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Trazado el gráfico de la función/ |sin(x-1)|/ln(x)^2

Gráfico de la función y = |sin(x-1)|/ln(x)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       |sin(x - 1)|
f(x) = ------------
            2      
         log (x)
f(x)=∣sin⁡(x−1)∣log⁡(x)2f{\left(x \right)} = \frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \right)}^{2}} 
f = Abs(sin(x - 1))/log(x)^2
Gráfico de la función
02468-8-6-4-2-1010025
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=1x_{1} = 1
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−1)∣log⁡(x)2=0\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \right)}^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1+πx_{1} = 1 + \pi
Solución numérica
x1=26.1327412287183x_{1} = 26.1327412287183
x2=7.28318530717959x_{2} = 7.28318530717959
x3=104.672557568463x_{3} = 104.672557568463
x4=35.5575191894877x_{4} = 35.5575191894877
x5=10.4247779607694x_{5} = 10.4247779607694
x6=82.6814089933346x_{6} = 82.6814089933346
x7=44.9822971502571x_{7} = 44.9822971502571
x8=95.2477796076938x_{8} = 95.2477796076938
x9=101.530964914873x_{9} = 101.530964914873
x10=98.3893722612836x_{10} = 98.3893722612836
x11=16.707963267949x_{11} = 16.707963267949
x12=51.2654824574367x_{12} = 51.2654824574367
x13=92.106186954104x_{13} = 92.106186954104
x14=79.5398163397448x_{14} = 79.5398163397448
x15=85.8230016469244x_{15} = 85.8230016469244
x16=0x_{16} = 0
x17=38.6991118430775x_{17} = 38.6991118430775
x18=13.5663706143592x_{18} = 13.5663706143592
x19=88.9645943005142x_{19} = 88.9645943005142
x20=63.8318530717959x_{20} = 63.8318530717959
x21=19.8495559215388x_{21} = 19.8495559215388
x22=60.6902604182061x_{22} = 60.6902604182061
x23=54.4070751110265x_{23} = 54.4070751110265
x24=76.398223686155x_{24} = 76.398223686155
x25=29.2743338823081x_{25} = 29.2743338823081
x26=4.14159265358979x_{26} = 4.14159265358979
x27=22.9911485751286x_{27} = 22.9911485751286
x28=57.5486677646163x_{28} = 57.5486677646163
x29=73.2566310325652x_{29} = 73.2566310325652
x30=41.8407044966673x_{30} = 41.8407044966673
x31=66.9734457253857x_{31} = 66.9734457253857
x32=32.4159265358979x_{32} = 32.4159265358979
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 Abs(sin(x - 1))/log(x)^2.
∣sin⁡(−1)∣log⁡(0)2\frac{\left|{\sin{\left(-1 \right)}}\right|}{\log{\left(0 \right)}^{2}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
cos⁡(x−1)sign⁡(sin⁡(x−1))log⁡(x)2−2∣sin⁡(x−1)∣xlog⁡(x)3=0\frac{\cos{\left(x - 1 \right)} \operatorname{sign}{\left(\sin{\left(x - 1 \right)} \right)}}{\log{\left(x \right)}^{2}} - \frac{2 \left|{\sin{\left(x - 1 \right)}}\right|}{x \log{\left(x \right)}^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=74.8212328267368x_{1} = 74.8212328267368
x2=99.9558233342634x_{2} = 99.9558233342634
x3=65.3953337580208x_{3} = 65.3953337580208
x4=33.9700244305033x_{4} = 33.9700244305033
x5=8.74897283044365x_{5} = 8.74897283044365
x6=15.0883640893372x_{6} = 15.0883640893372
x7=52.8267354764541x_{7} = 52.8267354764541
x8=21.3898344877303x_{8} = 21.3898344877303
x9=24.5364794123116x_{9} = 24.5364794123116
x10=81.1050028928673x_{10} = 81.1050028928673
x11=103.097576526141x_{11} = 103.097576526141
x12=71.6793037272081x_{12} = 71.6793037272081
x13=46.5419043406713x_{13} = 46.5419043406713
x14=68.537339261517x_{14} = 68.537339261517
x15=59.1111703198702x_{15} = 59.1111703198702
x16=77.9631312469326x_{16} = 77.9631312469326
x17=18.2410174144493x_{17} = 18.2410174144493
x18=84.2468510429452x_{18} = 84.2468510429452
x19=62.2532802911407x_{19} = 62.2532802911407
x20=55.9689931950121x_{20} = 55.9689931950121
x21=40.2564643725992x_{21} = 40.2564643725992
x22=49.684379969985x_{22} = 49.684379969985
x23=87.3886784686919x_{23} = 87.3886784686919
x24=30.8262080377715x_{24} = 30.8262080377715
x25=96.8140583410097x_{25} = 96.8140583410097
x26=93.6722802399529x_{26} = 93.6722802399529
x27=5.50237005910851x_{27} = 5.50237005910851
x28=27.6817841122258x_{28} = 27.6817841122258
x29=11.9280371818276x_{29} = 11.9280371818276
x30=37.1134053848121x_{30} = 37.1134053848121
x31=90.5304875282351x_{31} = 90.5304875282351
x32=43.3992790706471x_{32} = 43.3992790706471
Signos de extremos en los puntos:
(74.82123282673678, 0.0537043660965533)

