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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /   pi                                \      
       |   --                                |      
       |   2              2*x    2*x         |  -2*x
f(x) = \2*e   + cos(2*x)*e    + e   *sin(2*x)/*e
f(x)=((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2xf{\left(x \right)} = \left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x} 
f = (exp(2*x)*cos(2*x) + 2*exp(pi/2) + exp(2*x)*sin(2*x))*exp(-2*x)
Gráfico de la función
02468-8-6-4-2-1010-50000000005000000000
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:
((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x=0\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=26.3108384738145x_{1} = 26.3108384738145
x2=4.32029123301605x_{2} = 4.32029123301605
x3=86.0010988920206x_{3} = 86.0010988920206
x4=40.4480054149686x_{4} = 40.4480054149686
x5=46.7311907221482x_{5} = 46.7311907221482
x6=92.2842841992002x_{6} = 92.2842841992002
x7=54.5851723561227x_{7} = 54.5851723561227
x8=1.3940664696018x_{8} = 1.3940664696018
x9=34.164820107789x_{9} = 34.164820107789
x10=20.0276531666349x_{10} = 20.0276531666349
x11=70.2931356240716x_{11} = 70.2931356240716
x12=65.5807466436869x_{12} = 65.5807466436869
x13=90.7134878724053x_{13} = 90.7134878724053
x14=12.1736715325697x_{14} = 12.1736715325697
x15=68.7223392972767x_{15} = 68.7223392972767
x16=10.6028752079651x_{16} = 10.6028752079651
x17=84.4303025652257x_{17} = 84.4303025652257
x18=5.89046020684927x_{18} = 5.89046020684927
x19=48.3019870489431x_{19} = 48.3019870489431
x20=13.7444678594593x_{20} = 13.7444678594593
x21=76.5763209312512x_{21} = 76.5763209312512
x22=87.5718952188155x_{22} = 87.5718952188155
x23=21.5984494934298x_{23} = 21.5984494934298
x24=57.7267650097125x_{24} = 57.7267650097125
x25=71.8639319508665x_{25} = 71.8639319508665
x26=18.45685683984x_{26} = 18.45685683984
x27=24.7400421470196x_{27} = 24.7400421470196
x28=42.0188017417635x_{28} = 42.0188017417635
x29=78.1471172580461x_{29} = 78.1471172580461
x30=35.7356164345839x_{30} = 35.7356164345839
x31=100.138265833175x_{31} = 100.138265833175
x32=49.872783375738x_{32} = 49.872783375738
x33=64.009950316892x_{33} = 64.009950316892
x34=27.8816348006094x_{34} = 27.8816348006094
x35=43.5895980685584x_{35} = 43.5895980685584
x36=79.717913584841x_{36} = 79.717913584841
x37=93.8550805259951x_{37} = 93.8550805259951
x38=62.4391539900971x_{38} = 62.4391539900971
x39=56.1559686829176x_{39} = 56.1559686829176
x40=32.5940237809941x_{40} = 32.5940237809941
x41=98.5674695063798x_{41} = 98.5674695063798
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 (2*exp(pi/2) + cos(2*x)*exp(2*x) + exp(2*x)*sin(2*x))*exp(-2*x).
