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

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

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
02468-8-6-4-2-101005000
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)).
sin⁡(0)(−1)sin⁡(0)\frac{\sin{\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
cos⁡(x)+1x−sin⁡(x)+(x+sin⁡(x))(cos⁡(x)−1)(x−sin⁡(x))2=0\frac{\cos{\left(x \right)} + 1}{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=−64.3871195905574x_{1} = -64.3871195905574
x2=−61.2447302603744x_{2} = -61.2447302603744
x3=−26.6660542588127x_{3} = -26.6660542588127
x4=−86.3822220347287x_{4} = -86.3822220347287
x5=54.9596782878889x_{5} = 54.9596782878889
x6=−20.3713029592876x_{6} = -20.3713029592876
x7=95.8081387868617x_{7} = 95.8081387868617
x8=−36.1006222443756x_{8} = -36.1006222443756
x9=20.3713029592876x_{9} = 20.3713029592876
x10=−73.8138806006806x_{10} = -73.8138806006806
x11=−67.5294347771441x_{11} = -67.5294347771441
x12=70.6716857116195x_{12} = 70.6716857116195
x13=−92.6661922776228x_{13} = -92.6661922776228
x14=64.3871195905574x_{14} = 64.3871195905574
x15=26.6660542588127x_{15} = 26.6660542588127
x16=58.1022547544956x_{16} = 58.1022547544956
x17=−29.811598790893x_{17} = -29.811598790893
x18=32.9563890398225x_{18} = 32.9563890398225
x19=83.2401924707234x_{19} = 83.2401924707234
x20=23.519452498689x_{20} = 23.519452498689
x21=−7.72525183693771x_{21} = -7.72525183693771
x22=−4.49340945790906x_{22} = -4.49340945790906
x23=76.9560263103312x_{23} = 76.9560263103312
x24=89.5242209304172x_{24} = 89.5242209304172
x25=−45.5311340139913x_{25} = -45.5311340139913
x26=−83.2401924707234x_{26} = -83.2401924707234
x27=−14.0661939128315x_{27} = -14.0661939128315
x28=174.352656835193x_{28} = 174.352656835193
x29=7.72525183693771x_{29} = 7.72525183693771
x30=−80.0981286289451x_{30} = -80.0981286289451
x31=80.0981286289451x_{31} = 80.0981286289451
x32=−17.2207552719308x_{32} = -17.2207552719308
x33=−32.9563890398225x_{33} = -32.9563890398225
x34=17.2207552719308x_{34} = 17.2207552719308
x35=−48.6741442319544x_{35} = -48.6741442319544
x36=−10.9041216594289x_{36} = -10.9041216594289
x37=−39.2444323611642x_{37} = -39.2444323611642
x38=73.8138806006806x_{38} = 73.8138806006806
x39=98.9500628243319x_{39} = 98.9500628243319
x40=45.5311340139913x_{40} = 45.5311340139913
x41=29.811598790893x_{41} = 29.811598790893
x42=4.49340945790906x_{42} = 4.49340945790906
x43=−805.817274670293x_{43} = -805.817274670293
x44=10.9041216594289x_{44} = 10.9041216594289
x45=−42.3879135681319x_{45} = -42.3879135681319
x46=−23.519452498689x_{46} = -23.519452498689
x47=−98.9500628243319x_{47} = -98.9500628243319
x48=92.6661922776228x_{48} = 92.6661922776228
x49=48.6741442319544x_{49} = 48.6741442319544
x50=36.1006222443756x_{50} = 36.1006222443756
x51=14.0661939128315x_{51} = 14.0661939128315
x52=−76.9560263103312x_{52} = -76.9560263103312
x53=51.8169824872797x_{53} = 51.8169824872797
x54=−58.1022547544956x_{54} = -58.1022547544956
x55=86.3822220347287x_{55} = 86.3822220347287
x56=−89.5242209304172x_{56} = -89.5242209304172
x57=−95.8081387868617x_{57} = -95.8081387868617
x58=−51.8169824872797x_{58} = -51.8169824872797
x59=−70.6716857116195x_{59} = -70.6716857116195
x60=−54.9596782878889x_{60} = -54.9596782878889
x61=42.3879135681319x_{61} = 42.3879135681319
x62=67.5294347771441x_{62} = 67.5294347771441
x63=61.2447302603744x_{63} = 61.2447302603744
x64=39.2444323611642x_{64} = 39.2444323611642
Signos de extremos en los puntos:
(-64.38711959055742, 1.03154828675881)

