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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x - sin(x)
f(x) = ----------
       x + sin(x)
f(x)=x−sin⁡(x)x+sin⁡(x)f{\left(x \right)} = \frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \right)}} 
f = (x - sin(x))/(x + sin(x))
Gráfico de la función
02468-8-6-4-2-101002
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+sin⁡(x)=0\frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 + sin(x)).
(−1)sin⁡(0)sin⁡(0)\frac{\left(-1\right) \sin{\left(0 \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
1−cos⁡(x)x+sin⁡(x)+(x−sin⁡(x))(−cos⁡(x)−1)(x+sin⁡(x))2=0\frac{1 - \cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{\left(x - \sin{\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=17.2207552719308x_{1} = 17.2207552719308
x2=4.05735660895802⋅10−15x_{2} = 4.05735660895802 \cdot 10^{-15}
x3=−80.0981286289451x_{3} = -80.0981286289451
x4=−83.2401924707234x_{4} = -83.2401924707234
x5=98.9500628243319x_{5} = 98.9500628243319
x6=−45.5311340139913x_{6} = -45.5311340139913
x7=161.785840727966x_{7} = 161.785840727966
x8=−86.3822220347287x_{8} = -86.3822220347287
x9=7.72525183693771x_{9} = 7.72525183693771
x10=−4.49340945790906x_{10} = -4.49340945790906
x11=4.49340945790906x_{11} = 4.49340945790906
x12=39.2444323611642x_{12} = 39.2444323611642
x13=−70.6716857116195x_{13} = -70.6716857116195
x14=10.9041216594289x_{14} = 10.9041216594289
x15=−42.3879135681319x_{15} = -42.3879135681319
x16=80.0981286289451x_{16} = 80.0981286289451
x17=171.210958939446x_{17} = 171.210958939446
x18=−48.6741442319544x_{18} = -48.6741442319544
x19=89.5242209304172x_{19} = 89.5242209304172
x20=14.0661939128315x_{20} = 14.0661939128315
x21=−36.1006222443756x_{21} = -36.1006222443756
x22=102.091966464908x_{22} = 102.091966464908
x23=−95.8081387868617x_{23} = -95.8081387868617
x24=64.3871195905574x_{24} = 64.3871195905574
x25=61.2447302603744x_{25} = 61.2447302603744
x26=−54.9596782878889x_{26} = -54.9596782878889
x27=76.9560263103312x_{27} = 76.9560263103312
x28=−76.9560263103312x_{28} = -76.9560263103312
x29=−98.9500628243319x_{29} = -98.9500628243319
x30=−7.72525183693771x_{30} = -7.72525183693771
x31=−20.3713029592876x_{31} = -20.3713029592876
x32=−2.79413003615298⋅10−14x_{32} = -2.79413003615298 \cdot 10^{-14}
x33=−39.2444323611642x_{33} = -39.2444323611642
x34=−14.0661939128315x_{34} = -14.0661939128315
x35=−32.9563890398225x_{35} = -32.9563890398225
x36=−227.761076847648x_{36} = -227.761076847648
x37=73.8138806006806x_{37} = 73.8138806006806
x38=26.6660542588127x_{38} = 26.6660542588127
x39=54.9596782878889x_{39} = 54.9596782878889
x40=−26.6660542588127x_{40} = -26.6660542588127
x41=−61.2447302603744x_{41} = -61.2447302603744
x42=−67.5294347771441x_{42} = -67.5294347771441
x43=29.811598790893x_{43} = 29.811598790893
x44=51.8169824872797x_{44} = 51.8169824872797
x45=23.519452498689x_{45} = 23.519452498689
x46=−58.1022547544956x_{46} = -58.1022547544956
x47=67.5294347771441x_{47} = 67.5294347771441
x48=−10.9041216594289x_{48} = -10.9041216594289
x49=−89.5242209304172x_{49} = -89.5242209304172
x50=−13215.1094216545x_{50} = -13215.1094216545
x51=86.3822220347287x_{51} = 86.3822220347287
x52=−23.519452498689x_{52} = -23.519452498689
x53=−17.2207552719308x_{53} = -17.2207552719308
x54=58.1022547544956x_{54} = 58.1022547544956
x55=−92.6661922776228x_{55} = -92.6661922776228
x56=−29.811598790893x_{56} = -29.811598790893
x57=92.6661922776228x_{57} = 92.6661922776228
x58=−64.3871195905574x_{58} = -64.3871195905574
x59=127.226642643334x_{59} = 127.226642643334
x60=32.9563890398225x_{60} = 32.9563890398225
x61=20.3713029592876x_{61} = 20.3713029592876
x62=48.6741442319544x_{62} = 48.6741442319544
x63=45.5311340139913x_{63} = 45.5311340139913
x64=36.1006222443756x_{64} = 36.1006222443756
x65=70.6716857116195x_{65} = 70.6716857116195
x66=83.2401924707234x_{66} = 83.2401924707234
x67=95.8081387868617x_{67} = 95.8081387868617
x68=−73.8138806006806x_{68} = -73.8138806006806
x69=42.3879135681319x_{69} = 42.3879135681319
x70=−51.8169824872797x_{70} = -51.8169824872797
Signos de extremos en los puntos:
(17.22075527193077, 1.12307869868553)

