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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x + sin(x)
f(x) = -----------
        2         
       x  + cos(x)
f(x)=x+sin⁡(x)x2+cos⁡(x)f{\left(x \right)} = \frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} 
f = (x + sin(x))/(x^2 + cos(x))
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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)x2+cos⁡(x)=0\frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0x_{1} = 0
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^2 + cos(x)).
sin⁡(0)02+cos⁡(0)\frac{\sin{\left(0 \right)}}{0^{2} + \cos{\left(0 \right)}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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+sin⁡(x))(x+sin⁡(x))(x2+cos⁡(x))2+cos⁡(x)+1x2+cos⁡(x)=0\frac{\left(- 2 x + \sin{\left(x \right)}\right) \left(x + \sin{\left(x \right)}\right)}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)} + 1}{x^{2} + \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=43.9086304391986x_{1} = 43.9086304391986
x2=6.4735496375984x_{2} = 6.4735496375984
x3=−37.6131292591587x_{3} = -37.6131292591587
x4=−81.6965386967262x_{4} = -81.6965386967262
x5=100.543258652178x_{5} = 100.543258652178
x6=25.0034968231157x_{6} = 25.0034968231157
x7=−94.2608926838609x_{7} = -94.2608926838609
x8=75.3552861438656x_{8} = 75.3552861438656
x9=31.455217599475x_{9} = 31.455217599475
x10=−6.4735496375984x_{10} = -6.4735496375984
x11=−62.8515189153841x_{11} = -62.8515189153841
x12=5.73303466805554x_{12} = 5.73303466805554
x13=−75.3552861438656x_{13} = -75.3552861438656
x14=−18.6767199929799x_{14} = -18.6767199929799
x15=−87.9277947562266x_{15} = -87.9277947562266
x16=69.0681937171023x_{16} = 69.0681937171023
x17=−69.0681937171023x_{17} = -69.0681937171023
x18=−43.9086304391986x_{18} = -43.9086304391986
x19=−50.2900599933538x_{19} = -50.2900599933538
x20=87.9277947562266x_{20} = 87.9277947562266
x21=31.3126714581803x_{21} = 31.3126714581803
x22=−5.73303466805554x_{22} = -5.73303466805554
x23=50.2900599933538x_{23} = 50.2900599933538
x24=−18.9148816474771x_{24} = -18.9148816474771
x25=113.108263605015x_{25} = 113.108263605015
x26=37.6131292591587x_{26} = 37.6131292591587
x27=50.2010425988319x_{27} = 50.2010425988319
x28=−56.5705169339472x_{28} = -56.5705169339472
x29=56.5705169339472x_{29} = 56.5705169339472
x30=56.4913991620551x_{30} = 56.4913991620551
x31=−25.0034968231157x_{31} = -25.0034968231157
x32=−87.9786437060689x_{32} = -87.9786437060689
x33=−50.2010425988319x_{33} = -50.2010425988319
x34=−25.1818171106898x_{34} = -25.1818171106898
x35=−100.543258652178x_{35} = -100.543258652178
x36=−12.6638988770814x_{36} = -12.6638988770814
x37=−31.455217599475x_{37} = -31.455217599475
x38=75.4146136170676x_{38} = 75.4146136170676
x39=62.780318581017x_{39} = 62.780318581017
x40=−12.3048802608754x_{40} = -12.3048802608754
x41=−44.0103811318779x_{41} = -44.0103811318779
x42=81.6965386967262x_{42} = 81.6965386967262
x43=12.3048802608754x_{43} = 12.3048802608754
x44=−94.2134347195882x_{44} = -94.2134347195882
x45=−904.775107583154x_{45} = -904.775107583154
x46=12.6638988770814x_{46} = 12.6638988770814
x47=−69.1329174891759x_{47} = -69.1329174891759
x48=−81.6417767790886x_{48} = -81.6417767790886
x49=37.7318681711555x_{49} = 37.7318681711555
x50=−100.498767620427x_{50} = -100.498767620427
x51=94.2608926838609x_{51} = 94.2608926838609
x52=62.8515189153841x_{52} = 62.8515189153841
x53=44.0103811318779x_{53} = 44.0103811318779
x54=−56.4913991620551x_{54} = -56.4913991620551
x55=−31.3126714581803x_{55} = -31.3126714581803
x56=18.6767199929799x_{56} = 18.6767199929799
x57=25.1818171106898x_{57} = 25.1818171106898
x58=100.498767620427x_{58} = 100.498767620427
x59=−75.4146136170676x_{59} = -75.4146136170676
x60=87.9786437060689x_{60} = 87.9786437060689
x61=69.1329174891759x_{61} = 69.1329174891759
x62=81.6417767790886x_{62} = 81.6417767790886
x63=−62.780318581017x_{63} = -62.780318581017
x64=94.2134347195882x_{64} = 94.2134347195882
x65=18.9148816474771x_{65} = 18.9148816474771
x66=−37.7318681711555x_{66} = -37.7318681711555
Signos de extremos en los puntos:
(43.908630439198596, 0.0227246361278357)

(6.473549637598396, 0.155349866598708)

(-37.613129259158654, -0.0265070930364385)

(-81.69653869672617, -0.0122408536123202)

(100.54325865217778, 0.0099461999375388)

(25.003496823115675, 0.0397252350960918)

(-94.26089268386087, -0.0106091353599299)

(75.35528614386557, 0.0132605768352458)

(31.45521759947502, 0.0317988150447749)

(-6.473549637598396, -0.155349866598708)

(-62.85151891538414, -0.0159114653095468)

(5.7330346680555415, 0.15451359971509)

(-75.35528614386557, -0.0132605768352458)

(-18.676719992979912, -0.0529001724657087)

(-87.9277947562266, -0.0113667400750096)

(69.06819371710233, 0.0144655990958247)

(-69.06819371710233, -0.0144655990958247)

(-43.908630439198596, -0.0227246361278357)

(-50.290059993353765, -0.0198865014262232)

(87.9277947562266, 0.0113667400750096)

(31.31267145818032, 0.0317985707321494)

(-5.7330346680555415, -0.15451359971509)

(50.290059993353765, 0.0198865014262232)

(-18.914881647477134, -0.0529033319326675)

(113.10826360501456, 0.00884125014988744)

(37.613129259158654, 0.0265070930364385)

(50.2010425988319, 0.0198864781703916)

(-56.57051693394717, -0.0176783566023099)

(56.57051693394717, 0.0176783566023099)

(56.4913991620551, 0.0176783437002106)

(-25.003496823115675, -0.0397252350960918)

(-87.9786437060689, -0.0113667414907799)

(-50.2010425988319, -0.0198864781703916)

(-25.18181711068984, -0.0397259819833676)

(-100.54325865217778, -0.0099461999375388)

(-12.663898877081408, -0.0790810287310917)

(-31.45521759947502, -0.0317988150447749)

(75.4146136170676, 0.0132605798957128)

(62.780318581016964, 0.0159114576923506)

(-12.304880260875393, -0.0790567615089993)

(-44.01038113187789, -0.0227246814855013)

(81.69653869672617, 0.0122408536123202)

(12.304880260875393, 0.0790567615089993)

(-94.21343471958815, -0.010609134357267)

(-904.775107583154, -0.00110524131023034)

(12.663898877081408, 0.0790810287310917)

(-69.1329174891759, -0.014465603824878)

(-81.64177677908862, -0.0122408515614264)

(37.73186817115554, 0.0265071911277167)

(-100.49876762042679, -0.00994619921144368)

(94.26089268386087, 0.0106091353599299)

(62.85151891538414, 0.0159114653095468)

(44.01038113187789, 0.0227246814855013)

(-56.4913991620551, -0.0176783437002106)

(-31.31267145818032, -0.0317985707321494)

(18.676719992979912, 0.0529001724657087)

(25.18181711068984, 0.0397259819833676)

(100.49876762042679, 0.00994619921144368)

(-75.4146136170676, -0.0132605798957128)

(87.9786437060689, 0.0113667414907799)

(69.1329174891759, 0.014465603824878)

(81.64177677908862, 0.0122408515614264)

(-62.780318581016964, -0.0159114576923506)

(94.21343471958815, 0.010609134357267)

(18.914881647477134, 0.0529033319326675)

(-37.73186817115554, -0.0265071911277167)


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=43.9086304391986x_{1} = 43.9086304391986
x2=−81.6965386967262x_{2} = -81.6965386967262
x3=25.0034968231157x_{3} = 25.0034968231157
x4=−94.2608926838609x_{4} = -94.2608926838609
x5=−6.4735496375984x_{5} = -6.4735496375984
x6=−62.8515189153841x_{6} = -62.8515189153841
x7=5.73303466805554x_{7} = 5.73303466805554
x8=69.0681937171023x_{8} = 69.0681937171023
x9=−50.2900599933538x_{9} = -50.2900599933538
x10=31.3126714581803x_{10} = 31.3126714581803
x11=−18.9148816474771x_{11} = -18.9148816474771
x12=37.6131292591587x_{12} = 37.6131292591587
x13=50.2010425988319x_{13} = 50.2010425988319
x14=−56.5705169339472x_{14} = -56.5705169339472
x15=56.4913991620551x_{15} = 56.4913991620551
x16=−25.1818171106898x_{16} = -25.1818171106898
x17=−100.543258652178x_{17} = -100.543258652178
x18=−12.6638988770814x_{18} = -12.6638988770814
x19=−31.455217599475x_{19} = -31.455217599475
x20=62.780318581017x_{20} = 62.780318581017
x21=−44.0103811318779x_{21} = -44.0103811318779
x22=12.3048802608754x_{22} = 12.3048802608754
x23=18.6767199929799x_{23} = 18.6767199929799
x24=−75.4146136170676x_{24} = -75.4146136170676
x25=81.6417767790886x_{25} = 81.6417767790886
x26=94.2134347195882x_{26} = 94.2134347195882
x27=−37.7318681711555x_{27} = -37.7318681711555
Puntos máximos de la función:
x27=6.4735496375984x_{27} = 6.4735496375984
x27=−37.6131292591587x_{27} = -37.6131292591587
x27=100.543258652178x_{27} = 100.543258652178
x27=31.455217599475x_{27} = 31.455217599475
x27=−18.6767199929799x_{27} = -18.6767199929799
x27=−69.0681937171023x_{27} = -69.0681937171023
x27=−43.9086304391986x_{27} = -43.9086304391986
x27=−5.73303466805554x_{27} = -5.73303466805554
x27=50.2900599933538x_{27} = 50.2900599933538
x27=113.108263605015x_{27} = 113.108263605015
x27=56.5705169339472x_{27} = 56.5705169339472
x27=−25.0034968231157x_{27} = -25.0034968231157
x27=−50.2010425988319x_{27} = -50.2010425988319
x27=75.4146136170676x_{27} = 75.4146136170676
x27=−12.3048802608754x_{27} = -12.3048802608754
x27=81.6965386967262x_{27} = 81.6965386967262
x27=−94.2134347195882x_{27} = -94.2134347195882
x27=12.6638988770814x_{27} = 12.6638988770814
x27=−81.6417767790886x_{27} = -81.6417767790886
x27=37.7318681711555x_{27} = 37.7318681711555
x27=94.2608926838609x_{27} = 94.2608926838609
x27=62.8515189153841x_{27} = 62.8515189153841
x27=44.0103811318779x_{27} = 44.0103811318779
x27=−56.4913991620551x_{27} = -56.4913991620551
x27=−31.3126714581803x_{27} = -31.3126714581803
x27=25.1818171106898x_{27} = 25.1818171106898
x27=−62.780318581017x_{27} = -62.780318581017
x27=18.9148816474771x_{27} = 18.9148816474771
Decrece en los intervalos
[94.2134347195882,∞)\left[94.2134347195882, \infty\right)
Crece en los intervalos
(−∞,−100.543258652178]\left(-\infty, -100.543258652178\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x+sin⁡(x)x2+cos⁡(x))=0\lim_{x \to -\infty}\left(\frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(x+sin⁡(x)x2+cos⁡(x))=0\lim_{x \to \infty}\left(\frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x + sin(x))/(x^2 + cos(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x+sin⁡(x)x(x2+cos⁡(x)))=0\lim_{x \to -\infty}\left(\frac{x + \sin{\left(x \right)}}{x \left(x^{2} + \cos{\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(x2+cos⁡(x)))=0\lim_{x \to \infty}\left(\frac{x + \sin{\left(x \right)}}{x \left(x^{2} + \cos{\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)x2+cos⁡(x)=−x−sin⁡(x)x2+cos⁡(x)\frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} = \frac{- x - \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}}
- No
x+sin⁡(x)x2+cos⁡(x)=−−x−sin⁡(x)x2+cos⁡(x)\frac{x + \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} = - \frac{- x - \sin{\left(x \right)}}{x^{2} + \cos{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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