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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x - sin(x)
f(x) = ----------
       x - tan(x)
f(x)=x−sin⁡(x)x−tan⁡(x)f{\left(x \right)} = \frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}} 
f = (x - sin(x))/(x - tan(x))
Gráfico de la función
02468-8-6-4-2-1010-2020
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−sin⁡(x)x−tan⁡(x)=0\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−1.5707963267949x_{1} = -1.5707963267949
x2=1.5707963267949x_{2} = 1.5707963267949
x3=−7.85398163397448x_{3} = -7.85398163397448
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 - sin(x))/(x - tan(x)).
(−1)sin⁡(0)(−1)tan⁡(0)\frac{\left(-1\right) \sin{\left(0 \right)}}{\left(-1\right) \tan{\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
1−cos⁡(x)x−tan⁡(x)+(x−sin⁡(x))tan⁡2(x)(x−tan⁡(x))2=0\frac{1 - \cos{\left(x \right)}}{x - \tan{\left(x \right)}} + \frac{\left(x - \sin{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(x - \tan{\left(x \right)}\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−100.530964510497x_{1} = -100.530964510497
x2=−69.1150357337256x_{2} = -69.1150357337256
x3=69.1150388446184x_{3} = 69.1150388446184
x4=81.6814088654407x_{4} = 81.6814088654407
x5=−18.8495569354352x_{5} = -18.8495569354352
x6=87.9645891530623x_{6} = 87.9645891530623
x7=69.1150394922452x_{7} = 69.1150394922452
x8=−113.097336299759x_{8} = -113.097336299759
x9=81.6814098425437x_{9} = 81.6814098425437
x10=−12.5663701840803x_{10} = -12.5663701840803
x11=69.1150375367013x_{11} = 69.1150375367013
x12=−87.9645952173239x_{12} = -87.9645952173239
x13=−75.3982239004993x_{13} = -75.3982239004993
x14=100.530964745716x_{14} = 100.530964745716
x15=−31.4159267356987x_{15} = -31.4159267356987
x16=37.6991120578079x_{16} = 37.6991120578079
x17=43.9822971695403x_{17} = 43.9822971695403
x18=56.5486675846956x_{18} = 56.5486675846956
x19=62.8318526729861x_{19} = 62.8318526729861
x20=100.530967590802x_{20} = 100.530967590802
x21=62.8318524178403x_{21} = 62.8318524178403
x22=25.1327403665251x_{22} = 25.1327403665251
x23=−31.4159267389468x_{23} = -31.4159267389468
x24=−6.28318510764535x_{24} = -6.28318510764535
x25=−94.2477794289072x_{25} = -94.2477794289072
x26=43.9822973069874x_{26} = 43.9822973069874
x27=50.2654824463231x_{27} = 50.2654824463231
x28=−25.1327385834705x_{28} = -25.1327385834705
x29=87.9645878964094x_{29} = 87.9645878964094
x30=94.2477796093521x_{30} = 94.2477796093521
x31=113.097338200433x_{31} = 113.097338200433
x32=31.4159270078848x_{32} = 31.4159270078848
x33=−100.530967397304x_{33} = -100.530967397304
x34=62.8318570545501x_{34} = 62.8318570545501
x35=−62.8318521668007x_{35} = -62.8318521668007
x36=43.9822977582334x_{36} = 43.9822977582334
x37=−6.28318468774868x_{37} = -6.28318468774868
x38=−50.2654822672451x_{38} = -50.2654822672451
x39=−307.876075235129x_{39} = -307.876075235129
x40=−81.6814090386714x_{40} = -81.6814090386714
x41=75.3982213478593x_{41} = 75.3982213478593
x42=−56.5486673426872x_{42} = -56.5486673426872
x43=−56.5486711735805x_{43} = -56.5486711735805
x44=−87.9645943583029x_{44} = -87.9645943583029
x45=37.6991118536214x_{45} = 37.6991118536214
x46=−18.8495550098077x_{46} = -18.8495550098077
x47=−62.8318540980733x_{47} = -62.8318540980733
x48=−25.1327416744928x_{48} = -25.1327416744928
x49=31.4159241387587x_{49} = 31.4159241387587
x50=131.946894970856x_{50} = 131.946894970856
x51=81.6814030257389x_{51} = 81.6814030257389
x52=−43.9822971744607x_{52} = -43.9822971744607
x53=−69.1150388409748x_{53} = -69.1150388409748
x54=−257.610598459672x_{54} = -257.610598459672
x55=50.2654824337184x_{55} = 50.2654824337184
x56=−75.3982240263278x_{56} = -75.3982240263278
x57=37.6991063433335x_{57} = 37.6991063433335
x58=43.9822972992366x_{58} = 43.9822972992366
x59=−100.530968216638x_{59} = -100.530968216638
x60=87.9645943362011x_{60} = 87.9645943362011
x61=−37.6991118774875x_{61} = -37.6991118774875
x62=31.4159314451446x_{62} = 31.4159314451446
x63=6.28318528412783x_{63} = 6.28318528412783
x64=25.1327423142362x_{64} = 25.1327423142362
x65=18.8495555101384x_{65} = 18.8495555101384
x66=12.5663704242683x_{66} = 12.5663704242683
x67=75.3982241765023x_{67} = 75.3982241765023
x68=81.68140922005x_{68} = 81.68140922005
Signos de extremos en los puntos:
(-100.53096451049653, 1)

