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Trazado el gráfico de la función/ sin(x-lg(sqrt(x)-1))

Gráfico de la función y = sin(x-lg(sqrt(x)-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)=sin⁡(x−log⁡(x−1))f{\left(x \right)} = \sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} 
f = sin(x - log(sqrt(x) - 1))
Gráfico de la función
02468-8-6-4-2-10102-2
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⁡(x−log⁡(x−1))=0\sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=67.9535355301913x_{1} = 67.9535355301913
x2=6.75225766751991x_{2} = 6.75225766751991
x3=39.3618829021331x_{3} = 39.3618829021331
x4=71.1210119780842x_{4} = 71.1210119780842
x5=29.7685990204628x_{5} = 29.7685990204628
x6=52.0928564848743x_{6} = 52.0928564848743
x7=32.9724195969358x_{7} = 32.9724195969358
x8=90.1037541391495x_{8} = 90.1037541391495
x9=42.5496270483647x_{9} = 42.5496270483647
x10=26.5566478610746x_{10} = 26.5566478610746
x11=99.5842822133836x_{11} = 99.5842822133836
x12=23.3341554067006x_{12} = 23.3341554067006
x13=64.7847240562673x_{13} = 64.7847240562673
x14=80.6165885844527x_{14} = 80.6165885844527
x15=16.8405665731604x_{15} = 16.8405665731604
x16=77.4524380153677x_{16} = 77.4524380153677
x17=96.4247573560626x_{17} = 96.4247573560626
x18=86.9421780672637x_{18} = 86.9421780672637
x19=58.4424977392219x_{19} = 58.4424977392219
x20=102.743213036443x_{20} = 102.743213036443
x21=93.264596747894x_{21} = 93.264596747894
x22=61.6144340646518x_{22} = 61.6144340646518
x23=83.7798110450018x_{23} = 83.7798110450018
x24=2.70046081469023x_{24} = 2.70046081469023
x25=48.9146320274384x_{25} = 48.9146320274384
x26=55.2687171274405x_{26} = 55.2687171274405
x27=10.2111852794871x_{27} = 10.2111852794871
x28=20.0974552361018x_{28} = 20.0974552361018
x29=36.1697794443802x_{29} = 36.1697794443802
x30=45.7336979621476x_{30} = 45.7336979621476
x31=74.2872770436161x_{31} = 74.2872770436161
x32=13.5527098579846x_{32} = 13.5527098579846
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(sqrt(x) - 1)).
sin⁡(−log⁡(−1+0))\sin{\left(- \log{\left(-1 + \sqrt{0} \right)} \right)}
Resultado:
f(0)=−isinh⁡(π)f{\left(0 \right)} = - i \sinh{\left(\pi \right)}
Punto:
(0, -i*sinh(pi))
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
(1−12x(x−1))cos⁡(x−log⁡(x−1))=0\left(1 - \frac{1}{2 \sqrt{x} \left(\sqrt{x} - 1\right)}\right) \cos{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=120.108240385727x_{1} = 120.108240385727
x2=4.90734021945058x_{2} = 4.90734021945058
x3=75.8699893949772x_{3} = 75.8699893949772
x4=37.7664262991421x_{4} = 37.7664262991421
x5=34.5718231796106x_{5} = 34.5718231796106
x6=18.4720491933122x_{6} = 18.4720491933122
x7=1704.86852654851x_{7} = 1704.86852654851
x8=50.5040596960473x_{8} = 50.5040596960473
x9=15.2015018637899x_{9} = 15.2015018637899
x10=79.0346342123449x_{10} = 79.0346342123449
x11=66.3693051275948x_{11} = 66.3693051275948
x12=31.3714078944811x_{12} = 31.3714078944811
x13=72.7042888801966x_{13} = 72.7042888801966
x14=63.1997737330243x_{14} = 63.1997737330243
x15=28.1637686539645x_{15} = 28.1637686539645
x16=44.1420851794838x_{16} = 44.1420851794838
x17=91.6842637790294x_{17} = 91.6842637790294
x18=24.9469104360709x_{18} = 24.9469104360709
x19=101.16381943269x_{19} = 101.16381943269
x20=98.0045965472267x_{20} = 98.0045965472267
x21=60.0286832801772x_{21} = 60.0286832801772
x22=82.1983110791594x_{22} = 82.1983110791594
x23=56.8558517122253x_{23} = 56.8558517122253
x24=8.50433583871073x_{24} = 8.50433583871073
x25=40.9562529093523x_{25} = 40.9562529093523
x26=47.3245282138772x_{26} = 47.3245282138772
x27=1.86602540378444x_{27} = 1.86602540378444
x28=88.523061227302x_{28} = 88.523061227302
x29=53.6810632784904x_{29} = 53.6810632784904
x30=94.8447592978093x_{30} = 94.8447592978093
x31=11.8909804884821x_{31} = 11.8909804884821
x32=21.7178822993759x_{32} = 21.7178822993759
x33=85.3610972786191x_{33} = 85.3610972786191
x34=69.5374324845145x_{34} = 69.5374324845145
Signos de extremos en los puntos:
(120.10824038572679, -1)

