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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(x)*log(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=65.9734457253857x_{1} = 65.9734457253857
x2=69.1150383789755x_{2} = 69.1150383789755
x3=−25.1327412287183x_{3} = -25.1327412287183
x4=−21.9911485751286x_{4} = -21.9911485751286
x5=21.9911485751286x_{5} = 21.9911485751286
x6=−69.1150383789755x_{6} = -69.1150383789755
x7=−75.398223686155x_{7} = -75.398223686155
x8=138.230076757951x_{8} = 138.230076757951
x9=−12.5663706143592x_{9} = -12.5663706143592
x10=−65.9734457253857x_{10} = -65.9734457253857
x11=9.42477796076938x_{11} = 9.42477796076938
x12=97.3893722612836x_{12} = 97.3893722612836
x13=53.4070751110265x_{13} = 53.4070751110265
x14=−84.8230016469244x_{14} = -84.8230016469244
x15=−31.4159265358979x_{15} = -31.4159265358979
x16=6.28318530717959x_{16} = 6.28318530717959
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−40.8407044966673x_{18} = -40.8407044966673
x19=−34.5575191894877x_{19} = -34.5575191894877
x20=28.2743338823081x_{20} = 28.2743338823081
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=−59.6902604182061x_{22} = -59.6902604182061
x23=−53.4070751110265x_{23} = -53.4070751110265
x24=84.8230016469244x_{24} = 84.8230016469244
x25=−87.9645943005142x_{25} = -87.9645943005142
x26=−18.8495559215388x_{26} = -18.8495559215388
x27=59.6902604182061x_{27} = 59.6902604182061
x28=−6.28318530717959x_{28} = -6.28318530717959
x29=56.5486677646163x_{29} = 56.5486677646163
x30=25.1327412287183x_{30} = 25.1327412287183
x31=72.2566310325652x_{31} = 72.2566310325652
x32=−15.707963267949x_{32} = -15.707963267949
x33=78.5398163397448x_{33} = 78.5398163397448
x34=37.6991118430775x_{34} = 37.6991118430775
x35=−37.6991118430775x_{35} = -37.6991118430775
x36=−223.053078404875x_{36} = -223.053078404875
x37=−43.9822971502571x_{37} = -43.9822971502571
x38=47.1238898038469x_{38} = 47.1238898038469
x39=−28.2743338823081x_{39} = -28.2743338823081
x40=3.14159265358979x_{40} = 3.14159265358979
x41=−72.2566310325652x_{41} = -72.2566310325652
x42=91.106186954104x_{42} = 91.106186954104
x43=31.4159265358979x_{43} = 31.4159265358979
x44=−91.106186954104x_{44} = -91.106186954104
x45=43.9822971502571x_{45} = 43.9822971502571
x46=−47.1238898038469x_{46} = -47.1238898038469
x47=−50.2654824574367x_{47} = -50.2654824574367
x48=100.530964914873x_{48} = 100.530964914873
x49=−97.3893722612836x_{49} = -97.3893722612836
x50=81.6814089933346x_{50} = 81.6814089933346
x51=50.2654824574367x_{51} = 50.2654824574367
x52=94.2477796076938x_{52} = 94.2477796076938
x53=−100.530964914873x_{53} = -100.530964914873
x54=1x_{54} = 1
x55=12.5663706143592x_{55} = 12.5663706143592
x56=−78.5398163397448x_{56} = -78.5398163397448
x57=18.8495559215388x_{57} = 18.8495559215388
x58=34.5575191894877x_{58} = 34.5575191894877
x59=−62.8318530717959x_{59} = -62.8318530717959
x60=−56.5486677646163x_{60} = -56.5486677646163
x61=15.707963267949x_{61} = 15.707963267949
x62=87.9645943005142x_{62} = 87.9645943005142
x63=−81.6814089933346x_{63} = -81.6814089933346
x64=40.8407044966673x_{64} = 40.8407044966673
x65=−9.42477796076938x_{65} = -9.42477796076938
x66=−113.097335529233x_{66} = -113.097335529233
x67=62.8318530717959x_{67} = 62.8318530717959
x68=75.398223686155x_{68} = 75.398223686155
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(x)*log(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=32.9953908591221x_{1} = 32.9953908591221
x2=7.91497769383021x_{2} = 7.91497769383021
x3=26.7149311915258x_{3} = 26.7149311915258
x4=51.8411644567759x_{4} = 51.8411644567759
x5=86.3963937735675x_{5} = 86.3963937735675
x6=2.12761582523344x_{6} = 2.12761582523344
x7=45.5588408894342x_{7} = 45.5588408894342
x8=54.9824103570705x_{8} = 54.9824103570705
x9=89.5378754494563x_{9} = 89.5378754494563
x10=80.1134602593311x_{10} = 80.1134602593311
x11=17.2990352355066x_{11} = 17.2990352355066
x12=70.6891567862013x_{12} = 70.6891567862013
x13=95.8208633135828x_{13} = 95.8208633135828
x14=48.6999705880551x_{14} = 48.6999705880551
x15=39.2768442680313x_{15} = 39.2768442680313
x16=98.9623678062405x_{16} = 98.9623678062405
x17=64.406377021222x_{17} = 64.406377021222
x18=83.2549216304705x_{18} = 83.2549216304705
x19=23.5753663871051x_{19} = 23.5753663871051
x20=29.8549920106507x_{20} = 29.8549920106507
x21=11.0333063655933x_{21} = 11.0333063655933
x22=67.5477561419489x_{22} = 67.5477561419489
x23=73.8305759400225x_{23} = 73.8305759400225
x24=76.9720111193216x_{24} = 76.9720111193216
x25=14.1637961865355x_{25} = 14.1637961865355
x26=4.84255834039212x_{26} = 4.84255834039212
x27=61.2650231149052x_{27} = 61.2650231149052
x28=58.1236989891669x_{28} = 58.1236989891669
x29=20.4365678012128x_{29} = 20.4365678012128
x30=36.1360296011875x_{30} = 36.1360296011875
x31=42.4177914906586x_{31} = 42.4177914906586
x32=92.6793655993772x_{32} = 92.6793655993772
Signos de extremos en los puntos:
(32.99539085912214, 3.49623653326273)