(-69.11503573372556, 1)

(69.11503884461837, 1)

(81.68140886544074, 1)

(-18.84955693543524, 1)

(87.96458915306229, 1)

(69.11503949224522, 1)

(-113.09733629975894, 1)

(81.68140984254372, 1)

(-12.566370184080304, 1)

(69.11503753670127, 1)

(-87.96459521732386, 1)

(-75.3982239004993, 1)

(100.53096474571626, 1)

(-31.415926735698708, 1)

(37.699112057807874, 1)

(43.98229716954028, 1)

(56.54866758469556, 1)

(62.83185267298607, 1)

(100.53096759080243, 1)

(62.83185241784028, 1)

(25.132740366525116, 1)

(-31.415926738946762, 1)

(-6.283185107645348, 1)

(-94.24777942890718, 1)

(43.98229730698741, 1)

(50.265482446323084, 1)

(-25.132738583470527, 1)

(87.96458789640938, 1)

(94.24777960935205, 1)

(113.09733820043321, 1)

(31.41592700788479, 1)

(-100.53096739730367, 1)

(62.83185705455009, 1)

(-62.83185216680066, 1)

(43.98229775823344, 1)

(-6.283184687748678, 1)

(-50.265482267245105, 1)

(-307.8760752351294, 1)

(-81.68140903867145, 1)

(75.39822134785933, 1)

(-56.54866734268718, 1)

(-56.548671173580495, 1)

(-87.96459435830292, 1)

(37.69911185362138, 1)

(-18.849555009807688, 1)

(-62.8318540980733, 1)

(-25.132741674492816, 1)

(31.415924138758655, 1)

(131.94689497085557, 1)

(81.68140302573893, 1)

(-43.982297174460705, 1)

(-69.11503884097485, 1)

(-257.610598459672, 1)

(50.26548243371841, 1)

(-75.39822402632784, 1)

(37.69910634333351, 1)

(43.98229729923656, 1)

(-100.53096821663816, 1)

(87.96459433620113, 1)

(-37.69911187748746, 1)

(31.415931445144572, 1)

(6.283185284127832, 1)

(25.13274231423616, 1)

(18.849555510138448, 1)

(12.5663704242683, 1)

(75.39822417650228, 1)

(81.68140922004997, 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−sin⁡(x)x−tan⁡(x))y = \lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right)
True

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

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