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Trazado el gráfico de la función/ exp^x*cosx*1/x

Gráfico de la función y = exp^x*cosx*1/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x       
       E *cos(x)
f(x) = ---------
           x
f(x)=excos⁡(x)xf{\left(x \right)} = \frac{e^{x} \cos{\left(x \right)}}{x} 
f = (E^x*cos(x))/x
Gráfico de la función
02468-8-6-4-2-1010-20002000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
excos⁡(x)x=0\frac{e^{x} \cos{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Solución numérica
x1=1.5707963267949x_{1} = 1.5707963267949
x2=−39.2699081698724x_{2} = -39.2699081698724
x3=14.1371669411541x_{3} = 14.1371669411541
x4=−70.6858347057703x_{4} = -70.6858347057703
x5=32.9867228626928x_{5} = 32.9867228626928
x6=−61.261056745001x_{6} = -61.261056745001
x7=−105.243353895258x_{7} = -105.243353895258
x8=−36.1283155162826x_{8} = -36.1283155162826
x9=−48.6946861306418x_{9} = -48.6946861306418
x10=7.85398163397448x_{10} = 7.85398163397448
x11=−80.1106126665397x_{11} = -80.1106126665397
x12=−98.9601685880785x_{12} = -98.9601685880785
x13=−64.4026493985908x_{13} = -64.4026493985908
x14=−29.845130209103x_{14} = -29.845130209103
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=−86.3937979737193x_{16} = -86.3937979737193
x17=−10.9955742875643x_{17} = -10.9955742875643
x18=17.2787595947439x_{18} = 17.2787595947439
x19=−26.7035375555132x_{19} = -26.7035375555132
x20=−54.9778714378214x_{20} = -54.9778714378214
x21=−92.6769832808989x_{21} = -92.6769832808989
x22=4.71238898038469x_{22} = 4.71238898038469
x23=20.4203522483337x_{23} = 20.4203522483337
x24=−67.5442420521806x_{24} = -67.5442420521806
x25=−58.1194640914112x_{25} = -58.1194640914112
x26=−51.8362787842316x_{26} = -51.8362787842316
x27=−83.2522053201295x_{27} = -83.2522053201295
x28=−76.9690200129499x_{28} = -76.9690200129499
x29=−14.1371669411541x_{29} = -14.1371669411541
x30=−73.8274273593601x_{30} = -73.8274273593601
x31=−20.4203522483337x_{31} = -20.4203522483337
x32=−4.71238898038469x_{32} = -4.71238898038469
x33=−95.8185759344887x_{33} = -95.8185759344887
x34=−7.85398163397448x_{34} = -7.85398163397448
x35=−45.553093477052x_{35} = -45.553093477052
x36=29.845130209103x_{36} = 29.845130209103
x37=26.7035375555132x_{37} = 26.7035375555132
x38=−1.5707963267949x_{38} = -1.5707963267949
x39=−17.2787595947439x_{39} = -17.2787595947439
x40=−32.9867228626928x_{40} = -32.9867228626928
x41=23.5619449019235x_{41} = 23.5619449019235
x42=10.9955742875643x_{42} = 10.9955742875643
x43=−23.5619449019235x_{43} = -23.5619449019235
x44=−42.4115008234622x_{44} = -42.4115008234622
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 (E^x*cos(x))/x.
e0cos⁡(0)0\frac{e^{0} \cos{\left(0 \right)}}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−exsin⁡(x)+excos⁡(x)x−excos⁡(x)x2=0\frac{- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}}{x} - \frac{e^{x} \cos{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−68.3223752642964x_{1} = -68.3223752642964
x2=25.8984556983654x_{2} = 25.8984556983654
x3=−21.182694633356x_{3} = -21.182694633356
x4=−93.4570599146458x_{4} = -93.4570599146458
x5=−40.0429743654917x_{5} = -40.0429743654917
x6=−74.6061683782845x_{6} = -74.6061683782845
x7=−87.1734932209778x_{7} = -87.1734932209778
x8=−71.4642850900844x_{8} = -71.4642850900844
x9=−14.8900880847917x_{9} = -14.8900880847917
x10=−33.7575267296448x_{10} = -33.7575267296448
x11=−80.8898676363837x_{11} = -80.8898676363837
x12=−90.315283165105x_{12} = -90.315283165105
x13=−77.7480283259493x_{13} = -77.7480283259493
x14=−8.58439584086157x_{14} = -8.58439584086157
x15=−36.9003456156361x_{15} = -36.9003456156361
x16=22.7540827305528x_{16} = 22.7540827305528
x17=−58.8964444413399x_{17} = -58.8964444413399
x18=6.99171397294222x_{18} = 6.99171397294222
x19=32.1855459430969x_{19} = 32.1855459430969
x20=−49.4700786400964x_{20} = -49.4700786400964
x21=3.77551226807681x_{21} = 3.77551226807681
x22=−11.7401459632485x_{22} = -11.7401459632485
x23=−24.3272065905731x_{23} = -24.3272065905731
x24=−30.6144600567864x_{24} = -30.6144600567864
x25=−99.7405787929127x_{25} = -99.7405787929127
x26=−2.171150616426x_{26} = -2.171150616426
x27=13.3127649854021x_{27} = 13.3127649854021
x28=16.4620473680961x_{28} = 16.4620473680961
x29=−96.5988247504869x_{29} = -96.5988247504869
x30=29.0422159262667x_{30} = 29.0422159262667
x31=−65.1804350919889x_{31} = -65.1804350919889
x32=10.1584541766013x_{32} = 10.1584541766013
x33=−43.1854540292704x_{33} = -43.1854540292704
x34=−55.754381638728x_{34} = -55.754381638728
x35=−5.41343454978119x_{35} = -5.41343454978119
x36=−62.0384599985962x_{36} = -62.0384599985962
x37=19.6087940910157x_{37} = 19.6087940910157
x38=−46.3278146314103x_{38} = -46.3278146314103
x39=−27.4710620052193x_{39} = -27.4710620052193
x40=−84.0316886109242x_{40} = -84.0316886109242
x41=−18.0371914942542x_{41} = -18.0371914942542
x42=−52.6122632039851x_{42} = -52.6122632039851
Signos de extremos en los puntos:
(-68.32237526429635, -2.18631792775718e-32)

