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Trazado el gráfico de la función/ y=sinx*(logx+1)

Gráfico de la función y = y=sinx*(logx+1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=πx_{1} = \pi
x2=e−1x_{2} = e^{-1}
Solución numérica
x1=0.367879441171442x_{1} = 0.367879441171442
x2=65.9734457253857x_{2} = 65.9734457253857
x3=69.1150383789755x_{3} = 69.1150383789755
x4=−100.530964914873x_{4} = -100.530964914873
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=21.9911485751286x_{6} = 21.9911485751286
x7=−75.398223686155x_{7} = -75.398223686155
x8=−62.8318530717959x_{8} = -62.8318530717959
x9=−12.5663706143592x_{9} = -12.5663706143592
x10=−40.8407044966673x_{10} = -40.8407044966673
x11=9.42477796076938x_{11} = 9.42477796076938
x12=−6.28318530717959x_{12} = -6.28318530717959
x13=97.3893722612836x_{13} = 97.3893722612836
x14=53.4070751110265x_{14} = 53.4070751110265
x15=−50.2654824574367x_{15} = -50.2654824574367
x16=6.28318530717959x_{16} = 6.28318530717959
x17=−25.1327412287183x_{17} = -25.1327412287183
x18=−28.2743338823081x_{18} = -28.2743338823081
x19=−34.5575191894877x_{19} = -34.5575191894877
x20=28.2743338823081x_{20} = 28.2743338823081
x21=−53.4070751110265x_{21} = -53.4070751110265
x22=84.8230016469244x_{22} = 84.8230016469244
x23=−47.1238898038469x_{23} = -47.1238898038469
x24=−84.8230016469244x_{24} = -84.8230016469244
x25=59.6902604182061x_{25} = 59.6902604182061
x26=−97.3893722612836x_{26} = -97.3893722612836
x27=56.5486677646163x_{27} = 56.5486677646163
x28=25.1327412287183x_{28} = 25.1327412287183
x29=72.2566310325652x_{29} = 72.2566310325652
x30=−87.9645943005142x_{30} = -87.9645943005142
x31=−213.628300444106x_{31} = -213.628300444106
x32=78.5398163397448x_{32} = 78.5398163397448
x33=37.6991118430775x_{33} = 37.6991118430775
x34=285.884931476671x_{34} = 285.884931476671
x35=47.1238898038469x_{35} = 47.1238898038469
x36=3.14159265358979x_{36} = 3.14159265358979
x37=−81.6814089933346x_{37} = -81.6814089933346
x38=−91.106186954104x_{38} = -91.106186954104
x39=−15.707963267949x_{39} = -15.707963267949
x40=−94.2477796076938x_{40} = -94.2477796076938
x41=−3.14159265358979x_{41} = -3.14159265358979
x42=−37.6991118430775x_{42} = -37.6991118430775
x43=91.106186954104x_{43} = 91.106186954104
x44=31.4159265358979x_{44} = 31.4159265358979
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=43.9822971502571x_{46} = 43.9822971502571
x47=−18.8495559215388x_{47} = -18.8495559215388
x48=100.530964914873x_{48} = 100.530964914873
x49=−59.6902604182061x_{49} = -59.6902604182061
x50=−69.1150383789755x_{50} = -69.1150383789755
x51=81.6814089933346x_{51} = 81.6814089933346
x52=−113.097335529233x_{52} = -113.097335529233
x53=−78.5398163397448x_{53} = -78.5398163397448
x54=50.2654824574367x_{54} = 50.2654824574367
x55=94.2477796076938x_{55} = 94.2477796076938
x56=−72.2566310325652x_{56} = -72.2566310325652
x57=12.5663706143592x_{57} = 12.5663706143592
x58=−9.42477796076938x_{58} = -9.42477796076938
x59=18.8495559215388x_{59} = 18.8495559215388
x60=34.5575191894877x_{60} = 34.5575191894877
x61=15.707963267949x_{61} = 15.707963267949
x62=87.9645943005142x_{62} = 87.9645943005142
x63=−31.4159265358979x_{63} = -31.4159265358979
x64=40.8407044966673x_{64} = 40.8407044966673
x65=−56.5486677646163x_{65} = -56.5486677646163
x66=−43.9822971502571x_{66} = -43.9822971502571
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) + 1).
(log⁡(0)+1)sin⁡(0)\left(\log{\left(0 \right)} + 1\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)+1)cos⁡(x)+sin⁡(x)x=0\left(\log{\left(x \right)} + 1\right) \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=95.8204521084873x_{1} = 95.8204521084873
x2=64.4056553878471x_{2} = 64.4056553878471
x3=58.1228625440946x_{3} = 58.1228625440946
x4=51.8401771906044x_{4} = 51.8401771906044
x5=89.5374232382022x_{5} = 89.5374232382022
x6=17.2937764044618x_{6} = 17.2937764044618
x7=54.9815039255663x_{7} = 54.9815039255663
x8=4.79347157705818x_{8} = 4.79347157705818
x9=61.2642477618988x_{9} = 61.2642477618988
x10=83.2544206601072x_{10} = 83.2544206601072
x11=67.5470820571842x_{11} = 67.5470820571842
x12=32.9934636297173x_{12} = 32.9934636297173
x13=36.1343483683071x_{13} = 36.1343483683071
x14=98.9619747288706x_{14} = 98.9619747288706
x15=29.8527496343583x_{15} = 29.8527496343583
x16=11.0222526014154x_{16} = 11.0222526014154
x17=70.6885250388719x_{17} = 70.6885250388719
x18=7.89526518224072x_{18} = 7.89526518224072
x19=80.1129313258524x_{19} = 80.1129313258524
x20=39.2753595077596x_{20} = 39.2753595077596
x21=45.5576483965663x_{21} = 45.5576483965663
x22=14.1565167061041x_{22} = 14.1565167061041
x23=92.6789347457531x_{23} = 92.6789347457531
x24=1.88479134910863x_{24} = 1.88479134910863
x25=23.5721422140939x_{25} = 23.5721422140939
x26=20.432534864003x_{26} = 20.432534864003
x27=48.698889092841x_{27} = 48.698889092841
x28=86.3959182759432x_{28} = 86.3959182759432
x29=73.8299820941703x_{29} = 73.8299820941703
x30=26.7122735990589x_{30} = 26.7122735990589
x31=42.4164666736714x_{31} = 42.4164666736714
x32=76.9714513712894x_{32} = 76.9714513712894
Signos de extremos en los puntos:
(95.82045210848732, 5.56246635971863)

