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Trazado el gráfico de la función/ x^2/sinx

Gráfico de la función y = x^2/sinx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2  
         x   
f(x) = ------
       sin(x)
f(x)=x2sin⁡(x)f{\left(x \right)} = \frac{x^{2}}{\sin{\left(x \right)}} 
f = x^2/sin(x)
Gráfico de la función
02468-8-6-4-2-1010-5000050000
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:
x2sin⁡(x)=0\frac{x^{2}}{\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/sin(x).
02sin⁡(0)\frac{0^{2}}{\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
−x2cos⁡(x)sin⁡2(x)+2xsin⁡(x)=0- \frac{x^{2} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 x}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=29.7780674009765x_{1} = 29.7780674009765
x2=10.8126733338873x_{2} = 10.8126733338873
x3=23.4769601879883x_{3} = 23.4769601879883
x4=73.8003338423053x_{4} = 73.8003338423053
x5=36.0729289833362x_{5} = 36.0729289833362
x6=−17.1627513884202x_{6} = -17.1627513884202
x7=−36.0729289833362x_{7} = -36.0729289833362
x8=−80.0856445915527x_{8} = -80.0856445915527
x9=83.2281796214841x_{9} = 83.2281796214841
x10=−70.6575367178468x_{10} = -70.6575367178468
x11=−64.3715897831264x_{11} = -64.3715897831264
x12=−29.7780674009765x_{12} = -29.7780674009765
x13=−42.3643263176719x_{13} = -42.3643263176719
x14=17.1627513884202x_{14} = 17.1627513884202
x15=−92.6554012744443x_{15} = -92.6554012744443
x16=76.9430326079594x_{16} = 76.9430326079594
x17=−51.797686192112x_{17} = -51.797686192112
x18=−86.3706460958767x_{18} = -86.3706460958767
x19=89.5130512336412x_{19} = 89.5130512336412
x20=32.9260552340905x_{20} = 32.9260552340905
x21=−23.4769601879883x_{21} = -23.4769601879883
x22=58.0850454185866x_{22} = 58.0850454185866
x23=26.6285710115144x_{23} = 26.6285710115144
x24=−10.8126733338873x_{24} = -10.8126733338873
x25=−45.5091745543365x_{25} = -45.5091745543365
x26=−4.27478227145813x_{26} = -4.27478227145813
x27=−98.9399570606555x_{27} = -98.9399570606555
x28=70.6575367178468x_{28} = 70.6575367178468
x29=64.3715897831264x_{29} = 64.3715897831264
x30=−54.9414851392857x_{30} = -54.9414851392857
x31=−61.2284037765214x_{31} = -61.2284037765214
x32=45.5091745543365x_{32} = 45.5091745543365
x33=7.59654601975059x_{33} = 7.59654601975059
x34=−13.9952220914795x_{34} = -13.9952220914795
x35=95.7977016393173x_{35} = 95.7977016393173
x36=92.6554012744443x_{36} = 92.6554012744443
x37=13.9952220914795x_{37} = 13.9952220914795
x38=−48.6536023357065x_{38} = -48.6536023357065
x39=98.9399570606555x_{39} = 98.9399570606555
x40=54.9414851392857x_{40} = 54.9414851392857
x41=61.2284037765214x_{41} = 61.2284037765214
x42=20.3222538599925x_{42} = 20.3222538599925
x43=−89.5130512336412x_{43} = -89.5130512336412
x44=−67.5146275025823x_{44} = -67.5146275025823
x45=−20.3222538599925x_{45} = -20.3222538599925
x46=−83.2281796214841x_{46} = -83.2281796214841
x47=−58.0850454185866x_{47} = -58.0850454185866
x48=−39.2189565596149x_{48} = -39.2189565596149
x49=−32.9260552340905x_{49} = -32.9260552340905
x50=86.3706460958767x_{50} = 86.3706460958767
x51=51.797686192112x_{51} = 51.797686192112
x52=−76.9430326079594x_{52} = -76.9430326079594
x53=48.6536023357065x_{53} = 48.6536023357065
x54=42.3643263176719x_{54} = 42.3643263176719
x55=−26.6285710115144x_{55} = -26.6285710115144
x56=−7.59654601975059x_{56} = -7.59654601975059
x57=39.2189565596149x_{57} = 39.2189565596149
x58=−73.8003338423053x_{58} = -73.8003338423053
x59=80.0856445915527x_{59} = 80.0856445915527
x60=4.27478227145813x_{60} = 4.27478227145813
x61=67.5146275025823x_{61} = 67.5146275025823
x62=−95.7977016393173x_{62} = -95.7977016393173
Signos de extremos en los puntos:
(29.778067400976507, -888.731047740343)

(10.812673333887274, -118.89708454478)

(23.4769601879883, -553.164044116211)

(73.80033384230535, -5448.48890816148)

(36.07292898333623, -1303.25467081825)

(-17.162751388420226, 296.553291146996)