(e0⋅2sin⁡(0⋅2)+(e0⋅2cos⁡(0⋅2)+2eπ2))e−0\left(e^{0 \cdot 2} \sin{\left(0 \cdot 2 \right)} + \left(e^{0 \cdot 2} \cos{\left(0 \cdot 2 \right)} + 2 e^{\frac{\pi}{2}}\right)\right) e^{- 0}
Resultado:
f(0)=1+2eπ2f{\left(0 \right)} = 1 + 2 e^{\frac{\pi}{2}}
Punto:
(0, 1 + 2*exp(pi/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
−2((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x+4cos⁡(2x)=0- 2 \left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x} + 4 \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=96.2112750161874x_{1} = 96.2112750161874
x2=9.81747703236811x_{2} = 9.81747703236811
x3=22.3838476568273x_{3} = 22.3838476568273
x4=38.0918109247762x_{4} = 38.0918109247762
x5=8.24668094939388x_{5} = 8.24668094939388
x6=36.5210145979813x_{6} = 36.5210145979813
x7=69.5077374606742x_{7} = 69.5077374606742
x8=80.5033117482384x_{8} = 80.5033117482384
x9=47.5165888855456x_{9} = 47.5165888855456
x10=14.5298660228536x_{10} = 14.5298660228536
x11=50.6581815391354x_{11} = 50.6581815391354
x12=94.6404786893925x_{12} = 94.6404786893925
x13=39.6626072515711x_{13} = 39.6626072515711
x14=72.649330114264x_{14} = 72.649330114264
x15=30.2378292908018x_{15} = 30.2378292908018
x16=17.6714586764426x_{16} = 17.6714586764426
x17=44.3749962319558x_{17} = 44.3749962319558
x18=74.2201264410589x_{18} = 74.2201264410589
x19=67.9369411338793x_{19} = 67.9369411338793
x20=16.1006623496477x_{20} = 16.1006623496477
x21=25.5254403104171x_{21} = 25.5254403104171
x22=58.5121631731099x_{22} = 58.5121631731099
x23=82.0741080750334x_{23} = 82.0741080750334
x24=66.3661448070844x_{24} = 66.3661448070844
x25=53.7997741927252x_{25} = 53.7997741927252
x26=23.9546439836222x_{26} = 23.9546439836222
x27=2.02312219899996x_{27} = 2.02312219899996
x28=83.6449044018282x_{28} = 83.6449044018282
x29=3.53137861010007x_{29} = 3.53137861010007
x30=97.7820713429823x_{30} = 97.7820713429823
x31=89.9280897090078x_{31} = 89.9280897090078
x32=61.6537558266997x_{32} = 61.6537558266997
x33=31.8086256175967x_{33} = 31.8086256175967
x34=75.7909227678538x_{34} = 75.7909227678538
x35=45.9457925587507x_{35} = 45.9457925587507
x36=28.6670329640069x_{36} = 28.6670329640069
x37=91.4988860358027x_{37} = 91.4988860358027
x38=52.2289778659303x_{38} = 52.2289778659303
x39=6.67587898035914x_{39} = 6.67587898035914
x40=88.3572933822129x_{40} = 88.3572933822129
x41=60.0829594999048x_{41} = 60.0829594999048
Signos de extremos en los puntos:
                                                              pi 
                                                              -- 
                                                              2  
(96.21127501618741, -1.41421356237309 + 5.40727332689264e-84*e  )

