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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)             x
f(x) = ------ + (-1 - x)*e 
         2
f(x)=(−x−1)ex+cos⁡(x)2f{\left(x \right)} = \left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} 
f = (-x - 1)*exp(x) + cos(x)/2
Gráfico de la función
02468-8-6-4-2-1010-250000250000
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)ex+cos⁡(x)2=0\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−4.64213179049313x_{1} = -4.64213179049313
x2=−7.85927910942748x_{2} = -7.85927910942748
x3=−10.9952388191524x_{3} = -10.9952388191524
x4=−89.5353906273091x_{4} = -89.5353906273091
x5=−1449.84500963169x_{5} = -1449.84500963169
x6=−42.4115008234622x_{6} = -42.4115008234622
x7=−61.261056745001x_{7} = -61.261056745001
x8=−76.9690200129499x_{8} = -76.9690200129499
x9=−92.6769832808989x_{9} = -92.6769832808989
x10=−98.9601685880785x_{10} = -98.9601685880785
x11=−54.9778714378214x_{11} = -54.9778714378214
x12=−64.4026493985908x_{12} = -64.4026493985908
x13=−17.2787585747871x_{13} = -17.2787585747871
x14=−32.9867228626931x_{14} = -32.9867228626931
x15=−23.5619448992836x_{15} = -23.5619448992836
x16=−1.84093012704925x_{16} = -1.84093012704925
x17=−51.8362787842316x_{17} = -51.8362787842316
x18=−48.6946861306418x_{18} = -48.6946861306418
x19=−36.1283155162826x_{19} = -36.1283155162826
x20=−73.8274273593601x_{20} = -73.8274273593601
x21=−29.8451302090967x_{21} = -29.8451302090967
x22=−86.3937979737193x_{22} = -86.3937979737193
x23=−14.137185988325x_{23} = -14.137185988325
x24=−67.5442420521806x_{24} = -67.5442420521806
x25=−45.553093477052x_{25} = -45.553093477052
x26=−70.6858347057703x_{26} = -70.6858347057703
x27=−83.2522053201295x_{27} = -83.2522053201295
x28=−95.8185759344887x_{28} = -95.8185759344887
x29=−26.7035375556432x_{29} = -26.7035375556432
x30=−39.2699081698724x_{30} = -39.2699081698724
x31=−80.1106126665397x_{31} = -80.1106126665397
x32=−0.338417457221049x_{32} = -0.338417457221049
x33=−58.1194640914112x_{33} = -58.1194640914112
x34=−20.4203523009161x_{34} = -20.4203523009161
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 cos(x)/2 + (-1 - x)*exp(x).
(−1−0)e0+cos⁡(0)2\left(-1 - 0\right) e^{0} + \frac{\cos{\left(0 \right)}}{2}
Resultado:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
Punto:
(0, -1/2)
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
(−x−1)ex−ex−sin⁡(x)2=0\left(- x - 1\right) e^{x} - e^{x} - \frac{\sin{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=−37.6991118430775x_{5} = -37.6991118430775
x6=−81.6814089933346x_{6} = -81.6814089933346
x7=−84.8230016469244x_{7} = -84.8230016469244
x8=−113.097335529233x_{8} = -113.097335529233
x9=−87.9645943005142x_{9} = -87.9645943005142
x10=−100.530964914873x_{10} = -100.530964914873
x11=−18.8495557020762x_{11} = -18.8495557020762
x12=−9.42597507233309x_{12} = -9.42597507233309
x13=−0.949419181488109x_{13} = -0.949419181488109
x14=−15.7079673995606x_{14} = -15.7079673995606
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=−34.5575191894878x_{16} = -34.5575191894878
x17=−69.1150383789755x_{17} = -69.1150383789755
x18=−50.2654824574367x_{18} = -50.2654824574367
x19=−78.5398163397448x_{19} = -78.5398163397448
x20=−53.4070751110265x_{20} = -53.4070751110265
x21=−12.5662969123378x_{21} = -12.5662969123378
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=−232.477856365645x_{23} = -232.477856365645
x24=−65.9734457253857x_{24} = -65.9734457253857
x25=−31.4159265358966x_{25} = -31.4159265358966
x26=−3.23889333917348x_{26} = -3.23889333917348
x27=−28.2743338823358x_{27} = -28.2743338823358
x28=−47.1238898038469x_{28} = -47.1238898038469
x29=−75.398223686155x_{29} = -75.398223686155
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=−91.106186954104x_{31} = -91.106186954104
x32=−21.9911485863806x_{32} = -21.9911485863806
x33=−6.26698764944339x_{33} = -6.26698764944339
x34=−40.8407044966673x_{34} = -40.8407044966673
x35=−25.1327412281557x_{35} = -25.1327412281557
Signos de extremos en los puntos:
(-59.69026041820607, -0.5)