(-36.07292898333623, 1303.25467081825)

(-80.0856445915527, 6415.71015790972)

(83.22817962148409, 6928.92959446115)

(-70.65753671784677, -4994.48709459236)

(-64.37158978312642, -4145.70108877969)

(-29.778067400976507, 888.731047740343)

(-42.3643263176719, 1796.73503122035)

(17.162751388420226, -296.553291146996)

(-92.65540127444433, 8587.02315241872)

(76.94303260795941, 5922.22992919888)

(-51.79768619211198, -2684.99954997753)

(-86.37064609587671, 7461.88823899052)

(89.51305123364119, 8014.58609161146)

(32.926055234090526, 1086.12327186833)

(-23.4769601879883, 553.164044116211)

(58.08504541858663, 3375.87190883978)

(26.62857101151445, 711.077981489794)

(-10.812673333887274, 118.89708454478)

(-45.509174554336525, -2073.08400387105)

(-4.274782271458128, 20.1748726184708)

(-98.93995706065554, 9791.11489889754)

(70.65753671784677, 4994.48709459236)

(64.37158978312642, 4145.70108877969)

(-54.941485139285724, 3020.56612718289)

(-61.2284037765214, 3750.91689581782)

(45.509174554336525, 2073.08400387105)

(7.596546019750588, 59.6740054059227)

(-13.995222091479503, -197.856133293211)

(95.79770163931728, 9179.19942149173)

(92.65540127444433, -8587.02315241872)

(13.995222091479503, 197.856133293211)

(-48.653602335706516, 2369.1721760645)

(98.93995706065554, -9791.11489889754)

(54.941485139285724, -3020.56612718289)

(61.2284037765214, -3750.91689581782)

(20.32225385999246, 414.989182575231)

(-89.51305123364119, -8014.58609161146)

(-67.51462750258234, 4560.22448823757)

(-20.32225385999246, -414.989182575231)

(-83.22817962148409, -6928.92959446115)

(-58.08504541858663, -3375.87190883978)

(-39.21895655961492, -1540.12525502983)

(-32.926055234090526, -1086.12327186833)

(86.37064609587671, -7461.88823899052)

(51.79768619211198, 2684.99954997753)

(-76.94303260795941, -5922.22992919888)

(48.653602335706516, -2369.1721760645)

(42.3643263176719, -1796.73503122035)

(-26.62857101151445, -711.077981489794)

(-7.596546019750588, -59.6740054059227)

(39.21895655961492, 1540.12525502983)

(-73.80033384230535, 5448.48890816148)

(80.0856445915527, -6415.71015790972)

(4.274782271458128, -20.1748726184708)

(67.51462750258234, -4560.22448823757)

(-95.79770163931728, -9179.19942149173)


