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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x             
       e  - sin(x) + x
f(x) = ---------------
        x             
       e  - sin(x) - x
f(x)=x+(ex−sin⁡(x))−x+(ex−sin⁡(x))f{\left(x \right)} = \frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} 
f = (x + exp(x) - sin(x))/(-x + exp(x) - 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:
x+(ex−sin⁡(x))−x+(ex−sin⁡(x))=0\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−1.23498227923177x_{1} = -1.23498227923177
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 (exp(x) - sin(x) + x)/(exp(x) - sin(x) - x).
−sin⁡(0)+e0−0+(−sin⁡(0)+e0)\frac{- \sin{\left(0 \right)} + e^{0}}{- 0 + \left(- \sin{\left(0 \right)} + e^{0}\right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
ex−cos⁡(x)+1−x+(ex−sin⁡(x))+(x+(ex−sin⁡(x)))(−ex+cos⁡(x)+1)(−x+(ex−sin⁡(x)))2=0\frac{e^{x} - \cos{\left(x \right)} + 1}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} + \frac{\left(x + \left(e^{x} - \sin{\left(x \right)}\right)\right) \left(- e^{x} + \cos{\left(x \right)} + 1\right)}{\left(- x + \left(e^{x} - \sin{\left(x \right)}\right)\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=68.4026026533356x_{1} = 68.4026026533356
x2=55.6879491257269x_{2} = 55.6879491257269
x3=−633.02933999405x_{3} = -633.02933999405
x4=80x_{4} = 80
x5=−70.6716857116195x_{5} = -70.6716857116195
x6=66.5x_{6} = 66.5
x7=−92.6661922776228x_{7} = -92.6661922776228
x8=−86.3822220347287x_{8} = -86.3822220347287
x9=66.223996865801x_{9} = 66.223996865801
x10=−89.5242209304172x_{10} = -89.5242209304172
x11=74x_{11} = 74
x12=72x_{12} = 72
x13=−45.5311340139913x_{13} = -45.5311340139913
x14=65.6834795893247x_{14} = 65.6834795893247
x15=−80.0981286289451x_{15} = -80.0981286289451
x16=34.3069734414766x_{16} = 34.3069734414766
x17=49.7754698215909x_{17} = 49.7754698215909
x18=−14.0661930771291x_{18} = -14.0661930771291
x19=−58.1022547544956x_{19} = -58.1022547544956
x20=−42.3879135681319x_{20} = -42.3879135681319
x21=−23.5194524987527x_{21} = -23.5194524987527
x22=40.0202154992634x_{22} = 40.0202154992634
x23=−29.8115987908931x_{23} = -29.8115987908931
x24=63.5975419173648x_{24} = 63.5975419173648
x25=47.8119589618645x_{25} = 47.8119589618645
x26=45.8533704858379x_{26} = 45.8533704858379
x27=−20.3713029577941x_{27} = -20.3713029577941
x28=66.2441343042238x_{28} = 66.2441343042238
x29=86x_{29} = 86
x30=−73.8138806006806x_{30} = -73.8138806006806
x31=35.9035726630501x_{31} = 35.9035726630501
x32=−67.5294347771441x_{32} = -67.5294347771441
x33=−95.8081387868617x_{33} = -95.8081387868617
x34=78x_{34} = 78
x35=41.9557498041686x_{35} = 41.9557498041686
x36=67.3055555555556x_{36} = 67.3055555555556
x37=38.0970665022932x_{37} = 38.0970665022932
x38=−17.2207553071165x_{38} = -17.2207553071165
x39=66.3753305431907x_{39} = 66.3753305431907
x40=68x_{40} = 68
x41=90x_{41} = 90
x42=43.9008089821284x_{42} = 43.9008089821284
x43=70x_{43} = 70
x44=−7.72474872891224x_{44} = -7.72474872891224
x45=67.0420892015282x_{45} = 67.0420892015282
x46=65.710495061406x_{46} = 65.710495061406
x47=−32.9563890398225x_{47} = -32.9563890398225
x48=59.6426199192496x_{48} = 59.6426199192496
x49=61.6276851753556x_{49} = 61.6276851753556
x50=57.6644222037334x_{50} = 57.6644222037334
x51=−98.9500628243319x_{51} = -98.9500628243319
x52=53.7140644634518x_{52} = 53.7140644634518
x53=96x_{53} = 96
x54=36.190501914492x_{54} = 36.190501914492
x55=65.5294651682181x_{55} = 65.5294651682181
x56=−54.9596782878889x_{56} = -54.9596782878889
x57=98x_{57} = 98
x58=−39.2444323611642x_{58} = -39.2444323611642
x59=−83.2401924707234x_{59} = -83.2401924707234
x60=82x_{60} = 82
x61=−61.2447302603744x_{61} = -61.2447302603744
x62=88x_{62} = 88
x63=94x_{63} = 94
x64=−48.6741442319544x_{64} = -48.6741442319544
x65=−4.50721566039406x_{65} = -4.50721566039406
x66=32.4565264847902x_{66} = 32.4565264847902
x67=−64.3871195905574x_{67} = -64.3871195905574
x68=84x_{68} = 84
x69=51.7430578258417x_{69} = 51.7430578258417
x70=−26.6660542588099x_{70} = -26.6660542588099
x71=−76.9560263103312x_{71} = -76.9560263103312
x72=67.6734773926687x_{72} = 67.6734773926687
x73=−51.8169824872797x_{73} = -51.8169824872797
x74=−36.1006222443756x_{74} = -36.1006222443756
x75=92x_{75} = 92
x76=76x_{76} = 76
x77=−10.904141811359x_{77} = -10.904141811359
x78=65.687962582344x_{78} = 65.687962582344
x79=153.051567263916x_{79} = 153.051567263916
x80=100x_{80} = 100
Signos de extremos en los puntos:
(68.40260265333565, 1)