                                                            pi 
                                                            -- 
                                                            2  
(9.817477032368114, 1.41421356237309 + 5.93851411308751e-9*e  )

                                                             pi 
                                                             -- 
                                                             2  
(22.383847656827278, 1.4142135623731 + 7.22215746970667e-20*e  )

                                                             pi 
                                                             -- 
                                                             2  
(38.09181092477624, 1.41421356237309 + 1.64022495450558e-33*e  )

                                                             pi 
                                                             -- 
                                                             2  
(8.246680949393875, -1.41421356237294 + 1.37421262774078e-7*e  )

                                                              pi 
                                                              -- 
                                                              2  
(36.52101459798135, -1.41421356237309 + 3.79559415208275e-32*e  )

                                                            pi 
                                                            -- 
                                                            2  
(69.50773746067418, 1.4142135623731 + 8.46011639542855e-61*e  )

                                                             pi 
                                                             -- 
                                                             2  
(80.50331174823845, -1.4142135623731 + 2.38090387183114e-70*e  )

                                                            pi 
                                                            -- 
                                                            2  
(47.51658888554562, 1.4142135623731 + 1.06818208996232e-41*e  )

                                                               pi 
                                                               -- 
                                                               2  
(14.529866022853609, -1.41421356237309 + 4.79235214329284e-13*e  )

                                                            pi 
                                                            -- 
                                                            2  
(50.65818153913541, 1.4142135623731 + 1.99476888004079e-44*e  )

                                                            pi 
                                                            -- 
                                                            2  
(94.64047868939252, 1.4142135623731 + 1.25128050039047e-82*e  )

                                                             pi 
                                                             -- 
                                                             2  
(39.66260725157114, -1.4142135623731 + 7.08805471182034e-35*e  )

                                                             pi 
                                                             -- 
                                                             2  
(72.64933011426398, 1.41421356237309 + 1.57987828720465e-63*e  )

                                                             pi 
                                                             -- 
                                                             2  
(30.23782929080176, -1.4142135623731 + 1.08839160724354e-26*e  )

                                                               pi 
                                                               -- 
                                                               2  
(17.671458676442587, -1.41421356237309 + 8.94944317779201e-16*e  )

                                                             pi 
                                                             -- 
                                                             2  
(44.374996231955826, 1.4142135623731 + 5.72002595755828e-39*e  )

                                                              pi 
                                                              -- 
                                                              2  
(74.22012644105887, -1.41421356237309 + 6.82727311699711e-65*e  )

                                                             pi 
                                                             -- 
                                                             2  
(67.93694113387927, -1.4142135623731 + 1.95772953144152e-59*e  )

                                                              pi 
                                                              -- 
                                                              2  
(16.100662349647656, 1.41421356237309 + 2.07096313811822e-14*e  )

                                                             pi 
                                                             -- 
                                                             2  
(25.52544031041707, 1.41421356237309 + 1.34869654740543e-22*e  )

                                                            pi 
                                                            -- 
                                                            2  
(58.5121631731099, -1.4142135623731 + 3.00615116263196e-51*e  )

                                                             pi 
                                                             -- 
                                                             2  
(82.07410807503335, 1.41421356237309 + 1.02888185311208e-71*e  )

                                                             pi 
                                                             -- 
                                                             2  
(66.36614480708438, 1.41421356237309 + 4.53032173452027e-58*e  )

                                                             pi 
                                                             -- 
                                                             2  
(53.79977419272521, 1.41421356237309 + 3.72511664646944e-47*e  )

                                                              pi 
                                                              -- 
                                                              2  
(23.954643983622173, -1.4142135623731 + 3.12097722583997e-21*e  )

                                                             pi 
                                                             -- 
                                                             2  
(2.0231221989999573, -1.40416941423847 + 0.0349758583316791*e  )

                                                              pi 
                                                              -- 
                                                              2  
(83.64490440182824, -1.41421356237309 + 4.44620163034644e-73*e  )

                                                             pi 
                                                             -- 
                                                             2  
(3.5313786101000653, 1.41418955956451 + 0.00171282702999335*e  )

                                                            pi 
                                                            -- 
                                                            2  
(97.78207134298232, 1.41421356237309 + 2.3366946757821e-85*e  )

                                                              pi 
                                                              -- 
                                                              2  
(89.92808970900784, -1.41421356237309 + 1.55054272697524e-78*e  )

                                                              pi 
                                                              -- 
                                                              2  
(61.653755826699694, -1.4142135623731 + 5.61381513907253e-54*e  )

                                                              pi 
                                                              -- 
                                                              2  
(31.808625617596658, 1.41421356237309 + 4.70336659543979e-28*e  )

                                                            pi 
                                                            -- 
                                                            2  
(75.79092276785376, 1.4142135623731 + 2.95033222442366e-66*e  )

                                                               pi 
                                                               -- 
                                                               2  
(45.945792558750725, -1.41421356237309 + 2.47184734196578e-40*e  )

                                                              pi 
                                                              -- 
                                                              2  
(28.667032964006864, 1.41421356237309 + 2.51861356473193e-25*e  )

