Sr Examen

Otras calculadoras

(1-6*exp(-2*x)*sin(x)/5+2*cos(x)*exp(-2*x)/5)*exp(2*x)
Trazado el gráfico de la función/ (1-6*exp(-2*x)*sin(x)/5+2*cos(x)*exp(-2*x)/5)*exp(2*x)

Gráfico de la función y = (1-6*exp(-2*x)*sin(x)/5+2*cos(x)*exp(-2*x)/5)*exp(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /       -2*x                    -2*x\     
       |    6*e    *sin(x)   2*cos(x)*e    |  2*x
f(x) = |1 - -------------- + --------------|*e   
       \          5                5       /
f(x)=(e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2xf{\left(x \right)} = \left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x} 
f = ((exp(-2*x)*(2*cos(x)))/5 - (6*exp(-2*x))*sin(x)/5 + 1)*exp(2*x)
Gráfico de la función
02468-8-6-4-2-1010-500000000500000000
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:
(e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x=0\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−12.2446200599442x_{1} = -12.2446200599442
x2=−18.5278053671421x_{2} = -18.5278053671421
x3=−53.0853245566298x_{3} = -53.0853245566298
x4=−87.6428437461176x_{4} = -87.6428437461176
x5=−9.10302741617104x_{5} = -9.10302741617104
x6=−34.2357686350911x_{6} = -34.2357686350911
x7=−2.8226361222719x_{7} = -2.8226361222719
x8=−31.0941759815013x_{8} = -31.0941759815013
x9=−75.0764731317584x_{9} = -75.0764731317584
x10=−93.9260290532972x_{10} = -93.9260290532972
x11=−65.651695170989x_{11} = -65.651695170989
x12=−71.9348804781686x_{12} = -71.9348804781686
x13=−78.2180657853482x_{13} = -78.2180657853482
x14=−62.5101025173992x_{14} = -62.5101025173992
x15=−100.209214360477x_{15} = -100.209214360477
x16=−43.6605465958605x_{16} = -43.6605465958605
x17=−56.2269172102196x_{17} = -56.2269172102196
x18=−81.359658438938x_{18} = -81.359658438938
x19=−97.067621706887x_{19} = -97.067621706887
x20=−24.8109906743217x_{20} = -24.8109906743217
x21=−68.7932878245788x_{21} = -68.7932878245788
x22=−5.96142950581838x_{22} = -5.96142950581838
x23=−15.3862127135524x_{23} = -15.3862127135524
x24=−84.5012510925278x_{24} = -84.5012510925278
x25=−46.8021392494503x_{25} = -46.8021392494503
x26=−21.6693980207319x_{26} = -21.6693980207319
x27=−40.5189539422707x_{27} = -40.5189539422707
x28=−219.589735196889x_{28} = -219.589735196889
x29=−37.3773612886809x_{29} = -37.3773612886809
x30=−59.3685098638094x_{30} = -59.3685098638094
x31=−90.7844363997074x_{31} = -90.7844363997074
x32=−27.9525833279115x_{32} = -27.9525833279115
x33=−49.9437319030401x_{33} = -49.9437319030401
x34=−109.633992321246x_{34} = -109.633992321246
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 - (6*exp(-2*x))*sin(x)/5 + ((2*cos(x))*exp(-2*x))/5)*exp(2*x).
(e−0⋅2cos⁡(0)5+(−6e−0sin⁡(0)5+1))e0⋅2\left(\frac{e^{- 0} \cdot 2 \cos{\left(0 \right)}}{5} + \left(- \frac{6 e^{- 0} \sin{\left(0 \right)}}{5} + 1\right)\right) e^{0 \cdot 2}
Resultado:
f(0)=75f{\left(0 \right)} = \frac{7}{5}
Punto:
(0, 7/5)
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(e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x+(2e−2xsin⁡(x)−2e−2xcos⁡(x))e2x=02 \left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x} + \left(2 e^{- 2 x} \sin{\left(x \right)} - 2 e^{- 2 x} \cos{\left(x \right)}\right) e^{2 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−7.53223062609839x_{1} = -7.53223062609839
x2=−76.6472694585533x_{2} = -76.6472694585533
x3=−86.0720474193227x_{3} = -86.0720474193227
x4=−4.39088114237964x_{4} = -4.39088114237964
x5=−73.5056768049635x_{5} = -73.5056768049635
x6=−13.8154163867558x_{6} = -13.8154163867558
x7=−48.3729355762452x_{7} = -48.3729355762452
x8=−60.9393061906043x_{8} = -60.9393061906043
x9=−10.6738237340145x_{9} = -10.6738237340145
x10=−1.05689769016997x_{10} = -1.05689769016997
x11=−16.9570090403472x_{11} = -16.9570090403472
x12=−35.806564961886x_{12} = -35.806564961886
x13=−51.5145282298349x_{13} = -51.5145282298349
x14=−98.6384180336818x_{14} = -98.6384180336818
x15=−20.098601693937x_{15} = -20.098601693937
x16=−0.352584426420149x_{16} = -0.352584426420149
x17=−70.3640841513737x_{17} = -70.3640841513737
x18=−54.6561208834247x_{18} = -54.6561208834247
x19=−67.2224914977839x_{19} = -67.2224914977839
x20=−82.9304547657329x_{20} = -82.9304547657329
x21=−42.0897502690656x_{21} = -42.0897502690656
x22=−32.6649723082962x_{22} = -32.6649723082962
x23=−89.2136400729125x_{23} = -89.2136400729125
x24=−38.9481576154758x_{24} = -38.9481576154758
x25=−26.3817870011166x_{25} = -26.3817870011166
x26=−57.7977135370145x_{26} = -57.7977135370145
x27=−64.0808988441941x_{27} = -64.0808988441941
x28=−92.3552327265023x_{28} = -92.3552327265023
x29=−79.7888621121431x_{29} = -79.7888621121431
x30=−45.2313429226554x_{30} = -45.2313429226554
x31=−23.2401943475268x_{31} = -23.2401943475268
x32=−95.496825380092x_{32} = -95.496825380092
x33=−29.5233796547064x_{33} = -29.5233796547064
Signos de extremos en los puntos:
(-7.532230626098393, 1.26491135087281)