(7.914977693830208, 2.06490964318559)

(26.7149311915258, 3.28500939657186)

(51.84116445677586, 3.94813739322056)

(86.3963937735675, -4.45893091363236)

(2.127615825233441, 0.640951613895412)

(45.55884088943418, 3.81894162090863)

(54.98241035707053, -4.00697204664365)

(89.5378754494563, 4.49464784936066)

(80.11346025933112, -4.38342611095494)

(17.2990352355066, -2.85006479973796)

(70.6891567862013, 4.2582686940799)

(95.82086331358285, 4.56246850547861)

(48.69997058805509, -3.88562417153593)

(39.27684426803133, 3.67054684507133)

(98.96236780624047, -4.59472854333644)

(64.40637702122196, 4.16518371214019)

(83.25492163047046, 4.42189093263579)

(23.57536638710508, -3.15991774048714)

(29.854992010650733, -3.39618690740209)

(11.03330636559327, -2.39920964673997)

(67.54775614194894, -4.21280883436135)

(73.83057594002254, -4.30175163100997)

(76.9720111193216, 4.3434224340588)

(14.16379618653552, 2.6497493761583)

(4.8425583403921175, -1.56409787578554)

(61.26502311490521, -4.11517672431722)

(58.12369898916687, 4.06253705090375)

(20.43656780121277, 3.01692915004008)

(36.13602960118748, -3.58718368340644)

(42.417791490658566, -3.74749373479586)

(92.67936559937723, -4.52913300203105)


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

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 sin(x)*log(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
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