(55.68794912572692, 1)

(-633.0293399940497, -1.00316440610274)

(80, 1)

(-70.6716857116195, -0.972097731728565)

(66.5, 1)

(-92.66619227762284, -1.02181701056246)

(-86.38222203472871, -1.02342249220221)

(66.22399686580096, 1)

(-89.52422093041719, -0.977907828460357)

(74, 1)

(72, 1)

(-45.53113401399128, -0.957028166104047)

(65.68347958932469, 1)

(-80.09812862894512, -1.02528305302948)

(34.30697344147655, 1.00000000000009)

(49.77546982159091, 1)

(-14.066193077129078, -0.867564441384213)

(-58.10225475449559, -0.966165269591092)

(-42.38791356813192, -1.04830951977811)

(-23.519452498752702, -1.08872838161589)

(40.02021549926337, 1)

(-29.811598790893076, -1.06937611361913)

(63.59754191736475, 1)

(47.81195896186453, 1)

(45.85337048583789, 1)

(-20.371302957794114, -0.906523852218779)

(66.24413430422379, 1)

(86, 1)

(-73.81388060068065, -1.02746473545409)

(35.903572663050056, 1.00000000000002)

(-67.52943477714412, -1.03005853664807)

(-95.8081387868617, -0.979341693609794)

(78, 1)

(41.95574980416862, 1)

(67.30555555555556, 1)

(38.09706650229323, 1)

(-17.220755307116466, -1.12307869434072)

(66.3753305431907, 1)

(68, 1)

(90, 1)

(43.90080898212835, 1)

(70, 1)

(-7.724748728912244, -0.772371297588025)

(67.04208920152821, 1)

(65.71049506140604, 1)

(-32.95638903982247, -0.941127220242848)

(59.64261991924964, 1)

(61.62768517535564, 1)

(57.664422203733444, 1)

(-98.95006282433188, -1.02041751453922)

(53.71406446345178, 1)

(96, 1)

(36.190501914492025, 1.00000000000001)

(65.52946516821812, 1)

(-54.959678287888934, -1.0370584656736)

(98, 1)

(-39.24443236116419, -0.950319410117988)

(-83.2401924707234, -0.976260056542518)

(82, 1)

(-61.2447302603744, -1.03319342653464)

(88, 1)

(94, 1)

(-48.674144231954386, -1.0419424245269)

(-4.507215660394062, -1.5470113797114)

(32.456526484790174, 1.00000000000052)

(-64.38711959055742, -0.969416568120202)

(84, 1)

(51.74305782584166, 1)

(-26.666054258809947, -0.927758187267586)

(-76.95602631033118, -0.974346648664662)

(67.67347739266866, 1)

(-51.81698248727967, -0.96214030755708)

(-36.100622244375614, -1.05695657697143)

(92, 1)

(76, 1)