                                                             pi 
                                                             -- 
                                                             2  
(91.49888603580273, 1.41421356237309 + 6.70050266679953e-80*e  )

                                                            pi 
                                                            -- 
                                                            2  
(52.22897786593031, -1.4142135623731 + 8.6201779337199e-46*e  )

                                                            pi 
                                                            -- 
                                                            2  
(6.675878980359145, 1.41421356229036 + 3.18005908817298e-6*e  )

                                                            pi 
                                                            -- 
                                                            2  
(88.35729338221293, 1.4142135623731 + 3.58806326589261e-77*e  )

                                                              pi 
                                                              -- 
                                                              2  
(60.082959499904796, 1.41421356237309 + 1.29907570630521e-52*e  )


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=96.2112750161874x_{1} = 96.2112750161874
x2=8.24668094939388x_{2} = 8.24668094939388
x3=36.5210145979813x_{3} = 36.5210145979813
x4=80.5033117482384x_{4} = 80.5033117482384
x5=14.5298660228536x_{5} = 14.5298660228536
x6=39.6626072515711x_{6} = 39.6626072515711
x7=30.2378292908018x_{7} = 30.2378292908018
x8=17.6714586764426x_{8} = 17.6714586764426
x9=74.2201264410589x_{9} = 74.2201264410589
x10=67.9369411338793x_{10} = 67.9369411338793
x11=58.5121631731099x_{11} = 58.5121631731099
x12=23.9546439836222x_{12} = 23.9546439836222
x13=2.02312219899996x_{13} = 2.02312219899996
x14=83.6449044018282x_{14} = 83.6449044018282
x15=89.9280897090078x_{15} = 89.9280897090078
x16=61.6537558266997x_{16} = 61.6537558266997
x17=45.9457925587507x_{17} = 45.9457925587507
x18=52.2289778659303x_{18} = 52.2289778659303
Puntos máximos de la función:
x18=9.81747703236811x_{18} = 9.81747703236811
x18=22.3838476568273x_{18} = 22.3838476568273
x18=38.0918109247762x_{18} = 38.0918109247762
x18=69.5077374606742x_{18} = 69.5077374606742
x18=47.5165888855456x_{18} = 47.5165888855456
x18=50.6581815391354x_{18} = 50.6581815391354
x18=94.6404786893925x_{18} = 94.6404786893925
x18=72.649330114264x_{18} = 72.649330114264
x18=44.3749962319558x_{18} = 44.3749962319558
x18=16.1006623496477x_{18} = 16.1006623496477
x18=25.5254403104171x_{18} = 25.5254403104171
x18=82.0741080750334x_{18} = 82.0741080750334
x18=66.3661448070844x_{18} = 66.3661448070844
x18=53.7997741927252x_{18} = 53.7997741927252
x18=3.53137861010007x_{18} = 3.53137861010007
x18=97.7820713429823x_{18} = 97.7820713429823
x18=31.8086256175967x_{18} = 31.8086256175967
x18=75.7909227678538x_{18} = 75.7909227678538
x18=28.6670329640069x_{18} = 28.6670329640069
x18=91.4988860358027x_{18} = 91.4988860358027
x18=6.67587898035914x_{18} = 6.67587898035914
x18=88.3572933822129x_{18} = 88.3572933822129
x18=60.0829594999048x_{18} = 60.0829594999048
Decrece en los intervalos
[96.2112750161874,∞)\left[96.2112750161874, \infty\right)
Crece en los intervalos
(−∞,2.02312219899996]\left(-\infty, 2.02312219899996\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
4((e2xsin⁡(2x)+e2xcos⁡(2x)+2eπ2)e−2x−2sin⁡(2x)−2cos⁡(2x))=04 \left(\left(e^{2 x} \sin{\left(2 x \right)} + e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) e^{- 2 x} - 2 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=21.5984494934298x_{1} = 21.5984494934298
x2=93.8550805259951x_{2} = 93.8550805259951
x3=54.5851723561227x_{3} = 54.5851723561227
x4=34.164820107789x_{4} = 34.164820107789
x5=2.76245448537969x_{5} = 2.76245448537969
x6=18.45685683984x_{6} = 18.45685683984
x7=62.4391539900971x_{7} = 62.4391539900971
x8=98.5674695063798x_{8} = 98.5674695063798
x9=46.7311907221482x_{9} = 46.7311907221482
x10=76.5763209312512x_{10} = 76.5763209312512
x11=90.7134878724053x_{11} = 90.7134878724053
x12=86.0010988920206x_{12} = 86.0010988920206
x13=20.0276531666349x_{13} = 20.0276531666349
x14=13.7444678594514x_{14} = 13.7444678594514
x15=43.5895980685584x_{15} = 43.5895980685584
x16=92.2842841992002x_{16} = 92.2842841992002
x17=84.4303025652257x_{17} = 84.4303025652257
x18=27.8816348006094x_{18} = 27.8816348006094
x19=70.2931356240716x_{19} = 70.2931356240716
x20=12.1736715327512x_{20} = 12.1736715327512
x21=5.89051224140486x_{21} = 5.89051224140486
x22=68.7223392972767x_{22} = 68.7223392972767
x23=57.7267650097125x_{23} = 57.7267650097125
x24=35.7356164345839x_{24} = 35.7356164345839
x25=26.3108384738145x_{25} = 26.3108384738145
x26=40.4480054149686x_{26} = 40.4480054149686
x27=48.3019870489431x_{27} = 48.3019870489431
x28=71.8639319508665x_{28} = 71.8639319508665
x29=79.717913584841x_{29} = 79.717913584841
x30=49.872783375738x_{30} = 49.872783375738
x31=24.7400421470196x_{31} = 24.7400421470196
x32=10.602875203766x_{32} = 10.602875203766
x33=100.138265833175x_{33} = 100.138265833175
x34=4.31908711445437x_{34} = 4.31908711445437
x35=65.5807466436869x_{35} = 65.5807466436869
x36=32.5940237809941x_{36} = 32.5940237809941
x37=42.0188017417635x_{37} = 42.0188017417635
x38=87.5718952188155x_{38} = 87.5718952188155
x39=64.009950316892x_{39} = 64.009950316892
x40=56.1559686829176x_{40} = 56.1559686829176
x41=78.1471172580461x_{41} = 78.1471172580461