(64.40565538784706, 5.16517810954741)

(58.1228625440946, 5.06252985486046)

(51.84017719060436, 4.94812787006332)

(89.53742323820215, 5.49464532402323)

(17.29377640446183, -3.84991256413583)

(54.98150392556629, -5.00696380286788)

(4.793471577058176, -2.55882046224792)

(61.26424776189878, -5.11517039594561)

(83.25442066010716, 5.42188792386154)

(67.5470820571842, -5.21280384434829)

(32.99346362971729, 4.49620731888758)

(36.13434836830713, -4.58716041450293)

(98.96197472887062, -5.59472655728815)

(29.852749634358254, -4.39614933692888)

(11.022252601415369, -3.39870634958038)

(70.68852503887192, 5.25826422533101)

(7.895265182240717, 3.06365064257492)

(80.11293132585237, -5.3834228096576)

(39.27535950775963, 4.67052793960621)

(45.55764839656634, 4.81892853142903)

(14.156516706104114, 3.64949174675823)

(92.67893474575312, -5.52913067752638)

(1.8847913491086343, 1.55393532444818)

(23.572142214093933, -4.1598493094349)

(20.43253486400302, 4.01683037707484)

(48.698889092840965, -4.8856130663637)

(86.39591827594319, -5.45892816169326)

(73.82998209417029, -5.30174760911776)

(26.712273599058875, 4.28495962918833)

(42.416466673671366, -4.7474781155696)

(76.97145137128938, 5.34341879783482)


