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(x+sin(2*x))/sin(x)
Trazado el gráfico de la función/ (x+sin(2*x))/sin(x)

Gráfico de la función y = (x+sin(2*x))/sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x + sin(2*x)
f(x) = ------------
          sin(x)
f(x)=x+sin⁡(2x)sin⁡(x)f{\left(x \right)} = \frac{x + \sin{\left(2 x \right)}}{\sin{\left(x \right)}} 
f = (x + sin(2*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:
x+sin⁡(2x)sin⁡(x)=0\frac{x + \sin{\left(2 x \right)}}{\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(2*x))/sin(x).
sin⁡(0⋅2)sin⁡(0)\frac{\sin{\left(0 \cdot 2 \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
−(x+sin⁡(2x))cos⁡(x)sin⁡2(x)+2cos⁡(2x)+1sin⁡(x)=0- \frac{\left(x + \sin{\left(2 x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(2 x \right)} + 1}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=61.2773658575433x_{1} = 61.2773658575433
x2=−20.4689380267502x_{2} = -20.4689380267502
x3=−39.2953181686786x_{3} = -39.2953181686786
x4=23.604134582198x_{4} = 23.604134582198
x5=33.0169458205959x_{5} = 33.0169458205959
x6=−83.2642112429691x_{6} = -83.2642112429691
x7=−76.9820049516737x_{7} = -76.9820049516737
x8=20.4689380267502x_{8} = 20.4689380267502
x9=36.1559243301836x_{9} = 36.1559243301836
x10=−64.4181642526582x_{10} = -64.4181642526582
x11=−70.6999723871001x_{11} = -70.6999723871001
x12=4.90003208410804x_{12} = 4.90003208410804
x13=−42.4350358095921x_{13} = -42.4350358095921
x14=67.5590363627001x_{14} = 67.5590363627001
x15=92.6877692623911x_{15} = 92.6877692623911
x16=29.8785121081536x_{16} = 29.8785121081536
x17=−45.575010738978x_{17} = -45.575010738978
x18=76.9820049516737x_{18} = 76.9820049516737
x19=89.5465547553057x_{19} = 89.5465547553057
x20=−36.1559243301836x_{20} = -36.1559243301836
x21=64.4181642526582x_{21} = 64.4181642526582
x22=70.6999723871001x_{22} = 70.6999723871001
x23=−14.2067626065023x_{23} = -14.2067626065023
x24=−92.6877692623911x_{24} = -92.6877692623911
x25=−89.5465547553057x_{25} = -89.5465547553057
x26=−7.97511398695608x_{26} = -7.97511398695608
x27=−58.1366530875265x_{27} = -58.1366530875265
x28=−86.4053677142634x_{28} = -86.4053677142634
x29=−61.2773658575433x_{29} = -61.2773658575433
x30=−67.5590363627001x_{30} = -67.5590363627001
x31=−26.7408124348997x_{31} = -26.7408124348997
x32=−1.93718340864326x_{32} = -1.93718340864326
x33=17.3360027927035x_{33} = 17.3360027927035
x34=58.1366530875265x_{34} = 58.1366530875265
x35=54.9960405616261x_{35} = 54.9960405616261
x36=98.9702702267906x_{36} = 98.9702702267906
x37=−80.1230889308106x_{37} = -80.1230889308106
x38=−95.8290085380668x_{38} = -95.8290085380668
x39=45.575010738978x_{39} = 45.575010738978
x40=−73.8409641878896x_{40} = -73.8409641878896
x41=73.8409641878896x_{41} = 73.8409641878896
x42=95.8290085380668x_{42} = 95.8290085380668
x43=7.97511398695608x_{43} = 7.97511398695608
x44=−33.0169458205959x_{44} = -33.0169458205959
x45=39.2953181686786x_{45} = 39.2953181686786
x46=48.7151934688099x_{46} = 48.7151934688099
x47=14.2067626065023x_{47} = 14.2067626065023
x48=−54.9960405616261x_{48} = -54.9960405616261
x49=26.7408124348997x_{49} = 26.7408124348997
x50=86.4053677142634x_{50} = 86.4053677142634
x51=−51.8555464232494x_{51} = -51.8555464232494
x52=51.8555464232494x_{52} = 51.8555464232494
x53=−17.3360027927035x_{53} = -17.3360027927035
x54=83.2642112429691x_{54} = 83.2642112429691
x55=−29.8785121081536x_{55} = -29.8785121081536
x56=−23.604134582198x_{56} = -23.604134582198
x57=−48.7151934688099x_{57} = -48.7151934688099
x58=11.0841489258958x_{58} = 11.0841489258958
x59=1.93718340864326x_{59} = 1.93718340864326
x60=80.1230889308106x_{60} = 80.1230889308106
x61=−98.9702702267906x_{61} = -98.9702702267906
x62=−4.90003208410804x_{62} = -4.90003208410804
x63=−11.0841489258958x_{63} = -11.0841489258958
x64=42.4350358095921x_{64} = 42.4350358095921
Signos de extremos en los puntos:
(61.27736585754328, -61.2528994777449)