(-62.83185307179586, 0.5)

(-97.3893722612836, -0.5)

(-56.548667764616276, 0.5)

(-37.69911184307752, 0.500000000000002)

(-81.68140899333463, 0.5)

(-84.82300164692441, -0.5)

(-113.09733552923255, 0.5)

(-87.96459430051421, 0.5)

(-100.53096491487338, 0.5)

(-18.84955570207621, 0.500000116243677)

(-9.425975072333086, -0.499320483123082)

(-0.9494191814881093, 0.271504676449376)

(-15.707967399560632, -0.499997783488795)

(-94.2477796076938, 0.5)

(-34.55751918948779, -0.499999999999967)

(-69.11503837897546, 0.5)

(-50.26548245743669, 0.5)

(-78.53981633974483, -0.5)

(-53.40707511102649, -0.5)

(-12.566296912337776, 0.500040337252057)

(-43.982297150257104, 0.5)

(-232.4778563656447, 0.5)

(-65.97344572538566, -0.5)

(-31.415926535896595, 0.500000000000691)

(-3.238893339173482, -0.409854137097742)

(-28.274333882335757, -0.499999999985666)

(-47.1238898038469, -0.5)

(-75.39822368615503, 0.5)

(-72.25663103256524, -0.5)

(-91.106186954104, -0.5)

(-21.991148586380643, -0.499999994092527)

(-6.266987649443386, 0.50993082237036)

(-40.840704496667314, -0.5)

(-25.132741228155684, 0.500000000293492)


