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y=exp(x/2)-2*x*(sin(x))^2
Trazado el gráfico de la función/ y=exp(x/2)-2*x*(sin(x))^2

Gráfico de la función y = y=exp(x/2)-2*x*(sin(x))^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x              
        -              
        2          2   
f(x) = e  - 2*x*sin (x)
f(x)=−2xsin⁡2(x)+ex2f{\left(x \right)} = - 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}} 
f = -2*x*sin(x)^2 + exp(x/2)
Gráfico de la función
0.02.00.20.40.60.81.01.21.41.61.82.5-2.5
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:
−2xsin⁡2(x)+ex2=0- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−59.6902604496533x_{1} = -59.6902604496533
x2=1.09436047823253x_{2} = 1.09436047823253
x3=−87.964594358858x_{3} = -87.964594358858
x4=2.17002325143779x_{4} = 2.17002325143779
x5=−65.973445764769x_{5} = -65.973445764769
x6=−81.6814090380602x_{6} = -81.6814090380602
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 exp(x/2) - 2*x*sin(x)^2.
−0⋅2sin⁡2(0)+e02- 0 \cdot 2 \sin^{2}{\left(0 \right)} + e^{\frac{0}{2}}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
−4xsin⁡(x)cos⁡(x)+ex22−2sin⁡2(x)=0- 4 x \sin{\left(x \right)} \cos{\left(x \right)} + \frac{e^{\frac{x}{2}}}{2} - 2 \sin^{2}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−9.42489709601224x_{1} = -9.42489709601224
x2=0.322123013757735x_{2} = 0.322123013757735
x3=−53.4070751110265x_{3} = -53.4070751110265
x4=−94.2477796076938x_{4} = -94.2477796076938
x5=−29.8618724024448x_{5} = -29.8618724024448
x6=−39.282635752714x_{6} = -39.282635752714
x7=−37.6991118430991x_{7} = -37.6991118430991
x8=−67.5516436614121x_{8} = -67.5516436614121
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=−51.8459224452234x_{10} = -51.8459224452234
x11=−65.9734457253857x_{11} = -65.9734457253857
x12=−28.2743338855131x_{12} = -28.2743338855131
x13=−83.2582106616487x_{13} = -83.2582106616487
x14=−84.8230016469244x_{14} = -84.8230016469244
x15=−105.248104538899x_{15} = -105.248104538899
x16=−7.91675358472951x_{16} = -7.91675358472951
x17=−80.1168534696549x_{17} = -80.1168534696549
x18=−95.8237937978449x_{18} = -95.8237937978449
x19=−59.6902604182061x_{19} = -59.6902604182061
x20=−87.9645943005142x_{20} = -87.9645943005142
x21=−58.1280655761511x_{21} = -58.1280655761511
x22=−6.28404447572486x_{22} = -6.28404447572486
x23=−64.410411962776x_{23} = -64.410411962776
x24=−42.423286257699x_{24} = -42.423286257699
x25=3.33973504052274x_{25} = 3.33973504052274
x26=−75.398223686155x_{26} = -75.398223686155
x27=4.54932361086616x_{27} = 4.54932361086616
x28=−21.9911486704839x_{28} = -21.9911486704839
x29=−1.81231686241143x_{29} = -1.81231686241143
x30=−61.2692172687226x_{30} = -61.2692172687226
x31=−20.4448032446579x_{31} = -20.4448032446579
x32=−23.5831432702071x_{32} = -23.5831432702071
x33=−17.3076392795255x_{33} = -17.3076392795255
x34=−4.8135518668082x_{34} = -4.8135518668082
x35=−43.9822971502579x_{35} = -43.9822971502579
x36=−73.8341991854591x_{36} = -73.8341991854591
x37=−50.2654824574367x_{37} = -50.2654824574367
x38=−31.4159265364976x_{38} = -31.4159265364976
x39=−14.1724247146883x_{39} = -14.1724247146883
x40=−12.5663891899418x_{40} = -12.5663891899418
x41=−89.5409746049841x_{41} = -89.5409746049841
x42=1.69117540515033x_{42} = 1.69117540515033
x43=−97.3893722612836x_{43} = -97.3893722612836
x44=−36.142148896957x_{44} = -36.142148896957
x45=−45.5640665961994x_{45} = -45.5640665961994
x46=−86.3995849739529x_{46} = -86.3995849739529
x47=−72.2566310325652x_{47} = -72.2566310325652
x48=−15.7079663571662x_{48} = -15.7079663571662
Signos de extremos en los puntos:
(-9.424897096012243, 0.00898302346280056)

(0.3221230137577349, 1.11018857472654)

(-53.407075111026494, 2.52813925651777e-12)

(-94.2477796076938, 3.42259076367581e-21)

(-29.861872402444764, 59.7070060690897)

(-39.28263575271401, 78.5525453000074)

(-37.69911184309911, 6.51241213604474e-9)

(-67.5516436614121, 135.095885983915)

(-81.68140899333463, 1.8327676081836e-18)

(-51.845922445223415, 103.682201827358)