(-76.6472694585533, 1.26491106406735)

(-86.07204741932267, -1.26491106406735)

(-4.390881142379643, -1.26475751948553)

(-73.5056768049635, -1.26491106406735)

(-13.815416386755846, 1.26491106406835)

(-48.372935576245155, -1.26491106406735)

(-60.93930619060433, -1.26491106406735)

(-10.67382373401448, -1.26491106353176)

(-1.0568976901699663, 1.36241069617632)

(-16.957009040347224, -1.26491106406735)

(-35.80656496188598, -1.26491106406735)

(-51.514528229834944, 1.26491106406735)

(-98.63841803368184, -1.26491106406735)

(-20.098601693937013, 1.26491106406735)

(-0.3525844264201486, 1.28380778178455)

(-70.3640841513737, 1.26491106406735)

(-54.65612088342474, -1.26491106406735)

(-67.2224914977839, -1.26491106406735)

(-82.93045476573288, 1.26491106406735)

(-42.08975026906557, -1.26491106406735)

(-32.664972308296186, 1.26491106406735)

(-89.21364007291247, 1.26491106406735)

(-38.94815761547577, 1.26491106406735)

(-26.3817870011166, 1.26491106406735)

(-57.79771353701453, 1.26491106406735)

(-64.08089884419412, 1.26491106406735)

(-92.35523272650225, -1.26491106406735)

(-79.78886211214308, -1.26491106406735)

(-45.23134292265536, 1.26491106406735)

(-23.240194347526806, -1.26491106406735)

(-95.49682538009205, 1.26491106406735)

(-29.523379654706392, -1.26491106406735)


