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Trazado el gráfico de la función/ cos(x)+2^x

Gráfico de la función y = cos(x)+2^x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 x
f(x) = cos(x) + 2
f(x)=2x+cos⁡(x)f{\left(x \right)} = 2^{x} + \cos{\left(x \right)} 
f = 2^x + cos(x)
Gráfico de la función
02468-8-6-4-2-10102000-1000
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:
2x+cos⁡(x)=02^{x} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−48.6946861306418x_{1} = -48.6946861306418
x2=−7.85829106584085x_{2} = -7.85829106584085
x3=−20.420352960959x_{3} = -20.420352960959
x4=−70.6858347057703x_{4} = -70.6858347057703
x5=−42.411500823462x_{5} = -42.411500823462
x6=−61.261056745001x_{6} = -61.261056745001
x7=−80.1106126665397x_{7} = -80.1106126665397
x8=−98.9601685880785x_{8} = -98.9601685880785
x9=−64.4026493985908x_{9} = -64.4026493985908
x10=−45.553093477052x_{10} = -45.553093477052
x11=−14.1372224384715x_{11} = -14.1372224384715
x12=−89.5353906273091x_{12} = -89.5353906273091
x13=−86.3937979737193x_{13} = -86.3937979737193
x14=−32.9867228628103x_{14} = -32.9867228628103
x15=−54.9778714378214x_{15} = -54.9778714378214
x16=−92.6769832808989x_{16} = -92.6769832808989
x17=−353.429173528852x_{17} = -353.429173528852
x18=−67.5442420521806x_{18} = -67.5442420521806
x19=−58.1194640914112x_{19} = -58.1194640914112
x20=−51.8362787842316x_{20} = -51.8362787842316
x21=−83.2522053201295x_{21} = -83.2522053201295
x22=−76.9690200129499x_{22} = -76.9690200129499
x23=−73.8274273593601x_{23} = -73.8274273593601
x24=−10.9950843397481x_{24} = -10.9950843397481
x25=−23.5619448211725x_{25} = -23.5619448211725
x26=−1.85156267401784x_{26} = -1.85156267401784
x27=−39.2699081698739x_{27} = -39.2699081698739
x28=−29.8451302080662x_{28} = -29.8451302080662
x29=−4.67318386449327x_{29} = -4.67318386449327
x30=−36.1283155162693x_{30} = -36.1283155162693
x31=−17.2787533058109x_{31} = -17.2787533058109
x32=−95.8185759344887x_{32} = -95.8185759344887
x33=−26.7035375646635x_{33} = -26.7035375646635
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^x.
cos⁡(0)+20\cos{\left(0 \right)} + 2^{0}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 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
2xlog⁡(2)−sin⁡(x)=02^{x} \log{\left(2 \right)} - \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−21.9911487414047x_{1} = -21.9911487414047
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=−12.5662563251112x_{3} = -12.5662563251112
x4=−3.21624487559609x_{4} = -3.21624487559609
x5=−53.4070751110265x_{5} = -53.4070751110265
x6=−3.2162448755964x_{6} = -3.2162448755964
x7=−84.8230016469244x_{7} = -84.8230016469244
x8=−147.65485471872x_{8} = -147.65485471872
x9=−40.8407044966677x_{9} = -40.8407044966677
x10=−28.2743338844432x_{10} = -28.2743338844432
x11=−6.2742295985067x_{11} = -6.2742295985067
x12=−34.5575191895151x_{12} = -34.5575191895151
x13=−75.398223686155x_{13} = -75.398223686155
x14=−69.1150383789755x_{14} = -69.1150383789755
x15=−91.106186954104x_{15} = -91.106186954104
x16=−100.530964914873x_{16} = -100.530964914873
x17=−78.5398163397448x_{17} = -78.5398163397448
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=−31.415926535656x_{19} = -31.415926535656
x20=−9.42578577684859x_{20} = -9.42578577684859
x21=−113.097335529233x_{21} = -113.097335529233
x22=−81.6814089933346x_{22} = -81.6814089933346
x23=−65.9734457253857x_{23} = -65.9734457253857
x24=−43.9822971502571x_{24} = -43.9822971502571
x25=−56.5486677646163x_{25} = -56.5486677646163
x26=−59.6902604182061x_{26} = -59.6902604182061
x27=−62.8318530717959x_{27} = -62.8318530717959
x28=−97.3893722612836x_{28} = -97.3893722612836
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=−18.8495544541535x_{31} = -18.8495544541535
x32=−37.6991118430744x_{32} = -37.6991118430744
x33=−512.079602535136x_{33} = -512.079602535136
x34=−15.7079762174623x_{34} = -15.7079762174623
x35=−25.1327412098768x_{35} = -25.1327412098768
x36=−47.1238898038469x_{36} = -47.1238898038469
Signos de extremos en los puntos:
(-21.991148741404736, -0.999999760114159)

