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Trazado el gráfico de la función/ x*(sin(x)^(-1))

Gráfico de la función y = x*(sin(x)^(-1))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         x   
f(x) = ------
       sin(x)
f(x)=xsin⁡(x)f{\left(x \right)} = \frac{x}{\sin{\left(x \right)}} 
f = x/sin(x)
Gráfico de la función
02468-8-6-4-2-1010-50005000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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:
xsin⁡(x)=0\frac{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).
0sin⁡(0)\frac{0}{\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
−xcos⁡(x)sin⁡2(x)+1sin⁡(x)=0- \frac{x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−45.5311340139913x_{1} = -45.5311340139913
x2=67.5294347771441x_{2} = 67.5294347771441
x3=−14.0661939128315x_{3} = -14.0661939128315
x4=95.8081387868617x_{4} = 95.8081387868617
x5=4.49340945790906x_{5} = 4.49340945790906
x6=−58.1022547544956x_{6} = -58.1022547544956
x7=61.2447302603744x_{7} = 61.2447302603744
x8=29.811598790893x_{8} = 29.811598790893
x9=−23.519452498689x_{9} = -23.519452498689
x10=−48.6741442319544x_{10} = -48.6741442319544
x11=−39.2444323611642x_{11} = -39.2444323611642
x12=−89.5242209304172x_{12} = -89.5242209304172
x13=54.9596782878889x_{13} = 54.9596782878889
x14=−76.9560263103312x_{14} = -76.9560263103312
x15=86.3822220347287x_{15} = 86.3822220347287
x16=14.0661939128315x_{16} = 14.0661939128315
x17=−92.6661922776228x_{17} = -92.6661922776228
x18=4.93829501990806⋅10−17x_{18} = 4.93829501990806 \cdot 10^{-17}
x19=−98.9500628243319x_{19} = -98.9500628243319
x20=−51.8169824872797x_{20} = -51.8169824872797
x21=45.5311340139913x_{21} = 45.5311340139913
x22=−64.3871195905574x_{22} = -64.3871195905574
x23=23.519452498689x_{23} = 23.519452498689
x24=−42.3879135681319x_{24} = -42.3879135681319
x25=70.6716857116195x_{25} = 70.6716857116195
x26=58.1022547544956x_{26} = 58.1022547544956
x27=2.01537897347346⋅10−17x_{27} = 2.01537897347346 \cdot 10^{-17}
x28=92.6661922776228x_{28} = 92.6661922776228
x29=−95.8081387868617x_{29} = -95.8081387868617
x30=−73.8138806006806x_{30} = -73.8138806006806
x31=−67.5294347771441x_{31} = -67.5294347771441
x32=−80.0981286289451x_{32} = -80.0981286289451
x33=−54.9596782878889x_{33} = -54.9596782878889
x34=73.8138806006806x_{34} = 73.8138806006806
x35=−10.9041216594289x_{35} = -10.9041216594289
x36=−7.72525183693771x_{36} = -7.72525183693771
x37=42.3879135681319x_{37} = 42.3879135681319
x38=20.3713029592876x_{38} = 20.3713029592876
x39=32.9563890398225x_{39} = 32.9563890398225
x40=7.72525183693771x_{40} = 7.72525183693771
x41=−70.6716857116195x_{41} = -70.6716857116195
x42=48.6741442319544x_{42} = 48.6741442319544
x43=−32.9563890398225x_{43} = -32.9563890398225
x44=26.6660542588127x_{44} = 26.6660542588127
x45=64.3871195905574x_{45} = 64.3871195905574
x46=36.1006222443756x_{46} = 36.1006222443756
x47=−17.2207552719308x_{47} = -17.2207552719308
x48=−29.811598790893x_{48} = -29.811598790893
x49=17.2207552719308x_{49} = 17.2207552719308
x50=−20.3713029592876x_{50} = -20.3713029592876
x51=89.5242209304172x_{51} = 89.5242209304172
x52=98.9500628243319x_{52} = 98.9500628243319
x53=−61.2447302603744x_{53} = -61.2447302603744
x54=51.8169824872797x_{54} = 51.8169824872797
x55=83.2401924707234x_{55} = 83.2401924707234
x56=39.2444323611642x_{56} = 39.2444323611642
x57=−4.49340945790906x_{57} = -4.49340945790906
x58=−86.3822220347287x_{58} = -86.3822220347287
x59=−36.1006222443756x_{59} = -36.1006222443756
x60=−83.2401924707234x_{60} = -83.2401924707234
x61=−26.6660542588127x_{61} = -26.6660542588127
x62=80.0981286289451x_{62} = 80.0981286289451
x63=76.9560263103312x_{63} = 76.9560263103312
x64=10.9041216594289x_{64} = 10.9041216594289
Signos de extremos en los puntos:
(-45.53113401399128, 45.5421141867616)