(25.898455698365435, 4922094551.8402)

(-21.182694633356043, 2.05934409421649e-11)

(-93.45705991464582, -1.94385568258726e-43)

(-40.04297436549173, 7.09730657524411e-20)

(-74.60616837828448, -3.73897472577399e-35)

(-87.17349322097783, -1.11594439435482e-40)

(-71.46428509008439, 9.03260121741129e-34)

(-14.89008808479168, 1.56794826385968e-8)

(-33.7575267296448, 4.50790923313047e-17)

(-80.88986763638373, -6.43996205828751e-38)

(-90.31528316510503, 4.6546857765198e-42)

(-77.74802832594926, 1.55046887367563e-36)

(-8.584395840861575, 1.45337066444609e-5)

(-36.90034561563612, -1.78219039595166e-18)

(22.754082730552835, -242068503.87686)

(-58.89644444133988, 3.14273919682565e-28)

(6.991713972942219, 118.114727412863)

(32.185545943096855, 2121166417232.69)

(-49.47007864009639, -4.63629494943761e-24)

(3.775512268076811, -9.30868077852204)

(-11.740145963248548, -4.59891893913301e-7)

(-24.327206590573052, -7.74992513837921e-13)

(-30.614460056786445, -1.15020591421023e-15)

(-99.74057879291273, -3.40136346371062e-46)

(-2.1711506164259973, 0.0296749283468515)

(13.312764985402064, 33355.8555886438)

(16.46204736809609, -624539.306541806)

(-96.59882475048693, 8.12697125094014e-45)

(29.042215926266735, -101579204300.65)