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=17.2937764044618x_{1} = 17.2937764044618
x2=54.9815039255663x_{2} = 54.9815039255663
x3=4.79347157705818x_{3} = 4.79347157705818
x4=61.2642477618988x_{4} = 61.2642477618988
x5=67.5470820571842x_{5} = 67.5470820571842
x6=36.1343483683071x_{6} = 36.1343483683071
x7=98.9619747288706x_{7} = 98.9619747288706
x8=29.8527496343583x_{8} = 29.8527496343583
x9=11.0222526014154x_{9} = 11.0222526014154
x10=80.1129313258524x_{10} = 80.1129313258524
x11=92.6789347457531x_{11} = 92.6789347457531
x12=23.5721422140939x_{12} = 23.5721422140939
x13=48.698889092841x_{13} = 48.698889092841
x14=86.3959182759432x_{14} = 86.3959182759432
x15=73.8299820941703x_{15} = 73.8299820941703
x16=42.4164666736714x_{16} = 42.4164666736714
Puntos máximos de la función:
x16=95.8204521084873x_{16} = 95.8204521084873
x16=64.4056553878471x_{16} = 64.4056553878471
x16=58.1228625440946x_{16} = 58.1228625440946
x16=51.8401771906044x_{16} = 51.8401771906044
x16=89.5374232382022x_{16} = 89.5374232382022
x16=83.2544206601072x_{16} = 83.2544206601072
x16=32.9934636297173x_{16} = 32.9934636297173
x16=70.6885250388719x_{16} = 70.6885250388719
x16=7.89526518224072x_{16} = 7.89526518224072
x16=39.2753595077596x_{16} = 39.2753595077596
x16=45.5576483965663x_{16} = 45.5576483965663
x16=14.1565167061041x_{16} = 14.1565167061041
x16=1.88479134910863x_{16} = 1.88479134910863
x16=20.432534864003x_{16} = 20.432534864003
x16=26.7122735990589x_{16} = 26.7122735990589
x16=76.9714513712894x_{16} = 76.9714513712894
Decrece en los intervalos
[98.9619747288706,∞)\left[98.9619747288706, \infty\right)
Crece en los intervalos
(−∞,4.79347157705818]\left(-\infty, 4.79347157705818\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)+1)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2=0- \left(\log{\left(x \right)} + 1\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=53.4145959979388x_{1} = 53.4145959979388
x2=3.39170564491897x_{2} = 3.39170564491897
x3=65.9792867094135x_{3} = 65.9792867094135
x4=9.48931434968285x_{4} = 9.48931434968285
x5=69.1205644129687x_{5} = 69.1205644129687
x6=43.9917989710092x_{6} = 43.9917989710092
x7=37.7105645099626x_{7} = 37.7105645099626
x8=62.8380442014817x_{8} = 62.8380442014817
x9=84.8273350804536x_{9} = 84.8273350804536
x10=75.4032065385794x_{10} = 75.4032065385794
x11=81.6859404954632x_{11} = 81.6859404954632
x12=94.2516056780793x_{12} = 94.2516056780793
x13=56.5556904082389x_{13} = 56.5556904082389
x14=97.3930531659223x_{14} = 97.3930531659223
x15=59.6968429375231x_{15} = 59.6968429375231
x16=72.2618723774379x_{16} = 72.2618723774379
x17=28.2906074072024x_{17} = 28.2906074072024
x18=18.8764359491976x_{18} = 18.8764359491976
x19=25.1515531449089x_{19} = 25.1515531449089
x20=12.6111291852012x_{20} = 12.6111291852012
x21=78.5445635201864x_{21} = 78.5445635201864
x22=6.39143406973474x_{22} = 6.39143406973474
x23=87.968745251226x_{23} = 87.968745251226
x24=100.534510628858x_{24} = 100.534510628858
x25=50.2735716147382x_{25} = 50.2735716147382
x26=15.7417373661781x_{26} = 15.7417373661781
x27=91.1101692788815x_{27} = 91.1101692788815
x28=47.1326325876179x_{28} = 47.1326325876179
x29=31.4302290264216x_{29} = 31.4302290264216
x30=22.0133384825441x_{30} = 22.0133384825441
x31=34.5702507576831x_{31} = 34.5702507576831
x32=40.8510974885272x_{32} = 40.8510974885272

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
[97.3930531659223,∞)\left[97.3930531659223, \infty\right)
Convexa en los intervalos
(−∞,3.39170564491897]\left(-\infty, 3.39170564491897\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((log⁡(x)+1)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\log{\left(x \right)} + 1\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)+1)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(\log{\left(x \right)} + 1\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) + 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((log⁡(x)+1)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\log{\left(x \right)} + 1\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)+1)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(\log{\left(x \right)} + 1\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)+1)sin⁡(x)=−(log⁡(−x)+1)sin⁡(x)\left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} = - \left(\log{\left(- x \right)} + 1\right) \sin{\left(x \right)}
- No
(log⁡(x)+1)sin⁡(x)=(log⁡(−x)+1)sin⁡(x)\left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} = \left(\log{\left(- x \right)} + 1\right) \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
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