(-20.468938026750187, 20.3959877418391)

(-39.2953181686786, 39.2571929189699)

(23.60413458219802, -23.5408031583217)

(33.016945820595865, 32.9715941356851)

(-83.26421124296907, 83.246201277169)

(-76.98200495167366, 76.9625261753177)

(20.468938026750187, 20.3959877418391)

(36.15592433018363, -36.1144979601887)

(-64.41816425265819, 64.3948896376509)

(-70.6999723871001, 70.6787640991425)

(4.900032084108042, -4.61449316680033)

(-42.435035809592144, -42.3997251846591)

(67.55903636270008, -67.5368428733049)

(92.68776926239111, -92.6715895059293)

(29.878512108153572, -29.8284160199475)

(-45.57501073897802, 45.5421282670928)

(76.98200495167366, 76.9625261753177)

(89.54655475530569, 89.5298076936821)

(-36.15592433018363, -36.1144979601887)

(64.41816425265819, 64.3948896376509)

(70.6999723871001, 70.6787640991425)

(-14.206762606502263, 14.1021588144674)

(-92.68776926239111, -92.6715895059293)

(-89.54655475530569, 89.5298076936821)

(-7.975113986956083, 7.79231077920284)

(-58.1366530875265, 58.1108664195298)

(-86.40536771426336, -86.3880121355511)

(-61.27736585754328, -61.2528994777449)

(-67.55903636270008, -67.5368428733049)

(-26.7408124348997, 26.6848677639468)

(-1.9371834086432644, 1.35840983054794)

(17.336002792703507, -17.2500209082052)

(58.1366530875265, 58.1108664195298)

(54.996040561626074, -54.9687831276782)

(98.97027022679063, -98.9551171244958)

(-80.12308893081064, -80.1043733207136)

(-95.82900853806676, 95.8133589230563)

(45.57501073897802, 45.5421282670928)

(-73.84096418788961, -73.8206573948496)

(73.84096418788961, -73.8206573948496)

(95.82900853806676, 95.8133589230563)

(7.975113986956083, 7.79231077920284)

(-33.016945820595865, 32.9715941356851)

(39.2953181686786, 39.2571929189699)

(48.715193468809915, -48.6844270721817)

(14.206762606502263, 14.1021588144674)

(-54.996040561626074, -54.9687831276782)

(26.7408124348997, 26.6848677639468)

(86.40536771426336, -86.3880121355511)

(-51.85554642324937, 51.8266404947564)

(51.85554642324937, 51.8266404947564)

(-17.336002792703507, -17.2500209082052)

(83.26421124296907, 83.246201277169)

(-29.878512108153572, -29.8284160199475)

(-23.60413458219802, -23.5408031583217)

(-48.715193468809915, -48.6844270721817)

(11.08414892589577, -10.9508539438365)

(1.9371834086432644, 1.35840983054794)

(80.12308893081064, -80.1043733207136)

(-98.97027022679063, -98.9551171244958)

(-4.900032084108042, -4.61449316680033)

(-11.08414892589577, -10.9508539438365)

(42.435035809592144, -42.3997251846591)