(-65.97344572538566, 4.72115527932988e-15)

(-28.27433388551311, 7.24947251017935e-7)

(-83.25821066164869, 166.510416126162)

(-84.82300164692441, 3.8099496139816e-19)

(-105.24810453889911, 210.491458505634)

(-7.916753584729508, 15.7902941032982)

(-80.11685346965491, 160.227466298236)

(-95.82379379784489, 191.642369827041)

(-59.69026041820607, 1.09250803190593e-13)

(-87.96459430051421, 7.92010715765772e-20)

(-58.12806557615112, 116.247530091813)

(-6.284044475724856, 0.0432046356223051)

(-64.41041196277601, 128.813061673198)

(-42.423286257699004, 84.8347881730497)

(3.339735040522736, 5.05263933667331)

(-75.39822368615503, 4.24115118314635e-17)

(4.549323610866158, 0.865781515398343)

(-21.991148670483852, 1.67757811243109e-5)

(-1.8123168624114283, 3.82135376834411)

(-61.269217268722585, 122.530274376014)

(-20.444803244657862, 40.8652018063227)

(-23.583143270207053, 47.1451021303686)

(-17.30763927952554, 34.5865906485272)

(-4.813551866808199, 9.61902159877519)

(-43.98229715025791, 2.81426845748499e-10)

(-73.83419918545908, 147.661626751844)

(-50.265482457436725, 1.21615567094092e-11)

(-31.415926536497555, 1.50701727516415e-7)

(-14.172424714688313, 28.3104648049499)

(-12.566389189941818, 0.00186743405949238)

(-89.54097460498406, 179.076365348368)

(1.691175405150334, -1.00422667438438)

(-97.3893722612836, 7.114954274244e-22)

(-36.14214889695704, 72.2704661921991)

(-45.564066596199375, 91.1171609542022)

(-86.3995849739529, 172.793383076873)

(-72.25663103256524, 2.04019618357396e-16)

(-15.707966357166224, 0.000388202904115727)


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=−9.42489709601224x_{1} = -9.42489709601224
x2=−53.4070751110265x_{2} = -53.4070751110265
x3=−94.2477796076938x_{3} = -94.2477796076938
x4=−37.6991118430991x_{4} = -37.6991118430991
x5=−81.6814089933346x_{5} = -81.6814089933346
x6=−65.9734457253857x_{6} = -65.9734457253857
x7=−28.2743338855131x_{7} = -28.2743338855131
x8=−84.8230016469244x_{8} = -84.8230016469244
x9=−59.6902604182061x_{9} = -59.6902604182061
x10=−87.9645943005142x_{10} = -87.9645943005142
x11=−6.28404447572486x_{11} = -6.28404447572486
x12=−75.398223686155x_{12} = -75.398223686155
x13=4.54932361086616x_{13} = 4.54932361086616
x14=−21.9911486704839x_{14} = -21.9911486704839
x15=−43.9822971502579x_{15} = -43.9822971502579
x16=−50.2654824574367x_{16} = -50.2654824574367
x17=−31.4159265364976x_{17} = -31.4159265364976
x18=−12.5663891899418x_{18} = -12.5663891899418
x19=1.69117540515033x_{19} = 1.69117540515033
x20=−97.3893722612836x_{20} = -97.3893722612836
x21=−72.2566310325652x_{21} = -72.2566310325652
x22=−15.7079663571662x_{22} = -15.7079663571662
Puntos máximos de la función:
x22=0.322123013757735x_{22} = 0.322123013757735
x22=−29.8618724024448x_{22} = -29.8618724024448
x22=−39.282635752714x_{22} = -39.282635752714
x22=−67.5516436614121x_{22} = -67.5516436614121
x22=−51.8459224452234x_{22} = -51.8459224452234
x22=−83.2582106616487x_{22} = -83.2582106616487
x22=−105.248104538899x_{22} = -105.248104538899
x22=−7.91675358472951x_{22} = -7.91675358472951
x22=−80.1168534696549x_{22} = -80.1168534696549
x22=−95.8237937978449x_{22} = -95.8237937978449
x22=−58.1280655761511x_{22} = -58.1280655761511
x22=−64.410411962776x_{22} = -64.410411962776
x22=−42.423286257699x_{22} = -42.423286257699
x22=3.33973504052274x_{22} = 3.33973504052274
x22=−1.81231686241143x_{22} = -1.81231686241143
x22=−61.2692172687226x_{22} = -61.2692172687226
x22=−20.4448032446579x_{22} = -20.4448032446579
x22=−23.5831432702071x_{22} = -23.5831432702071
x22=−17.3076392795255x_{22} = -17.3076392795255
x22=−4.8135518668082x_{22} = -4.8135518668082
x22=−73.8341991854591x_{22} = -73.8341991854591
x22=−14.1724247146883x_{22} = -14.1724247146883
x22=−89.5409746049841x_{22} = -89.5409746049841
x22=−36.142148896957x_{22} = -36.142148896957
x22=−45.5640665961994x_{22} = -45.5640665961994
x22=−86.3995849739529x_{22} = -86.3995849739529
Decrece en los intervalos
[4.54932361086616,∞)\left[4.54932361086616, \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
4xsin⁡2(x)−4xcos⁡2(x)+ex24−8sin⁡(x)cos⁡(x)=04 x \sin^{2}{\left(x \right)} - 4 x \cos^{2}{\left(x \right)} + \frac{e^{\frac{x}{2}}}{4} - 8 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−11.8231548031724x_{1} = -11.8231548031724
x2=−77.760847792972x_{2} = -77.760847792972
x3=−5.58602087316209x_{3} = -5.58602087316209
x4=−82.4728694594266x_{4} = -82.4728694594266
x5=6.99155527501611x_{5} = 6.99155527501611
x6=−24.367850390908x_{6} = -24.367850390908
x7=−76.1901839979235x_{7} = -76.1901839979235
x8=−47.9197205706165x_{8} = -47.9197205706165
x9=0.0210642961476989x_{9} = 0.0210642961476989
x10=−32.2168395519636x_{10} = -32.2168395519636
x11=3.99364100206226x_{11} = 3.99364100206226
x12=−46.3492776216984x_{12} = -46.3492776216984
x13=−69.9075883539626x_{13} = -69.9075883539626
x14=−33.7869153353869x_{14} = -33.7869153353869
x15=−93.4677306800165x_{15} = -93.4677306800165
x16=−66.766332133246x_{16} = -66.766332133246
x17=−90.3263240494369x_{17} = -90.3263240494369
x18=−55.7722336752062x_{18} = -55.7722336752062
x19=−18.0917663423891x_{19} = -18.0917663423891
x20=−19.660364151425x_{20} = -19.660364151425
x21=9.76299870182123x_{21} = 9.76299870182123
x22=−98.1798629425939x_{22} = -98.1798629425939
x23=−1.15392192631066x_{23} = -1.15392192631066
x24=−60.4839244878466x_{24} = -60.4839244878466
x25=2.58203362264665x_{25} = 2.58203362264665
x26=−79.3315168346756x_{26} = -79.3315168346756
x27=−21.2292853497857x_{27} = -21.2292853497857
x28=−41.6381085824895x_{28} = -41.6381085824895
x29=−13.3890464105107x_{29} = -13.3890464105107
x30=−49.4901859325761x_{30} = -49.4901859325761
x31=−57.3427845371101x_{31} = -57.3427845371101
x32=5.67817379048019x_{32} = 5.67817379048019
x33=−35.3570550332928x_{33} = -35.3570550332928
x34=−40.0677825970357x_{34} = -40.0677825970357
x35=−63.6251091208926x_{35} = -63.6251091208926
x36=−68.3369563786298x_{36} = -68.3369563786298
x37=−27.507104838212x_{37} = -27.507104838212
x38=−85.6142396947314x_{38} = -85.6142396947314
x39=−38.4974949445873x_{39} = -38.4974949445873
x40=−62.0545116429054x_{40} = -62.0545116429054
x41=−4.04904361838806x_{41} = -4.04904361838806
x42=−71.4782275499213x_{42} = -71.4782275499213
x43=−54.2016970313842x_{43} = -54.2016970313842
x44=−99.7505790857949x_{44} = -99.7505790857949
x45=−91.8970257752571x_{45} = -91.8970257752571
x46=−10.2587793196844x_{46} = -10.2587793196844
x47=−58.9133484807877x_{47} = -58.9133484807877
x48=−84.0435524991391x_{48} = -84.0435524991391
x49=−25.9374070295191x_{49} = -25.9374070295191

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
[9.76299870182123,∞)\left[9.76299870182123, \infty\right)
Convexa en los intervalos
(−∞,−98.1798629425939]\left(-\infty, -98.1798629425939\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2xsin⁡2(x)+ex2)=⟨0,∞⟩\lim_{x \to -\infty}\left(- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}}\right) = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨0,∞⟩y = \left\langle 0, \infty\right\rangle
lim⁡x→∞(−2xsin⁡2(x)+ex2)=∞\lim_{x \to \infty}\left(- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{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 exp(x/2) - 2*x*sin(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−2xsin⁡2(x)+ex2x)=⟨−2,0⟩\lim_{x \to -\infty}\left(\frac{- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}}}{x}\right) = \left\langle -2, 0\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−2,0⟩xy = \left\langle -2, 0\right\rangle x
lim⁡x→∞(−2xsin⁡2(x)+ex2x)=∞\lim_{x \to \infty}\left(\frac{- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{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:
−2xsin⁡2(x)+ex2=2xsin⁡2(x)+e−x2- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}} = 2 x \sin^{2}{\left(x \right)} + e^{- \frac{x}{2}}
- No
−2xsin⁡2(x)+ex2=−2xsin⁡2(x)−e−x2- 2 x \sin^{2}{\left(x \right)} + e^{\frac{x}{2}} = - 2 x \sin^{2}{\left(x \right)} - e^{- \frac{x}{2}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = y=exp(x/2)-2*x*(sin(x))^2
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