(4.057356608958019e-15, 0)

(-80.09812862894512, 1.02528305302948)

(-83.2401924707234, 0.976260056542518)

(98.95006282433188, 1.02041751453922)

(-45.53113401399128, 0.957028166104047)

(161.78584072796568, 1.01243866717085)

(-86.38222203472871, 1.02342249220221)

(7.725251836937707, 0.772461097913349)

(-4.493409457909064, 1.55504077855261)

(4.493409457909064, 1.55504077855261)

(39.24443236116419, 0.950319410117988)

(-70.6716857116195, 0.972097731728565)

(10.904121659428899, 1.20100745196584)

(-42.38791356813192, 1.04830951977811)

(80.09812862894512, 1.02528305302948)

(171.21095893944562, 0.988386534184906)

(-48.674144231954386, 1.0419424245269)

(89.52422093041719, 0.977907828460357)

(14.066193912831473, 0.86756453787246)

(-36.10062224437561, 1.05695657697143)

(102.09196646490764, 0.98060076781689)

(-95.8081387868617, 0.979341693609794)

(64.38711959055742, 0.969416568120202)

(61.2447302603744, 1.03319342653464)

(-54.959678287888934, 1.0370584656736)

(76.95602631033118, 0.974346648664662)

(-76.95602631033118, 0.974346648664662)

(-98.95006282433188, 1.02041751453922)

(-7.725251836937707, 0.772461097913349)

(-20.37130295928756, 0.906523852345629)

(-2.7941300361529808e-14, 0)

(-39.24443236116419, 0.950319410117988)

(-14.066193912831473, 0.86756453787246)

(-32.956389039822476, 0.941127220242848)

(-227.7610768476483, 0.99125733785207)

(73.81388060068065, 1.02746473545409)

(26.666054258812675, 0.927758187267769)

(54.959678287888934, 1.0370584656736)

(-26.666054258812675, 0.927758187267769)

(-61.2447302603744, 1.03319342653464)

(-67.52943477714412, 1.03005853664807)

(29.81159879089296, 1.06937611361914)

(51.81698248727967, 0.96214030755708)

(23.519452498689006, 1.08872838162155)

(-58.10225475449559, 0.966165269591092)

(67.52943477714412, 1.03005853664807)

(-10.904121659428899, 1.20100745196584)

(-89.52422093041719, 0.977907828460357)

(-13215.109421654506, 0.999848669534243)

(86.38222203472871, 1.02342249220221)

(-23.519452498689006, 1.08872838162155)

(-17.22075527193077, 1.12307869868553)

(58.10225475449559, 0.966165269591092)

(-92.66619227762284, 1.02181701056246)

(-29.81159879089296, 1.06937611361914)

(92.66619227762284, 1.02181701056246)

(-64.38711959055742, 0.969416568120202)

(127.22664264333433, 0.984403095456837)

(32.956389039822476, 0.941127220242848)

(20.37130295928756, 0.906523852345629)

(48.674144231954386, 1.0419424245269)

(45.53113401399128, 0.957028166104047)

(36.10062224437561, 1.05695657697143)

(70.6716857116195, 0.972097731728565)

(83.2401924707234, 0.976260056542518)

(95.8081387868617, 0.979341693609794)

(-73.81388060068065, 1.02746473545409)

(42.38791356813192, 1.04830951977811)

(-51.81698248727967, 0.96214030755708)


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=4.05735660895802⋅10−15x_{1} = 4.05735660895802 \cdot 10^{-15}
x2=−83.2401924707234x_{2} = -83.2401924707234
x3=−45.5311340139913x_{3} = -45.5311340139913
x4=7.72525183693771x_{4} = 7.72525183693771
x5=39.2444323611642x_{5} = 39.2444323611642
x6=−70.6716857116195x_{6} = -70.6716857116195
x7=171.210958939446x_{7} = 171.210958939446
x8=89.5242209304172x_{8} = 89.5242209304172
x9=14.0661939128315x_{9} = 14.0661939128315
x10=102.091966464908x_{10} = 102.091966464908
x11=−95.8081387868617x_{11} = -95.8081387868617
x12=64.3871195905574x_{12} = 64.3871195905574
x13=76.9560263103312x_{13} = 76.9560263103312
x14=−76.9560263103312x_{14} = -76.9560263103312
x15=−7.72525183693771x_{15} = -7.72525183693771
x16=−20.3713029592876x_{16} = -20.3713029592876
x17=−2.79413003615298⋅10−14x_{17} = -2.79413003615298 \cdot 10^{-14}
x18=−39.2444323611642x_{18} = -39.2444323611642
x19=−14.0661939128315x_{19} = -14.0661939128315
x20=−32.9563890398225x_{20} = -32.9563890398225
x21=−227.761076847648x_{21} = -227.761076847648
x22=26.6660542588127x_{22} = 26.6660542588127
x23=−26.6660542588127x_{23} = -26.6660542588127
x24=51.8169824872797x_{24} = 51.8169824872797
x25=−58.1022547544956x_{25} = -58.1022547544956
x26=−89.5242209304172x_{26} = -89.5242209304172
x27=−13215.1094216545x_{27} = -13215.1094216545
x28=58.1022547544956x_{28} = 58.1022547544956
x29=−64.3871195905574x_{29} = -64.3871195905574
x30=127.226642643334x_{30} = 127.226642643334
x31=32.9563890398225x_{31} = 32.9563890398225
x32=20.3713029592876x_{32} = 20.3713029592876
x33=45.5311340139913x_{33} = 45.5311340139913
x34=70.6716857116195x_{34} = 70.6716857116195
x35=83.2401924707234x_{35} = 83.2401924707234
x36=95.8081387868617x_{36} = 95.8081387868617
x37=−51.8169824872797x_{37} = -51.8169824872797
Puntos máximos de la función:
x37=17.2207552719308x_{37} = 17.2207552719308
x37=−80.0981286289451x_{37} = -80.0981286289451
x37=98.9500628243319x_{37} = 98.9500628243319
x37=161.785840727966x_{37} = 161.785840727966
x37=−86.3822220347287x_{37} = -86.3822220347287
x37=−4.49340945790906x_{37} = -4.49340945790906
x37=4.49340945790906x_{37} = 4.49340945790906
x37=10.9041216594289x_{37} = 10.9041216594289
x37=−42.3879135681319x_{37} = -42.3879135681319
x37=80.0981286289451x_{37} = 80.0981286289451
x37=−48.6741442319544x_{37} = -48.6741442319544
x37=−36.1006222443756x_{37} = -36.1006222443756
x37=61.2447302603744x_{37} = 61.2447302603744
x37=−54.9596782878889x_{37} = -54.9596782878889
x37=−98.9500628243319x_{37} = -98.9500628243319
x37=73.8138806006806x_{37} = 73.8138806006806
x37=54.9596782878889x_{37} = 54.9596782878889
x37=−61.2447302603744x_{37} = -61.2447302603744
x37=−67.5294347771441x_{37} = -67.5294347771441
x37=29.811598790893x_{37} = 29.811598790893
x37=23.519452498689x_{37} = 23.519452498689
x37=67.5294347771441x_{37} = 67.5294347771441
x37=−10.9041216594289x_{37} = -10.9041216594289
x37=86.3822220347287x_{37} = 86.3822220347287
x37=−23.519452498689x_{37} = -23.519452498689
x37=−17.2207552719308x_{37} = -17.2207552719308
x37=−92.6661922776228x_{37} = -92.6661922776228
x37=−29.811598790893x_{37} = -29.811598790893
x37=92.6661922776228x_{37} = 92.6661922776228
x37=48.6741442319544x_{37} = 48.6741442319544
x37=36.1006222443756x_{37} = 36.1006222443756
x37=−73.8138806006806x_{37} = -73.8138806006806
x37=42.3879135681319x_{37} = 42.3879135681319
Decrece en los intervalos
[171.210958939446,∞)\left[171.210958939446, \infty\right)
Crece en los intervalos
(−∞,−13215.1094216545]\left(-\infty, -13215.1094216545\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x−sin⁡(x)x+sin⁡(x))=1\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \right)}}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(x−sin⁡(x)x+sin⁡(x))=1\lim_{x \to \infty}\left(\frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \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 (x - sin(x))/(x + sin(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x−sin⁡(x)x(x+sin⁡(x)))=0\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x \left(x + \sin{\left(x \right)}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(x−sin⁡(x)x(x+sin⁡(x)))=0\lim_{x \to \infty}\left(\frac{x - \sin{\left(x \right)}}{x \left(x + \sin{\left(x \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−sin⁡(x)x+sin⁡(x)=−x+sin⁡(x)−x−sin⁡(x)\frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \right)}} = \frac{- x + \sin{\left(x \right)}}{- x - \sin{\left(x \right)}}
- No
x−sin⁡(x)x+sin⁡(x)=−−x+sin⁡(x)−x−sin⁡(x)\frac{x - \sin{\left(x \right)}}{x + \sin{\left(x \right)}} = - \frac{- x + \sin{\left(x \right)}}{- x - \sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (x-sin(x))/(x+sin(x))
    © Sr Examen – Калькулятор