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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=−17.2787606146987x_{1} = -17.2787606146987
x2=−32.9867228626925x_{2} = -32.9867228626925
x3=−98.9601685880785x_{3} = -98.9601685880785
x4=−7.84863578206022x_{4} = -7.84863578206022
x5=−4.77608862105439x_{5} = -4.77608862105439
x6=−64.4026493985908x_{6} = -64.4026493985908
x7=−67.5442420521806x_{7} = -67.5442420521806
x8=−51.8362787842316x_{8} = -51.8362787842316
x9=−86.3937979737193x_{9} = -86.3937979737193
x10=−26.7035375553833x_{10} = -26.7035375553833
x11=−29.8451302091093x_{11} = -29.8451302091093
x12=−45.553093477052x_{12} = -45.553093477052
x13=−89.5353906273091x_{13} = -89.5353906273091
x14=−10.9959095535381x_{14} = -10.9959095535381
x15=−58.1194640914112x_{15} = -58.1194640914112
x16=−36.1283155162826x_{16} = -36.1283155162826
x17=−61.261056745001x_{17} = -61.261056745001
x18=−20.4203521957512x_{18} = -20.4203521957512
x19=−73.8274273593601x_{19} = -73.8274273593601
x20=−14.1371478933127x_{20} = -14.1371478933127
x21=−284.314135149876x_{21} = -284.314135149876
x22=−1.37879548460342x_{22} = -1.37879548460342
x23=−48.6946861306418x_{23} = -48.6946861306418
x24=−23.5619449045633x_{24} = -23.5619449045633
x25=−54.9778714378214x_{25} = -54.9778714378214
x26=−95.8185759344887x_{26} = -95.8185759344887
x27=−83.2522053201295x_{27} = -83.2522053201295
x28=−76.9690200129499x_{28} = -76.9690200129499
x29=−42.4115008234622x_{29} = -42.4115008234622
x30=−70.6858347057703x_{30} = -70.6858347057703
x31=−92.6769832808989x_{31} = -92.6769832808989
x32=−80.1106126665397x_{32} = -80.1106126665397
x33=−39.2699081698724x_{33} = -39.2699081698724
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).
cos⁡(0)2+e0\frac{\cos{\left(0 \right)}}{2} + e^{0}
Resultado:
f(0)=32f{\left(0 \right)} = \frac{3}{2}
Punto:
(0, 3/2)
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=−56.5486677646163x_{1} = -56.5486677646163
x2=−9.42357836392699x_{2} = -9.42357836392699
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=−37.6991118430775x_{4} = -37.6991118430775
x5=−31.4159265358993x_{5} = -31.4159265358993
x6=−25.132741229281x_{6} = -25.132741229281
x7=−18.8495561410012x_{7} = -18.8495561410012
x8=−50.2654824574367x_{8} = -50.2654824574367
x9=−94.2477796076938x_{9} = -94.2477796076938
x10=−21.9911485638765x_{10} = -21.9911485638765
x11=−75.398223686155x_{11} = -75.398223686155
x12=−28.2743338822805x_{12} = -28.2743338822805
x13=−53.4070751110265x_{13} = -53.4070751110265
x14=−113.097335529233x_{14} = -113.097335529233
x15=−47.1238898038469x_{15} = -47.1238898038469
x16=−43.9822971502571x_{16} = -43.9822971502571
x17=−62.8318530717959x_{17} = -62.8318530717959
x18=−65.9734457253857x_{18} = -65.9734457253857
x19=−59.6902604182061x_{19} = -59.6902604182061
x20=−91.106186954104x_{20} = -91.106186954104
x21=−3.04193881531373x_{21} = -3.04193881531373
x22=−100.530964914873x_{22} = -100.530964914873
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=−69.1150383789755x_{24} = -69.1150383789755
x25=−84.8230016469244x_{25} = -84.8230016469244
x26=−15.7079591363057x_{26} = -15.7079591363057
x27=−78.5398163397448x_{27} = -78.5398163397448
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=−81.6814089933346x_{29} = -81.6814089933346
x30=−12.5664443065461x_{30} = -12.5664443065461
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=−232.477856365645x_{32} = -232.477856365645
x33=−34.5575191894877x_{33} = -34.5575191894877
x34=−6.29899042622584x_{34} = -6.29899042622584
Signos de extremos en los puntos:
(-56.548667764616276, 0.5)

(-9.423578363926987, -0.500680234901783)

(-97.3893722612836, -0.5)

(-37.699111843077524, 0.499999999999998)

(-31.415926535899267, 0.499999999999309)

(-25.132741229281006, 0.499999999706508)

(-18.84955614100122, 0.499999883756347)

(-50.26548245743669, 0.5)

(-94.2477796076938, 0.5)

(-21.99114856387646, -0.500000005907473)

(-75.39822368615503, 0.5)

(-28.274333882280523, -0.500000000014334)

(-53.40707511102649, -0.5)

(-113.09733552923255, 0.5)

(-47.1238898038469, -0.5)

(-43.982297150257104, 0.5)

(-62.83185307179586, 0.5)

(-65.97344572538566, -0.5)

(-59.69026041820607, -0.5)

(-91.106186954104, -0.5)

(-3.0419388153137312, -0.595006057568337)

(-100.53096491487338, 0.5)

(-40.840704496667314, -0.5)

(-69.11503837897546, 0.5)

(-84.82300164692441, -0.5)

