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-3*sin(x)+4*cos(x)+(-1-x)*exp(-2*x)
Trazado el gráfico de la función/ -3*sin(x)+4*cos(x)+(-1-x)*exp(-2*x)

Gráfico de la función y = -3*sin(x)+4*cos(x)+(-1-x)*exp(-2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                        -2*x
f(x) = -3*sin(x) + 4*cos(x) + (-1 - x)*e
f(x)=(−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x))f{\left(x \right)} = \left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) 
f = (-x - 1)*exp(-2*x) - 3*sin(x) + 4*cos(x)
Gráfico de la función
02468-8-6-4-2-1010-50000000005000000000
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−1)e−2x+(−3sin⁡(x)+4cos⁡(x))=0\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=88.8918895185158x_{1} = 88.8918895185158
x2=82.6087042113362x_{2} = 82.6087042113362
x3=51.1927776754383x_{3} = 51.1927776754383
x4=54.3343703290281x_{4} = 54.3343703290281
x5=29.2016291003098x_{5} = 29.2016291003098
x6=26.06003644672x_{6} = 26.06003644672
x7=104.599852786465x_{7} = 104.599852786465
x8=22.9184437931302x_{8} = 22.9184437931302
x9=57.4759629826179x_{9} = 57.4759629826179
x10=19.7768511395404x_{10} = 19.7768511395404
x11=16.6352584859506x_{11} = 16.6352584859506
x12=92.0334821721056x_{12} = 92.0334821721056
x13=13.4936658323553x_{13} = 13.4936658323553
x14=76.3255189041567x_{14} = 76.3255189041567
x15=7.21047962887687x_{15} = 7.21047962887687
x16=66.9007409433873x_{16} = 66.9007409433873
x17=44.9095923682587x_{17} = 44.9095923682587
x18=95.1750748256954x_{18} = 95.1750748256954
x19=32.3432217538995x_{19} = 32.3432217538995
x20=98.3166674792852x_{20} = 98.3166674792852
x21=73.1839262505669x_{21} = 73.1839262505669
x22=4.06918402703664x_{22} = 4.06918402703664
x23=0.860701728002154x_{23} = 0.860701728002154
x24=41.7679997146689x_{24} = 41.7679997146689
x25=10.3520731810852x_{25} = 10.3520731810852
x26=70.0423335969771x_{26} = 70.0423335969771
x27=63.7591482897975x_{27} = 63.7591482897975
x28=48.0511850218485x_{28} = 48.0511850218485
x29=60.6175556362077x_{29} = 60.6175556362077
x30=35.4848144074893x_{30} = 35.4848144074893
x31=38.6264070610791x_{31} = 38.6264070610791
x32=79.4671115577464x_{32} = 79.4671115577464
x33=85.750296864926x_{33} = 85.750296864926
x34=0.860701728002154x_{34} = 0.860701728002154
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 -3*sin(x) + 4*cos(x) + (-1 - x)*exp(-2*x).
(−1−0)e−0+(−3sin⁡(0)+4cos⁡(0))\left(-1 - 0\right) e^{- 0} + \left(- 3 \sin{\left(0 \right)} + 4 \cos{\left(0 \right)}\right)
Resultado:
f(0)=3f{\left(0 \right)} = 3
Punto:
(0, 3)
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
−2(−x−1)e−2x−4sin⁡(x)−3cos⁡(x)−e−2x=0- 2 \left(- x - 1\right) e^{- 2 x} - 4 \sin{\left(x \right)} - 3 \cos{\left(x \right)} - e^{- 2 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=62.1883519630026x_{1} = 62.1883519630026
x2=18.2060548127455x_{2} = 18.2060548127455
x3=43.3387960414638x_{3} = 43.3387960414638
x4=5.63971521638853x_{4} = 5.63971521638853
x5=103.02905645967x_{5} = 103.02905645967
x6=68.4715372701822x_{6} = 68.4715372701822
x7=52.7635740022332x_{7} = 52.7635740022332
x8=8.78127676440765x_{8} = 8.78127676440765
x9=65.3299446165924x_{9} = 65.3299446165924
x10=74.7547225773617x_{10} = 74.7547225773617
x11=90.4626858453107x_{11} = 90.4626858453107
x12=33.9140180806944x_{12} = 33.9140180806944
x13=15.0644621591552x_{13} = 15.0644621591552
x14=21.3476474663353x_{14} = 21.3476474663353
x15=143.869760956337x_{15} = 143.869760956337
x16=37.0556107342842x_{16} = 37.0556107342842
x17=96.7458711524903x_{17} = 96.7458711524903
x18=40.197203387874x_{18} = 40.197203387874
x19=46.4803886950536x_{19} = 46.4803886950536
x20=87.3210931917209x_{20} = 87.3210931917209
x21=59.0467593094128x_{21} = 59.0467593094128
x22=77.8963152309515x_{22} = 77.8963152309515
x23=99.8874638060801x_{23} = 99.8874638060801
x24=30.7724254271046x_{24} = 30.7724254271046
x25=2.48986826930282x_{25} = 2.48986826930282
x26=55.905166655823x_{26} = 55.905166655823
x27=71.613129923772x_{27} = 71.613129923772
x28=11.9228695057848x_{28} = 11.9228695057848
x29=84.1795005381311x_{29} = 84.1795005381311
x30=27.6308327735149x_{30} = 27.6308327735149
x31=49.6219813486434x_{31} = 49.6219813486434
x32=93.6042784989005x_{32} = 93.6042784989005
x33=81.0379078845413x_{33} = 81.0379078845413
x34=24.4892401199251x_{34} = 24.4892401199251
Signos de extremos en los puntos:
(62.18835196300258, 5)