(67.52943477714412, -67.5368385499393)

(-14.066193912831473, 14.1016953304692)

(95.8081387868617, 95.8133574080491)

(4.493409457909064, -4.6033388487517)

(-58.10225475449559, 58.1108596353238)

(61.2447302603744, -61.2528936840213)

(29.81159879089296, -29.8283660710601)

(-23.519452498689006, -23.5407018977364)

(-48.674144231954386, -48.6844155424824)

(-39.24443236116419, 39.2571709544892)

(-89.52422093041719, 89.5298058369287)

(54.959678287888934, -54.9687751137703)

(-76.95602631033118, 76.9625232530508)

(86.38222203472871, -86.3880100688583)

(14.066193912831473, 14.1016953304692)

(-92.66619227762284, -92.6715878316184)

(4.938295019908061e-17, 1)

(-98.95006282433188, -98.9551157492084)

(-51.81698248727967, 51.8266309351384)

(45.53113401399128, 45.5421141867616)

(-64.38711959055742, 64.3948846506362)

(23.519452498689006, -23.5407018977364)

(-42.38791356813192, -42.399707742618)

(70.6716857116195, 70.67876032672)

(58.10225475449559, 58.1108596353238)

(2.0153789734734588e-17, 1)

(92.66619227762284, -92.6715878316184)

(-95.8081387868617, 95.8133574080491)

(-73.81388060068065, -73.8206540836068)

(-67.52943477714412, -67.5368385499393)

(-80.09812862894512, -80.1043707288125)

(-54.959678287888934, -54.9687751137703)

(73.81388060068065, -73.8206540836068)

(-10.904121659428899, -10.9498798698263)

(-7.725251836937707, 7.78970576749272)

(42.38791356813192, -42.399707742618)

(20.37130295928756, 20.3958325218432)

(32.956389039822476, 32.9715571143392)

(7.725251836937707, 7.78970576749272)

(-70.6716857116195, 70.67876032672)

(48.674144231954386, -48.6844155424824)

(-32.956389039822476, 32.9715571143392)

(26.666054258812675, 26.6847981018021)

(64.38711959055742, 64.3948846506362)

(36.10062224437561, -36.1144697653324)

(-17.22075527193077, -17.2497655675586)

(-29.81159879089296, -29.8283660710601)

(17.22075527193077, -17.2497655675586)

(-20.37130295928756, 20.3958325218432)

(89.52422093041719, 89.5298058369287)

(98.95006282433188, -98.9551157492084)

(-61.2447302603744, -61.2528936840213)

(51.81698248727967, 51.8266309351384)

(83.2401924707234, 83.2461989676591)

(39.24443236116419, 39.2571709544892)

(-4.493409457909064, -4.6033388487517)

(-86.38222203472871, -86.3880100688583)

(-36.10062224437561, -36.1144697653324)

(-83.2401924707234, 83.2461989676591)

(-26.666054258812675, 26.6847981018021)

(80.09812862894512, -80.1043707288125)

(76.95602631033118, 76.9625232530508)

(10.904121659428899, -10.9498798698263)