(-15.70795913630565, -0.500002216519741)

(-78.53981633974483, -0.5)

(-87.96459430051421, 0.5)

(-81.68140899333463, 0.5)

(-12.566444306546071, 0.499959665463574)

(-72.25663103256524, -0.5)

(-232.4778563656447, 0.5)

(-34.557519189487664, -0.500000000000033)

(-6.298990426225845, 0.490197160739698)


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=−9.42357836392699x_{1} = -9.42357836392699
x2=−97.3893722612836x_{2} = -97.3893722612836
x3=−21.9911485638765x_{3} = -21.9911485638765
x4=−28.2743338822805x_{4} = -28.2743338822805
x5=−53.4070751110265x_{5} = -53.4070751110265
x6=−47.1238898038469x_{6} = -47.1238898038469
x7=−65.9734457253857x_{7} = -65.9734457253857
x8=−59.6902604182061x_{8} = -59.6902604182061
x9=−91.106186954104x_{9} = -91.106186954104
x10=−3.04193881531373x_{10} = -3.04193881531373
x11=−40.8407044966673x_{11} = -40.8407044966673
x12=−84.8230016469244x_{12} = -84.8230016469244
x13=−15.7079591363057x_{13} = -15.7079591363057
x14=−78.5398163397448x_{14} = -78.5398163397448
x15=−72.2566310325652x_{15} = -72.2566310325652
x16=−34.5575191894877x_{16} = -34.5575191894877
Puntos máximos de la función:
x16=−56.5486677646163x_{16} = -56.5486677646163
x16=−37.6991118430775x_{16} = -37.6991118430775
x16=−31.4159265358993x_{16} = -31.4159265358993
x16=−25.132741229281x_{16} = -25.132741229281
x16=−18.8495561410012x_{16} = -18.8495561410012
x16=−50.2654824574367x_{16} = -50.2654824574367
x16=−94.2477796076938x_{16} = -94.2477796076938
x16=−75.398223686155x_{16} = -75.398223686155
x16=−113.097335529233x_{16} = -113.097335529233
x16=−43.9822971502571x_{16} = -43.9822971502571
x16=−62.8318530717959x_{16} = -62.8318530717959
x16=−100.530964914873x_{16} = -100.530964914873
x16=−69.1150383789755x_{16} = -69.1150383789755
x16=−87.9645943005142x_{16} = -87.9645943005142
x16=−81.6814089933346x_{16} = -81.6814089933346
x16=−12.5664443065461x_{16} = -12.5664443065461
x16=−232.477856365645x_{16} = -232.477856365645
x16=−6.29899042622584x_{16} = -6.29899042622584
Decrece en los intervalos
[−3.04193881531373,∞)\left[-3.04193881531373, \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=−98.9601685880785x_{1} = -98.9601685880785
x2=−4.68122187186061x_{2} = -4.68122187186061
x3=−20.4203522955009x_{3} = -20.4203522955009
x4=−39.2699081698724x_{4} = -39.2699081698724
x5=−23.5619448995176x_{5} = -23.5619448995176
x6=−64.4026493985908x_{6} = -64.4026493985908
x7=−67.5442420521806x_{7} = -67.5442420521806
x8=−51.8362787842316x_{8} = -51.8362787842316
x9=−86.3937979737193x_{9} = -86.3937979737193
x10=−89.5353906273091x_{10} = -89.5353906273091
x11=−45.553093477052x_{11} = -45.553093477052
x12=−26.7035375556331x_{12} = -26.7035375556331
x13=−58.1194640914112x_{13} = -58.1194640914112
x14=−61.261056745001x_{14} = -61.261056745001
x15=−73.8274273593601x_{15} = -73.8274273593601
x16=−205.774318810131x_{16} = -205.774318810131
x17=−10.9953059605593x_{17} = -10.9953059605593
x18=−7.85773907765605x_{18} = -7.85773907765605
x19=−36.1283155162826x_{19} = -36.1283155162826
x20=−48.6946861306418x_{20} = -48.6946861306418
x21=−54.9778714378214x_{21} = -54.9778714378214
x22=−14.1371830886339x_{22} = -14.1371830886339
x23=−95.8185759344887x_{23} = -95.8185759344887
x24=−17.2787587000985x_{24} = -17.2787587000985
x25=−29.8451302090972x_{25} = -29.8451302090972
x26=−83.2522053201295x_{26} = -83.2522053201295
x27=−42.4115008234622x_{27} = -42.4115008234622
x28=−76.9690200129499x_{28} = -76.9690200129499
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=−92.6769832808989x_{30} = -92.6769832808989
x31=−80.1106126665397x_{31} = -80.1106126665397
x32=−32.9867228626931x_{32} = -32.9867228626931

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
[−4.68122187186061,∞)\left[-4.68122187186061, \infty\right)
Convexa en los intervalos
(−∞,−205.774318810131]\left(-\infty, -205.774318810131\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=(1−x)e−x+cos⁡(x)2\left(x + 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} = \left(1 - x\right) e^{- x} + \frac{\cos{\left(x \right)}}{2}
- No
(x+1)ex+cos⁡(x)2=−(1−x)e−x−cos⁡(x)2\left(x + 1\right) e^{x} + \frac{\cos{\left(x \right)}}{2} = - \left(1 - x\right) e^{- x} - \frac{\cos{\left(x \right)}}{2}
- No
es decir, función
no es
par ni impar
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