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=−86.0720474193227x_{1} = -86.0720474193227
x2=−4.39088114237964x_{2} = -4.39088114237964
x3=−73.5056768049635x_{3} = -73.5056768049635
x4=−48.3729355762452x_{4} = -48.3729355762452
x5=−60.9393061906043x_{5} = -60.9393061906043
x6=−10.6738237340145x_{6} = -10.6738237340145
x7=−16.9570090403472x_{7} = -16.9570090403472
x8=−35.806564961886x_{8} = -35.806564961886
x9=−98.6384180336818x_{9} = -98.6384180336818
x10=−0.352584426420149x_{10} = -0.352584426420149
x11=−54.6561208834247x_{11} = -54.6561208834247
x12=−67.2224914977839x_{12} = -67.2224914977839
x13=−42.0897502690656x_{13} = -42.0897502690656
x14=−92.3552327265023x_{14} = -92.3552327265023
x15=−79.7888621121431x_{15} = -79.7888621121431
x16=−23.2401943475268x_{16} = -23.2401943475268
x17=−29.5233796547064x_{17} = -29.5233796547064
Puntos máximos de la función:
x17=−7.53223062609839x_{17} = -7.53223062609839
x17=−76.6472694585533x_{17} = -76.6472694585533
x17=−13.8154163867558x_{17} = -13.8154163867558
x17=−1.05689769016997x_{17} = -1.05689769016997
x17=−51.5145282298349x_{17} = -51.5145282298349
x17=−20.098601693937x_{17} = -20.098601693937
x17=−70.3640841513737x_{17} = -70.3640841513737
x17=−82.9304547657329x_{17} = -82.9304547657329
x17=−32.6649723082962x_{17} = -32.6649723082962
x17=−89.2136400729125x_{17} = -89.2136400729125
x17=−38.9481576154758x_{17} = -38.9481576154758
x17=−26.3817870011166x_{17} = -26.3817870011166
x17=−57.7977135370145x_{17} = -57.7977135370145
x17=−64.0808988441941x_{17} = -64.0808988441941
x17=−45.2313429226554x_{17} = -45.2313429226554
x17=−95.496825380092x_{17} = -95.496825380092
Decrece en los intervalos
[−0.352584426420149,∞)\left[-0.352584426420149, \infty\right)
Crece en los intervalos
(−∞,−98.6384180336818]\left(-\infty, -98.6384180336818\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
2(2(5−6e−2xsin⁡(x)+2e−2xcos⁡(x))e2x5+3sin⁡(x)−cos⁡(x))=02 \left(\frac{2 \left(5 - 6 e^{- 2 x} \sin{\left(x \right)} + 2 e^{- 2 x} \cos{\left(x \right)}\right) e^{2 x}}{5} + 3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−5.96145573954006x_{1} = -5.96145573954006
x2=−18.5278053671421x_{2} = -18.5278053671421
x3=−53.0853245566298x_{3} = -53.0853245566298
x4=−87.6428437461176x_{4} = -87.6428437461176
x5=−34.2357686350911x_{5} = -34.2357686350911
x6=−9.10302736717952x_{6} = -9.10302736717952
x7=−31.0941759815013x_{7} = -31.0941759815013
x8=−75.0764731317584x_{8} = -75.0764731317584
x9=−93.9260290532972x_{9} = -93.9260290532972
x10=−65.651695170989x_{10} = -65.651695170989
x11=−15.3862127135522x_{11} = -15.3862127135522
x12=−71.9348804781686x_{12} = -71.9348804781686
x13=−78.2180657853482x_{13} = -78.2180657853482
x14=−62.5101025173992x_{14} = -62.5101025173992
x15=−100.209214360477x_{15} = -100.209214360477
x16=−43.6605465958605x_{16} = -43.6605465958605
x17=−56.2269172102196x_{17} = -56.2269172102196
x18=−81.359658438938x_{18} = -81.359658438938
x19=−97.067621706887x_{19} = -97.067621706887
x20=−24.8109906743217x_{20} = -24.8109906743217
x21=−68.7932878245788x_{21} = -68.7932878245788
x22=−84.5012510925278x_{22} = -84.5012510925278
x23=−46.8021392494503x_{23} = -46.8021392494503
x24=−21.6693980207319x_{24} = -21.6693980207319
x25=−0.665950401177449x_{25} = -0.665950401177449
x26=−40.5189539422707x_{26} = -40.5189539422707
x27=−12.2446200600357x_{27} = -12.2446200600357
x28=−37.3773612886809x_{28} = -37.3773612886809
x29=−219.589735196889x_{29} = -219.589735196889
x30=−59.3685098638094x_{30} = -59.3685098638094
x31=−90.7844363997074x_{31} = -90.7844363997074
x32=−2.8083416444192x_{32} = -2.8083416444192
x33=−27.9525833279115x_{33} = -27.9525833279115
x34=−49.9437319030401x_{34} = -49.9437319030401
x35=−109.633992321246x_{35} = -109.633992321246

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
[−0.665950401177449,∞)\left[-0.665950401177449, \infty\right)
Convexa en los intervalos
(−∞,−219.589735196889]\left(-\infty, -219.589735196889\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞((e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x)y = \lim_{x \to -\infty}\left(\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x}\right)
lim⁡x→∞((e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x)=∞\lim_{x \to \infty}\left(\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 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 (1 - (6*exp(-2*x))*sin(x)/5 + ((2*cos(x))*exp(-2*x))/5)*exp(2*x), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞((e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2xx)y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x}}{x}\right)
lim⁡x→∞((e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2xx)=∞\lim_{x \to \infty}\left(\frac{\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 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:
(e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x=(6e2xsin⁡(x)5+2e2xcos⁡(x)5+1)e−2x\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x} = \left(\frac{6 e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5} + 1\right) e^{- 2 x}
- No
(e−2x2cos⁡(x)5+(−6e−2xsin⁡(x)5+1))e2x=−(6e2xsin⁡(x)5+2e2xcos⁡(x)5+1)e−2x\left(\frac{e^{- 2 x} 2 \cos{\left(x \right)}}{5} + \left(- \frac{6 e^{- 2 x} \sin{\left(x \right)}}{5} + 1\right)\right) e^{2 x} = - \left(\frac{6 e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5} + 1\right) e^{- 2 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (1-6*exp(-2*x)*sin(x)/5+2*cos(x)*exp(-2*x)/5)*exp(2*x)
    © Sr Examen – Калькулятор