(-10.904141811359024, -1.20100336861316)

(65.68796258234401, 1)

(153.05156726391584, 1)

(100, 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=−633.02933999405x_{1} = -633.02933999405
x2=−92.6661922776228x_{2} = -92.6661922776228
x3=−86.3822220347287x_{3} = -86.3822220347287
x4=72x_{4} = 72
x5=−80.0981286289451x_{5} = -80.0981286289451
x6=−42.3879135681319x_{6} = -42.3879135681319
x7=−23.5194524987527x_{7} = -23.5194524987527
x8=−29.8115987908931x_{8} = -29.8115987908931
x9=63.5975419173648x_{9} = 63.5975419173648
x10=−73.8138806006806x_{10} = -73.8138806006806
x11=−67.5294347771441x_{11} = -67.5294347771441
x12=−17.2207553071165x_{12} = -17.2207553071165
x13=90x_{13} = 90
x14=65.710495061406x_{14} = 65.710495061406
x15=61.6276851753556x_{15} = 61.6276851753556
x16=−98.9500628243319x_{16} = -98.9500628243319
x17=−54.9596782878889x_{17} = -54.9596782878889
x18=−61.2447302603744x_{18} = -61.2447302603744
x19=−48.6741442319544x_{19} = -48.6741442319544
x20=−4.50721566039406x_{20} = -4.50721566039406
x21=−36.1006222443756x_{21} = -36.1006222443756
x22=−10.904141811359x_{22} = -10.904141811359
Puntos máximos de la función:
x22=−70.6716857116195x_{22} = -70.6716857116195
x22=−89.5242209304172x_{22} = -89.5242209304172
x22=−45.5311340139913x_{22} = -45.5311340139913
x22=−14.0661930771291x_{22} = -14.0661930771291
x22=−58.1022547544956x_{22} = -58.1022547544956
x22=−20.3713029577941x_{22} = -20.3713029577941
x22=−95.8081387868617x_{22} = -95.8081387868617
x22=38.0970665022932x_{22} = 38.0970665022932
x22=−7.72474872891224x_{22} = -7.72474872891224
x22=−32.9563890398225x_{22} = -32.9563890398225
x22=−39.2444323611642x_{22} = -39.2444323611642
x22=−83.2401924707234x_{22} = -83.2401924707234
x22=−64.3871195905574x_{22} = -64.3871195905574
x22=−26.6660542588099x_{22} = -26.6660542588099
x22=−76.9560263103312x_{22} = -76.9560263103312
x22=−51.8169824872797x_{22} = -51.8169824872797
Decrece en los intervalos
[90,∞)\left[90, \infty\right)
Crece en los intervalos
(−∞,−633.02933999405]\left(-\infty, -633.02933999405\right]
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+(ex−sin⁡(x))−x+(ex−sin⁡(x)))y = \lim_{x \to -\infty}\left(\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)}\right)
lim⁡x→∞(x+(ex−sin⁡(x))−x+(ex−sin⁡(x)))=1\lim_{x \to \infty}\left(\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (exp(x) - sin(x) + x)/(exp(x) - sin(x) - 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+(ex−sin⁡(x))x(−x+(ex−sin⁡(x))))y = x \lim_{x \to -\infty}\left(\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{x \left(- x + \left(e^{x} - \sin{\left(x \right)}\right)\right)}\right)
lim⁡x→∞(x+(ex−sin⁡(x))x(−x+(ex−sin⁡(x))))=0\lim_{x \to \infty}\left(\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{x \left(- x + \left(e^{x} - \sin{\left(x \right)}\right)\right)}\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:
x+(ex−sin⁡(x))−x+(ex−sin⁡(x))=−x+sin⁡(x)+e−xx+sin⁡(x)+e−x\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} = \frac{- x + \sin{\left(x \right)} + e^{- x}}{x + \sin{\left(x \right)} + e^{- x}}
- No
x+(ex−sin⁡(x))−x+(ex−sin⁡(x))=−−x+sin⁡(x)+e−xx+sin⁡(x)+e−x\frac{x + \left(e^{x} - \sin{\left(x \right)}\right)}{- x + \left(e^{x} - \sin{\left(x \right)}\right)} = - \frac{- x + \sin{\left(x \right)} + e^{- x}}{x + \sin{\left(x \right)} + e^{- x}}
- No
es decir, función
no es
par ni impar
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