823081
x30=94.2477796076938x_{30} = 94.2477796076938
x31=−87.9645943005142x_{31} = -87.9645943005142
x32=69.1150383789755x_{32} = 69.1150383789755
x33=−65.9734457253857x_{33} = -65.9734457253857
x34=37.6991118430775x_{34} = 37.6991118430775
x35=−21.9911485751286x_{35} = -21.9911485751286
x36=6.28318530717959x_{36} = 6.28318530717959
x37=−59.6902604182061x_{37} = -59.6902604182061
x38=−40.8407044966673x_{38} = -40.8407044966673
x39=131.946891450771x_{39} = 131.946891450771
x40=−97.3893722612836x_{40} = -97.3893722612836
x41=0x_{41} = 0
x42=−31.4159265358979x_{42} = -31.4159265358979
x43=−53.4070751110265x_{43} = -53.4070751110265
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=53.4070751110265x_{45} = 53.4070751110265
x46=−81.6814089933346x_{46} = -81.6814089933346
x47=−25.1327412287183x_{47} = -25.1327412287183
x48=−100.530964914873x_{48} = -100.530964914873
x49=81.6814089933346x_{49} = 81.6814089933346
x50=−72.2566310325652x_{50} = -72.2566310325652
x51=−9.42477796076938x_{51} = -9.42477796076938
x52=100.530964914873x_{52} = 100.530964914873
x53=47.1238898038469x_{53} = 47.1238898038469
x54=25.1327412287183x_{54} = 25.1327412287183
x55=−50.2654824574367x_{55} = -50.2654824574367
x56=78.5398163397448x_{56} = 78.5398163397448
x57=−15.707963267949x_{57} = -15.707963267949
x58=12.5663706143592x_{58} = 12.5663706143592
x59=87.9645943005142x_{59} = 87.9645943005142
x60=18.8495559215388x_{60} = 18.8495559215388
x61=−18.8495559215388x_{61} = -18.8495559215388
x62=43.9822971502571x_{62} = 43.9822971502571
x63=56.5486677646163x_{63} = 56.5486677646163
x64=5x_{64} = 5
x65=65.9734457253857x_{65} = 65.9734457253857
x66=−78.5398163397448x_{66} = -78.5398163397448
x67=75.398223686155x_{67} = 75.398223686155
x68=−56.5486677646163x_{68} = -56.5486677646163
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 - 4)/log(x))*(x^2 - 9))*(x - 5))*sin(x).
(−5)log⁡(−4+02)log⁡(0)(−9+02)sin⁡(0)\left(-5\right) \frac{\log{\left(-4 + 0^{2} \right)}}{\log{\left(0 \right)}} \left(-9 + 0^{2}\right) \sin{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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→−∞(log⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \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⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \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 - 4)/log(x))*(x^2 - 9))*(x - 5))*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x−5)(x2−9)log⁡(x2−4)sin⁡(x)xlog⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\frac{\left(x - 5\right) \left(x^{2} - 9\right) \log{\left(x^{2} - 4 \right)} \sin{\left(x \right)}}{x \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→∞((x−5)(x2−9)log⁡(x2−4)sin⁡(x)xlog⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(x - 5\right) \left(x^{2} - 9\right) \log{\left(x^{2} - 4 \right)} \sin{\left(x \right)}}{x \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:
log⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x)=−(−x−5)(x2−9)log⁡(x2−4)sin⁡(x)log⁡(−x)\frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \sin{\left(x \right)} = - \frac{\left(- x - 5\right) \left(x^{2} - 9\right) \log{\left(x^{2} - 4 \right)} \sin{\left(x \right)}}{\log{\left(- x \right)}}
- No
log⁡(x2−4)log⁡(x)(x2−9)(x−5)sin⁡(x)=(−x−5)(x2−9)log⁡(x2−4)sin⁡(x)log⁡(−x)\frac{\log{\left(x^{2} - 4 \right)}}{\log{\left(x \right)}} \left(x^{2} - 9\right) \left(x - 5\right) \sin{\left(x \right)} = \frac{\left(- x - 5\right) \left(x^{2} - 9\right) \log{\left(x^{2} - 4 \right)} \sin{\left(x \right)}}{\log{\left(- x \right)}}
- No
es decir, función
no es
par ni impar
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