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Trazado el gráfico de la función/ 4*sin(x)+x^2*exp(x)-2*x*exp(x)+cos(x)+exp(x)

Gráfico de la función y = 4*sin(x)+x^2*exp(x)-2*x*exp(x)+cos(x)+exp(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   2  x        x             x
f(x) = 4*sin(x) + x *e  - 2*x*e  + cos(x) + e
f(x)=((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+exf{\left(x \right)} = \left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x} 
f = -2*x*exp(x) + x^2*exp(x) + 4*sin(x) + cos(x) + exp(x)
Gráfico de la función
02468-8-6-4-2-1010-20000002000000
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:
((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+ex=0\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−9.66801008136603x_{1} = -9.66801008136603
x2=−44.227275813384x_{2} = -44.227275813384
x3=−6.54796638582494x_{3} = -6.54796638582494
x4=−88.2095729636411x_{4} = -88.2095729636411
x5=−85.0679803100513x_{5} = -85.0679803100513
x6=−97.6343509244105x_{6} = -97.6343509244105
x7=−0.591958682307481x_{7} = -0.591958682307481
x8=−91.3511656172309x_{8} = -91.3511656172309
x9=−22.2361272094101x_{9} = -22.2361272094101
x10=−34.8024978526144x_{10} = -34.8024978526144
x11=−69.3600170421023x_{11} = -69.3600170421023
x12=−56.7936464277431x_{12} = -56.7936464277431
x13=−3.21243190972053x_{13} = -3.21243190972053
x14=−19.0945350838721x_{14} = -19.0945350838721
x15=−100.775943578x_{15} = -100.775943578
x16=−31.6609051990294x_{16} = -31.6609051990294
x17=−66.2184243885125x_{17} = -66.2184243885125
x18=−47.3688684669738x_{18} = -47.3688684669738
x19=−12.8114755481786x_{19} = -12.8114755481786
x20=−28.5193125453481x_{20} = -28.5193125453481
x21=−15.9529337087594x_{21} = -15.9529337087594
x22=−63.0768317349227x_{22} = -63.0768317349227
x23=−37.9440905062044x_{23} = -37.9440905062044
x24=−75.6432023492819x_{24} = -75.6432023492819
x25=−78.7847950028717x_{25} = -78.7847950028717
x26=−41.0856831597942x_{26} = -41.0856831597942
x27=−25.3777198934516x_{27} = -25.3777198934516
x28=−132.191870113898x_{28} = -132.191870113898
x29=−59.9352390813329x_{29} = -59.9352390813329
x30=−53.6520537741534x_{30} = -53.6520537741534
x31=−94.4927582708207x_{31} = -94.4927582708207
x32=−50.5104611205636x_{32} = -50.5104611205636
x33=−72.5016096956921x_{33} = -72.5016096956921
x34=−81.9263876564615x_{34} = -81.9263876564615
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 4*sin(x) + x^2*exp(x) - 2*x*exp(x) + cos(x) + exp(x).
(((4sin⁡(0)+02e0)−0⋅2e0)+cos⁡(0))+e0\left(\left(\left(4 \sin{\left(0 \right)} + 0^{2} e^{0}\right) - 0 \cdot 2 e^{0}\right) + \cos{\left(0 \right)}\right) + e^{0}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
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
x2ex−ex−sin⁡(x)+4cos⁡(x)=0x^{2} e^{x} - e^{x} - \sin{\left(x \right)} + 4 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4.91618334634381x_{1} = -4.91618334634381
x2=−67.7892207153074x_{2} = -67.7892207153074
x3=−96.0635545976156x_{3} = -96.0635545976156
x4=−58.364442754538x_{4} = -58.364442754538
x5=−42.6564794865891x_{5} = -42.6564794865891
x6=−64.6476280617176x_{6} = -64.6476280617176
x7=−168.320185630181x_{7} = -168.320185630181
x8=−83.4971839832564x_{8} = -83.4971839832564
x9=−17.5237364375398x_{9} = -17.5237364375398
x10=−20.6653310209577x_{10} = -20.6653310209577
x11=−33.2317015258207x_{11} = -33.2317015258207
x12=−39.5148868329993x_{12} = -39.5148868329993
x13=−102.346739904795x_{13} = -102.346739904795
x14=−99.2051472512053x_{14} = -99.2051472512053
x15=−55.2228501009482x_{15} = -55.2228501009482
x16=−70.9308133688972x_{16} = -70.9308133688972
x17=−26.9485162189882x_{17} = -26.9485162189882
x18=−14.3821739326225x_{18} = -14.3821739326225
x19=−92.9219619440258x_{19} = -92.9219619440258
x20=−77.2139986760768x_{20} = -77.2139986760768
x21=−61.5060354081278x_{21} = -61.5060354081278
x22=−23.8069235587668x_{22} = -23.8069235587668
x23=−48.9396647937687x_{23} = -48.9396647937687
x24=−1.91107003039074x_{24} = -1.91107003039074
x25=−89.780369290436x_{25} = -89.780369290436
x26=−11.2401536204521x_{26} = -11.2401536204521
x27=−86.6387766368462x_{27} = -86.6387766368462
x28=−8.103703659797x_{28} = -8.103703659797
x29=−30.0901088722111x_{29} = -30.0901088722111
x30=−45.7980721401789x_{30} = -45.7980721401789
x31=−36.3732941794094x_{31} = -36.3732941794094
x32=−74.072406022487x_{32} = -74.072406022487
x33=−52.0812574473585x_{33} = -52.0812574473585
x34=−80.3555913296666x_{34} = -80.3555913296666
Signos de extremos en los puntos:
(-4.916183346343812, 4.37606488465944)

(-67.78922071530742, 4.12310562561766)

(-96.06355459761556, -4.12310562561766)

(-58.36444275453804, -4.12310562561766)

(-42.65647948658907, 4.12310562561766)

(-64.64762806171763, -4.12310562561766)

(-168.32018563018082, 4.12310562561766)

(-83.49718398325639, -4.12310562561766)

(-17.52373643753977, 4.12311403946826)

(-20.66533102095769, -4.12310512823407)

(-33.23170152582068, -4.12310562561333)

(-39.51488683299928, -4.12310562561765)

(-102.34673990479514, -4.12310562561766)

(-99.20514725120535, 4.12310562561766)

(-55.22285010094824, 4.12310562561766)

(-70.93081336889722, -4.12310562561766)

(-26.948516218988168, -4.12310562407196)

(-14.38217393262253, -4.12297136704973)

(-92.92196194402577, 4.12310562561766)

(-77.2139986760768, -4.12310562561766)

(-61.50603540812783, 4.12310562561766)

(-23.806923558766844, 4.12310565379697)

(-48.93966479376866, 4.12310562561766)

(-1.9110700303907386, -2.85085869972302)

(-89.78036929043597, -4.12310562561766)

(-11.24015362045213, 4.12507335102373)

(-86.63877663684617, 4.12310562561766)

(-8.103703659797, -4.09799570003066)

(-30.090108872211143, 4.12310562570032)

(-45.79807214017887, -4.12310562561766)

(-36.373294179409434, 4.12310562561788)

(-74.072406022487, 4.12310562561766)

(-52.08125744735845, -4.12310562561766)

(-80.3555913296666, 4.12310562561766)


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=−96.0635545976156x_{1} = -96.0635545976156
x2=−58.364442754538x_{2} = -58.364442754538
x3=−64.6476280617176x_{3} = -64.6476280617176
x4=−83.4971839832564x_{4} = -83.4971839832564
x5=−20.6653310209577x_{5} = -20.6653310209577
x6=−33.2317015258207x_{6} = -33.2317015258207
x7=−39.5148868329993x_{7} = -39.5148868329993
x8=−102.346739904795x_{8} = -102.346739904795
x9=−70.9308133688972x_{9} = -70.9308133688972
x10=−26.9485162189882x_{10} = -26.9485162189882
x11=−14.3821739326225x_{11} = -14.3821739326225
x12=−77.2139986760768x_{12} = -77.2139986760768
x13=−1.91107003039074x_{13} = -1.91107003039074
x14=−89.780369290436x_{14} = -89.780369290436
x15=−8.103703659797x_{15} = -8.103703659797
x16=−45.7980721401789x_{16} = -45.7980721401789
x17=−52.0812574473585x_{17} = -52.0812574473585
Puntos máximos de la función:
x17=−4.91618334634381x_{17} = -4.91618334634381
x17=−67.7892207153074x_{17} = -67.7892207153074
x17=−42.6564794865891x_{17} = -42.6564794865891
x17=−168.320185630181x_{17} = -168.320185630181
x17=−17.5237364375398x_{17} = -17.5237364375398
x17=−99.2051472512053x_{17} = -99.2051472512053
x17=−55.2228501009482x_{17} = -55.2228501009482
x17=−92.9219619440258x_{17} = -92.9219619440258
x17=−61.5060354081278x_{17} = -61.5060354081278
x17=−23.8069235587668x_{17} = -23.8069235587668
x17=−48.9396647937687x_{17} = -48.9396647937687
x17=−11.2401536204521x_{17} = -11.2401536204521
x17=−86.6387766368462x_{17} = -86.6387766368462
x17=−30.0901088722111x_{17} = -30.0901088722111
x17=−36.3732941794094x_{17} = -36.3732941794094
x17=−74.072406022487x_{17} = -74.072406022487
x17=−80.3555913296666x_{17} = -80.3555913296666
Decrece en los intervalos
[−1.91107003039074,∞)\left[-1.91107003039074, \infty\right)
Crece en los intervalos
(−∞,−102.346739904795]\left(-\infty, -102.346739904795\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
x2ex+2xex−ex−4sin⁡(x)−cos⁡(x)=0x^{2} e^{x} + 2 x e^{x} - e^{x} - 4 \sin{\left(x \right)} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−91.3511656172309x_{1} = -91.3511656172309
x2=−47.3688684669738x_{2} = -47.3688684669738
x3=−9.67087653629034x_{3} = -9.67087653629034
x4=−69.3600170421023x_{4} = -69.3600170421023
x5=−85.0679803100513x_{5} = -85.0679803100513
x6=−63.0768317349227x_{6} = -63.0768317349227
x7=−66.2184243885125x_{7} = -66.2184243885125
x8=−50.5104611205636x_{8} = -50.5104611205636
x9=−94.4927582708207x_{9} = -94.4927582708207
x10=−2642.32440033214x_{10} = -2642.32440033214
x11=−15.9529482704968x_{11} = -15.9529482704968
x12=−97.6343509244105x_{12} = -97.6343509244105
x13=−78.7847950028717x_{13} = -78.7847950028717
x14=−88.2095729636411x_{14} = -88.2095729636411
x15=−34.8024978526148x_{15} = -34.8024978526148
x16=−53.6520537741534x_{16} = -53.6520537741534
x17=−100.775943578x_{17} = -100.775943578
x18=0.881890321767176x_{18} = 0.881890321767176
x19=−44.227275813384x_{19} = -44.227275813384
x20=−75.6432023492819x_{20} = -75.6432023492819
x21=−0.504793508736992x_{21} = -0.504793508736992
x22=−12.8112582363087x_{22} = -12.8112582363087
x23=−72.5016096956921x_{23} = -72.5016096956921
x24=−28.5193125455104x_{24} = -28.5193125455104
x25=−81.9263876564615x_{25} = -81.9263876564615
x26=−25.3777198904778x_{26} = -25.3777198904778
x27=−41.0856831597942x_{27} = -41.0856831597942
x28=−22.236127262242x_{28} = -22.236127262242
x29=−19.0945341823578x_{29} = -19.0945341823578
x30=−6.51797536227247x_{30} = -6.51797536227247
x31=−59.9352390813329x_{31} = -59.9352390813329
x32=−56.7936464277431x_{32} = -56.7936464277431
x33=−37.9440905062044x_{33} = -37.9440905062044
x34=−3.41715012977162x_{34} = -3.41715012977162
x35=−132.191870113898x_{35} = -132.191870113898
x36=−31.6609051990208x_{36} = -31.6609051990208

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
[0.881890321767176,∞)\left[0.881890321767176, \infty\right)
Convexa en los intervalos
(−∞,−2642.32440033214]\left(-\infty, -2642.32440033214\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+ex)=⟨−5,5⟩\lim_{x \to -\infty}\left(\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x}\right) = \left\langle -5, 5\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−5,5⟩y = \left\langle -5, 5\right\rangle
lim⁡x→∞(((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+ex)=∞\lim_{x \to \infty}\left(\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 4*sin(x) + x^2*exp(x) - 2*x*exp(x) + cos(x) + exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+exx)=0\lim_{x \to -\infty}\left(\frac{\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+exx)=∞\lim_{x \to \infty}\left(\frac{\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x}}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+ex=x2e−x+2xe−x−4sin⁡(x)+cos⁡(x)+e−x\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x} = x^{2} e^{- x} + 2 x e^{- x} - 4 \sin{\left(x \right)} + \cos{\left(x \right)} + e^{- x}
- No
((−2xex+(x2ex+4sin⁡(x)))+cos⁡(x))+ex=−x2e−x−2xe−x+4sin⁡(x)−cos⁡(x)−e−x\left(\left(- 2 x e^{x} + \left(x^{2} e^{x} + 4 \sin{\left(x \right)}\right)\right) + \cos{\left(x \right)}\right) + e^{x} = - x^{2} e^{- x} - 2 x e^{- x} + 4 \sin{\left(x \right)} - \cos{\left(x \right)} - e^{- x}
- No
es decir, función
no es
par ni impar
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