(4.907340219450576, -1)

(75.86998939497717, -1)

(37.76642629914207, -1)

(34.57182317961065, 1)

(18.472049193312156, -1)

(1704.8685265485078, -1)

(50.50405969604729, -1)

(15.201501863789906, 1)

(79.03463421234486, 1)

(66.36930512759479, 1)

(31.37140789448108, -1)

(72.70428888019663, 1)

(63.19977373302428, -1)

(28.163768653964475, 1)

(44.14208517948375, -1)

(91.68426377902935, 1)

(24.94691043607088, -1)

(101.16381943268998, -1)

(98.00459654722668, 1)

(60.02868328017717, 1)

(82.19831107915941, -1)

(56.85585171222525, -1)

(8.504335838710732, 1)

(40.956252909352294, 1)

(47.324528213877215, 1)

(1.8660254037844386, 0.267227464877216)

(88.523061227302, -1)

(53.68106327849037, 1)

(94.84475929780929, -1)

(11.890980488482121, -1)

(21.717882299375876, 1)

(85.36109727861913, 1)

(69.53743248451453, -1)


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=120.108240385727x_{1} = 120.108240385727
x2=4.90734021945058x_{2} = 4.90734021945058
x3=75.8699893949772x_{3} = 75.8699893949772
x4=37.7664262991421x_{4} = 37.7664262991421
x5=18.4720491933122x_{5} = 18.4720491933122
x6=1704.86852654851x_{6} = 1704.86852654851
x7=50.5040596960473x_{7} = 50.5040596960473
x8=31.3714078944811x_{8} = 31.3714078944811
x9=63.1997737330243x_{9} = 63.1997737330243
x10=44.1420851794838x_{10} = 44.1420851794838
x11=24.9469104360709x_{11} = 24.9469104360709
x12=101.16381943269x_{12} = 101.16381943269
x13=82.1983110791594x_{13} = 82.1983110791594
x14=56.8558517122253x_{14} = 56.8558517122253
x15=88.523061227302x_{15} = 88.523061227302
x16=94.8447592978093x_{16} = 94.8447592978093
x17=11.8909804884821x_{17} = 11.8909804884821
x18=69.5374324845145x_{18} = 69.5374324845145
Puntos máximos de la función:
x18=34.5718231796106x_{18} = 34.5718231796106
x18=15.2015018637899x_{18} = 15.2015018637899
x18=79.0346342123449x_{18} = 79.0346342123449
x18=66.3693051275948x_{18} = 66.3693051275948
x18=72.7042888801966x_{18} = 72.7042888801966
x18=28.1637686539645x_{18} = 28.1637686539645
x18=91.6842637790294x_{18} = 91.6842637790294
x18=98.0045965472267x_{18} = 98.0045965472267
x18=60.0286832801772x_{18} = 60.0286832801772
x18=8.50433583871073x_{18} = 8.50433583871073
x18=40.9562529093523x_{18} = 40.9562529093523
x18=47.3245282138772x_{18} = 47.3245282138772
x18=1.86602540378444x_{18} = 1.86602540378444
x18=53.6810632784904x_{18} = 53.6810632784904
x18=21.7178822993759x_{18} = 21.7178822993759
x18=85.3610972786191x_{18} = 85.3610972786191
Decrece en los intervalos
[1704.86852654851,∞)\left[1704.86852654851, \infty\right)
Crece en los intervalos
(−∞,4.90734021945058]\left(-\infty, 4.90734021945058\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞sin⁡(x−log⁡(x−1))=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x−log⁡(x−1))=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x - log(sqrt(x) - 1)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x−log⁡(x−1))x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x−log⁡(x−1))x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \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⁡(x−log⁡(x−1))=−sin⁡(x+log⁡(−x−1))\sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = - \sin{\left(x + \log{\left(\sqrt{- x} - 1 \right)} \right)}
- No
sin⁡(x−log⁡(x−1))=sin⁡(x+log⁡(−x−1))\sin{\left(x - \log{\left(\sqrt{x} - 1 \right)} \right)} = \sin{\left(x + \log{\left(\sqrt{- x} - 1 \right)} \right)}
- No
es decir, función
no es
par ni impar
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