(99.95582333426341, 0.0471615289270383)

(65.3953337580208, 0.0572192450200257)

(33.970024430503315, 0.0804457874275903)

(8.748972830443648, 0.211401245166425)

(15.088364089337157, 0.135608619115604)

(52.8267354764541, 0.0635407020232425)

(21.38983448773027, 0.106543662036093)

(24.536479412311593, 0.0976147664597613)

(81.10500289286729, 0.0517521330416725)

(103.09757652614121, 0.0465339662244336)

(71.67930372720808, 0.0547882171803239)

(46.54190434067129, 0.0678001265257218)

(68.53733926151696, 0.0559560879891657)

(59.11117031987017, 0.0600880679184576)

(77.96313124693256, 0.0526950307994656)

(18.2410174144493, 0.118520988468029)

(84.24685104294522, 0.0508687584792063)

(62.25328029114071, 0.0585911559209487)

(55.96899319501206, 0.0617297815507547)

(40.256464372599154, 0.07322649717077)

(49.684379969984995, 0.065551292986626)

(87.38867846869192, 0.0500389510351725)

(30.82620803777149, 0.08506444643842)

(96.81405834100973, 0.0478225393193714)

(93.67228023995294, 0.0485200582895605)

(5.5023700591085065, 0.336365094272206)

(27.681784112225824, 0.0906605694013076)

(11.928037181827644, 0.162365470792692)

(37.11340538481211, 0.0765562510991704)

(90.53048752823514, 0.0492575497699442)

(43.39927907064712, 0.0703368384581488)


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x32=74.8212328267368x_{32} = 74.8212328267368
x32=99.9558233342634x_{32} = 99.9558233342634
x32=65.3953337580208x_{32} = 65.3953337580208
x32=33.9700244305033x_{32} = 33.9700244305033
x32=8.74897283044365x_{32} = 8.74897283044365
x32=15.0883640893372x_{32} = 15.0883640893372
x32=52.8267354764541x_{32} = 52.8267354764541
x32=21.3898344877303x_{32} = 21.3898344877303
x32=24.5364794123116x_{32} = 24.5364794123116
x32=81.1050028928673x_{32} = 81.1050028928673
x32=103.097576526141x_{32} = 103.097576526141
x32=71.6793037272081x_{32} = 71.6793037272081
x32=46.5419043406713x_{32} = 46.5419043406713
x32=68.537339261517x_{32} = 68.537339261517
x32=59.1111703198702x_{32} = 59.1111703198702
x32=77.9631312469326x_{32} = 77.9631312469326
x32=18.2410174144493x_{32} = 18.2410174144493
x32=84.2468510429452x_{32} = 84.2468510429452
x32=62.2532802911407x_{32} = 62.2532802911407
x32=55.9689931950121x_{32} = 55.9689931950121
x32=40.2564643725992x_{32} = 40.2564643725992
x32=49.684379969985x_{32} = 49.684379969985
x32=87.3886784686919x_{32} = 87.3886784686919
x32=30.8262080377715x_{32} = 30.8262080377715
x32=96.8140583410097x_{32} = 96.8140583410097
x32=93.6722802399529x_{32} = 93.6722802399529
x32=5.50237005910851x_{32} = 5.50237005910851
x32=27.6817841122258x_{32} = 27.6817841122258
x32=11.9280371818276x_{32} = 11.9280371818276
x32=37.1134053848121x_{32} = 37.1134053848121
x32=90.5304875282351x_{32} = 90.5304875282351
x32=43.3992790706471x_{32} = 43.3992790706471
Decrece en los intervalos
(−∞,5.50237005910851]\left(-\infty, 5.50237005910851\right]
Crece en los intervalos
[103.097576526141,∞)\left[103.097576526141, \infty\right)
Asíntotas verticales
Hay:
x1=1x_{1} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(∣sin⁡(x−1)∣log⁡(x)2)=0\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \right)}^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(∣sin⁡(x−1)∣log⁡(x)2)=0\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \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 Abs(sin(x - 1))/log(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(∣sin⁡(x−1)∣xlog⁡(x)2)=0\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{x \log{\left(x \right)}^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(∣sin⁡(x−1)∣xlog⁡(x)2)=0\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{x \log{\left(x \right)}^{2}}\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−1)∣log⁡(x)2=∣sin⁡(x+1)∣log⁡(−x)2\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \right)}^{2}} = \frac{\left|{\sin{\left(x + 1 \right)}}\right|}{\log{\left(- x \right)}^{2}}
- No
∣sin⁡(x−1)∣log⁡(x)2=−∣sin⁡(x+1)∣log⁡(−x)2\frac{\left|{\sin{\left(x - 1 \right)}}\right|}{\log{\left(x \right)}^{2}} = - \frac{\left|{\sin{\left(x + 1 \right)}}\right|}{\log{\left(- x \right)}^{2}}
- No
es decir, función
no es
par ni impar
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