(-61.2447302603744, 0.967872979364599)

(-26.666054258812675, 1.07786707110069)

(-86.38222203472871, 0.977113565139839)

(54.959678287888934, 0.964265789345318)

(-20.37130295928756, 1.10311493449676)

(95.8081387868617, 1.02109407423885)

(-36.10062224437561, 0.946112661378551)

(20.37130295928756, 1.10311493449676)

(-73.81388060068065, 0.973269413045158)

(-67.52943477714412, 0.970818613138353)

(70.6716857116195, 1.02870315129922)

(-92.66619227762284, 0.978648808605709)

(64.38711959055742, 1.03154828675881)

(26.666054258812675, 1.07786707110069)

(58.10225475449559, 1.0350196094538)

(-29.81159879089296, 0.935124683695855)

(32.956389039822476, 1.06255560193229)

(83.2401924707234, 1.02431723319866)

(23.519452498689006, 0.9185027384981)

(-7.725251836937707, 1.29456357440045)

(-4.493409457909064, 0.643069952757621)

(76.95602631033118, 1.02632877258879)

(89.52422093041719, 1.02259126156544)

(-45.53113401399128, 1.04490132622835)

(-83.2401924707234, 1.02431723319866)

(-14.066193912831473, 1.15265200033684)

(174.35265683519268, 0.988594599428852)

(7.725251836937707, 1.29456357440045)

(-80.09812862894512, 0.975340416527152)

(80.09812862894512, 0.975340416527152)

(-17.22075527193077, 0.89040955114759)

(-32.956389039822476, 1.06255560193229)

(17.22075527193077, 0.89040955114759)

(-48.674144231954386, 0.959745928815648)

(-10.904121659428899, 0.832634300780709)

(-39.24443236116419, 1.05227778087353)

(73.81388060068065, 0.973269413045158)

(98.95006282433188, 0.97999101911884)

(45.53113401399128, 1.04490132622835)

(29.81159879089296, 0.935124683695855)

(4.493409457909064, 0.643069952757621)

(-805.8172746702927, 1.00248503420716)

(10.904121659428899, 0.832634300780709)

(-42.38791356813192, 0.953916740364682)

(-23.519452498689006, 0.9185027384981)

(-98.95006282433188, 0.97999101911884)

(92.66619227762284, 0.978648808605709)

(48.674144231954386, 0.959745928815648)

(36.10062224437561, 0.946112661378551)

(14.066193912831473, 1.15265200033684)

(-76.95602631033118, 1.02632877258879)

(51.81698248727967, 1.03934945053809)

(-58.10225475449559, 1.0350196094538)

(86.38222203472871, 0.977113565139839)

(-89.52422093041719, 1.02259126156544)

(-95.8081387868617, 1.02109407423885)

(-51.81698248727967, 1.03934945053809)

(-70.6716857116195, 1.02870315129922)

(-54.959678287888934, 0.964265789345318)

(42.38791356813192, 0.953916740364682)

(67.52943477714412, 0.970818613138353)

(61.2447302603744, 0.967872979364599)

(39.24443236116419, 1.05227778087353)


