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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = log(x)*sin(x)
f(x)=log⁡(x)sin⁡(x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} 
f = log(x)*sin(x)
Gráfico de la función
02468-8-6-4-2-10105-5
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)sin⁡(x)=0\log{\left(x \right)} \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=−40.8407044966673x_{3} = -40.8407044966673
x4=−31.4159265358979x_{4} = -31.4159265358979
x5=53.4070751110265x_{5} = 53.4070751110265
x6=−75.398223686155x_{6} = -75.398223686155
x7=−12.5663706143592x_{7} = -12.5663706143592
x8=−53.4070751110265x_{8} = -53.4070751110265
x9=1x_{9} = 1
x10=−100.530964914873x_{10} = -100.530964914873
x11=−87.9645943005142x_{11} = -87.9645943005142
x12=−9.42477796076938x_{12} = -9.42477796076938
x13=−91.106186954104x_{13} = -91.106186954104
x14=28.2743338823081x_{14} = 28.2743338823081
x15=−34.5575191894877x_{15} = -34.5575191894877
x16=62.8318530717959x_{16} = 62.8318530717959
x17=9.42477796076938x_{17} = 9.42477796076938
x18=50.2654824574367x_{18} = 50.2654824574367
x19=−97.3893722612836x_{19} = -97.3893722612836
x20=−69.1150383789755x_{20} = -69.1150383789755
x21=−6.28318530717959x_{21} = -6.28318530717959
x22=138.230076757951x_{22} = 138.230076757951
x23=−84.8230016469244x_{23} = -84.8230016469244
x24=34.5575191894877x_{24} = 34.5575191894877
x25=−3.14159265358979x_{25} = -3.14159265358979
x26=65.9734457253857x_{26} = 65.9734457253857
x27=−81.6814089933346x_{27} = -81.6814089933346
x28=3.14159265358979x_{28} = 3.14159265358979
x29=−65.9734457253857x_{29} = -65.9734457253857
x30=−18.8495559215388x_{30} = -18.8495559215388
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=−56.5486677646163x_{32} = -56.5486677646163
x33=78.5398163397448x_{33} = 78.5398163397448
x34=59.6902604182061x_{34} = 59.6902604182061
x35=−47.1238898038469x_{35} = -47.1238898038469
x36=100.530964914873x_{36} = 100.530964914873
x37=21.9911485751286x_{37} = 21.9911485751286
x38=91.106186954104x_{38} = 91.106186954104
x39=84.8230016469244x_{39} = 84.8230016469244
x40=−62.8318530717959x_{40} = -62.8318530717959
x41=−28.2743338823081x_{41} = -28.2743338823081
x42=−37.6991118430775x_{42} = -37.6991118430775
x43=−21.9911485751286x_{43} = -21.9911485751286
x44=12.5663706143592x_{44} = 12.5663706143592
x45=−94.2477796076938x_{45} = -94.2477796076938
x46=15.707963267949x_{46} = 15.707963267949
x47=18.8495559215388x_{47} = 18.8495559215388
x48=40.8407044966673x_{48} = 40.8407044966673
x49=−43.9822971502571x_{49} = -43.9822971502571
x50=−15.707963267949x_{50} = -15.707963267949
x51=81.6814089933346x_{51} = 81.6814089933346
x52=−78.5398163397448x_{52} = -78.5398163397448
x53=94.2477796076938x_{53} = 94.2477796076938
x54=87.9645943005142x_{54} = 87.9645943005142
x55=−223.053078404875x_{55} = -223.053078404875
x56=47.1238898038469x_{56} = 47.1238898038469
x57=6.28318530717959x_{57} = 6.28318530717959
x58=97.3893722612836x_{58} = 97.3893722612836
x59=−25.1327412287183x_{59} = -25.1327412287183
x60=37.6991118430775x_{60} = 37.6991118430775
x61=−113.097335529233x_{61} = -113.097335529233
x62=43.9822971502571x_{62} = 43.9822971502571
x63=56.5486677646163x_{63} = 56.5486677646163
x64=25.1327412287183x_{64} = 25.1327412287183
x65=75.398223686155x_{65} = 75.398223686155
x66=72.2566310325652x_{66} = 72.2566310325652
x67=−50.2654824574367x_{67} = -50.2654824574367
x68=−59.6902604182061x_{68} = -59.6902604182061
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)*sin(x).
log⁡(0)sin⁡(0)\log{\left(0 \right)} \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)cos⁡(x)+sin⁡(x)x=0\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.2990352355066x_{1} = 17.2990352355066
x2=64.406377021222x_{2} = 64.406377021222
x3=76.9720111193216x_{3} = 76.9720111193216
x4=92.6793655993772x_{4} = 92.6793655993772
x5=4.84255834039212x_{5} = 4.84255834039212
x6=2.12761582523344x_{6} = 2.12761582523344
x7=20.4365678012128x_{7} = 20.4365678012128
x8=39.2768442680313x_{8} = 39.2768442680313
x9=70.6891567862013x_{9} = 70.6891567862013
x10=45.5588408894342x_{10} = 45.5588408894342
x11=58.1236989891669x_{11} = 58.1236989891669
x12=42.4177914906586x_{12} = 42.4177914906586
x13=36.1360296011875x_{13} = 36.1360296011875
x14=14.1637961865355x_{14} = 14.1637961865355
x15=61.2650231149052x_{15} = 61.2650231149052
x16=67.5477561419489x_{16} = 67.5477561419489
x17=51.8411644567759x_{17} = 51.8411644567759
x18=23.5753663871051x_{18} = 23.5753663871051
x19=83.2549216304705x_{19} = 83.2549216304705
x20=48.6999705880551x_{20} = 48.6999705880551
x21=86.3963937735675x_{21} = 86.3963937735675
x22=32.9953908591221x_{22} = 32.9953908591221
x23=29.8549920106507x_{23} = 29.8549920106507
x24=7.91497769383021x_{24} = 7.91497769383021
x25=89.5378754494563x_{25} = 89.5378754494563
x26=26.7149311915258x_{26} = 26.7149311915258
x27=11.0333063655933x_{27} = 11.0333063655933
x28=98.9623678062405x_{28} = 98.9623678062405
x29=73.8305759400225x_{29} = 73.8305759400225
x30=95.8208633135828x_{30} = 95.8208633135828
x31=80.1134602593311x_{31} = 80.1134602593311
x32=54.9824103570705x_{32} = 54.9824103570705
Signos de extremos en los puntos:
(17.2990352355066, -2.85006479973796)

(64.40637702122196, 4.16518371214019)

(76.9720111193216, 4.3434224340588)

(92.67936559937723, -4.52913300203105)

(4.8425583403921175, -1.56409787578554)

(2.127615825233441, 0.640951613895412)

(20.43656780121277, 3.01692915004008)

(39.27684426803133, 3.67054684507133)

(70.6891567862013, 4.2582686940799)

(45.55884088943418, 3.81894162090863)

(58.12369898916687, 4.06253705090375)

(42.417791490658566, -3.74749373479586)

(36.13602960118748, -3.58718368340644)

(14.16379618653552, 2.6497493761583)

(61.26502311490521, -4.11517672431722)

(67.54775614194894, -4.21280883436135)

(51.84116445677586, 3.94813739322056)

(23.57536638710508, -3.15991774048714)

(83.25492163047046, 4.42189093263579)

(48.69997058805509, -3.88562417153593)

(86.3963937735675, -4.45893091363236)

(32.99539085912214, 3.49623653326273)

(29.854992010650733, -3.39618690740209)

(7.914977693830208, 2.06490964318559)

(89.5378754494563, 4.49464784936066)

(26.7149311915258, 3.28500939657186)

(11.03330636559327, -2.39920964673997)

(98.96236780624047, -4.59472854333644)

(73.83057594002254, -4.30175163100997)

(95.82086331358285, 4.56246850547861)

(80.11346025933112, -4.38342611095494)

(54.98241035707053, -4.00697204664365)


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=17.2990352355066x_{1} = 17.2990352355066
x2=92.6793655993772x_{2} = 92.6793655993772
x3=4.84255834039212x_{3} = 4.84255834039212
x4=42.4177914906586x_{4} = 42.4177914906586
x5=36.1360296011875x_{5} = 36.1360296011875
x6=61.2650231149052x_{6} = 61.2650231149052
x7=67.5477561419489x_{7} = 67.5477561419489
x8=23.5753663871051x_{8} = 23.5753663871051
x9=48.6999705880551x_{9} = 48.6999705880551
x10=86.3963937735675x_{10} = 86.3963937735675
x11=29.8549920106507x_{11} = 29.8549920106507
x12=11.0333063655933x_{12} = 11.0333063655933
x13=98.9623678062405x_{13} = 98.9623678062405
x14=73.8305759400225x_{14} = 73.8305759400225
x15=80.1134602593311x_{15} = 80.1134602593311
x16=54.9824103570705x_{16} = 54.9824103570705
Puntos máximos de la función:
x16=64.406377021222x_{16} = 64.406377021222
x16=76.9720111193216x_{16} = 76.9720111193216
x16=2.12761582523344x_{16} = 2.12761582523344
x16=20.4365678012128x_{16} = 20.4365678012128
x16=39.2768442680313x_{16} = 39.2768442680313
x16=70.6891567862013x_{16} = 70.6891567862013
x16=45.5588408894342x_{16} = 45.5588408894342
x16=58.1236989891669x_{16} = 58.1236989891669
x16=14.1637961865355x_{16} = 14.1637961865355
x16=51.8411644567759x_{16} = 51.8411644567759
x16=83.2549216304705x_{16} = 83.2549216304705
x16=32.9953908591221x_{16} = 32.9953908591221
x16=7.91497769383021x_{16} = 7.91497769383021
x16=89.5378754494563x_{16} = 89.5378754494563
x16=26.7149311915258x_{16} = 26.7149311915258
x16=95.8208633135828x_{16} = 95.8208633135828
Decrece en los intervalos
[98.9623678062405,∞)\left[98.9623678062405, \infty\right)
Crece en los intervalos
(−∞,4.84255834039212]\left(-\infty, 4.84255834039212\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
−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2=0- \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=56.5574301916107x_{1} = 56.5574301916107
x2=65.9806806486246x_{2} = 65.9806806486246
x3=12.6285861720285x_{3} = 12.6285861720285
x4=75.4043590276847x_{4} = 75.4043590276847
x5=28.2954682170335x_{5} = 28.2954682170335
x6=69.1218687386001x_{6} = 69.1218687386001
x7=22.0204948431363x_{7} = 22.0204948431363
x8=31.4343721697806x_{8} = 31.4343721697806
x9=87.9696723207031x_{9} = 87.9696723207031
x10=97.3938570020224x_{10} = 97.3938570020224
x11=84.8283108211935x_{11} = 84.8283108211935
x12=6.4461035560751x_{12} = 6.4461035560751
x13=59.6984521889897x_{13} = 59.6984521889897
x14=3.53961476088587x_{14} = 3.53961476088587
x15=47.1349005959502x_{15} = 47.1349005959502
x16=78.5456512461642x_{16} = 78.5456512461642
x17=94.2524472357136x_{17} = 94.2524472357136
x18=37.7137169986599x_{18} = 37.7137169986599
x19=18.8855464491534x_{19} = 18.8855464491534
x20=15.7539096110127x_{20} = 15.7539096110127
x21=9.51732588699837x_{21} = 9.51732588699837
x22=53.4164858945863x_{22} = 53.4164858945863
x23=72.2630966850528x_{23} = 72.2630966850528
x24=81.686969567961x_{24} = 81.686969567961
x25=34.5738406188022x_{25} = 34.5738406188022
x26=50.2756356438169x_{26} = 50.2756356438169
x27=43.9943085957168x_{27} = 43.9943085957168
x28=100.535279615268x_{28} = 100.535279615268
x29=91.1110517789567x_{29} = 91.1110517789567
x30=25.1573740446396x_{30} = 25.1573740446396
x31=40.8538969938589x_{31} = 40.8538969938589
x32=62.8395390693532x_{32} = 62.8395390693532

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
[97.3938570020224,∞)\left[97.3938570020224, \infty\right)
Convexa en los intervalos
(−∞,3.53961476088587]\left(-\infty, 3.53961476088587\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \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⁡(x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \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)*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \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)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \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)sin⁡(x)=−log⁡(−x)sin⁡(x)\log{\left(x \right)} \sin{\left(x \right)} = - \log{\left(- x \right)} \sin{\left(x \right)}
- No
log⁡(x)sin⁡(x)=log⁡(−x)sin⁡(x)\log{\left(x \right)} \sin{\left(x \right)} = \log{\left(- x \right)} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = log(x)*sin(x)
    © Sr Examen – Калькулятор