(-65.1804350919889, 5.30314020107426e-31)

(10.158454176601259, -1886.93443153188)

(-43.185454029270375, -2.84390654620895e-21)

(-55.75438163872798, -7.6822989861738e-27)

(-5.413434549781193, -0.000530974345268083)

(-62.03845999859621, -1.28932705678727e-29)

(19.60879409101571, 12136519.8730325)

(-46.32781463141033, 1.14562400114401e-22)

(-27.471062005219288, 2.966033348239e-14)

(-84.03168861092416, 2.67891659017783e-39)

(-18.037191494254227, -5.59539057148987e-10)

(-52.61226320398511, 1.88388921924828e-25)


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=−68.3223752642964x_{1} = -68.3223752642964
x2=−93.4570599146458x_{2} = -93.4570599146458
x3=−74.6061683782845x_{3} = -74.6061683782845
x4=−87.1734932209778x_{4} = -87.1734932209778
x5=−80.8898676363837x_{5} = -80.8898676363837
x6=−36.9003456156361x_{6} = -36.9003456156361
x7=22.7540827305528x_{7} = 22.7540827305528
x8=−49.4700786400964x_{8} = -49.4700786400964
x9=3.77551226807681x_{9} = 3.77551226807681
x10=−11.7401459632485x_{10} = -11.7401459632485
x11=−24.3272065905731x_{11} = -24.3272065905731
x12=−30.6144600567864x_{12} = -30.6144600567864
x13=−99.7405787929127x_{13} = -99.7405787929127
x14=16.4620473680961x_{14} = 16.4620473680961
x15=29.0422159262667x_{15} = 29.0422159262667
x16=10.1584541766013x_{16} = 10.1584541766013
x17=−43.1854540292704x_{17} = -43.1854540292704
x18=−55.754381638728x_{18} = -55.754381638728
x19=−5.41343454978119x_{19} = -5.41343454978119
x20=−62.0384599985962x_{20} = -62.0384599985962
x21=−18.0371914942542x_{21} = -18.0371914942542
Puntos máximos de la función:
x21=25.8984556983654x_{21} = 25.8984556983654
x21=−21.182694633356x_{21} = -21.182694633356
x21=−40.0429743654917x_{21} = -40.0429743654917
x21=−71.4642850900844x_{21} = -71.4642850900844
x21=−14.8900880847917x_{21} = -14.8900880847917
x21=−33.7575267296448x_{21} = -33.7575267296448
x21=−90.315283165105x_{21} = -90.315283165105
x21=−77.7480283259493x_{21} = -77.7480283259493
x21=−8.58439584086157x_{21} = -8.58439584086157
x21=−58.8964444413399x_{21} = -58.8964444413399
x21=6.99171397294222x_{21} = 6.99171397294222
x21=32.1855459430969x_{21} = 32.1855459430969
x21=−2.171150616426x_{21} = -2.171150616426
x21=13.3127649854021x_{21} = 13.3127649854021
x21=−96.5988247504869x_{21} = -96.5988247504869
x21=−65.1804350919889x_{21} = -65.1804350919889
x21=19.6087940910157x_{21} = 19.6087940910157
x21=−46.3278146314103x_{21} = -46.3278146314103
x21=−27.4710620052193x_{21} = -27.4710620052193
x21=−84.0316886109242x_{21} = -84.0316886109242
x21=−52.6122632039851x_{21} = -52.6122632039851
Decrece en los intervalos
[29.0422159262667,∞)\left[29.0422159262667, \infty\right)
Crece en los intervalos
(−∞,−99.7405787929127]\left(-\infty, -99.7405787929127\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
2(−sin⁡(x)+sin⁡(x)−cos⁡(x)x+cos⁡(x)x2)exx=0\frac{2 \left(- \sin{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}}\right) e^{x}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−72.2427897046973x_{1} = -72.2427897046973
x2=21.945612879981x_{2} = 21.945612879981
x3=−28.2389365752603x_{3} = -28.2389365752603
x4=−50.2455828375744x_{4} = -50.2455828375744
x5=−53.3883466217256x_{5} = -53.3883466217256
x6=−56.5309801938186x_{6} = -56.5309801938186
x7=−66.444462944599x_{7} = -66.444462944599
x8=−62.8159348889734x_{8} = -62.8159348889734
x9=9.31786646179107x_{9} = 9.31786646179107
x10=−65.9582857893902x_{10} = -65.9582857893902
x11=−37.672573565113x_{11} = -37.672573565113
x12=−75.3849592185347x_{12} = -75.3849592185347
x13=6.12125046689807x_{13} = 6.12125046689807
x14=−87.9532251106725x_{14} = -87.9532251106725
x15=12.4864543952238x_{15} = 12.4864543952238
x16=−12.4864543952238x_{16} = -12.4864543952238
x17=−69.100567727981x_{17} = -69.100567727981
x18=−78.5270825679419x_{18} = -78.5270825679419
x19=−43.9595528888955x_{19} = -43.9595528888955
x20=−84.811211299318x_{20} = -84.811211299318
x21=18.7964043662102x_{21} = 18.7964043662102
x22=−44.4316477041399x_{22} = -44.4316477041399
x23=−97.3791034786112x_{23} = -97.3791034786112
x24=−91.0952098694071x_{24} = -91.0952098694071
x25=−40.8162093266346x_{25} = -40.8162093266346
x26=−25.0929104121121x_{26} = -25.0929104121121
x27=−31.3840740178899x_{27} = -31.3840740178899
x28=25.0929104121121x_{28} = 25.0929104121121
x29=−100.521017074687x_{29} = -100.521017074687
x30=2.79838604578389x_{30} = 2.79838604578389
x31=34.5285657554621x_{31} = 34.5285657554621
x32=−9.31786646179107x_{32} = -9.31786646179107
x33=−34.5285657554621x_{33} = -34.5285657554621
x34=−18.7964043662102x_{34} = -18.7964043662102
x35=−21.945612879981x_{35} = -21.945612879981
x36=15.644128370333x_{36} = 15.644128370333
x37=−91.75x_{37} = -91.75
x38=31.3840740178899x_{38} = 31.3840740178899
x39=−15.644128370333x_{39} = -15.644128370333
x40=−47.1026627703624x_{40} = -47.1026627703624
x41=−94.2371684817036x_{41} = -94.2371684817036
x42=−81.6691650818489x_{42} = -81.6691650818489
x43=−6.12125046689807x_{43} = -6.12125046689807
x44=28.2389365752603x_{44} = 28.2389365752603
x45=−2.79838604578389x_{45} = -2.79838604578389
x46=−59.6735041304405x_{46} = -59.6735041304405
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2(−sin⁡(x)+sin⁡(x)−cos⁡(x)x+cos⁡(x)x2)exx)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(- \sin{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}}\right) e^{x}}{x}\right) = -\infty
lim⁡x→0+(2(−sin⁡(x)+sin⁡(x)−cos⁡(x)x+cos⁡(x)x2)exx)=∞\lim_{x \to 0^+}\left(\frac{2 \left(- \sin{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}}\right) e^{x}}{x}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[34.5285657554621,∞)\left[34.5285657554621, \infty\right)
Convexa en los intervalos
(−∞,−100.521017074687]\left(-\infty, -100.521017074687\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(excos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(excos⁡(x)x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (E^x*cos(x))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(excos⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(excos⁡(x)x2)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{e^{x} \cos{\left(x \right)}}{x^{2}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
excos⁡(x)x=−e−xcos⁡(x)x\frac{e^{x} \cos{\left(x \right)}}{x} = - \frac{e^{- x} \cos{\left(x \right)}}{x}
- No
excos⁡(x)x=e−xcos⁡(x)x\frac{e^{x} \cos{\left(x \right)}}{x} = \frac{e^{- x} \cos{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
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