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Trazado el gráfico de la función/ (x^2+2cosx)/sinx

Gráfico de la función y = (x^2+2cosx)/sinx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        2           
       x  + 2*cos(x)
f(x) = -------------
           sin(x)
f(x)=x2+2cos⁡(x)sin⁡(x)f{\left(x \right)} = \frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}} 
f = (x^2 + 2*cos(x))/sin(x)
Gráfico de la función
10.000010.010010.001010.002010.003010.004010.005010.006010.007010.008010.0090-180.74-180.72
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:
x2+2cos⁡(x)sin⁡(x)=0\frac{x^{2} + 2 \cos{\left(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^2 + 2*cos(x))/sin(x).
02+2cos⁡(0)sin⁡(0)\frac{0^{2} + 2 \cos{\left(0 \right)}}{\sin{\left(0 \right)}}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
2x−2sin⁡(x)sin⁡(x)−(x2+2cos⁡(x))cos⁡(x)sin⁡2(x)=0\frac{2 x - 2 \sin{\left(x \right)}}{\sin{\left(x \right)}} - \frac{\left(x^{2} + 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−92.6551683094389x_{1} = -92.6551683094389
x2=−26.6313909606767x_{2} = -26.6313909606767
x3=98.9397527512558x_{3} = 98.9397527512558
x4=32.9278998310383x_{4} = 32.9278998310383
x5=1.2849319665371x_{5} = 1.2849319665371
x6=80.0853327571602x_{6} = 80.0853327571602
x7=17.155956342572x_{7} = 17.155956342572
x8=−7.63088453995118x_{8} = -7.63088453995118
x9=39.2202567558535x_{9} = 39.2202567558535
x10=−61.2278702799341x_{10} = -61.2278702799341
x11=26.6313909606767x_{11} = 26.6313909606767
x12=−14.0054180069591x_{12} = -14.0054180069591
x13=−32.9278998310383x_{13} = -32.9278998310383
x14=42.363211889599x_{14} = 42.363211889599
x15=10.7955143777089x_{15} = 10.7955143777089
x16=−23.4733304235899x_{16} = -23.4733304235899
x17=89.5133008395388x_{17} = 89.5133008395388
x18=36.0713918773106x_{18} = 36.0713918773106
x19=−39.2202567558535x_{19} = -39.2202567558535
x20=45.5101401905028x_{20} = 45.5101401905028
x21=−45.5101401905028x_{21} = -45.5101401905028
x22=−1.2849319665371x_{22} = -1.2849319665371
x23=−98.9397527512558x_{23} = -98.9397527512558
x24=−48.6527574169176x_{24} = -48.6527574169176
x25=23.4733304235899x_{25} = 23.4733304235899
x26=−89.5133008395388x_{26} = -89.5133008395388
x27=−29.775811593805x_{27} = -29.775811593805
x28=61.2278702799341x_{28} = 61.2278702799341
x29=92.6551683094389x_{29} = 92.6551683094389
x30=−36.0713918773106x_{30} = -36.0713918773106
x31=95.7979195697262x_{31} = 95.7979195697262
x32=−95.7979195697262x_{32} = -95.7979195697262
x33=20.3270942073626x_{33} = 20.3270942073626
x34=−83.2284683476736x_{34} = -83.2284683476736
x35=−80.0853327571602x_{35} = -80.0853327571602
x36=−70.6579373151886x_{36} = -70.6579373151886
x37=29.775811593805x_{37} = 29.775811593805
x38=−58.0856381970501x_{38} = -58.0856381970501
x39=−86.3703779936973x_{39} = -86.3703779936973
x40=51.7984316046612x_{40} = 51.7984316046612
x41=−73.7999666296462x_{41} = -73.7999666296462
x42=−54.9408225573151x_{42} = -54.9408225573151
x43=−76.9433704296783x_{43} = -76.9433704296783
x44=54.9408225573151x_{44} = 54.9408225573151
x45=−64.3720724360942x_{45} = -64.3720724360942
x46=−51.7984316046612x_{46} = -51.7984316046612
x47=70.6579373151886x_{47} = 70.6579373151886
x48=64.3720724360942x_{48} = 64.3720724360942
x49=14.0054180069591x_{49} = 14.0054180069591
x50=67.5141887296183x_{50} = 67.5141887296183
x51=−20.3270942073626x_{51} = -20.3270942073626
x52=−10.7955143777089x_{52} = -10.7955143777089
x53=48.6527574169176x_{53} = 48.6527574169176
x54=−17.155956342572x_{54} = -17.155956342572
x55=−67.5141887296183x_{55} = -67.5141887296183
x56=−4.15988080716962x_{56} = -4.15988080716962
x57=7.63088453995118x_{57} = 7.63088453995118
x58=−42.363211889599x_{58} = -42.363211889599
x59=58.0856381970501x_{59} = 58.0856381970501
x60=73.7999666296462x_{60} = 73.7999666296462
x61=86.3703779936973x_{61} = 86.3703779936973
x62=83.2284683476736x_{62} = 83.2284683476736
x63=4.15988080716962x_{63} = 4.15988080716962
x64=76.9433704296783x_{64} = 76.9433704296783
Signos de extremos en los puntos:
(-92.65516830943886, 8586.97974862893)