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=−59.6902604182061x_{1} = -59.6902604182061
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−84.8230016469244x_{3} = -84.8230016469244
x4=−9.42597507233309x_{4} = -9.42597507233309
x5=−15.7079673995606x_{5} = -15.7079673995606
x6=−34.5575191894878x_{6} = -34.5575191894878
x7=−78.5398163397448x_{7} = -78.5398163397448
x8=−53.4070751110265x_{8} = -53.4070751110265
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=−3.23889333917348x_{10} = -3.23889333917348
x11=−28.2743338823358x_{11} = -28.2743338823358
x12=−47.1238898038469x_{12} = -47.1238898038469
x13=−72.2566310325652x_{13} = -72.2566310325652
x14=−91.106186954104x_{14} = -91.106186954104
x15=−21.9911485863806x_{15} = -21.9911485863806
x16=−40.8407044966673x_{16} = -40.8407044966673
Puntos máximos de la función:
x16=−62.8318530717959x_{16} = -62.8318530717959
x16=−56.5486677646163x_{16} = -56.5486677646163
x16=−37.6991118430775x_{16} = -37.6991118430775
x16=−81.6814089933346x_{16} = -81.6814089933346
x16=−113.097335529233x_{16} = -113.097335529233
x16=−87.9645943005142x_{16} = -87.9645943005142
x16=−100.530964914873x_{16} = -100.530964914873
x16=−18.8495557020762x_{16} = -18.8495557020762
x16=−0.949419181488109x_{16} = -0.949419181488109
x16=−94.2477796076938x_{16} = -94.2477796076938
x16=−69.1150383789755x_{16} = -69.1150383789755
x16=−50.2654824574367x_{16} = -50.2654824574367
x16=−12.5662969123378x_{16} = -12.5662969123378
x16=−43.9822971502571x_{16} = -43.9822971502571
x16=−232.477856365645x_{16} = -232.477856365645
x16=−31.4159265358966x_{16} = -31.4159265358966
x16=−75.398223686155x_{16} = -75.398223686155
x16=−6.26698764944339x_{16} = -6.26698764944339
x16=−25.1327412281557x_{16} = -25.1327412281557
Decrece en los intervalos
[−3.23889333917348,∞)\left[-3.23889333917348, \infty\right)
Crece en los intervalos
(−∞,−97.3893722612836]\left(-\infty, -97.3893722612836\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
−((x+1)ex+2ex+cos⁡(x)2)=0- (\left(x + 1\right) e^{x} + 2 e^{x} + \frac{\cos{\left(x \right)}}{2}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−36.1283155162826x_{1} = -36.1283155162826
x2=−26.7035375553934x_{2} = -26.7035375553934
x3=−10.9958424886394x_{3} = -10.9958424886394
x4=−89.5353906273091x_{4} = -89.5353906273091
x5=−42.4115008234622x_{5} = -42.4115008234622
x6=−61.261056745001x_{6} = -61.261056745001
x7=−76.9690200129499x_{7} = -76.9690200129499
x8=−92.6769832808989x_{8} = -92.6769832808989
x9=−98.9601685880785x_{9} = -98.9601685880785
x10=−54.9778714378214x_{10} = -54.9778714378214
x11=−64.4026493985908x_{11} = -64.4026493985908
x12=−51.8362787842316x_{12} = -51.8362787842316
x13=−23.5619449043293x_{13} = -23.5619449043293
x14=−29.8451302091089x_{14} = -29.8451302091089
x15=−48.6946861306418x_{15} = -48.6946861306418
x16=−73.8274273593601x_{16} = -73.8274273593601
x17=−20.4203522011664x_{17} = -20.4203522011664
x18=−4.74276832118964x_{18} = -4.74276832118964
x19=−86.3937979737193x_{19} = -86.3937979737193
x20=−67.5442420521806x_{20} = -67.5442420521806
x21=−356.570766182442x_{21} = -356.570766182442
x22=−45.553093477052x_{22} = -45.553093477052
x23=−70.6858347057703x_{23} = -70.6858347057703
x24=−83.2522053201295x_{24} = -83.2522053201295
x25=−95.8185759344887x_{25} = -95.8185759344887
x26=−39.2699081698724x_{26} = -39.2699081698724
x27=−7.8502016359633x_{27} = -7.8502016359633
x28=−32.9867228626925x_{28} = -32.9867228626925
x29=−17.2787604893877x_{29} = -17.2787604893877
x30=−1.90366454954695x_{30} = -1.90366454954695
x31=−80.1106126665397x_{31} = -80.1106126665397
x32=−14.1371507931996x_{32} = -14.1371507931996
x33=−58.1194640914112x_{33} = -58.1194640914112

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
[−4.74276832118964,∞)\left[-4.74276832118964, \infty\right)
Convexa en los intervalos
(−∞,−356.570766182442]\left(-\infty, -356.570766182442\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−x−1)ex+cos⁡(x)2)=⟨−12,12⟩\lim_{x \to -\infty}\left(\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
lim⁡x→∞((−x−1)ex+cos⁡(x)2)=−∞\lim_{x \to \infty}\left(\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2}\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 cos(x)/2 + (-1 - x)*exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((−x−1)ex+cos⁡(x)2x)=0\lim_{x \to -\infty}\left(\frac{\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((−x−1)ex+cos⁡(x)2x)=−∞\lim_{x \to \infty}\left(\frac{\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2}}{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:
(−x−1)ex+cos⁡(x)2=(x−1)e−x+cos⁡(x)2\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} = \left(x - 1\right) e^{- x} + \frac{\cos{\left(x \right)}}{2}
- No
(−x−1)ex+cos⁡(x)2=−(x−1)e−x−cos⁡(x)2\left(- x - 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} = - \left(x - 1\right) e^{- x} - \frac{\cos{\left(x \right)}}{2}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = cos(x)/2+(-1-x)*exp(x)
    © Sr Examen – Калькулятор