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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                               x
f(x) = (-2*cos(x) - x*sin(x))*e
f(x)=(−xsin⁡(x)−2cos⁡(x))exf{\left(x \right)} = \left(- x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} 
f = (-x*sin(x) - 2*cos(x))*exp(x)
Gráfico de la función
02468-8-6-4-2-1010-200000200000
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:
(−xsin⁡(x)−2cos⁡(x))ex=0\left(- x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−9.21096438740149x_{1} = -9.21096438740149
x2=−34.4996123350132x_{2} = -34.4996123350132
x3=−72.2289483771681x_{3} = -72.2289483771681
x4=−5.95939190757933x_{4} = -5.95939190757933
x5=21.9000773156394x_{5} = 21.9000773156394
x6=−75.3716947511882x_{6} = -75.3716947511882
x7=28.2035393053095x_{7} = 28.2035393053095
x8=−65.9431258539286x_{8} = -65.9431258539286
x9=−94.2265573558031x_{9} = -94.2265573558031
x10=−62.8000167068325x_{10} = -62.8000167068325
x11=15.5802941824244x_{11} = 15.5802941824244
x12=−91.75x_{12} = -91.75
x13=−84.7994209518635x_{13} = -84.7994209518635
x14=−18.7432530945386x_{14} = -18.7432530945386
x15=−56.5132926241755x_{15} = -56.5132926241755
x16=−78.5143487963623x_{16} = -78.5143487963623
x17=−12.4065403639626x_{17} = -12.4065403639626
x18=−47.75x_{18} = -47.75
x19=−50.2256832197934x_{19} = -50.2256832197934
x20=−28.2035393053095x_{20} = -28.2035393053095
x21=12.4065403639626x_{21} = 12.4065403639626
x22=−15.5802941824244x_{22} = -15.5802941824244
x23=2.45871417599962x_{23} = 2.45871417599962
x24=−53.3696181339615x_{24} = -53.3696181339615
x25=−21.9000773156394x_{25} = -21.9000773156394
x26=−87.9418559209576x_{26} = -87.9418559209576
x27=5.95939190757933x_{27} = 5.95939190757933
x28=−25.053079662454x_{28} = -25.053079662454
x29=25.053079662454x_{29} = 25.053079662454
x30=−31.3522215217643x_{30} = -31.3522215217643
x31=−100.511069234565x_{31} = -100.511069234565
x32=−40.7917141624847x_{32} = -40.7917141624847
x33=−47.0814357397523x_{33} = -47.0814357397523
x34=−37.6460352959305x_{34} = -37.6460352959305
x35=18.7432530945386x_{35} = 18.7432530945386
x36=−69.0860970774096x_{36} = -69.0860970774096
x37=9.21096438740149x_{37} = 9.21096438740149
x38=−2.45871417599962x_{38} = -2.45871417599962
x39=−91.0842327848165x_{39} = -91.0842327848165
x40=−59.6567478435559x_{40} = -59.6567478435559
x41=−43.9368086315937x_{41} = -43.9368086315937
x42=−81.6569211705466x_{42} = -81.6569211705466
x43=−97.3688346960149x_{43} = -97.3688346960149
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 (-2*cos(x) - x*sin(x))*exp(x).
(−2cos⁡(0)−0sin⁡(0))e0\left(- 2 \cos{\left(0 \right)} - 0 \sin{\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
(−xsin⁡(x)−2cos⁡(x))ex+(−xcos⁡(x)+sin⁡(x))ex=0\left(- x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} + \left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−101.301483443117x_{1} = -101.301483443117
x2=21.136533975368x_{2} = 21.136533975368
x3=52.5934323788799x_{3} = 52.5934323788799
x4=−66.7362015328148x_{4} = -66.7362015328148
x5=17.9831803577612x_{5} = 17.9831803577612
x6=−104.443525820276x_{6} = -104.443525820276
x7=−13.234518967478x_{7} = -13.234518967478
x8=−60.4506432185942x_{8} = -60.4506432185942
x9=−79.3061827617297x_{9} = -79.3061827617297
x10=−32.1539727052608x_{10} = -32.1539727052608
x11=−69.878819478905x_{11} = -69.878819478905
x12=−82.4485050228275x_{12} = -82.4485050228275
x13=−22.709109301662x_{13} = -22.709109301662
x14=−63.5934814660579x_{14} = -63.5934814660579
x15=−57.3076671756744x_{15} = -57.3076671756744
x16=1.77284537202095x_{16} = 1.77284537202095
x17=−25.8590577697517x_{17} = -25.8590577697517
x18=−3.45762927914644x_{18} = -3.45762927914644
x19=−47.8776380334436x_{19} = -47.8776380334436
x20=27.4352917154902x_{20} = 27.4352917154902
x21=−85.5907734027611x_{21} = -85.5907734027611
x22=5.24227625159811x_{22} = 5.24227625159811
x23=−6.8361253524594x_{23} = -6.8361253524594
x24=24.2869009417436x_{24} = 24.2869009417436
x25=−38.4449996784093x_{25} = -38.4449996784093
x26=−88.7329936581891x_{26} = -88.7329936581891
x27=−19.5564023459924x_{27} = -19.5564023459924
x28=−76.1637999064041x_{28} = -76.1637999064041
x29=−54.1645291264666x_{29} = -54.1645291264666
x30=14.8249965889442x_{30} = 14.8249965889442
x31=−91.8751707538451x_{31} = -91.8751707538451
x32=−16.3992960566677x_{32} = -16.3992960566677
x33=8.47375692649236x_{33} = 8.47375692649236
x34=−98.1594121296962x_{34} = -98.1594121296962
x35=−95.0173089941994x_{35} = -95.0173089941994
x36=−29.0071621707914x_{36} = -29.0071621707914
x37=−10.0544523473053x_{37} = -10.0544523473053
x38=11.6582194214432x_{38} = 11.6582194214432
x39=−73.0213485801257x_{39} = -73.0213485801257
x40=−35.2998403716107x_{40} = -35.2998403716107
x41=−44.733797581381x_{41} = -44.733797581381
x42=−41.5896132875985x_{42} = -41.5896132875985
x43=−51.0211988336933x_{43} = -51.0211988336933
Signos de extremos en los puntos:
(-101.30148344311704, -7.28804382062422e-43)

(21.13653397536802, -22117887701.9285)

(52.5934323788799, -2.55555566187774e+24)

(-66.73620153281477, 4.94347256593906e-28)

(17.983180357761203, 801104933.43779)

(-104.44352582027601, 3.24517464393413e-44)

(-13.234518967477959, -1.74654212658824e-5)

(-60.45064321859424, 2.40554761739842e-25)

(-79.30618276172967, 2.03873450145918e-33)

(-32.15397270526078, -2.5094894980679e-13)

(-69.87881947890499, -2.23377731123586e-29)

(-82.4485050228275, -9.15026729207009e-35)

(-22.709109301661965, 2.25738654239487e-9)

(-63.59348146605792, -1.091741077811e-26)

(-57.30766717567436, -5.2870440964363e-24)

(1.772845372020951, -7.86243543366342)

(-25.85905776975167, -1.09817047251221e-10)

(-3.4576292791464414, 0.0937439764769793)

(-47.8776380334436, 5.51244444259474e-20)

(27.435291715490195, -15684620629094.6)

(-85.59077340276107, 4.10116198155916e-36)

(5.2422762515981125, 664.213854766819)

(-6.836125352459403, -0.00568523251375146)

(24.286900941743593, 594771730220.334)

(-38.44499967840931, -5.54379496340967e-16)

(-88.73299365818906, -1.835787043832e-37)

(-19.556402345992414, -4.56781297584888e-8)

(-76.16379990640405, -4.53565928150833e-32)

(-54.164529126466604, 1.15877171002088e-22)

(14.824996588944192, -27948067.7349385)

(-91.87517075384513, 8.20763087183536e-39)

(-16.39929605666771, 9.05580940875288e-7)

(8.47375692649236, -27460.9630783351)

(-98.15941212969621, 1.63524441525757e-41)

(-95.01730899419941, -3.66545285779068e-40)

(-29.007162170791386, 5.27620574510492e-12)

(-10.054452347305288, 0.000324058805787881)

(11.658219421443249, 920542.203423491)

(-73.02134858012569, 1.00744203847326e-30)

(-35.29984037161067, 1.18364312001276e-14)

(-44.73379758138097, -1.1954877830457e-18)

(-41.589613287598475, 2.58104656547556e-17)

(-51.021198833693305, -2.53180213022537e-21)


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=−101.301483443117x_{1} = -101.301483443117
x2=21.136533975368x_{2} = 21.136533975368
x3=52.5934323788799x_{3} = 52.5934323788799
x4=−13.234518967478x_{4} = -13.234518967478
x5=−32.1539727052608x_{5} = -32.1539727052608
x6=−69.878819478905x_{6} = -69.878819478905
x7=−82.4485050228275x_{7} = -82.4485050228275
x8=−63.5934814660579x_{8} = -63.5934814660579
x9=−57.3076671756744x_{9} = -57.3076671756744
x10=1.77284537202095x_{10} = 1.77284537202095
x11=−25.8590577697517x_{11} = -25.8590577697517
x12=27.4352917154902x_{12} = 27.4352917154902
x13=−6.8361253524594x_{13} = -6.8361253524594
x14=−38.4449996784093x_{14} = -38.4449996784093
x15=−88.7329936581891x_{15} = -88.7329936581891
x16=−19.5564023459924x_{16} = -19.5564023459924
x17=−76.1637999064041x_{17} = -76.1637999064041
x18=14.8249965889442x_{18} = 14.8249965889442
x19=8.47375692649236x_{19} = 8.47375692649236
x20=−95.0173089941994x_{20} = -95.0173089941994
x21=−44.733797581381x_{21} = -44.733797581381
x22=−51.0211988336933x_{22} = -51.0211988336933
Puntos máximos de la función:
x22=−66.7362015328148x_{22} = -66.7362015328148
x22=17.9831803577612x_{22} = 17.9831803577612
x22=−104.443525820276x_{22} = -104.443525820276
x22=−60.4506432185942x_{22} = -60.4506432185942
x22=−79.3061827617297x_{22} = -79.3061827617297
x22=−22.709109301662x_{22} = -22.709109301662
x22=−3.45762927914644x_{22} = -3.45762927914644
x22=−47.8776380334436x_{22} = -47.8776380334436
x22=−85.5907734027611x_{22} = -85.5907734027611
x22=5.24227625159811x_{22} = 5.24227625159811
x22=24.2869009417436x_{22} = 24.2869009417436
x22=−54.1645291264666x_{22} = -54.1645291264666
x22=−91.8751707538451x_{22} = -91.8751707538451
x22=−16.3992960566677x_{22} = -16.3992960566677
x22=−98.1594121296962x_{22} = -98.1594121296962
x22=−29.0071621707914x_{22} = -29.0071621707914
x22=−10.0544523473053x_{22} = -10.0544523473053
x22=11.6582194214432x_{22} = 11.6582194214432
x22=−73.0213485801257x_{22} = -73.0213485801257
x22=−35.2998403716107x_{22} = -35.2998403716107
x22=−41.5896132875985x_{22} = -41.5896132875985
Decrece en los intervalos
[52.5934323788799,∞)\left[52.5934323788799, \infty\right)
Crece en los intervalos
(−∞,−101.301483443117]\left(-\infty, -101.301483443117\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
(−2xcos⁡(x)+2sin⁡(x)−2cos⁡(x))ex=0\left(- 2 x \cos{\left(x \right)} + 2 \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.2239416992227x_{1} = 17.2239416992227
x2=−95.808028695694x_{2} = -95.808028695694
x3=−32.9554394950363x_{3} = -32.9554394950363
x4=26.667409674396x_{4} = 26.667409674396
x5=−67.5292121928599x_{5} = -67.5292121928599
x6=−89.5240947482585x_{6} = -89.5240947482585
x7=10.9118204503674x_{7} = 10.9118204503674
x8=−76.9558552315675x_{8} = -76.9558552315675
x9=−70.6714826164982x_{9} = -70.6714826164982
x10=23.5211864259599x_{10} = 23.5211864259599
x11=−42.3873435477815x_{11} = -42.3873435477815
x12=−80.0979707906851x_{12} = -80.0979707906851
x13=−20.3687685412497x_{13} = -20.3687685412497
x14=14.0709110733623x_{14} = 14.0709110733623
x15=−26.6645930692902x_{15} = -26.6645930692902
x16=−23.5175642862085x_{16} = -23.5175642862085
x17=−10.8948536303641x_{17} = -10.8948536303641
x18=−14.0607507547713x_{18} = -14.0607507547713
x19=−92.6660745522577x_{19} = -92.6660745522577
x20=−64.3868745704895x_{20} = -64.3868745704895
x21=−4.42859686541094x_{21} = -4.42859686541094
x22=−58.1019533456208x_{22} = -58.1019533456208
x23=−83.2400463931404x_{23} = -83.2400463931404
x24=4.53360450162482x_{24} = 4.53360450162482
x25=−17.2171745487221x_{25} = -17.2171745487221
x26=−7.705951184346x_{26} = -7.705951184346
x27=−45.5306408032197x_{27} = -45.5306408032197
x28=1.13226772527289x_{28} = 1.13226772527289
x29=−39.2437660743313x_{29} = -39.2437660743313
x30=−61.2444592323791x_{30} = -61.2444592323791
x31=7.74006134563749x_{31} = 7.74006134563749
x32=−105.23376036896x_{32} = -105.23376036896
x33=−54.9593410866482x_{33} = -54.9593410866482
x34=−29.8104344913478x_{34} = -29.8104344913478
x35=−36.0998330569238x_{35} = -36.0998330569238
x36=−86.3820864505975x_{36} = -86.3820864505975
x37=−48.6737132861388x_{37} = -48.6737132861388
x38=20.3735996512825x_{38} = 20.3735996512825
x39=−98.9499596481x_{39} = -98.9499596481
x40=−73.8136945424488x_{40} = -73.8136945424488
x41=−51.8166027158994x_{41} = -51.8166027158994

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
[26.667409674396,∞)\left[26.667409674396, \infty\right)
Convexa en los intervalos
(−∞,−95.808028695694]\left(-\infty, -95.808028695694\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−xsin⁡(x)−2cos⁡(x))ex)=0\lim_{x \to -\infty}\left(\left(- x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((−xsin⁡(x)−2cos⁡(x))ex)y = \lim_{x \to \infty}\left(\left(- x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-2*cos(x) - x*sin(x))*exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−xsin⁡(x)−2cos⁡(x))exx)=0\lim_{x \to -\infty}\left(\frac{\left(- x \sin{\left(x \right)} - 2 \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
True

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