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=−61.2447302603744x_{1} = -61.2447302603744
x2=−86.3822220347287x_{2} = -86.3822220347287
x3=54.9596782878889x_{3} = 54.9596782878889
x4=−36.1006222443756x_{4} = -36.1006222443756
x5=−73.8138806006806x_{5} = -73.8138806006806
x6=−67.5294347771441x_{6} = -67.5294347771441
x7=−92.6661922776228x_{7} = -92.6661922776228
x8=−29.811598790893x_{8} = -29.811598790893
x9=23.519452498689x_{9} = 23.519452498689
x10=−4.49340945790906x_{10} = -4.49340945790906
x11=174.352656835193x_{11} = 174.352656835193
x12=−80.0981286289451x_{12} = -80.0981286289451
x13=80.0981286289451x_{13} = 80.0981286289451
x14=−17.2207552719308x_{14} = -17.2207552719308
x15=17.2207552719308x_{15} = 17.2207552719308
x16=−48.6741442319544x_{16} = -48.6741442319544
x17=−10.9041216594289x_{17} = -10.9041216594289
x18=73.8138806006806x_{18} = 73.8138806006806
x19=98.9500628243319x_{19} = 98.9500628243319
x20=29.811598790893x_{20} = 29.811598790893
x21=4.49340945790906x_{21} = 4.49340945790906
x22=10.9041216594289x_{22} = 10.9041216594289
x23=−42.3879135681319x_{23} = -42.3879135681319
x24=−23.519452498689x_{24} = -23.519452498689
x25=−98.9500628243319x_{25} = -98.9500628243319
x26=92.6661922776228x_{26} = 92.6661922776228
x27=48.6741442319544x_{27} = 48.6741442319544
x28=36.1006222443756x_{28} = 36.1006222443756
x29=86.3822220347287x_{29} = 86.3822220347287
x30=−54.9596782878889x_{30} = -54.9596782878889
x31=42.3879135681319x_{31} = 42.3879135681319
x32=67.5294347771441x_{32} = 67.5294347771441
x33=61.2447302603744x_{33} = 61.2447302603744
Puntos máximos de la función:
x33=−64.3871195905574x_{33} = -64.3871195905574
x33=−26.6660542588127x_{33} = -26.6660542588127
x33=−20.3713029592876x_{33} = -20.3713029592876
x33=95.8081387868617x_{33} = 95.8081387868617
x33=20.3713029592876x_{33} = 20.3713029592876
x33=70.6716857116195x_{33} = 70.6716857116195
x33=64.3871195905574x_{33} = 64.3871195905574
x33=26.6660542588127x_{33} = 26.6660542588127
x33=58.1022547544956x_{33} = 58.1022547544956
x33=32.9563890398225x_{33} = 32.9563890398225
x33=83.2401924707234x_{33} = 83.2401924707234
x33=−7.72525183693771x_{33} = -7.72525183693771
x33=76.9560263103312x_{33} = 76.9560263103312
x33=89.5242209304172x_{33} = 89.5242209304172
x33=−45.5311340139913x_{33} = -45.5311340139913
x33=−83.2401924707234x_{33} = -83.2401924707234
x33=−14.0661939128315x_{33} = -14.0661939128315
x33=7.72525183693771x_{33} = 7.72525183693771
x33=−32.9563890398225x_{33} = -32.9563890398225
x33=−39.2444323611642x_{33} = -39.2444323611642
x33=45.5311340139913x_{33} = 45.5311340139913
x33=−805.817274670293x_{33} = -805.817274670293
x33=14.0661939128315x_{33} = 14.0661939128315
x33=−76.9560263103312x_{33} = -76.9560263103312
x33=51.8169824872797x_{33} = 51.8169824872797
x33=−58.1022547544956x_{33} = -58.1022547544956
x33=−89.5242209304172x_{33} = -89.5242209304172
x33=−95.8081387868617x_{33} = -95.8081387868617
x33=−51.8169824872797x_{33} = -51.8169824872797
x33=−70.6716857116195x_{33} = -70.6716857116195
x33=39.2444323611642x_{33} = 39.2444323611642
Decrece en los intervalos
[174.352656835193,∞)\left[174.352656835193, \infty\right)
Crece en los intervalos
(−∞,−98.9500628243319]\left(-\infty, -98.9500628243319\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
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