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Trazado el gráfico de la función/ (exp(-x)*sin(x))/sqrt(x)

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        -x       
       e  *sin(x)
f(x) = ----------
           ___   
         \/ x
f(x)=e−xsin⁡(x)xf{\left(x \right)} = \frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}} 
f = (exp(-x)*sin(x))/sqrt(x)
Gráfico de la función
02468-8-6-4-2-10100.5-0.5
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:
e−xsin⁡(x)x=0\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=−12.5663706143592x_{3} = -12.5663706143592
x4=75.398223686155x_{4} = 75.398223686155
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=59.6902604182061x_{6} = 59.6902604182061
x7=72.2566310325652x_{7} = 72.2566310325652
x8=91.106186954104x_{8} = 91.106186954104
x9=−6.28318530717959x_{9} = -6.28318530717959
x10=6.28318530717959x_{10} = 6.28318530717959
x11=62.8318530717959x_{11} = 62.8318530717959
x12=−25.1327412287183x_{12} = -25.1327412287183
x13=94.2477796076938x_{13} = 94.2477796076938
x14=−9.42477796076938x_{14} = -9.42477796076938
x15=65.9734457253857x_{15} = 65.9734457253857
x16=106.814150222053x_{16} = 106.814150222053
x17=25.1327412287183x_{17} = 25.1327412287183
x18=21.9911485751286x_{18} = 21.9911485751286
x19=87.9645943005142x_{19} = 87.9645943005142
x20=43.9822971502571x_{20} = 43.9822971502571
x21=97.3893722612836x_{21} = 97.3893722612836
x22=−31.4159265358979x_{22} = -31.4159265358979
x23=18.8495559215388x_{23} = 18.8495559215388
x24=78.5398163397448x_{24} = 78.5398163397448
x25=−18.8495559215388x_{25} = -18.8495559215388
x26=53.4070751110265x_{26} = 53.4070751110265
x27=47.1238898038469x_{27} = 47.1238898038469
x28=12.5663706143592x_{28} = 12.5663706143592
x29=81.6814089933346x_{29} = 81.6814089933346
x30=34.5575191894877x_{30} = 34.5575191894877
x31=−15.707963267949x_{31} = -15.707963267949
x32=50.2654824574367x_{32} = 50.2654824574367
x33=−3.14159265358979x_{33} = -3.14159265358979
x34=−28.2743338823081x_{34} = -28.2743338823081
x35=9.42477796076938x_{35} = 9.42477796076938
x36=−21.9911485751286x_{36} = -21.9911485751286
x37=56.5486677646163x_{37} = 56.5486677646163
x38=15.707963267949x_{38} = 15.707963267949
x39=84.8230016469244x_{39} = 84.8230016469244
x40=37.6991118430775x_{40} = 37.6991118430775
x41=100.530964914873x_{41} = 100.530964914873
x42=69.1150383789755x_{42} = 69.1150383789755
x43=28.2743338823081x_{43} = 28.2743338823081
x44=40.8407044966673x_{44} = 40.8407044966673
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)*sin(x))/sqrt(x).
e−0sin⁡(0)0\frac{e^{- 0} \sin{\left(0 \right)}}{\sqrt{0}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−e−xsin⁡(x)+e−xcos⁡(x)x−e−xsin⁡(x)2x32=0\frac{- e^{- x} \sin{\left(x \right)} + e^{- x} \cos{\left(x \right)}}{\sqrt{x}} - \frac{e^{- x} \sin{\left(x \right)}}{2 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=22.7656850079973x_{1} = 22.7656850079973
x2=57.3297241491537x_{2} = 57.3297241491537
x3=41.6201318876339x_{3} = 41.6201318876339
x4=63.6133366460702x_{4} = 63.6133366460702
x5=−21.1938142799481x_{5} = -21.1938142799481
x6=113.880543222024x_{6} = 113.880543222024
x7=3.8663316382073x_{7} = 3.8663316382073
x8=60.4715414498334x_{8} = 60.4715414498334
x9=25.9085825903261x_{9} = 25.9085825903261
x10=13.3333659917582x_{10} = 13.3333659917582
x11=88.7471833936367x_{11} = 88.7471833936367
x12=−8.60948255111037x_{12} = -8.60948255111037
x13=−5.44974485732153x_{13} = -5.44974485732153
x14=113.825673545378x_{14} = 113.825673545378
x15=79.3220727076741x_{15} = 79.3220727076741
x16=7.0342764376962x_{16} = 7.0342764376962
x17=−27.4797548438953x_{17} = -27.4797548438953
x18=−2.23063642276895x_{18} = -2.23063642276895
x19=−24.3369643800052x_{19} = -24.3369643800052
x20=−14.905508326343x_{20} = -14.905508326343
x21=73.0386180383092x_{21} = 73.0386180383092
x22=69.8968726067014x_{22} = 69.8968726067014
x23=19.6223744703054x_{23} = 19.6223744703054
x24=76.1803509093378x_{24} = 76.1803509093378
x25=98.172230353587x_{25} = 98.172230353587
x26=44.7621413137309x_{26} = 44.7621413137309
x27=85.6054879487811x_{27} = 85.6054879487811
x28=47.9040963475716x_{28} = 47.9040963475716
x29=−18.0501138275121x_{29} = -18.0501138275121
x30=10.1862256842502x_{30} = 10.1862256842502
x31=32.1936191763481x_{31} = 32.1936191763481
x32=91.8888718285112x_{32} = 91.8888718285112
x33=101.31390157875x_{33} = 101.31390157875
x34=128.005904833072x_{34} = 128.005904833072
x35=54.1878809163396x_{35} = 54.1878809163396
x36=29.0512001790114x_{36} = 29.0512001790114
x37=−11.7592542155948x_{37} = -11.7592542155948
x38=35.3358922125353x_{38} = 35.3358922125353
x39=82.4637846951237x_{39} = 82.4637846951237
x40=−30.6222973733396x_{40} = -30.6222973733396
x41=0.435858890573625x_{41} = 0.435858890573625
x42=16.4784179147545x_{42} = 16.4784179147545
x43=38.4780548274411x_{43} = 38.4780548274411
x44=66.7551128475339x_{44} = 66.7551128475339
x45=51.0460069857244x_{45} = 51.0460069857244
x46=95.0305539468591x_{46} = 95.0305539468591
Signos de extremos en los puntos:
(22.765685007997305, -1.90136048353024e-11)

(57.329724149153684, 1.17603001965326e-26)

(41.620131887633946, -9.1586411512486e-20)

(63.61333664607024, 2.08488966454994e-29)

(-21.193814279948125, 248805192.95401*I)

(113.88054322202419, 2.30470489105291e-51)

(3.8663316382073045, -0.00705826905916923)

(60.471541449833396, -4.94831444238962e-28)

(25.90858259032609, 7.70226285204766e-13)

(13.33336599175816, 3.07798628684476e-7)

(88.74718339363669, 2.14670458294408e-40)

(-8.609482551110368, 1360.3503436859*I)

(-5.449744857321526, -73.7876689255173*I)

(113.82567354537848, 2.29747616376516e-51)

(79.32207270767411, -2.81371914000362e-36)

(7.0342764376962, 0.00022672803885392)

(-27.479754843895293, 117013316129.03*I)

(-2.2306364227689457, 4.92283640466589*I)

(-24.336964380005227, -5373064829.23067*I)

(-14.905508326342956, 553954.824335417*I)

(73.03861803830924, -1.5701944454017e-33)

(69.89687260670136, 3.71429785492543e-32)

(19.622374470305402, 4.73901083394631e-10)

(76.1803509093378, 6.64404064194979e-35)

(98.17223035358695, -1.64712594809598e-44)

(44.762141313730936, 3.81639296324583e-21)

(85.6054879487811, -5.0579539023613e-39)

(47.90409634757164, -1.59421735917212e-22)

(-18.050113827512142, -11649953.6731109*I)

(10.18622568425015, -8.1470805137006e-6)

(32.19361917634808, 1.29037682423588e-15)

(91.88887182851121, -9.11679080429682e-42)

(101.31390157874985, 7.00665017748097e-46)

(128.0059048330722, 1.62033902308878e-57)

(54.18788091633964, -2.79919160633266e-25)

(29.051200179011364, -3.14332940182675e-14)

(-11.759254215594837, -26946.632137189*I)

(35.335892212535256, -5.32257091006969e-17)

(82.46378469512366, 1.19253217940794e-37)

(-30.622297373339553, -2565111834082.7*I)

(0.4358588905736254, 0.413564260102904)

(16.478417914754502, -1.19661219755555e-8)

(38.47805482744107, 2.20419461931286e-18)

(66.75511284753388, -8.79506705757247e-31)

(51.04600698572436, 6.67387418710409e-24)

(95.03055394685913, 3.87405402589361e-43)


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=22.7656850079973x_{1} = 22.7656850079973
x2=41.6201318876339x_{2} = 41.6201318876339
x3=3.8663316382073x_{3} = 3.8663316382073
x4=60.4715414498334x_{4} = 60.4715414498334
x5=79.3220727076741x_{5} = 79.3220727076741
x6=73.0386180383092x_{6} = 73.0386180383092
x7=98.172230353587x_{7} = 98.172230353587
x8=85.6054879487811x_{8} = 85.6054879487811
x9=47.9040963475716x_{9} = 47.9040963475716
x10=10.1862256842502x_{10} = 10.1862256842502
x11=91.8888718285112x_{11} = 91.8888718285112
x12=54.1878809163396x_{12} = 54.1878809163396
x13=29.0512001790114x_{13} = 29.0512001790114
x14=35.3358922125353x_{14} = 35.3358922125353
x15=16.4784179147545x_{15} = 16.4784179147545
x16=66.7551128475339x_{16} = 66.7551128475339
Puntos máximos de la función:
x16=57.3297241491537x_{16} = 57.3297241491537
x16=63.6133366460702x_{16} = 63.6133366460702
x16=113.880543222024x_{16} = 113.880543222024
x16=25.9085825903261x_{16} = 25.9085825903261
x16=13.3333659917582x_{16} = 13.3333659917582
x16=88.7471833936367x_{16} = 88.7471833936367
x16=7.0342764376962x_{16} = 7.0342764376962
x16=69.8968726067014x_{16} = 69.8968726067014
x16=19.6223744703054x_{16} = 19.6223744703054
x16=76.1803509093378x_{16} = 76.1803509093378
x16=44.7621413137309x_{16} = 44.7621413137309
x16=32.1936191763481x_{16} = 32.1936191763481
x16=101.31390157875x_{16} = 101.31390157875
x16=82.4637846951237x_{16} = 82.4637846951237
x16=0.435858890573625x_{16} = 0.435858890573625
x16=38.4780548274411x_{16} = 38.4780548274411
x16=51.0460069857244x_{16} = 51.0460069857244
x16=95.0305539468591x_{16} = 95.0305539468591
Decrece en los intervalos
[98.172230353587,∞)\left[98.172230353587, \infty\right)
Crece en los intervalos
(−∞,3.8663316382073]\left(-\infty, 3.8663316382073\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
(−2cos⁡(x)+sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2)e−xx=0\frac{\left(- 2 \cos{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}\right) e^{- x}}{\sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−23.5409388524024x_{1} = -23.5409388524024
x2=48.6843640566062x_{2} = 48.6843640566062
x3=89.5297904445622x_{3} = 89.5297904445622
x4=70.6787357147007x_{4} = 70.6787357147007
x5=98.9551031336913x_{5} = 98.9551031336913
x6=4.59882548982395x_{6} = 4.59882548982395
x7=−10.9510381672693x_{7} = -10.9510381672693
x8=67.5368116154608x_{8} = 67.5368116154608
x9=42.399640104995x_{9} = 42.399640104995
x10=83.2461811814626x_{10} = 83.2461811814626
x11=−17.2502148123565x_{11} = -17.2502148123565
x12=54.9687346157917x_{12} = 54.9687346157917
x13=58.110823356587x_{13} = 58.110823356587
x14=80.1043515306431x_{14} = 80.1043515306431
x15=45.5420554488498x_{15} = 45.5420554488498
x16=92.6715734588638x_{16} = 92.6715734588638
x17=61.2528609979107x_{17} = 61.2528609979107
x18=39.2570922296848x_{18} = 39.2570922296848
x19=102.09685203708x_{19} = 102.09685203708
x20=−29.8285121391728x_{20} = -29.8285121391728
x21=7.78793937856499x_{21} = 7.78793937856499
x22=76.9625024678429x_{22} = 76.9625024678429
x23=−1.31295596697718x_{23} = -1.31295596697718
x24=17.2493732203163x_{24} = 17.2493732203163
x25=−20.3961505746626x_{25} = -20.3961505746626
x26=26.6846301150466x_{26} = 26.6846301150466
x27=51.8265854366237x_{27} = 51.8265854366237
x28=36.1143769854381x_{28} = 36.1143769854381
x29=20.3955488771286x_{29} = 20.3955488771286
x30=−14.1023776014424x_{30} = -14.1023776014424
x31=23.5404873162478x_{31} = 23.5404873162478
x32=−7.79209306262557x_{32} = -7.79209306262557
x33=−26.6849814463173x_{33} = -26.6849814463173
x34=73.8206315061098x_{34} = 73.8206315061098
x35=86.3879935444025x_{35} = 86.3879935444025
x36=10.9489444336304x_{36} = 10.9489444336304
x37=1.07757085627999x_{37} = 1.07757085627999
x38=64.3948550487468x_{38} = 64.3948550487468
x39=95.8133439568503x_{39} = 95.8133439568503
x40=−4.61089469861488x_{40} = -4.61089469861488
x41=32.9714461491181x_{41} = 32.9714461491181
x42=29.828230997683x_{42} = 29.828230997683
x43=64.1744024104206x_{43} = 64.1744024104206
x44=14.1011172682879x_{44} = 14.1011172682879
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−((−2cos⁡(x)+sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2)e−xx)=−∞i\lim_{x \to 0^-}\left(\frac{\left(- 2 \cos{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}\right) e^{- x}}{\sqrt{x}}\right) = - \infty i
lim⁡x→0+((−2cos⁡(x)+sin⁡(x)−cos⁡(x)x+3sin⁡(x)4x2)e−xx)=−∞\lim_{x \to 0^+}\left(\frac{\left(- 2 \cos{\left(x \right)} + \frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{4 x^{2}}\right) e^{- x}}{\sqrt{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
[102.09685203708,∞)\left[102.09685203708, \infty\right)
Convexa en los intervalos
(−∞,1.07757085627999]\left(-\infty, 1.07757085627999\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→−∞(e−xsin⁡(x)x)=∞isin⁡(p)\lim_{x \to -\infty}\left(\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}}\right) = \infty i \sin{\left(p \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=∞isin⁡(p)y = \infty i \sin{\left(p \right)}
lim⁡x→∞(e−xsin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (exp(-x)*sin(x))/sqrt(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(e−xsin⁡(x)xx)=−∞isin⁡(p)\lim_{x \to -\infty}\left(\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x} x}\right) = - \infty i \sin{\left(p \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=−∞ixsin⁡(p)y = - \infty i x \sin{\left(p \right)}
lim⁡x→∞(e−xsin⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x} x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
e−xsin⁡(x)x=−exsin⁡(x)−x\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}} = - \frac{e^{x} \sin{\left(x \right)}}{\sqrt{- x}}
- No
e−xsin⁡(x)x=exsin⁡(x)−x\frac{e^{- x} \sin{\left(x \right)}}{\sqrt{x}} = \frac{e^{x} \sin{\left(x \right)}}{\sqrt{- x}}
- No
es decir, función
no es
par ni impar
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