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=−20.4689380267502x_{1} = -20.4689380267502
x2=−39.2953181686786x_{2} = -39.2953181686786
x3=33.0169458205959x_{3} = 33.0169458205959
x4=−83.2642112429691x_{4} = -83.2642112429691
x5=−76.9820049516737x_{5} = -76.9820049516737
x6=20.4689380267502x_{6} = 20.4689380267502
x7=−64.4181642526582x_{7} = -64.4181642526582
x8=−70.6999723871001x_{8} = -70.6999723871001
x9=−45.575010738978x_{9} = -45.575010738978
x10=76.9820049516737x_{10} = 76.9820049516737
x11=89.5465547553057x_{11} = 89.5465547553057
x12=64.4181642526582x_{12} = 64.4181642526582
x13=70.6999723871001x_{13} = 70.6999723871001
x14=−14.2067626065023x_{14} = -14.2067626065023
x15=−89.5465547553057x_{15} = -89.5465547553057
x16=−7.97511398695608x_{16} = -7.97511398695608
x17=−58.1366530875265x_{17} = -58.1366530875265
x18=−26.7408124348997x_{18} = -26.7408124348997
x19=−1.93718340864326x_{19} = -1.93718340864326
x20=58.1366530875265x_{20} = 58.1366530875265
x21=−95.8290085380668x_{21} = -95.8290085380668
x22=45.575010738978x_{22} = 45.575010738978
x23=95.8290085380668x_{23} = 95.8290085380668
x24=7.97511398695608x_{24} = 7.97511398695608
x25=−33.0169458205959x_{25} = -33.0169458205959
x26=39.2953181686786x_{26} = 39.2953181686786
x27=14.2067626065023x_{27} = 14.2067626065023
x28=26.7408124348997x_{28} = 26.7408124348997
x29=−51.8555464232494x_{29} = -51.8555464232494
x30=51.8555464232494x_{30} = 51.8555464232494
x31=83.2642112429691x_{31} = 83.2642112429691
x32=1.93718340864326x_{32} = 1.93718340864326
Puntos máximos de la función:
x32=61.2773658575433x_{32} = 61.2773658575433
x32=23.604134582198x_{32} = 23.604134582198
x32=36.1559243301836x_{32} = 36.1559243301836
x32=4.90003208410804x_{32} = 4.90003208410804
x32=−42.4350358095921x_{32} = -42.4350358095921
x32=67.5590363627001x_{32} = 67.5590363627001
x32=92.6877692623911x_{32} = 92.6877692623911
x32=29.8785121081536x_{32} = 29.8785121081536
x32=−36.1559243301836x_{32} = -36.1559243301836
x32=−92.6877692623911x_{32} = -92.6877692623911
x32=−86.4053677142634x_{32} = -86.4053677142634
x32=−61.2773658575433x_{32} = -61.2773658575433
x32=−67.5590363627001x_{32} = -67.5590363627001
x32=17.3360027927035x_{32} = 17.3360027927035
x32=54.9960405616261x_{32} = 54.9960405616261
x32=98.9702702267906x_{32} = 98.9702702267906
x32=−80.1230889308106x_{32} = -80.1230889308106
x32=−73.8409641878896x_{32} = -73.8409641878896
x32=73.8409641878896x_{32} = 73.8409641878896
x32=48.7151934688099x_{32} = 48.7151934688099
x32=−54.9960405616261x_{32} = -54.9960405616261
x32=86.4053677142634x_{32} = 86.4053677142634
x32=−17.3360027927035x_{32} = -17.3360027927035
x32=−29.8785121081536x_{32} = -29.8785121081536
x32=−23.604134582198x_{32} = -23.604134582198
x32=−48.7151934688099x_{32} = -48.7151934688099
x32=11.0841489258958x_{32} = 11.0841489258958
x32=80.1230889308106x_{32} = 80.1230889308106
x32=−98.9702702267906x_{32} = -98.9702702267906
x32=−4.90003208410804x_{32} = -4.90003208410804
x32=−11.0841489258958x_{32} = -11.0841489258958
x32=42.4350358095921x_{32} = 42.4350358095921
Decrece en los intervalos
[95.8290085380668,∞)\left[95.8290085380668, \infty\right)
Crece en los intervalos
(−∞,−95.8290085380668]\left(-\infty, -95.8290085380668\right]
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→−∞(x+sin⁡(2x)sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x + \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x+sin⁡(2x)sin⁡(x))y = \lim_{x \to \infty}\left(\frac{x + \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x + sin(2*x))/sin(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⁡(2x)xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x + \sin{\left(2 x \right)}}{x \sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(x+sin⁡(2x)xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{x + \sin{\left(2 x \right)}}{x \sin{\left(x \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⁡(2x)sin⁡(x)=−−x−sin⁡(2x)sin⁡(x)\frac{x + \sin{\left(2 x \right)}}{\sin{\left(x \right)}} = - \frac{- x - \sin{\left(2 x \right)}}{\sin{\left(x \right)}}
- No
x+sin⁡(2x)sin⁡(x)=−x−sin⁡(2x)sin⁡(x)\frac{x + \sin{\left(2 x \right)}}{\sin{\left(x \right)}} = \frac{- x - \sin{\left(2 x \right)}}{\sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (x+sin(2*x))/sin(x)
    © Sr Examen – Калькулятор