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=−45.5311340139913x_{1} = -45.5311340139913
x2=−14.0661939128315x_{2} = -14.0661939128315
x3=95.8081387868617x_{3} = 95.8081387868617
x4=−58.1022547544956x_{4} = -58.1022547544956
x5=−39.2444323611642x_{5} = -39.2444323611642
x6=−89.5242209304172x_{6} = -89.5242209304172
x7=−76.9560263103312x_{7} = -76.9560263103312
x8=14.0661939128315x_{8} = 14.0661939128315
x9=4.93829501990806⋅10−17x_{9} = 4.93829501990806 \cdot 10^{-17}
x10=−51.8169824872797x_{10} = -51.8169824872797
x11=45.5311340139913x_{11} = 45.5311340139913
x12=−64.3871195905574x_{12} = -64.3871195905574
x13=70.6716857116195x_{13} = 70.6716857116195
x14=58.1022547544956x_{14} = 58.1022547544956
x15=2.01537897347346⋅10−17x_{15} = 2.01537897347346 \cdot 10^{-17}
x16=−95.8081387868617x_{16} = -95.8081387868617
x17=−7.72525183693771x_{17} = -7.72525183693771
x18=20.3713029592876x_{18} = 20.3713029592876
x19=32.9563890398225x_{19} = 32.9563890398225
x20=7.72525183693771x_{20} = 7.72525183693771
x21=−70.6716857116195x_{21} = -70.6716857116195
x22=−32.9563890398225x_{22} = -32.9563890398225
x23=26.6660542588127x_{23} = 26.6660542588127
x24=64.3871195905574x_{24} = 64.3871195905574
x25=−20.3713029592876x_{25} = -20.3713029592876
x26=89.5242209304172x_{26} = 89.5242209304172
x27=51.8169824872797x_{27} = 51.8169824872797
x28=83.2401924707234x_{28} = 83.2401924707234
x29=39.2444323611642x_{29} = 39.2444323611642
x30=−83.2401924707234x_{30} = -83.2401924707234
x31=−26.6660542588127x_{31} = -26.6660542588127
x32=76.9560263103312x_{32} = 76.9560263103312
Puntos máximos de la función:
x32=67.5294347771441x_{32} = 67.5294347771441
x32=4.49340945790906x_{32} = 4.49340945790906
x32=61.2447302603744x_{32} = 61.2447302603744
x32=29.811598790893x_{32} = 29.811598790893
x32=−23.519452498689x_{32} = -23.519452498689
x32=−48.6741442319544x_{32} = -48.6741442319544
x32=54.9596782878889x_{32} = 54.9596782878889
x32=86.3822220347287x_{32} = 86.3822220347287
x32=−92.6661922776228x_{32} = -92.6661922776228
x32=−98.9500628243319x_{32} = -98.9500628243319
x32=23.519452498689x_{32} = 23.519452498689
x32=−42.3879135681319x_{32} = -42.3879135681319
x32=92.6661922776228x_{32} = 92.6661922776228
x32=−73.8138806006806x_{32} = -73.8138806006806
x32=−67.5294347771441x_{32} = -67.5294347771441
x32=−80.0981286289451x_{32} = -80.0981286289451
x32=−54.9596782878889x_{32} = -54.9596782878889
x32=73.8138806006806x_{32} = 73.8138806006806
x32=−10.9041216594289x_{32} = -10.9041216594289
x32=42.3879135681319x_{32} = 42.3879135681319
x32=48.6741442319544x_{32} = 48.6741442319544
x32=36.1006222443756x_{32} = 36.1006222443756
x32=−17.2207552719308x_{32} = -17.2207552719308
x32=−29.811598790893x_{32} = -29.811598790893
x32=17.2207552719308x_{32} = 17.2207552719308
x32=98.9500628243319x_{32} = 98.9500628243319
x32=−61.2447302603744x_{32} = -61.2447302603744
x32=−4.49340945790906x_{32} = -4.49340945790906
x32=−86.3822220347287x_{32} = -86.3822220347287
x32=−36.1006222443756x_{32} = -36.1006222443756
x32=80.0981286289451x_{32} = 80.0981286289451
x32=10.9041216594289x_{32} = 10.9041216594289
Decrece en los intervalos
[95.8081387868617,∞)\left[95.8081387868617, \infty\right)
Crece en los intervalos
(−∞,−95.8081387868617]\left(-\infty, -95.8081387868617\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
x(1+2cos⁡2(x)sin⁡2(x))−2cos⁡(x)sin⁡(x)sin⁡(x)=0\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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→−∞(xsin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xsin⁡(x))y = \lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x/sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞1sin⁡(x)=⟨−∞,∞⟩\lim_{x \to -\infty} \frac{1}{\sin{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞1sin⁡(x)=⟨−∞,∞⟩\lim_{x \to \infty} \frac{1}{\sin{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
xsin⁡(x)=xsin⁡(x)\frac{x}{\sin{\left(x \right)}} = \frac{x}{\sin{\left(x \right)}}
- Sí
xsin⁡(x)=−xsin⁡(x)\frac{x}{\sin{\left(x \right)}} = - \frac{x}{\sin{\left(x \right)}}
- No
es decir, función
es
par
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