(18.206054812745474, 5)

(43.33879604146382, 5)

(5.6397152163885265, 4.99991613755639)

(103.02905645966989, -5)

(68.47153727018217, 5)

(52.7635740022332, -5)

(8.781276764407652, -5.00000023071478)

(65.32994461659237, -5)

(74.75472257736175, 5)

(90.46268584531072, -5)

(33.91401808069444, -5)

(15.06446215915517, -5.00000000000132)

(21.347647466335268, -5)

(143.86976095633722, 5)

(37.05561073428424, 5)

(96.74587115249031, -5)

(40.197203387874026, -5)

(46.48038869505361, -5)

(87.32109319172092, 5)

(59.046759309412785, -5)

(77.89631523095154, -5)

(99.8874638060801, 5)

(30.772425427104647, 5)

(2.489868269302824, -5.02382683926261)

(55.905166655822995, 5)

(71.61312992377196, -5)

(11.922869505784771, 4.99999999943077)

(84.17950053813114, -5)

(27.630832773514854, -5)

(49.62198134864341, 5)

(93.60427849890051, 5)

(81.03790788454134, 5)

(24.48924011992506, 5)


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=103.02905645967x_{1} = 103.02905645967
x2=52.7635740022332x_{2} = 52.7635740022332
x3=8.78127676440765x_{3} = 8.78127676440765
x4=65.3299446165924x_{4} = 65.3299446165924
x5=90.4626858453107x_{5} = 90.4626858453107
x6=33.9140180806944x_{6} = 33.9140180806944
x7=15.0644621591552x_{7} = 15.0644621591552
x8=21.3476474663353x_{8} = 21.3476474663353
x9=96.7458711524903x_{9} = 96.7458711524903
x10=40.197203387874x_{10} = 40.197203387874
x11=46.4803886950536x_{11} = 46.4803886950536
x12=59.0467593094128x_{12} = 59.0467593094128
x13=77.8963152309515x_{13} = 77.8963152309515
x14=2.48986826930282x_{14} = 2.48986826930282
x15=71.613129923772x_{15} = 71.613129923772
x16=84.1795005381311x_{16} = 84.1795005381311
x17=27.6308327735149x_{17} = 27.6308327735149
Puntos máximos de la función:
x17=62.1883519630026x_{17} = 62.1883519630026
x17=18.2060548127455x_{17} = 18.2060548127455
x17=43.3387960414638x_{17} = 43.3387960414638
x17=5.63971521638853x_{17} = 5.63971521638853
x17=68.4715372701822x_{17} = 68.4715372701822
x17=74.7547225773617x_{17} = 74.7547225773617
x17=143.869760956337x_{17} = 143.869760956337
x17=37.0556107342842x_{17} = 37.0556107342842
x17=87.3210931917209x_{17} = 87.3210931917209
x17=99.8874638060801x_{17} = 99.8874638060801
x17=30.7724254271046x_{17} = 30.7724254271046
x17=55.905166655823x_{17} = 55.905166655823
x17=11.9228695057848x_{17} = 11.9228695057848
x17=49.6219813486434x_{17} = 49.6219813486434
x17=93.6042784989005x_{17} = 93.6042784989005
x17=81.0379078845413x_{17} = 81.0379078845413
x17=24.4892401199251x_{17} = 24.4892401199251
Decrece en los intervalos
[103.02905645967,∞)\left[103.02905645967, \infty\right)
Crece en los intervalos
(−∞,2.48986826930282]\left(-\infty, 2.48986826930282\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
−4(x+1)e−2x+3sin⁡(x)−4cos⁡(x)+4e−2x=0- 4 \left(x + 1\right) e^{- 2 x} + 3 \sin{\left(x \right)} - 4 \cos{\left(x \right)} + 4 e^{- 2 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=79.4671115577464x_{1} = 79.4671115577464
