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(x-tan(x))/(x-sin(x))
Trazado el gráfico de la función/ (x-tan(x))/(x-sin(x))

Gráfico de la función y = (x-tan(x))/(x-sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x - tan(x)
f(x) = ----------
       x - sin(x)
f(x)=x−tan⁡(x)x−sin⁡(x)f{\left(x \right)} = \frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}} 
f = (x - tan(x))/(x - sin(x))
Gráfico de la función
02468-8-6-4-2-1010-200200
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:
x−tan⁡(x)x−sin⁡(x)=0\frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−7.72525183693771x_{1} = -7.72525183693771
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 (x - tan(x))/(x - sin(x)).
(−1)tan⁡(0)(−1)sin⁡(0)\frac{\left(-1\right) \tan{\left(0 \right)}}{\left(-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
−tan⁡2(x)x−sin⁡(x)+(x−tan⁡(x))(cos⁡(x)−1)(x−sin⁡(x))2=0- \frac{\tan^{2}{\left(x \right)}}{x - \sin{\left(x \right)}} + \frac{\left(x - \tan{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right)}{\left(x - \sin{\left(x \right)}\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−207.345108301703x_{1} = -207.345108301703
x2=−75.398223904069x_{2} = -75.398223904069
x3=−18.8495568615716x_{3} = -18.8495568615716
x4=−25.1327416654918x_{4} = -25.1327416654918
x5=81.681406428613x_{5} = 81.681406428613
x6=−12.5663734439644x_{6} = -12.5663734439644
x7=−81.6814090386687x_{7} = -81.6814090386687
x8=−75.3982239001244x_{8} = -75.3982239001244
x9=−62.8318521465565x_{9} = -62.8318521465565
x10=56.5486675844073x_{10} = 56.5486675844073
x11=−31.4159240793588x_{11} = -31.4159240793588
x12=69.1150394658599x_{12} = 69.1150394658599
x13=−37.6991118774846x_{13} = -37.6991118774846
x14=56.5486678803449x_{14} = 56.5486678803449
x15=75.3982241725269x_{15} = 75.3982241725269
x16=−100.530964508641x_{16} = -100.530964508641
x17=−62.8318540736204x_{17} = -62.8318540736204
x18=−75.3982237958595x_{18} = -75.3982237958595
x19=81.6814087168407x_{19} = 81.6814087168407
x20=62.8318567779893x_{20} = 62.8318567779893
x21=62.8318526701554x_{21} = 62.8318526701554
x22=−94.2477778453903x_{22} = -94.2477778453903
x23=−6.28318510384512x_{23} = -6.28318510384512
x24=94.247779609352x_{24} = 94.247779609352
x25=−94.2477794287357x_{25} = -94.2477794287357
x26=−69.1150388373004x_{26} = -69.1150388373004
x27=−25.1327381393735x_{27} = -25.1327381393735
x28=−56.5486673389673x_{28} = -56.5486673389673
x29=81.6814092196325x_{29} = 81.6814092196325
x30=50.2654824463229x_{30} = 50.2654824463229
x31=−62.8318552647775x_{31} = -62.8318552647775
x32=−12.5663701652999x_{32} = -12.5663701652999
x33=81.6814103897661x_{33} = 81.6814103897661
x34=31.4159238543107x_{34} = 31.4159238543107
x35=37.6991118280756x_{35} = 37.6991118280756
x36=25.132740318945x_{36} = 25.132740318945
x37=−87.9645943583008x_{37} = -87.9645943583008
x38=87.9645943361994x_{38} = 87.9645943361994
x39=31.4159269993762x_{39} = 31.4159269993762
x40=12.5663725284885x_{40} = 12.5663725284885
x41=31.4159275367066x_{41} = 31.4159275367066
x42=−50.2654822668523x_{42} = -50.2654822668523
x43=−43.9822967170422x_{43} = -43.9822967170422
x44=1256.63704090224x_{44} = 1256.63704090224
x45=100.530964745584x_{45} = 100.530964745584
x46=−18.8495549354584x_{46} = -18.8495549354584
x47=12.5663704226957x_{47} = 12.5663704226957
x48=75.3982279048483x_{48} = 75.3982279048483
x49=43.9822952350179x_{49} = 43.9822952350179
x50=81.6814110944089x_{50} = 81.6814110944089
x51=37.6991120570478x_{51} = 37.6991120570478
x52=−100.530965088797x_{52} = -100.530965088797
x53=94.2477798470804x_{53} = 94.2477798470804
x54=−94.2477804030121x_{54} = -94.2477804030121
x55=37.6991096880266x_{55} = 37.6991096880266
x56=−37.6991112814928x_{56} = -37.6991112814928
x57=43.9822971695394x_{57} = 43.9822971695394
x58=43.9822964021506x_{58} = 43.9822964021506
x59=−31.4159267381921x_{59} = -31.4159267381921
x60=69.1150375210825x_{60} = 69.1150375210825
x61=75.3982212426238x_{61} = 75.3982212426238
x62=18.8495592252303x_{62} = 18.8495592252303
x63=18.8495554995137x_{63} = 18.8495554995137
x64=−56.5486709387627x_{64} = -56.5486709387627
x65=6.2831852841214x_{65} = 6.2831852841214
x66=−69.1150355891385x_{66} = -69.1150355891385
x67=−56.5486712509204x_{67} = -56.5486712509204
x68=−37.6991115785943x_{68} = -37.6991115785943
x69=−43.9822971744598x_{69} = -43.9822971744598
x70=25.1327416787285x_{70} = 25.1327416787285
x71=100.530967635014x_{71} = 100.530967635014
x72=25.1327422488485x_{72} = 25.1327422488485
x73=−81.6814093919831x_{73} = -81.6814093919831
Signos de extremos en los puntos:
(-207.34510830170288, 1)