(-50.26548245743669, 1)

(-12.566256325111233, 1.00016487799986)

(-3.216244875596087, -0.889614433250578)

(-53.40707511102649, -1)

(-3.2162448755964004, -0.889614433250578)

(-84.82300164692441, -1)

(-147.6548547187203, -1)

(-40.84070449666766, -0.999999999999492)

(-28.274333884443163, -0.999999996919811)

(-6.274229598506704, 1.01288008168756)

(-34.55751918951514, -0.99999999996045)

(-75.39822368615503, 1)

(-69.11503837897546, 1)

(-91.106186954104, -1)

(-100.53096491487338, 1)

(-78.53981633974483, -1)

(-94.2477796076938, 1)

(-31.415926535656002, 1.00000000034903)

(-9.425785776848588, -0.998545521139948)

(-113.09733552923255, 1)

(-81.68140899333463, 1)

(-65.97344572538566, -1)

(-43.98229715025707, 1.00000000000006)

(-56.548667764616276, 1)

(-59.69026041820607, -1)

(-62.83185307179586, 1)

(-97.3893722612836, -1)

(-87.96459430051421, 1)

(-72.25663103256524, -1)

(-18.849554454153456, 1.00000211698843)

(-37.69911184307441, 1.00000000000448)

(-512.0796025351362, -1)

(-15.707976217462338, -0.999981317717432)

(-25.1327412098768, 1.0000000271826)

(-47.1238898038469, -0.999999999999993)


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=−21.9911487414047x_{1} = -21.9911487414047
x2=−3.21624487559609x_{2} = -3.21624487559609
x3=−53.4070751110265x_{3} = -53.4070751110265
x4=−3.2162448755964x_{4} = -3.2162448755964
x5=−84.8230016469244x_{5} = -84.8230016469244
x6=−147.65485471872x_{6} = -147.65485471872
x7=−40.8407044966677x_{7} = -40.8407044966677
x8=−28.2743338844432x_{8} = -28.2743338844432
x9=−34.5575191895151x_{9} = -34.5575191895151
x10=−91.106186954104x_{10} = -91.106186954104
x11=−78.5398163397448x_{11} = -78.5398163397448
x12=−9.42578577684859x_{12} = -9.42578577684859
x13=−65.9734457253857x_{13} = -65.9734457253857
x14=−59.6902604182061x_{14} = -59.6902604182061
x15=−97.3893722612836x_{15} = -97.3893722612836
x16=−72.2566310325652x_{16} = -72.2566310325652
x17=−512.079602535136x_{17} = -512.079602535136
x18=−15.7079762174623x_{18} = -15.7079762174623
x19=−47.1238898038469x_{19} = -47.1238898038469
Puntos máximos de la función:
x19=−50.2654824574367x_{19} = -50.2654824574367
x19=−12.5662563251112x_{19} = -12.5662563251112
x19=−6.2742295985067x_{19} = -6.2742295985067
x19=−75.398223686155x_{19} = -75.398223686155
x19=−69.1150383789755x_{19} = -69.1150383789755
x19=−100.530964914873x_{19} = -100.530964914873
x19=−94.2477796076938x_{19} = -94.2477796076938
x19=−31.415926535656x_{19} = -31.415926535656
x19=−113.097335529233x_{19} = -113.097335529233
x19=−81.6814089933346x_{19} = -81.6814089933346
x19=−43.9822971502571x_{19} = -43.9822971502571
x19=−56.5486677646163x_{19} = -56.5486677646163
x19=−62.8318530717959x_{19} = -62.8318530717959
x19=−87.9645943005142x_{19} = -87.9645943005142
x19=−18.8495544541535x_{19} = -18.8495544541535
x19=−37.6991118430744x_{19} = -37.6991118430744
x19=−25.1327412098768x_{19} = -25.1327412098768
Decrece en los intervalos
[−3.21624487559609,∞)\left[-3.21624487559609, \infty\right)
Crece en los intervalos
(−∞,−512.079602535136]\left(-\infty, -512.079602535136\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
2xlog⁡(2)2−cos⁡(x)=02^{x} \log{\left(2 \right)}^{2} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−86.3937979737193x_{1} = -86.3937979737193
x2=−98.9601685880785x_{2} = -98.9601685880785
x3=−95.8185759344887x_{3} = -95.8185759344887
x4=−64.4026493985908x_{4} = -64.4026493985908
x5=−48.6946861306418x_{5} = -48.6946861306418
x6=−67.5442420521806x_{6} = -67.5442420521806
x7=−10.9958095661604x_{7} = -10.9958095661604
x8=−80.1106126665397x_{8} = -80.1106126665397
x9=−39.2699081698717x_{9} = -39.2699081698717
x10=−76.9690200129499x_{10} = -76.9690200129499
x11=−7.85190196975803x_{11} = -7.85190196975803
x12=−17.2787626162612x_{12} = -17.2787626162612
x13=−4.73048803293588x_{13} = -4.73048803293588
x14=−14.1371402757821x_{14} = -14.1371402757821
x15=−26.703537551117x_{15} = -26.703537551117
x16=−58.1194640914112x_{16} = -58.1194640914112
x17=0.686705468273149x_{17} = 0.686705468273149
x18=−36.128315516289x_{18} = -36.128315516289
x19=−42.4115008234623x_{19} = -42.4115008234623
x20=−70.6858347057703x_{20} = -70.6858347057703
x21=−23.5619449407205x_{21} = -23.5619449407205
x22=−83.2522053201295x_{22} = -83.2522053201295
x23=−89.5353906273091x_{23} = -89.5353906273091
x24=−32.9867228626364x_{24} = -32.9867228626364
x25=−20.4203519059504x_{25} = -20.4203519059504
x26=−1.38589876000339x_{26} = -1.38589876000339
x27=−92.6769832808989x_{27} = -92.6769832808989
x28=−51.8362787842316x_{28} = -51.8362787842316
x29=−73.8274273593601x_{29} = -73.8274273593601
x30=−54.9778714378214x_{30} = -54.9778714378214
x31=−45.553093477052x_{31} = -45.553093477052
x32=−61.261056745001x_{32} = -61.261056745001
x33=−29.8451302096012x_{33} = -29.8451302096012

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
[0.686705468273149,∞)\left[0.686705468273149, \infty\right)
Convexa en los intervalos
(−∞,−98.9601685880785]\left(-\infty, -98.9601685880785\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(2x+cos⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(2^{x} + \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(2x+cos⁡(x))=∞\lim_{x \to \infty}\left(2^{x} + \cos{\left(x \right)}\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^x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2x+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{2^{x} + \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(2x+cos⁡(x)x)=∞\lim_{x \to \infty}\left(\frac{2^{x} + \cos{\left(x \right)}}{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:
2x+cos⁡(x)=cos⁡(x)+2−x2^{x} + \cos{\left(x \right)} = \cos{\left(x \right)} + 2^{- x}
- No
2x+cos⁡(x)=−cos⁡(x)−2−x2^{x} + \cos{\left(x \right)} = - \cos{\left(x \right)} - 2^{- x}
- No
es decir, función
no es
par ni impar
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