(-26.631390960676704, -711.225360426234)

(98.93975275125575, -9791.07426594427)

(32.92789983103829, 1086.24290487152)

(1.2849319665370957, 2.3087155001914)

(80.08533275716019, -6415.65989935066)

(17.155956342571987, -296.313339112184)

(-7.630884539951176, -60.1639505608096)

(39.220256755853505, 1540.22594297569)

(-61.22787027993408, 3750.85103259194)

(26.631390960676704, 711.225360426234)

(-14.00541800695914, -198.131545970685)

(-32.92789983103829, -1086.24290487152)

(42.36321188959896, -1796.63949522568)

(10.795514377708916, -118.509382882767)

(-23.473330423589946, 552.990007818584)

(89.51330083953881, 8014.63052810254)

(36.07139187731056, -1303.14224246637)

(-39.220256755853505, -1540.22594297569)

(45.510140190502824, 2073.17093074854)

(-45.510140190502824, -2073.17093074854)

(-1.2849319665370957, -2.3087155001914)

(-98.93975275125575, 9791.07426594427)

(-48.652757416917595, 2369.08911585739)

(23.473330423589946, -552.990007818584)

(-89.51330083953881, -8014.63052810254)

(-29.77581159380503, 888.594454590108)

(61.22787027993408, -3750.85103259194)

(92.65516830943886, -8586.97974862893)

(-36.07139187731056, 1303.14224246637)

(95.79791956972616, 9179.24095812185)

(-95.79791956972616, -9179.24095812185)

(20.327094207362574, 415.181124677421)

(-83.2284683476736, -6928.97736621397)

(-80.08533275716019, 6415.65989935066)

(-70.65793731518859, -4994.54330476316)

(29.77581159380503, -888.594454590108)

(-58.08563819705012, -3375.9401799037)

(-86.3703779936973, 7461.84165871348)

(51.79843160466122, 2685.07602698745)

(-73.79996662964622, 5448.43434038109)

(-54.9408225573151, 3020.4926589874)

(-76.94337042967828, -5922.28157766336)

(54.9408225573151, -3020.4926589874)

(-64.37207243609416, -4145.76274487732)

(-51.79843160466122, -2685.07602698745)

(70.65793731518859, 4994.54330476316)

(64.37207243609416, 4145.76274487732)

(14.00541800695914, 198.131545970685)

(67.51418872961827, -4560.16480265732)

(-20.327094207362574, -415.181124677421)

(-10.795514377708916, 118.509382882767)

(48.652757416917595, -2369.08911585739)

(-17.155956342571987, 296.313339112184)

(-67.51418872961827, 4560.16480265732)

(-4.1598808071696185, 19.0962798149071)

(7.630884539951176, 60.1639505608096)

(-42.36321188959896, 1796.63949522568)

(58.08563819705012, 3375.9401799037)

(73.79996662964622, -5448.43434038109)

(86.3703779936973, -7461.84165871348)

(83.2284683476736, 6928.97736621397)

(4.1598808071696185, -19.0962798149071)

(76.94337042967828, 5922.28157766336)