(-75.39822390406897, 1)

(-18.849556861571607, 1)

(-25.132741665491807, 1)

(81.68140642861297, 1)

(-12.566373443964437, 1)

(-81.68140903866866, 1)

(-75.39822390012436, 1)

(-62.831852146556535, 1)

(56.5486675844073, 1)

(-31.41592407935884, 1)

(69.11503946585992, 1)

(-37.699111877484555, 1)

(56.548667880344865, 1)

(75.39822417252688, 1)

(-100.53096450864139, 1)

(-62.83185407362036, 1)

(-75.39822379585945, 1)

(81.68140871684069, 1)

(62.831856777989266, 1)

(62.831852670155435, 1)

(-94.24777784539033, 1)

(-6.283185103845123, 1)

(94.24777960935204, 1)

(-94.2477794287357, 1)

(-69.11503883730045, 1)

(-25.132738139373547, 1)

(-56.548667338967256, 1)

(81.68140921963246, 1)

(50.265482446322935, 1)

(-62.83185526477752, 1)

(-12.566370165299931, 1)

(81.68141038976609, 1)

(31.415923854310673, 1)

(37.69911182807561, 1)

(25.132740318945018, 1)

(-87.96459435830079, 1)

(87.96459433619935, 1)

(31.415926999376214, 1)

(12.566372528488479, 1)

(31.41592753670663, 1)

(-50.26548226685232, 1)

(-43.98229671704224, 1)

(1256.63704090224, 1)

(100.53096474558367, 1)

(-18.84955493545835, 1)

(12.566370422695671, 1)

(75.39822790484831, 1)

(43.9822952350179, 1)

(81.68141109440889, 1)

(37.69911205704782, 1)

(-100.53096508879712, 1)

(94.2477798470804, 1)

(-94.24778040301214, 1)

(37.6991096880266, 1)

(-37.6991112814928, 1)

(43.982297169539414, 1)

(43.98229640215063, 1)

(-31.41592673819211, 1)

(69.11503752108251, 1)

(75.39822124262379, 1)

(18.849559225230276, 1)

(18.849555499513663, 1)

(-56.54867093876273, 1)

(6.283185284121399, 1)

(-69.11503558913849, 1)

(-56.548671250920414, 1)

(-37.69911157859432, 1)

(-43.9822971744598, 1)

(25.13274167872845, 1)

(100.53096763501418, 1)

(25.13274224884854, 1)

(-81.68140939198305, 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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(x−tan⁡(x)x−sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x−tan⁡(x)x−sin⁡(x))y = \lim_{x \to \infty}\left(\frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x - tan(x))/(x - sin(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(x−tan⁡(x)x(x−sin⁡(x)))y = x \lim_{x \to -\infty}\left(\frac{x - \tan{\left(x \right)}}{x \left(x - \sin{\left(x \right)}\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x−tan⁡(x)x(x−sin⁡(x)))y = x \lim_{x \to \infty}\left(\frac{x - \tan{\left(x \right)}}{x \left(x - \sin{\left(x \right)}\right)}\right)
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:
x−tan⁡(x)x−sin⁡(x)=−x+tan⁡(x)−x+sin⁡(x)\frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}} = \frac{- x + \tan{\left(x \right)}}{- x + \sin{\left(x \right)}}
- No
x−tan⁡(x)x−sin⁡(x)=−−x+tan⁡(x)−x+sin⁡(x)\frac{x - \tan{\left(x \right)}}{x - \sin{\left(x \right)}} = - \frac{- x + \tan{\left(x \right)}}{- x + \sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (x-tan(x))/(x-sin(x))
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