x2=92.0334821721056x_{2} = 92.0334821721056
x3=48.0511850218485x_{3} = 48.0511850218485
x4=88.8918895185158x_{4} = 88.8918895185158
x5=73.1839262505669x_{5} = 73.1839262505669
x6=4.06793485442827x_{6} = 4.06793485442827
x7=7.21048367371124x_{7} = 7.21048367371124
x8=51.1927776754383x_{8} = 51.1927776754383
x9=54.3343703290281x_{9} = 54.3343703290281
x10=38.6264070610791x_{10} = 38.6264070610791
x11=70.0423335969771x_{11} = 70.0423335969771
x12=26.06003644672x_{12} = 26.06003644672
x13=95.1750748256954x_{13} = 95.1750748256954
x14=63.7591482897975x_{14} = 63.7591482897975
x15=16.6352584859505x_{15} = 16.6352584859505
x16=44.9095923682587x_{16} = 44.9095923682587
x17=35.4848144074893x_{17} = 35.4848144074893
x18=60.6175556362077x_{18} = 60.6175556362077
x19=66.9007409433873x_{19} = 66.9007409433873
x20=10.3520731703295x_{20} = 10.3520731703295
x21=82.6087042113362x_{21} = 82.6087042113362
x22=76.3255189041567x_{22} = 76.3255189041567
x23=57.4759629826179x_{23} = 57.4759629826179
x24=32.3432217538995x_{24} = 32.3432217538995
x25=19.7768511395404x_{25} = 19.7768511395404
x26=104.599852786465x_{26} = 104.599852786465
x27=98.3166674792852x_{27} = 98.3166674792852
x28=41.7679997146689x_{28} = 41.7679997146689
x29=1.03226516573201x_{29} = 1.03226516573201
x30=29.2016291003098x_{30} = 29.2016291003098
x31=13.4936658323813x_{31} = 13.4936658323813
x32=22.9184437931302x_{32} = 22.9184437931302
x33=−0.475843991727831x_{33} = -0.475843991727831
x34=85.750296864926x_{34} = 85.750296864926

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
[95.1750748256954,∞)\left[95.1750748256954, \infty\right)
Convexa en los intervalos
(−∞,1.03226516573201]\left(-\infty, 1.03226516573201\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x)))=∞\lim_{x \to -\infty}\left(\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x)))=⟨−7,7⟩\lim_{x \to \infty}\left(\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)\right) = \left\langle -7, 7\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−7,7⟩y = \left\langle -7, 7\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -3*sin(x) + 4*cos(x) + (-1 - x)*exp(-2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x))x)=−∞\lim_{x \to -\infty}\left(\frac{\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞((−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)}{x}\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−1)e−2x+(−3sin⁡(x)+4cos⁡(x))=(x−1)e2x+3sin⁡(x)+4cos⁡(x)\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) = \left(x - 1\right) e^{2 x} + 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}
- No
(−x−1)e−2x+(−3sin⁡(x)+4cos⁡(x))=−(x−1)e2x−3sin⁡(x)−4cos⁡(x)\left(- x - 1\right) e^{- 2 x} + \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) = - \left(x - 1\right) e^{2 x} - 3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = -3*sin(x)+4*cos(x)+(-1-x)*exp(-2*x)
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