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=−92.6551683094389x_{1} = -92.6551683094389
x2=32.9278998310383x_{2} = 32.9278998310383
x3=1.2849319665371x_{3} = 1.2849319665371
x4=39.2202567558535x_{4} = 39.2202567558535
x5=−61.2278702799341x_{5} = -61.2278702799341
x6=26.6313909606767x_{6} = 26.6313909606767
x7=−23.4733304235899x_{7} = -23.4733304235899
x8=89.5133008395388x_{8} = 89.5133008395388
x9=45.5101401905028x_{9} = 45.5101401905028
x10=−98.9397527512558x_{10} = -98.9397527512558
x11=−48.6527574169176x_{11} = -48.6527574169176
x12=−29.775811593805x_{12} = -29.775811593805
x13=−36.0713918773106x_{13} = -36.0713918773106
x14=95.7979195697262x_{14} = 95.7979195697262
x15=20.3270942073626x_{15} = 20.3270942073626
x16=−80.0853327571602x_{16} = -80.0853327571602
x17=−86.3703779936973x_{17} = -86.3703779936973
x18=51.7984316046612x_{18} = 51.7984316046612
x19=−73.7999666296462x_{19} = -73.7999666296462
x20=−54.9408225573151x_{20} = -54.9408225573151
x21=70.6579373151886x_{21} = 70.6579373151886
x22=64.3720724360942x_{22} = 64.3720724360942
x23=14.0054180069591x_{23} = 14.0054180069591
x24=−10.7955143777089x_{24} = -10.7955143777089
x25=−17.155956342572x_{25} = -17.155956342572
x26=−67.5141887296183x_{26} = -67.5141887296183
x27=−4.15988080716962x_{27} = -4.15988080716962
x28=7.63088453995118x_{28} = 7.63088453995118
x29=−42.363211889599x_{29} = -42.363211889599
x30=58.0856381970501x_{30} = 58.0856381970501
x31=83.2284683476736x_{31} = 83.2284683476736
x32=76.9433704296783x_{32} = 76.9433704296783
Puntos máximos de la función:
x32=−26.6313909606767x_{32} = -26.6313909606767
x32=98.9397527512558x_{32} = 98.9397527512558
x32=80.0853327571602x_{32} = 80.0853327571602
x32=17.155956342572x_{32} = 17.155956342572
x32=−7.63088453995118x_{32} = -7.63088453995118
x32=−14.0054180069591x_{32} = -14.0054180069591
x32=−32.9278998310383x_{32} = -32.9278998310383
x32=42.363211889599x_{32} = 42.363211889599
x32=10.7955143777089x_{32} = 10.7955143777089
x32=36.0713918773106x_{32} = 36.0713918773106
x32=−39.2202567558535x_{32} = -39.2202567558535
x32=−45.5101401905028x_{32} = -45.5101401905028
x32=−1.2849319665371x_{32} = -1.2849319665371
x32=23.4733304235899x_{32} = 23.4733304235899
x32=−89.5133008395388x_{32} = -89.5133008395388
x32=61.2278702799341x_{32} = 61.2278702799341
x32=92.6551683094389x_{32} = 92.6551683094389
x32=−95.7979195697262x_{32} = -95.7979195697262
x32=−83.2284683476736x_{32} = -83.2284683476736
x32=−70.6579373151886x_{32} = -70.6579373151886
x32=29.775811593805x_{32} = 29.775811593805
x32=−58.0856381970501x_{32} = -58.0856381970501
x32=−76.9433704296783x_{32} = -76.9433704296783
x32=54.9408225573151x_{32} = 54.9408225573151
x32=−64.3720724360942x_{32} = -64.3720724360942
x32=−51.7984316046612x_{32} = -51.7984316046612
x32=67.5141887296183x_{32} = 67.5141887296183
x32=−20.3270942073626x_{32} = -20.3270942073626
x32=48.6527574169176x_{32} = 48.6527574169176
x32=73.7999666296462x_{32} = 73.7999666296462
x32=86.3703779936973x_{32} = 86.3703779936973
x32=4.15988080716962x_{32} = 4.15988080716962
Decrece en los intervalos
[95.7979195697262,∞)\left[95.7979195697262, \infty\right)
Crece en los intervalos
(−∞,−98.9397527512558]\left(-\infty, -98.9397527512558\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
(1+2cos⁡2(x)sin⁡2(x))(x2+2cos⁡(x))−4(x−sin⁡(x))cos⁡(x)sin⁡(x)−2cos⁡(x)+2sin⁡(x)=0\frac{\left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(x^{2} + 2 \cos{\left(x \right)}\right) - \frac{4 \left(x - \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - 2 \cos{\left(x \right)} + 2}{\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→−∞(x2+2cos⁡(x)sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x^{2} + 2 \cos{\left(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→∞(x2+2cos⁡(x)sin⁡(x))y = \lim_{x \to \infty}\left(\frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x^2 + 2*cos(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→−∞(x2+2cos⁡(x)xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x^{2} + 2 \cos{\left(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→∞(x2+2cos⁡(x)xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{x^{2} + 2 \cos{\left(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:
x2+2cos⁡(x)sin⁡(x)=−x2+2cos⁡(x)sin⁡(x)\frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}} = - \frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
x2+2cos⁡(x)sin⁡(x)=x2+2cos⁡(x)sin⁡(x)\frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}} = \frac{x^{2} + 2 \cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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