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=−17.1627513884202x_{1} = -17.1627513884202
x2=−36.0729289833362x_{2} = -36.0729289833362
x3=−80.0856445915527x_{3} = -80.0856445915527
x4=83.2281796214841x_{4} = 83.2281796214841
x5=−29.7780674009765x_{5} = -29.7780674009765
x6=−42.3643263176719x_{6} = -42.3643263176719
x7=−92.6554012744443x_{7} = -92.6554012744443
x8=76.9430326079594x_{8} = 76.9430326079594
x9=−86.3706460958767x_{9} = -86.3706460958767
x10=89.5130512336412x_{10} = 89.5130512336412
x11=32.9260552340905x_{11} = 32.9260552340905
x12=−23.4769601879883x_{12} = -23.4769601879883
x13=58.0850454185866x_{13} = 58.0850454185866
x14=26.6285710115144x_{14} = 26.6285710115144
x15=−10.8126733338873x_{15} = -10.8126733338873
x16=−4.27478227145813x_{16} = -4.27478227145813
x17=−98.9399570606555x_{17} = -98.9399570606555
x18=70.6575367178468x_{18} = 70.6575367178468
x19=64.3715897831264x_{19} = 64.3715897831264
x20=−54.9414851392857x_{20} = -54.9414851392857
x21=−61.2284037765214x_{21} = -61.2284037765214
x22=45.5091745543365x_{22} = 45.5091745543365
x23=7.59654601975059x_{23} = 7.59654601975059
x24=95.7977016393173x_{24} = 95.7977016393173
x25=13.9952220914795x_{25} = 13.9952220914795
x26=−48.6536023357065x_{26} = -48.6536023357065
x27=20.3222538599925x_{27} = 20.3222538599925
x28=−67.5146275025823x_{28} = -67.5146275025823
x29=51.797686192112x_{29} = 51.797686192112
x30=39.2189565596149x_{30} = 39.2189565596149
x31=−73.8003338423053x_{31} = -73.8003338423053
Puntos máximos de la función:
x31=29.7780674009765x_{31} = 29.7780674009765
x31=10.8126733338873x_{31} = 10.8126733338873
x31=23.4769601879883x_{31} = 23.4769601879883
x31=73.8003338423053x_{31} = 73.8003338423053
x31=36.0729289833362x_{31} = 36.0729289833362
x31=−70.6575367178468x_{31} = -70.6575367178468
x31=−64.3715897831264x_{31} = -64.3715897831264
x31=17.1627513884202x_{31} = 17.1627513884202
x31=−51.797686192112x_{31} = -51.797686192112
x31=−45.5091745543365x_{31} = -45.5091745543365
x31=−13.9952220914795x_{31} = -13.9952220914795
x31=92.6554012744443x_{31} = 92.6554012744443
x31=98.9399570606555x_{31} = 98.9399570606555
x31=54.9414851392857x_{31} = 54.9414851392857
x31=61.2284037765214x_{31} = 61.2284037765214
x31=−89.5130512336412x_{31} = -89.5130512336412
x31=−20.3222538599925x_{31} = -20.3222538599925
x31=−83.2281796214841x_{31} = -83.2281796214841
x31=−58.0850454185866x_{31} = -58.0850454185866
x31=−39.2189565596149x_{31} = -39.2189565596149
x31=−32.9260552340905x_{31} = -32.9260552340905
x31=86.3706460958767x_{31} = 86.3706460958767
x31=−76.9430326079594x_{31} = -76.9430326079594
x31=48.6536023357065x_{31} = 48.6536023357065
x31=42.3643263176719x_{31} = 42.3643263176719
x31=−26.6285710115144x_{31} = -26.6285710115144
x31=−7.59654601975059x_{31} = -7.59654601975059
x31=80.0856445915527x_{31} = 80.0856445915527
x31=4.27478227145813x_{31} = 4.27478227145813
x31=67.5146275025823x_{31} = 67.5146275025823
x31=−95.7977016393173x_{31} = -95.7977016393173
Decrece en los intervalos
[95.7977016393173,∞)\left[95.7977016393173, \infty\right)
Crece en los intervalos
(−∞,−98.9399570606555]\left(-\infty, -98.9399570606555\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
x2(1+2cos⁡2(x)sin⁡2(x))−4xcos⁡(x)sin⁡(x)+2sin⁡(x)=0\frac{x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{4 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.96097948476164⋅10−15x_{1} = -1.96097948476164 \cdot 10^{-15}
x2=1.1380968926558⋅10−16x_{2} = 1.1380968926558 \cdot 10^{-16}
x3=6.55263526323316⋅10−15x_{3} = 6.55263526323316 \cdot 10^{-15}
x4=1.26872880450359⋅10−14x_{4} = 1.26872880450359 \cdot 10^{-14}
x5=2.61071439835042⋅10−18x_{5} = 2.61071439835042 \cdot 10^{-18}
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979

lim⁡x→0−(x2(1+2cos⁡2(x)sin⁡2(x))−4xcos⁡(x)sin⁡(x)+2sin⁡(x))=0\lim_{x \to 0^-}\left(\frac{x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{4 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2}{\sin{\left(x \right)}}\right) = 0
lim⁡x→0+(x2(1+2cos⁡2(x)sin⁡2(x))−4xcos⁡(x)sin⁡(x)+2sin⁡(x))=0\lim_{x \to 0^+}\left(\frac{x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{4 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2}{\sin{\left(x \right)}}\right) = 0
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→3.14159265358979−(x2(1+2cos⁡2(x)sin⁡2(x))−4xcos⁡(x)sin⁡(x)+2sin⁡(x))=1.07472459029048⋅1049\lim_{x \to 3.14159265358979^-}\left(\frac{x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{4 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2}{\sin{\left(x \right)}}\right) = 1.07472459029048 \cdot 10^{49}
lim⁡x→3.14159265358979+(x2(1+2cos⁡2(x)sin⁡2(x))−4xcos⁡(x)sin⁡(x)+2sin⁡(x))=1.07472459029048⋅1049\lim_{x \to 3.14159265358979^+}\left(\frac{x^{2} \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{4 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2}{\sin{\left(x \right)}}\right) = 1.07472459029048 \cdot 10^{49}
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[1.26872880450359⋅10−14,∞)\left[1.26872880450359 \cdot 10^{-14}, \infty\right)
Convexa en los intervalos
(−∞,−1.96097948476164⋅10−15]\left(-\infty, -1.96097948476164 \cdot 10^{-15}\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→−∞(x2sin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x^{2}}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x2sin⁡(x))y = \lim_{x \to \infty}\left(\frac{x^{2}}{\sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^2/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→−∞(xsin⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(xsin⁡(x))y = x \lim_{x \to \infty}\left(\frac{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:
x2sin⁡(x)=−x2sin⁡(x)\frac{x^{2}}{\sin{\left(x \right)}} = - \frac{x^{2}}{\sin{\left(x \right)}}
- No
x2sin⁡(x)=x2sin⁡(x)\frac{x^{2}}{\sin{\left(x \right)}} = \frac{x^{2}}{\sin{\left(x \right)}}
- Sí
es decir, función
es
impar
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