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Trazado el gráfico de la función/ x*e^x*sin(x)

Gráfico de la función y = x*e^x*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          x       
f(x) = x*E *sin(x)
f(x)=exxsin⁡(x)f{\left(x \right)} = e^{x} x \sin{\left(x \right)} 
f = (E^x*x)*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-200000200000
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:
exxsin⁡(x)=0e^{x} x \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=−91.106186954104x_{1} = -91.106186954104
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=12.5663706143592x_{4} = 12.5663706143592
x5=21.9911485751286x_{5} = 21.9911485751286
x6=−69.1150383789755x_{6} = -69.1150383789755
x7=−100.530964914873x_{7} = -100.530964914873
x8=3.14159265358979x_{8} = 3.14159265358979
x9=−3.14159265358979x_{9} = -3.14159265358979
x10=−15.707963267949x_{10} = -15.707963267949
x11=−25.1327412287183x_{11} = -25.1327412287183
x12=−53.4070751110265x_{12} = -53.4070751110265
x13=−72.2566310325652x_{13} = -72.2566310325652
x14=−81.6814089933346x_{14} = -81.6814089933346
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=18.8495559215388x_{16} = 18.8495559215388
x17=−65.9734457253857x_{17} = -65.9734457253857
x18=−40.8407044966673x_{18} = -40.8407044966673
x19=9.42477796076938x_{19} = 9.42477796076938
x20=0x_{20} = 0
x21=−62.8318530717959x_{21} = -62.8318530717959
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=−56.5486677646163x_{23} = -56.5486677646163
x24=15.707963267949x_{24} = 15.707963267949
x25=−18.8495559215388x_{25} = -18.8495559215388
x26=6.28318530717959x_{26} = 6.28318530717959
x27=25.1327412287183x_{27} = 25.1327412287183
x28=−47.1238898038469x_{28} = -47.1238898038469
x29=−34.5575191894877x_{29} = -34.5575191894877
x30=−97.3893722612836x_{30} = -97.3893722612836
x31=−50.2654824574367x_{31} = -50.2654824574367
x32=−75.398223686155x_{32} = -75.398223686155
x33=−9.42477796076938x_{33} = -9.42477796076938
x34=−87.9645943005142x_{34} = -87.9645943005142
x35=−6.28318530717959x_{35} = -6.28318530717959
x36=−78.5398163397448x_{36} = -78.5398163397448
x37=−37.6991118430775x_{37} = -37.6991118430775
x38=−43.9822971502571x_{38} = -43.9822971502571
x39=28.2743338823081x_{39} = 28.2743338823081
x40=−31.4159265358979x_{40} = -31.4159265358979
x41=−12.5663706143592x_{41} = -12.5663706143592
x42=−84.8230016469244x_{42} = -84.8230016469244
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 (x*E^x)*sin(x).
0e0sin⁡(0)0 e^{0} \sin{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
xexcos⁡(x)+(ex+xex)sin⁡(x)=0x e^{x} \cos{\left(x \right)} + \left(e^{x} + x e^{x}\right) \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−10.2613537997497x_{1} = -10.2613537997497
x2=−76.1902276260569x_{2} = -76.1902276260569
x3=−79.3315570557189x_{3} = -79.3315570557189
x4=−19.6610427777074x_{4} = -19.6610427777074
x5=14.9549060107419x_{5} = 14.9549060107419
x6=2.52025542588455x_{6} = 2.52025542588455
x7=0x_{7} = 0
x8=−22.7989655466536x_{8} = -22.7989655466536
x9=−98.1798891409031x_{9} = -98.1798891409031
x10=−35.3572605830509x_{10} = -35.3572605830509
x11=−47.919831688167x_{11} = -47.919831688167
x12=−1.32842662320324x_{12} = -1.32842662320324
x13=21.2287573315256x_{13} = 21.2287573315256
x14=−41.6382561985925x_{14} = -41.6382561985925
x15=27.5067866382649x_{15} = 27.5067866382649
x16=−82.4729066574052x_{16} = -82.4729066574052
x17=−95.038466574223x_{17} = -95.038466574223
x18=−25.9377926548812x_{18} = -25.9377926548812
x19=18.0910459439885x_{19} = 18.0910459439885
x20=−7.14370111864923x_{20} = -7.14370111864923
x21=−38.4976679493542x_{21} = -38.4976679493542
x22=−51.0607693885089x_{22} = -51.0607693885089
x23=−44.7789868716883x_{23} = -44.7789868716883
x24=8.69371127497694x_{24} = 8.69371127497694
x25=−32.2170877665294x_{25} = -32.2170877665294
x26=−69.9076402356465x_{26} = -69.9076402356465
x27=11.821529576608x_{27} = 11.821529576608
x28=5.57984147784894x_{28} = 5.57984147784894
x29=99.7505542137211x_{29} = 99.7505542137211
x30=−16.5245534289837x_{30} = -16.5245534289837
x31=−85.6142741980587x_{31} = -85.6142741980587
x32=−91.8970556986738x_{32} = -91.8970556986738
x33=−60.4839939455455x_{33} = -60.4839939455455
x34=−101.321322306037x_{34} = -101.321322306037
x35=−73.0489209881083x_{35} = -73.0489209881083
x36=−63.6251718400706x_{36} = -63.6251718400706
x37=−88.7556577624704x_{37} = -88.7556577624704
x38=−13.3905374849088x_{38} = -13.3905374849088
x39=−57.3428618803743x_{39} = -57.3428618803743
x40=−54.2017836830983x_{40} = -54.2017836830983
x41=24.3674469641161x_{41} = 24.3674469641161
x42=−110.745676333198x_{42} = -110.745676333198
x43=−4.06628479803594x_{43} = -4.06628479803594
x44=−66.7663890488394x_{44} = -66.7663890488394
x45=−29.0772267100229x_{45} = -29.0772267100229
Signos de extremos en los puntos:
(-10.261353799749651, -0.000266296800871636)

(-76.19022762605688, 4.41813339775209e-32)

(-79.33155705571886, -1.98797376433644e-33)

(-19.661042777707443, 4.12523562512903e-8)

(14.954906010741901, 31958913.4976957)

(2.5202554258845455, 18.2386348237996)

(0, 0)

(-22.79896554665356, -2.06755224340277e-9)

(-98.17988914090311, -1.60226793501503e-41)

(-35.35726058305085, -1.11851323250263e-14)

(-47.91983168816704, -5.28704425121392e-20)

(-1.3284266232032362, 0.341606750164678)

(21.228757331525596, 24306166818.6128)

(-41.63825619859252, -2.45995527603649e-17)

(27.506786638264945, 16868555328127.2)

(-82.47290665740518, 8.93101713670501e-35)

(-95.03846657422295, 3.58911698121622e-40)

(-25.937792654881243, 1.01659391562998e-10)

(18.091045943988497, -894937992.939162)

(-7.143701118649228, 0.00427784504503054)

(-38.497667949354245, 5.26301431017594e-16)

(-51.06076938850889, 2.43452599030039e-21)

(-44.77898687168834, 1.14325014007464e-18)

(8.69371127497694, 34625.3431755488)

(-32.217087766529446, 2.35834240961195e-13)

(-69.90764023564653, 2.17076765714376e-29)

(11.821529576607984, -1091027.02462304)

(5.579841477848944, -956.460754111719)

(99.75055421372109, -1.47007236922402e+45)

(-16.524553428983673, -8.0209323639257e-7)

(-85.61427419805874, -4.00645815139304e-36)

(-91.89705569867381, -8.0309199941571e-39)

(-60.48399394554552, -2.32729075447104e-25)

(-101.32132230603702, 7.14558506812486e-43)

(-73.04892098810835, -9.80230129103682e-31)

(-63.62517184007065, 1.05795136201781e-26)

(-88.75565776247038, 1.79487895886271e-37)

(-13.390537484908817, 1.50330464355879e-5)

(-57.34286188037428, 5.1057847642861e-24)

(-54.20178368309829, -1.11678368891239e-22)

(24.367446964116052, -645704548597.118)

(-110.74567633319846, -6.30284174732226e-47)

(-4.0662847980359444, -0.0556509462046117)

(-66.76638904883937, -4.79756393484494e-28)

(-29.077226710022934, -4.92521636405751e-12)


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=−10.2613537997497x_{1} = -10.2613537997497
x2=−79.3315570557189x_{2} = -79.3315570557189
x3=0x_{3} = 0
x4=−22.7989655466536x_{4} = -22.7989655466536
x5=−98.1798891409031x_{5} = -98.1798891409031
x6=−35.3572605830509x_{6} = -35.3572605830509
x7=−47.919831688167x_{7} = -47.919831688167
x8=−41.6382561985925x_{8} = -41.6382561985925
x9=18.0910459439885x_{9} = 18.0910459439885
x10=11.821529576608x_{10} = 11.821529576608
x11=5.57984147784894x_{11} = 5.57984147784894
x12=99.7505542137211x_{12} = 99.7505542137211
x13=−16.5245534289837x_{13} = -16.5245534289837
x14=−85.6142741980587x_{14} = -85.6142741980587
x15=−91.8970556986738x_{15} = -91.8970556986738
x16=−60.4839939455455x_{16} = -60.4839939455455
x17=−73.0489209881083x_{17} = -73.0489209881083
x18=−54.2017836830983x_{18} = -54.2017836830983
x19=24.3674469641161x_{19} = 24.3674469641161
x20=−110.745676333198x_{20} = -110.745676333198
x21=−4.06628479803594x_{21} = -4.06628479803594
x22=−66.7663890488394x_{22} = -66.7663890488394
x23=−29.0772267100229x_{23} = -29.0772267100229
Puntos máximos de la función:
x23=−76.1902276260569x_{23} = -76.1902276260569
x23=−19.6610427777074x_{23} = -19.6610427777074
x23=14.9549060107419x_{23} = 14.9549060107419
x23=2.52025542588455x_{23} = 2.52025542588455
x23=−1.32842662320324x_{23} = -1.32842662320324
x23=21.2287573315256x_{23} = 21.2287573315256
x23=27.5067866382649x_{23} = 27.5067866382649
x23=−82.4729066574052x_{23} = -82.4729066574052
x23=−95.038466574223x_{23} = -95.038466574223
x23=−25.9377926548812x_{23} = -25.9377926548812
x23=−7.14370111864923x_{23} = -7.14370111864923
x23=−38.4976679493542x_{23} = -38.4976679493542
x23=−51.0607693885089x_{23} = -51.0607693885089
x23=−44.7789868716883x_{23} = -44.7789868716883
x23=8.69371127497694x_{23} = 8.69371127497694
x23=−32.2170877665294x_{23} = -32.2170877665294
x23=−69.9076402356465x_{23} = -69.9076402356465
x23=−101.321322306037x_{23} = -101.321322306037
x23=−63.6251718400706x_{23} = -63.6251718400706
x23=−88.7556577624704x_{23} = -88.7556577624704
x23=−13.3905374849088x_{23} = -13.3905374849088
x23=−57.3428618803743x_{23} = -57.3428618803743
Decrece en los intervalos
[99.7505542137211,∞)\left[99.7505542137211, \infty\right)
Crece en los intervalos
(−∞,−110.745676333198]\left(-\infty, -110.745676333198\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
(−xsin⁡(x)+2(x+1)cos⁡(x)+(x+2)sin⁡(x))ex=0\left(- x \sin{\left(x \right)} + 2 \left(x + 1\right) \cos{\left(x \right)} + \left(x + 2\right) \sin{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1.9025816596713x_{1} = 1.9025816596713
x2=−36.1567518873749x_{2} = -36.1567518873749
x3=−76.9821802337515x_{3} = -76.9821802337515
x4=−45.5755235510474x_{4} = -45.5755235510474
x5=−61.2776451215302x_{5} = -61.2776451215302
x6=−64.418416382217x_{6} = -64.418416382217
x7=−89.5466836249472x_{7} = -89.5466836249472
x8=−33.0179451984154x_{8} = -33.0179451984154
x9=29.8775049299126x_{9} = 29.8775049299126
x10=11.0781798144108x_{10} = 11.0781798144108
x11=−95.8291208275139x_{11} = -95.8291208275139
x12=−86.4055062856094x_{12} = -86.4055062856094
x13=−83.2643606537498x_{13} = -83.2643606537498
x14=−73.8411549997741x_{14} = -73.8411549997741
x15=−26.74236450316x_{15} = -26.74236450316
x16=−42.4356299587436x_{16} = -42.4356299587436
x17=20.4669019371238x_{17} = 20.4669019371238
x18=−58.1369641096559x_{18} = -58.1369641096559
x19=−23.6061518286588x_{19} = -23.6061518286588
x20=−67.5592651262192x_{20} = -67.5592651262192
x21=−105.252945656203x_{21} = -105.252945656203
x22=−29.8797427274828x_{22} = -29.8797427274828
x23=23.6025687023826x_{23} = 23.6025687023826
x24=−98.9703754007943x_{24} = -98.9703754007943
x25=−2.24679137687774x_{25} = -2.24679137687774
x26=7.96506651296683x_{26} = 7.96506651296683
x27=−48.7156405519083x_{27} = -48.7156405519083
x28=−39.2960146150878x_{28} = -39.2960146150878
x29=−54.9963890778611x_{29} = -54.9963890778611
x30=4.88082214577343x_{30} = 4.88082214577343
x31=−4.95975747525199x_{31} = -4.95975747525199
x32=−92.6878894142842x_{32} = -92.6878894142842
x33=−80.1232505037716x_{33} = -80.1232505037716
x34=−70.7001808865412x_{34} = -70.7001808865412
x35=−7.99595954344623x_{35} = -7.99595954344623
x36=−11.0943177411687x_{36} = -11.0943177411687
x37=−14.2127076381121x_{37} = -14.2127076381121
x38=26.7395715348192x_{38} = 26.7395715348192
x39=−51.8559396371055x_{39} = -51.8559396371055
x40=−17.3398833066804x_{40} = -17.3398833066804
x41=−20.4716638479466x_{41} = -20.4716638479466
x42=17.3332512943446x_{42} = 17.3332512943446
x43=14.2028494649391x_{43} = 14.2028494649391
x44=−0.47973100728041x_{44} = -0.47973100728041

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
[29.8775049299126,∞)\left[29.8775049299126, \infty\right)
Convexa en los intervalos
(−∞,−105.252945656203]\left(-\infty, -105.252945656203\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(exxsin⁡(x))=0\lim_{x \to -\infty}\left(e^{x} x \sin{\left(x \right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(exxsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(e^{x} x \sin{\left(x \right)}\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 (x*E^x)*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(exsin⁡(x))=0\lim_{x \to -\infty}\left(e^{x} \sin{\left(x \right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(exsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(e^{x} \sin{\left(x \right)}\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:
exxsin⁡(x)=xe−xsin⁡(x)e^{x} x \sin{\left(x \right)} = x e^{- x} \sin{\left(x \right)}
- No
exxsin⁡(x)=−xe−xsin⁡(x)e^{x} x \sin{\left(x \right)} = - x e^{- x} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
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