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Trazado el gráfico de la función/ lambertw(exp(-x)/x)

Gráfico de la función y = lambertw(exp(-x)/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        / -x\
        |e  |
f(x) = W|---|
        \ x /
f(x)=W(e−xx)f{\left(x \right)} = W\left(\frac{e^{- x}}{x}\right) 
f = LambertW(exp(-x)/x)
Gráfico de la función
02468-8-6-4-2-10100.02.5
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:
W(e−xx)=0W\left(\frac{e^{- x}}{x}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=42.5196428555452x_{1} = 42.5196428555452
x2=62.7797667134846x_{2} = 62.7797667134846
x3=48.6281937301383x_{3} = 48.6281937301383
x4=118.988395027578x_{4} = 118.988395027578
x5=98.9445847998326x_{5} = 98.9445847998326
x6=66.8087344899079x_{6} = 66.8087344899079
x7=70.8337593963567x_{7} = 70.8337593963567
x8=112.977059649009x_{8} = 112.977059649009
x9=68.8216870569048x_{9} = 68.8216870569048
x10=50.6566729420972x_{10} = 50.6566729420972
x11=90.9209018586394x_{11} = 90.9209018586394
x12=94.933310222745x_{12} = 94.933310222745
x13=54.705479530997x_{13} = 54.705479530997
x14=38.4184849954022x_{14} = 38.4184849954022
x15=36.3545103631678x_{15} = 36.3545103631678
x16=58.7458256964343x_{16} = 58.7458256964343
x17=25.6972250147378x_{17} = 25.6972250147378
x18=46.5963357507647x_{18} = 46.5963357507647
x19=116.98475932027x_{19} = 116.98475932027
x20=76.8655172918778x_{20} = 76.8655172918778
x21=74.8556029651331x_{21} = 74.8556029651331
x22=108.968730358115x_{22} = 108.968730358115
x23=72.8450392511437x_{23} = 72.8450392511437
x24=64.794800314078x_{24} = 64.794800314078
x25=40.4728301100237x_{25} = 40.4728301100237
x26=56.7265636263096x_{26} = 56.7265636263096
x27=30.0660853130855x_{27} = 30.0660853130855
x28=44.5604368263924x_{28} = 44.5604368263924
x29=96.9390792363674x_{29} = 96.9390792363674
x30=60.7634959137344x_{30} = 60.7634959137344
x31=100.949844652299x_{31} = 100.949844652299
x32=92.9272582589556x_{32} = 92.9272582589556
x33=106.964304849234x_{33} = 106.964304849234
x34=104.959690522986x_{34} = 104.959690522986
x35=120.991898414683x_{35} = 120.991898414683
x36=86.9071782608046x_{36} = 86.9071782608046
x37=32.1841394504197x_{37} = 32.1841394504197
x38=82.8919167465527x_{38} = 82.8919167465527
x39=27.9115050708033x_{39} = 27.9115050708033
x40=78.8748408669233x_{40} = 78.8748408669233
x41=114.98098362059x_{41} = 114.98098362059
x42=84.8997555854761x_{42} = 84.8997555854761
x43=110.972978457067x_{43} = 110.972978457067
x44=80.8836254150038x_{44} = 80.8836254150038
x45=88.9142172877899x_{45} = 88.9142172877899
x46=102.954874967394x_{46} = 102.954874967394
x47=52.6822955724978x_{47} = 52.6822955724978
x48=34.2778979009726x_{48} = 34.2778979009726
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 LambertW(exp(-x)/x).
W(e−00)W\left(\frac{e^{- 0}}{0}\right)
Resultado:
f(0)=W(∞~)f{\left(0 \right)} = W\left(\tilde{\infty}\right)
Punto:
(0, LambertW(±oo))
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
x(−e−xx−e−xx2)exW(e−xx)W(e−xx)+1=0\frac{x \left(- \frac{e^{- x}}{x} - \frac{e^{- x}}{x^{2}}\right) e^{x} W\left(\frac{e^{- x}}{x}\right)}{W\left(\frac{e^{- x}}{x}\right) + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=38.390297963018x_{1} = 38.390297963018
x2=88.9108332646095x_{2} = 88.9108332646095
x3=96.9362952671738x_{3} = 96.9362952671738
x4=102.952441679599x_{4} = 102.952441679599
x5=68.8155492631461x_{5} = 68.8155492631461
x6=110.970917957504x_{6} = 110.970917957504
x7=120.990194329818x_{7} = 120.990194329818
x8=100.947302258915x_{8} = 100.947302258915
x9=50.6435433637218x_{9} = 50.6435433637218
x10=106.96206962061x_{10} = 106.96206962061
x11=56.7167264749022x_{11} = 56.7167264749022
x12=66.8021451899636x_{12} = 66.8021451899636
x13=78.8703856528871x_{13} = 78.8703856528871
x14=−1x_{14} = -1
x15=25.5811688759785x_{15} = 25.5811688759785
x16=40.4486194018046x_{16} = 40.4486194018046
x17=86.9036136400823x_{17} = 86.9036136400823
x18=116.98292472987x_{18} = 116.98292472987
x19=54.6947010181784x_{19} = 54.6947010181784
x20=48.6135800530457x_{20} = 48.6135800530457
x21=34.2378524089663x_{21} = 34.2378524089663
x22=52.670430358141x_{22} = 52.670430358141
x23=94.9303922615188x_{23} = 94.9303922615188
x24=44.5419536864735x_{24} = 44.5419536864735
x25=36.321213637371x_{25} = 36.321213637371
x26=98.9419257516645x_{26} = 98.9419257516645
x27=62.772108347413x_{27} = 62.772108347413
x28=72.8396743022252x_{28} = 72.8396743022252
x29=58.7368097609923x_{29} = 58.7368097609923
x30=114.979078058674x_{30} = 114.979078058674
x31=60.7552009428775x_{31} = 60.7552009428775
x32=90.9176849605592x_{32} = 90.9176849605592
x33=104.957359432185x_{33} = 104.957359432185
x34=64.7877070206403x_{34} = 64.7877070206403
x35=30.0035943508711x_{35} = 30.0035943508711
x36=118.986627502015x_{36} = 118.986627502015
x37=42.4985959002091x_{37} = 42.4985959002091
x38=27.8288766375404x_{38} = 27.8288766375404
x39=108.966585172022x_{39} = 108.966585172022
x40=80.8794220504032x_{40} = 80.8794220504032
x41=84.8959953951563x_{41} = 84.8959953951563
x42=76.8607866615301x_{42} = 76.8607866615301
x43=82.8879442986464x_{43} = 82.8879442986464
x44=46.579962571028x_{44} = 46.579962571028
x45=92.9241963289231x_{45} = 92.9241963289231
x46=74.8505702854613x_{46} = 74.8505702854613
x47=70.8280277395619x_{47} = 70.8280277395619
x48=32.1348639353965x_{48} = 32.1348639353965
x49=112.975078898168x_{49} = 112.975078898168
Signos de extremos en los puntos:
(38.39029796301798, 5.53457087286961e-19)

(88.91083326460945, 2.73880461704969e-41)

(96.93629526717378, 8.21514912942539e-45)

(102.95244167959889, 1.88662692868056e-47)

(68.81554926314612, 1.88842658288787e-32)

(110.97091795750407, 5.76412646616186e-51)

(120.99019432981764, 2.3543773112306e-55)

(100.94730225891524, 1.42905501612803e-46)

(50.64354336372176, 2.00108017268806e-24)

(106.96206962060974, 3.29407583617902e-49)

(56.71672647490217, 4.11649416031037e-27)

(66.8021451899636, 1.45682221603634e-31)

(78.87038565288708, 7.08129620510859e-37)

(-1, 0.394979082707293 + 1.78818804138363*I)

(25.581168875978488, 3.03612139198865e-13)

(40.44861940180457, 6.70631718850359e-20)

(86.90361364008226, 2.08546210624092e-40)

(116.98292472986961, 1.33917963948612e-53)

(54.69470101817838, 3.22439252594356e-26)

(48.6135800530457, 1.58720430359471e-23)

(34.23785240896633, 3.94623651741001e-17)

(52.67043035814098, 2.53487103608113e-25)

(94.93039226151878, 6.23518294437212e-44)

(44.541953686473484, 1.01602884743247e-21)

(36.32121363737098, 4.63166339529762e-18)

(98.94192575166446, 1.08314664336778e-45)

(62.772108347413024, 8.7227416873179e-30)

(72.83967430222516, 3.18979954459285e-34)

(58.736809760992344, 5.27250841681264e-28)

(114.97907805867402, 1.01065293788101e-52)

(60.755200942877465, 6.77279642654136e-29)

(90.91768496055923, 3.60000202840912e-42)

(104.9573594321853, 2.49221248588298e-48)

(64.78770702064027, 1.12606571592338e-30)

(30.00359435087106, 3.1076439223969e-15)

(118.9866275020151, 1.77527689296683e-54)

(42.498595900209146, 8.217121908105e-21)

(27.828876637540393, 2.94833455946144e-14)

(108.96658517202219, 4.35632233603432e-50)

(80.87942205040315, 9.26137063506555e-38)

(84.89599539515635, 1.58946313440439e-39)

(76.86078666153011, 5.42100306774707e-36)

(82.88794429864642, 1.21263976608657e-38)

(46.57996257102804, 1.26584389942384e-22)

(92.92419632892314, 4.73593229790692e-43)

(74.85057028546126, 4.15542304683041e-35)

(70.82802773956195, 2.45229797724113e-33)

(32.134863935396474, 3.44373282926734e-16)

(112.97507889816829, 7.63069312969556e-52)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
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→−∞W(e−xx)=−∞\lim_{x \to -\infty} W\left(\frac{e^{- x}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞W(e−xx)=0\lim_{x \to \infty} W\left(\frac{e^{- x}}{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 LambertW(exp(-x)/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(W(e−xx)x)=∞\lim_{x \to -\infty}\left(\frac{W\left(\frac{e^{- x}}{x}\right)}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(W(e−xx)x)=0\lim_{x \to \infty}\left(\frac{W\left(\frac{e^{- x}}{x}\right)}{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:
W(e−xx)=W(−exx)W\left(\frac{e^{- x}}{x}\right) = W\left(- \frac{e^{x}}{x}\right)
- No
W(e−xx)=−W(−exx)W\left(\frac{e^{- x}}{x}\right) = - W\left(- \frac{e^{x}}{x}\right)
- No
es decir, función
no es
par ni impar
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