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
[98.5674695063798,∞)\left[98.5674695063798, \infty\right)
Convexa en los intervalos
(−∞,4.31908711445437]\left(-\infty, 4.31908711445437\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x)=∞\lim_{x \to -\infty}\left(\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x)y = \lim_{x \to \infty}\left(\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (2*exp(pi/2) + cos(2*x)*exp(2*x) + exp(2*x)*sin(2*x))*exp(-2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2xx)y = x \lim_{x \to \infty}\left(\frac{\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x}}{x}\right)
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:
((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x=(2eπ2−e−2xsin⁡(2x)+e−2xcos⁡(2x))e2x\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x} = \left(2 e^{\frac{\pi}{2}} - e^{- 2 x} \sin{\left(2 x \right)} + e^{- 2 x} \cos{\left(2 x \right)}\right) e^{2 x}
- No
((e2xcos⁡(2x)+2eπ2)+e2xsin⁡(2x))e−2x=−(2eπ2−e−2xsin⁡(2x)+e−2xcos⁡(2x))e2x\left(\left(e^{2 x} \cos{\left(2 x \right)} + 2 e^{\frac{\pi}{2}}\right) + e^{2 x} \sin{\left(2 x \right)}\right) e^{- 2 x} = - \left(2 e^{\frac{\pi}{2}} - e^{- 2 x} \sin{\left(2 x \right)} + e^{- 2 x} \cos{\left(2 x \right)}\right) e^{2 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор