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Trazado el gráfico de la función/ sin(1/x)/(lnxln^2(lnx))

Gráfico de la función y = sin(1/x)/(lnxln^2(lnx))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 /1\      
              sin|-|      
                 \x/      
f(x) = -------------------
                 2        
       log(x)*log (log(x))
f(x)=sin⁡(1x)log⁡(x)log⁡(log⁡(x))2f{\left(x \right)} = \frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} 
f = sin(1/x)/((log(x)*log(log(x))^2))
Gráfico de la función
10020030040050060070080090010000.00000.0010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=1x_{2} = 1
x3=2.71828182845905x_{3} = 2.71828182845905
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:
sin⁡(1x)log⁡(x)log⁡(log⁡(x))2=0\frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1πx_{1} = \frac{1}{\pi}
Solución numérica
x1=0.318309886183791x_{1} = 0.318309886183791
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 sin(1/x)/((log(x)*log(log(x))^2)).
sin⁡(10)log⁡(0)log⁡(log⁡(0))2\frac{\sin{\left(\frac{1}{0} \right)}}{\log{\left(0 \right)} \log{\left(\log{\left(0 \right)} \right)}^{2}}
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⁡(log⁡(x))2x−2log⁡(log⁡(x))x)sin⁡(1x)log⁡(x)2log⁡(log⁡(x))4−1log⁡(x)log⁡(log⁡(x))2cos⁡(1x)x2=0\frac{\left(- \frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{x} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{x}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)}^{2} \log{\left(\log{\left(x \right)} \right)}^{4}} - \frac{\frac{1}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} \cos{\left(\frac{1}{x} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=123936.081390546x_{1} = 123936.081390546
x2=1.18486335120708x_{2} = 1.18486335120708
x3=150684.75506836x_{3} = 150684.75506836
x4=115153.501298535x_{4} = 115153.501298535
x5=155195.855429689x_{5} = 155195.855429689
x6=164259.479644699x_{6} = 164259.479644699
x7=173375.795091768x_{7} = 173375.795091768
x8=146188.012534746x_{8} = 146188.012534746
x9=187143.038744899x_{9} = 187143.038744899
x10=128353.856807181x_{10} = 128353.856807181
x11=182542.014130585x_{11} = 182542.014130585
x12=141706.069594359x_{12} = 141706.069594359
x13=177952.830257034x_{13} = 177952.830257034
x14=191755.610273719x_{14} = 191755.610273719
x15=159720.895631442x_{15} = 159720.895631442
x16=168811.231757655x_{16} = 168811.231757655
x17=119535.750763085x_{17} = 119535.750763085
x18=137239.393814113x_{18} = 137239.393814113
x19=196379.448105231x_{19} = 196379.448105231
x20=132788.480817713x_{20} = 132788.480817713
Signos de extremos en los puntos:
(123936.08139054646, 1.13511973473116e-7)

(1.1848633512070803, 1.39963198816373)

(150684.75506836042, 9.0611152595887e-8)

(115153.50129853451, 1.23570171838804e-7)

(155195.85542968876, 8.75854918714103e-8)

(164259.4796446994, 8.20478696165653e-8)

(173375.79509176788, 7.71069615054369e-8)

(146188.01253474577, 9.38288221231438e-8)

(187143.03874489915, 7.06262941164073e-8)

(128353.85680718144, 1.09014413525719e-7)

(182542.01413058455, 7.26746883621352e-8)

(141706.06959435856, 9.72564495504268e-8)

(177952.83025703442, 7.48320720862356e-8)

(191755.6102737185, 6.86791587895291e-8)

(159720.8956314423, 8.47358352230513e-8)

(168811.23175765487, 7.95087590585813e-8)

(119535.75076308475, 1.18351105121923e-7)

(137239.39381411337, 1.009142270632e-7)

(196379.44810523078, 6.68262501732177e-8)

(132788.48081771337, 1.04824944677654e-7)


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=1.18486335120708x_{1} = 1.18486335120708
La función no tiene puntos máximos
Decrece en los intervalos
[1.18486335120708,∞)\left[1.18486335120708, \infty\right)
Crece en los intervalos
(−∞,1.18486335120708]\left(-\infty, 1.18486335120708\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
((1+2log⁡(log⁡(x)))(log⁡(log⁡(x))+2)log⁡(x)+log⁡(log⁡(x))+2log⁡(x)+2(log⁡(log⁡(x))+2)log⁡(x)log⁡(log⁡(x))+log⁡(log⁡(x))2+2log⁡(log⁡(x))−2log⁡(log⁡(x))log⁡(x)−2log⁡(x)log⁡(log⁡(x)))sin⁡(1x)log⁡(x)log⁡(log⁡(x))+2(log⁡(log⁡(x))+2)cos⁡(1x)xlog⁡(x)log⁡(log⁡(x))+2cos⁡(1x)−sin⁡(1x)xxx2log⁡(x)log⁡(log⁡(x))2=0\frac{\frac{\left(\frac{\left(1 + \frac{2}{\log{\left(\log{\left(x \right)} \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)} + 2}{\log{\left(x \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2} + 2 \log{\left(\log{\left(x \right)} \right)} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{\log{\left(x \right)}} - \frac{2}{\log{\left(x \right)}}}{\log{\left(\log{\left(x \right)} \right)}}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right) \cos{\left(\frac{1}{x} \right)}}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x}}{x^{2} \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=4734.21640764846x_{1} = 4734.21640764846
x2=8525.74928207858x_{2} = 8525.74928207858
x3=22272.0085483679x_{3} = 22272.0085483679
x4=11982.735077658x_{4} = 11982.735077658
x5=14506.7974696496x_{5} = 14506.7974696496
x6=17066.6666097022x_{6} = 17066.6666097022
x7=17582.3926928376x_{7} = 17582.3926928376
x8=12484.3311405173x_{8} = 12484.3311405173
x9=6130.39430398357x_{9} = 6130.39430398357
x10=18617.2767763802x_{10} = 18617.2767763802
x11=16552.134633935x_{11} = 16552.134633935
x12=16038.83725317x_{12} = 16038.83725317
x13=20177.6717745926x_{13} = 20177.6717745926
x14=13492.4801077979x_{14} = 13492.4801077979
x15=19136.3647917006x_{15} = 19136.3647917006
x16=20699.8313456432x_{16} = 20699.8313456432
x17=12987.6027704594x_{17} = 12987.6027704594
x18=4277.55217499511x_{18} = 4277.55217499511
x19=10488.7888169327x_{19} = 10488.7888169327
x20=19656.5066180118x_{20} = 19656.5066180118
x21=8040.93988735038x_{21} = 8040.93988735038
x22=10984.8783301856x_{22} = 10984.8783301856
x23=9502.77552018325x_{23} = 9502.77552018325
x24=6603.3184748473x_{24} = 6603.3184748473
x25=21222.9578666788x_{25} = 21222.9578666788
x26=23324.6294399768x_{26} = 23324.6294399768
x27=15526.8175173399x_{27} = 15526.8175173399
x28=21747.0252214732x_{28} = 21747.0252214732
x29=13998.8983980117x_{29} = 13998.8983980117
x30=9994.71898554314x_{30} = 9994.71898554314
x31=7079.53315738084x_{31} = 7079.53315738084
x32=7558.80791323059x_{32} = 7558.80791323059
x33=18099.2747432156x_{33} = 18099.2747432156
x34=5661.0244280765x_{34} = 5661.0244280765
x35=22797.884154918x_{35} = 22797.884154918
x36=5195.51341811941x_{36} = 5195.51341811941
x37=11482.8901493896x_{37} = 11482.8901493896
x38=9013.07575527668x_{38} = 9013.07575527668
x39=15016.1212799033x_{39} = 15016.1212799033
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=0x_{1} = 0
x2=1x_{2} = 1
x3=2.71828182845905x_{3} = 2.71828182845905

True

True

- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión
lim⁡x→1−(((1+2log⁡(log⁡(x)))(log⁡(log⁡(x))+2)log⁡(x)+log⁡(log⁡(x))+2log⁡(x)+2(log⁡(log⁡(x))+2)log⁡(x)log⁡(log⁡(x))+log⁡(log⁡(x))2+2log⁡(log⁡(x))−2log⁡(log⁡(x))log⁡(x)−2log⁡(x)log⁡(log⁡(x)))sin⁡(1x)log⁡(x)log⁡(log⁡(x))+2(log⁡(log⁡(x))+2)cos⁡(1x)xlog⁡(x)log⁡(log⁡(x))+2cos⁡(1x)−sin⁡(1x)xxx2log⁡(x)log⁡(log⁡(x))2)=−∞\lim_{x \to 1^-}\left(\frac{\frac{\left(\frac{\left(1 + \frac{2}{\log{\left(\log{\left(x \right)} \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)} + 2}{\log{\left(x \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2} + 2 \log{\left(\log{\left(x \right)} \right)} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{\log{\left(x \right)}} - \frac{2}{\log{\left(x \right)}}}{\log{\left(\log{\left(x \right)} \right)}}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right) \cos{\left(\frac{1}{x} \right)}}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x}}{x^{2} \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = -\infty
lim⁡x→1+(((1+2log⁡(log⁡(x)))(log⁡(log⁡(x))+2)log⁡(x)+log⁡(log⁡(x))+2log⁡(x)+2(log⁡(log⁡(x))+2)log⁡(x)log⁡(log⁡(x))+log⁡(log⁡(x))2+2log⁡(log⁡(x))−2log⁡(log⁡(x))log⁡(x)−2log⁡(x)log⁡(log⁡(x)))sin⁡(1x)log⁡(x)log⁡(log⁡(x))+2(log⁡(log⁡(x))+2)cos⁡(1x)xlog⁡(x)log⁡(log⁡(x))+2cos⁡(1x)−sin⁡(1x)xxx2log⁡(x)log⁡(log⁡(x))2)=∞\lim_{x \to 1^+}\left(\frac{\frac{\left(\frac{\left(1 + \frac{2}{\log{\left(\log{\left(x \right)} \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)} + 2}{\log{\left(x \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2} + 2 \log{\left(\log{\left(x \right)} \right)} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{\log{\left(x \right)}} - \frac{2}{\log{\left(x \right)}}}{\log{\left(\log{\left(x \right)} \right)}}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right) \cos{\left(\frac{1}{x} \right)}}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x}}{x^{2} \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = \infty
- los límites no son iguales, signo
x2=1x_{2} = 1
- es el punto de flexión
lim⁡x→2.71828182845905−(((1+2log⁡(log⁡(x)))(log⁡(log⁡(x))+2)log⁡(x)+log⁡(log⁡(x))+2log⁡(x)+2(log⁡(log⁡(x))+2)log⁡(x)log⁡(log⁡(x))+log⁡(log⁡(x))2+2log⁡(log⁡(x))−2log⁡(log⁡(x))log⁡(x)−2log⁡(x)log⁡(log⁡(x)))sin⁡(1x)log⁡(x)log⁡(log⁡(x))+2(log⁡(log⁡(x))+2)cos⁡(1x)xlog⁡(x)log⁡(log⁡(x))+2cos⁡(1x)−sin⁡(1x)xxx2log⁡(x)log⁡(log⁡(x))2)=∞\lim_{x \to 2.71828182845905^-}\left(\frac{\frac{\left(\frac{\left(1 + \frac{2}{\log{\left(\log{\left(x \right)} \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)} + 2}{\log{\left(x \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2} + 2 \log{\left(\log{\left(x \right)} \right)} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{\log{\left(x \right)}} - \frac{2}{\log{\left(x \right)}}}{\log{\left(\log{\left(x \right)} \right)}}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right) \cos{\left(\frac{1}{x} \right)}}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x}}{x^{2} \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = \infty
lim⁡x→2.71828182845905+(((1+2log⁡(log⁡(x)))(log⁡(log⁡(x))+2)log⁡(x)+log⁡(log⁡(x))+2log⁡(x)+2(log⁡(log⁡(x))+2)log⁡(x)log⁡(log⁡(x))+log⁡(log⁡(x))2+2log⁡(log⁡(x))−2log⁡(log⁡(x))log⁡(x)−2log⁡(x)log⁡(log⁡(x)))sin⁡(1x)log⁡(x)log⁡(log⁡(x))+2(log⁡(log⁡(x))+2)cos⁡(1x)xlog⁡(x)log⁡(log⁡(x))+2cos⁡(1x)−sin⁡(1x)xxx2log⁡(x)log⁡(log⁡(x))2)=∞\lim_{x \to 2.71828182845905^+}\left(\frac{\frac{\left(\frac{\left(1 + \frac{2}{\log{\left(\log{\left(x \right)} \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)} + 2}{\log{\left(x \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right)}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{\log{\left(\log{\left(x \right)} \right)}^{2} + 2 \log{\left(\log{\left(x \right)} \right)} - \frac{2 \log{\left(\log{\left(x \right)} \right)}}{\log{\left(x \right)}} - \frac{2}{\log{\left(x \right)}}}{\log{\left(\log{\left(x \right)} \right)}}\right) \sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \left(\log{\left(\log{\left(x \right)} \right)} + 2\right) \cos{\left(\frac{1}{x} \right)}}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x}}{x^{2} \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = \infty
- los límites son iguales, es decir omitimos el punto correspondiente

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:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=1x_{2} = 1
x3=2.71828182845905x_{3} = 2.71828182845905
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(1x)log⁡(x)log⁡(log⁡(x))2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(1x)log⁡(x)log⁡(log⁡(x))2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(1/x)/((log(x)*log(log(x))^2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1log⁡(x)log⁡(log⁡(x))2sin⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} \sin{\left(\frac{1}{x} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1log⁡(x)log⁡(log⁡(x))2sin⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} \sin{\left(\frac{1}{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:
sin⁡(1x)log⁡(x)log⁡(log⁡(x))2=−sin⁡(1x)log⁡(−x)log⁡(log⁡(−x))2\frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} = - \frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(- x \right)} \log{\left(\log{\left(- x \right)} \right)}^{2}}
- No
sin⁡(1x)log⁡(x)log⁡(log⁡(x))2=sin⁡(1x)log⁡(−x)log⁡(log⁡(−x))2\frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}^{2}} = \frac{\sin{\left(\frac{1}{x} \right)}}{\log{\left(- x \right)} \log{\left(\log{\left(- x \right)} \right)}^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор