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Trazado el gráfico de la función/ (-1-x*cos(x)+sin(x))*exp(x)

Gráfico de la función y = (-1-x*cos(x)+sin(x))*exp(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                 x
f(x) = (-1 - x*cos(x) + sin(x))*e
f(x)=((−xcos⁡(x)−1)+sin⁡(x))exf{\left(x \right)} = \left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x} 
f = (-x*cos(x) - 1 + sin(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:
((−xcos⁡(x)−1)+sin⁡(x))ex=0\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=7.85398163397448x_{1} = 7.85398163397448
x2=−76.9430282181184x_{2} = -76.9430282181184
x3=−86.3937979737193x_{3} = -86.3937979737193
x4=−67.5442420521806x_{4} = -67.5442420521806
x5=−4.71238898038469x_{5} = -4.71238898038469
x6=29.7779917432681x_{6} = 29.7779917432681
x7=−98.9601685880785x_{7} = -98.9601685880785
x8=10.8111042087213x_{8} = 10.8111042087213
x9=26.7035375555132x_{9} = 26.7035375555132
x10=−83.2281761528687x_{10} = -83.2281761528687
x11=−45.5091533451563x_{11} = -45.5091533451563
x12=−48.6946861306418x_{12} = -48.6946861306418
x13=−17.2787595947439x_{13} = -17.2787595947439
x14=20.4203522483337x_{14} = 20.4203522483337
x15=−26.6284652377851x_{15} = -26.6284652377851
x16=−13.9944961126907x_{16} = -13.9944961126907
x17=−89.5130484454873x_{17} = -89.5130484454873
x18=14.1371669411541x_{18} = 14.1371669411541
x19=−64.3715822869017x_{19} = -64.3715822869017
x20=−105.243353895258x_{20} = -105.243353895258
x21=−7.59205618191083x_{21} = -7.59205618191083
x22=4.2502319840436x_{22} = 4.2502319840436
x23=−23.5619449019235x_{23} = -23.5619449019235
x24=17.1623570970183x_{24} = 17.1623570970183
x25=−36.1283155162826x_{25} = -36.1283155162826
x26=−51.7976718062027x_{26} = -51.7976718062027
x27=−32.9259992567895x_{27} = -32.9259992567895
x28=−10.9955742875643x_{28} = -10.9955742875643
x29=1.5707963267949x_{29} = 1.5707963267949
x30=−73.8274273593601x_{30} = -73.8274273593601
x31=−92.6769832808989x_{31} = -92.6769832808989
x32=−58.0850352160434x_{32} = -58.0850352160434
x33=23.4768059032848x_{33} = 23.4768059032848
x34=−39.2189234266452x_{34} = -39.2189234266452
x35=−54.9778714378214x_{35} = -54.9778714378214
x36=−20.3220161353369x_{36} = -20.3220161353369
x37=−70.6575310493539x_{37} = -70.6575310493539
x38=−61.261056745001x_{38} = -61.261056745001
x39=−42.4115008234622x_{39} = -42.4115008234622
x40=−80.1106126665397x_{40} = -80.1106126665397
x41=−29.845130209103x_{41} = -29.845130209103
x42=−95.7976993646524x_{42} = -95.7976993646524
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 (-1 - x*cos(x) + sin(x))*exp(x).
((−1−0cos⁡(0))+sin⁡(0))e0\left(\left(-1 - 0 \cos{\left(0 \right)}\right) + \sin{\left(0 \right)}\right) e^{0}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
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
xexsin⁡(x)+((−xcos⁡(x)−1)+sin⁡(x))ex=0x e^{x} \sin{\left(x \right)} + \left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=3.63614085442611x_{1} = 3.63614085442611
x2=−40.0247669690155x_{2} = -40.0247669690155
x3=−202.570295845803x_{3} = -202.570295845803
x4=−84.0231512440298x_{4} = -84.0231512440298
x5=−62.0498198248722x_{5} = -62.0498198248722
x6=−5.53898388760792x_{6} = -5.53898388760792
x7=−77.7387900851067x_{7} = -77.7387900851067
x8=−33.735803492797x_{8} = -33.735803492797
x9=−93.4646092619029x_{9} = -93.4646092619029
x10=10.096318237648x_{10} = 10.096318237648
x11=−55.7670170583429x_{11} = -55.7670170583429
x12=−87.1815854381936x_{12} = -87.1815854381936
x13=−11.799310071313x_{13} = -11.799310071313
x14=−21.1472952972999x_{14} = -21.1472952972999
x15=−99.7476535314933x_{15} = -99.7476535314933
x16=−80.8985868536874x_{16} = -80.8985868536874
x17=16.422042703626x_{17} = 16.422042703626
x18=−96.5914121108004x_{18} = -96.5914121108004
x19=13.3667088973172x_{19} = 13.3667088973172
x20=−43.2017492208242x_{20} = -43.2017492208242
x21=−71.4542205950111x_{21} = -71.4542205950111
x22=−68.332693454702x_{22} = -68.332693454702
x23=−27.444139007049x_{23} = -27.444139007049
x24=22.7245770102362x_{24} = 22.7245770102362
x25=−18.0759435437534x_{25} = -18.0759435437534
x26=−46.3121433230492x_{26} = -46.3121433230492
x27=−90.3073478196549x_{27} = -90.3073478196549
x28=−65.1693820090398x_{28} = -65.1693820090398
x29=19.6452362607766x_{29} = 19.6452362607766
x30=−74.615619927002x_{30} = -74.615619927002
x31=0.934204800997839x_{31} = 0.934204800997839
x32=−8.48841166585914x_{32} = -8.48841166585914
x33=−49.4843124205006x_{33} = -49.4843124205006
x34=−58.8841873774204x_{34} = -58.8841873774204
x35=−36.9194006481444x_{35} = -36.9194006481444
x36=−24.3560254935667x_{36} = -24.3560254935667
x37=25.9259772956177x_{37} = 25.9259772956177
x38=7.0958770591975x_{38} = 7.0958770591975
x39=−52.5985077508649x_{39} = -52.5985077508649
x40=−30.6374009034184x_{40} = -30.6374009034184
x41=−14.8384022309755x_{41} = -14.8384022309755
Signos de extremos en los puntos:
(3.6361408544261073, 65.4870584045966)

(-40.02476696901548, -1.20821553003001e-16)

(-202.57029584580343, 1.12665411056714e-87)

(-84.02315124402982, -1.94661340509247e-35)

(-62.04981982487223, 4.93020389892824e-26)

(-5.538983887607923, 0.0147473932851821)

(-77.73879008510667, -9.6666818750798e-33)

(-33.73580349279697, -5.51462302336914e-14)

(-93.46460926190288, 1.6904349124844e-39)

(10.096318237648006, 152357.254347151)

(-55.76701705834285, 2.37094250361599e-23)

(-87.18158543819362, 8.440888079831e-37)

(-11.799310071313025, 6.14970915133252e-5)

(-21.147295297299948, -1.03405481883767e-8)

(-99.74765353149333, 3.36995427235159e-42)

(-80.89858685368739, 4.19270622633503e-34)

(16.42204270362597, 145753604.01587)

(-96.59141211080038, -7.77488320524273e-41)

(13.366708897317233, -6123360.56004408)

(-43.20174922082419, 5.25518923039174e-18)

(-71.45422059501108, -4.77093194429667e-30)

(-68.33269345470197, 1.01454608419165e-28)

(-27.444139007048975, -2.44177047942839e-11)

(22.724577010236207, 112551998397.141)

(-18.07594354375344, 1.78288830130343e-7)

(-46.312143323049206, -2.58951349979634e-19)

(-90.30734781965488, -3.899283665924e-38)

(-65.16938200903975, -2.3376661856707e-27)

(19.64523626077659, -4775071191.47761)

(-74.615619927002, 2.06988283804643e-31)

(0.9342048009978386, -1.91199399612607)

(-8.488411665859138, -0.00140725538768078)

(-49.48431242050056, 1.12549787524123e-20)

(-58.88418737742038, -1.13553857333513e-24)

(-36.91940064814441, 2.40084293545839e-15)

(-24.356025493566655, 4.51440260886129e-10)

(25.925977295617667, -3358199036136.65)

(7.095877059197498, -6219.09223170797)

(-52.59850775086492, -5.45819933528738e-22)

(-30.63740090341845, 1.06433283419363e-12)

(-14.838402230975538, -4.07633931747515e-6)


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=−40.0247669690155x_{1} = -40.0247669690155
x2=−84.0231512440298x_{2} = -84.0231512440298
x3=−77.7387900851067x_{3} = -77.7387900851067
x4=−33.735803492797x_{4} = -33.735803492797
x5=−21.1472952972999x_{5} = -21.1472952972999
x6=−96.5914121108004x_{6} = -96.5914121108004
x7=13.3667088973172x_{7} = 13.3667088973172
x8=−71.4542205950111x_{8} = -71.4542205950111
x9=−27.444139007049x_{9} = -27.444139007049
x10=−46.3121433230492x_{10} = -46.3121433230492
x11=−90.3073478196549x_{11} = -90.3073478196549
x12=−65.1693820090398x_{12} = -65.1693820090398
x13=19.6452362607766x_{13} = 19.6452362607766
x14=0.934204800997839x_{14} = 0.934204800997839
x15=−8.48841166585914x_{15} = -8.48841166585914
x16=−58.8841873774204x_{16} = -58.8841873774204
x17=25.9259772956177x_{17} = 25.9259772956177
x18=7.0958770591975x_{18} = 7.0958770591975
x19=−52.5985077508649x_{19} = -52.5985077508649
x20=−14.8384022309755x_{20} = -14.8384022309755
Puntos máximos de la función:
x20=3.63614085442611x_{20} = 3.63614085442611
x20=−62.0498198248722x_{20} = -62.0498198248722
x20=−5.53898388760792x_{20} = -5.53898388760792
x20=−93.4646092619029x_{20} = -93.4646092619029
x20=10.096318237648x_{20} = 10.096318237648
x20=−55.7670170583429x_{20} = -55.7670170583429
x20=−87.1815854381936x_{20} = -87.1815854381936
x20=−11.799310071313x_{20} = -11.799310071313
x20=−99.7476535314933x_{20} = -99.7476535314933
x20=−80.8985868536874x_{20} = -80.8985868536874
x20=16.422042703626x_{20} = 16.422042703626
x20=−43.2017492208242x_{20} = -43.2017492208242
x20=−68.332693454702x_{20} = -68.332693454702
x20=22.7245770102362x_{20} = 22.7245770102362
x20=−18.0759435437534x_{20} = -18.0759435437534
x20=−74.615619927002x_{20} = -74.615619927002
x20=−49.4843124205006x_{20} = -49.4843124205006
x20=−36.9194006481444x_{20} = -36.9194006481444
x20=−24.3560254935667x_{20} = -24.3560254935667
x20=−30.6374009034184x_{20} = -30.6374009034184
Decrece en los intervalos
[25.9259772956177,∞)\left[25.9259772956177, \infty\right)
Crece en los intervalos
(−∞,−96.5914121108004]\left(-\infty, -96.5914121108004\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
(2xsin⁡(x)+2sin⁡(x)−1)ex=0\left(2 x \sin{\left(x \right)} + 2 \sin{\left(x \right)} - 1\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−97.3841846649539x_{1} = -97.3841846649539
x2=15.6779791238938x_{2} = 15.6779791238938
x3=−100.535988244743x_{3} = -100.535988244743
x4=25.1518614916041x_{4} = 25.1518614916041
x5=−28.255988260673x_{5} = -28.255988260673
x6=−94.253141383054x_{6} = -94.253141383054
x7=−75.4049437203121x_{7} = -75.4049437203121
x8=−87.9703434183712x_{8} = -87.9703434183712
x9=−56.5576675439777x_{9} = -56.5576675439777
x10=−6.37632030431633x_{10} = -6.37632030431633
x11=18.8747161679964x_{11} = 18.8747161679964
x12=3.01679147585534x_{12} = 3.01679147585534
x13=−69.122378176892x_{13} = -69.122378176892
x14=0.372795476130402x_{14} = 0.372795476130402
x15=−37.7127315177142x_{15} = -37.7127315177142
x16=−47.1130466730195x_{16} = -47.1130466730195
x17=−81.6876057715788x_{17} = -81.6876057715788
x18=−50.2756296353467x_{18} = -50.2756296353467
x19=−78.5333674586749x_{19} = -78.5333674586749
x20=−31.4323571559532x_{20} = -31.4323571559532
x21=−34.5426122253177x_{21} = -34.5426122253177
x22=−18.8775276420917x_{22} = -18.8775276420917
x23=−40.8281502321218x_{23} = -40.8281502321218
x24=9.37657383317963x_{24} = 9.37657383317963
x25=−91.1006375753231x_{25} = -91.1006375753231
x26=−12.6094522938474x_{26} = -12.6094522938474
x27=−84.8170362371704x_{27} = -84.8170362371704
x28=−21.9672996575815x_{28} = -21.9672996575815
x29=−53.397532532124x_{29} = -53.397532532124
x30=−59.681739776997x_{30} = -59.681739776997
x31=−25.1534436893781x_{31} = -25.1534436893781
x32=6.35125346958337x_{32} = 6.35125346958337
x33=−72.249613393028x_{33} = -72.249613393028
x34=21.9693787440177x_{34} = 21.9693787440177
x35=−65.9657492862344x_{35} = -65.9657492862344
x36=−43.9939269618736x_{36} = -43.9939269618736
x37=12.6031351284827x_{37} = 12.6031351284827
x38=−15.6738825259471x_{38} = -15.6738825259471
x39=−1.50121290983133x_{39} = -1.50121290983133
x40=−9.36496922731976x_{40} = -9.36496922731976
x41=−62.839938549558x_{41} = -62.839938549558

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
[25.1518614916041,∞)\left[25.1518614916041, \infty\right)
Convexa en los intervalos
(−∞,−97.3841846649539]\left(-\infty, -97.3841846649539\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((−xcos⁡(x)−1)+sin⁡(x))ex)=0\lim_{x \to -\infty}\left(\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\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→∞(((−xcos⁡(x)−1)+sin⁡(x))ex)y = \lim_{x \to \infty}\left(\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-1 - x*cos(x) + sin(x))*exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−xcos⁡(x)−1)+sin⁡(x))exx)=0\lim_{x \to -\infty}\left(\frac{\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\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→∞(((−xcos⁡(x)−1)+sin⁡(x))exx)y = x \lim_{x \to \infty}\left(\frac{\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\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:
((−xcos⁡(x)−1)+sin⁡(x))ex=(xcos⁡(x)−sin⁡(x)−1)e−x\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x} = \left(x \cos{\left(x \right)} - \sin{\left(x \right)} - 1\right) e^{- x}
- No
((−xcos⁡(x)−1)+sin⁡(x))ex=−(xcos⁡(x)−sin⁡(x)−1)e−x\left(\left(- x \cos{\left(x \right)} - 1\right) + \sin{\left(x \right)}\right) e^{x} = - \left(x \cos{\left(x \right)} - \sin{\left(x \right)} - 1\right) e^{- x}
- No
es decir, función
no es
par ni impar
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