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Trazado el gráfico de la función/ (x*(-2)+2*log(x))*exp(-x^2+log(x)^2)/x

Gráfico de la función y = (x*(-2)+2*log(x))*exp(-x^2+log(x)^2)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                               2      2   
                            - x  + log (x)
       (x*(-2) + 2*log(x))*e              
f(x) = -----------------------------------
                        x
f(x)=((−2)x+2log⁡(x))e−x2+log⁡(x)2xf{\left(x \right)} = \frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} 
f = (((-2)*x + 2*log(x))*exp(-x^2 + log(x)^2))/x
Gráfico de la función
02468-8-6-4-2-1010-10000001000000
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:
((−2)x+2log⁡(x))e−x2+log⁡(x)2x=0\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=13.0679806245012x_{1} = 13.0679806245012
x2=100.254985764063x_{2} = 100.254985764063
x3=36.530139428267x_{3} = 36.530139428267
x4=−14.5841091214985x_{4} = -14.5841091214985
x5=96.3556019626852x_{5} = 96.3556019626852
x6=−98x_{6} = -98
x7=88.3651722359887x_{7} = 88.3651722359887
x8=66.4034031498912x_{8} = 66.4034031498912
x9=24.6680154837385x_{9} = 24.6680154837385
x10=7.77864479086517x_{10} = 7.77864479086517
x11=6.34252010689127x_{11} = 6.34252010689127
x12=18.8038660380685x_{12} = 18.8038660380685
x13=60.418670523396x_{13} = 60.418670523396
x14=78.3798867063687x_{14} = 78.3798867063687
x15=34.5464512545871x_{15} = 34.5464512545871
x16=38.5155259457939x_{16} = 38.5155259457939
x17=90.3626204676866x_{17} = 90.3626204676866
x18=16.8709071443127x_{18} = 16.8709071443127
x19=54.4373102234958x_{19} = 54.4373102234958
x20=50.4522052304797x_{20} = 50.4522052304797
x21=−54.1817616927656x_{21} = -54.1817616927656
x22=9.44049737574337x_{22} = 9.44049737574337
x23=22.7052841024948x_{23} = 22.7052841024948
x24=44.4795849104216x_{24} = 44.4795849104216
x25=48.4605769288869x_{25} = 48.4605769288869
x26=72.390669230937x_{26} = 72.390669230937
x27=26.636362247018x_{27} = 26.636362247018
x28=74.3868815680711x_{28} = 74.3868815680711
x29=80.3766506038623x_{29} = 80.3766506038623
x30=14.9561446945893x_{30} = 14.9561446945893
x31=58.4244576986122x_{31} = 58.4244576986122
x32=30.5855054963056x_{32} = 30.5855054963056
x33=28.6091488812077x_{33} = 28.6091488812077
x34=40.5023591548354x_{34} = 40.5023591548354
x35=76.3832924627972x_{35} = 76.3832924627972
x36=56.430655817604x_{36} = 56.430655817604
x37=32.5647748438127x_{37} = 32.5647748438127
x38=42.4904346960968x_{38} = 42.4904346960968
x39=62.4132547599208x_{39} = 62.4132547599208
x40=64.4081757858712x_{40} = 64.4081757858712
x41=82.3735717912955x_{41} = 82.3735717912955
x42=84.3706390750664x_{42} = 84.3706390750664
x43=11.2207281863269x_{43} = 11.2207281863269
x44=92.360179254039x_{44} = 92.360179254039
x45=46.4696709665382x_{45} = 46.4696709665382
x46=68.3989099622339x_{46} = 68.3989099622339
x47=20.7497953202703x_{47} = 20.7497953202703
x48=−100x_{48} = -100
x49=86.3678423015666x_{49} = 86.3678423015666
x50=−42.1127721591923x_{50} = -42.1127721591923
x51=98.2550927573974x_{51} = 98.2550927573974
x52=52.4444731777806x_{52} = 52.4444731777806
x53=94.3578417364215x_{53} = 94.3578417364215
x54=70.3946723891429x_{54} = 70.3946723891429
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*(-2) + 2*log(x))*exp(-x^2 + log(x)^2))/x.
(2log⁡(0)+(−2)0)elog⁡(0)2−020\frac{\left(2 \log{\left(0 \right)} + \left(-2\right) 0\right) e^{\log{\left(0 \right)}^{2} - 0^{2}}}{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
(−2+2x)e−x2+log⁡(x)2+(−2x+2log⁡(x)x)((−2)x+2log⁡(x))e−x2+log⁡(x)2x−((−2)x+2log⁡(x))e−x2+log⁡(x)2x2=0\frac{\left(-2 + \frac{2}{x}\right) e^{- x^{2} + \log{\left(x \right)}^{2}} + \left(- 2 x + \frac{2 \log{\left(x \right)}}{x}\right) \left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} - \frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=20.7503963186919x_{1} = 20.7503963186919
x2=−100x_{2} = -100
x3=50.4522452203035x_{3} = 50.4522452203035
x4=38.5156164448738x_{4} = 38.5156164448738
x5=34.5465771541859x_{5} = 34.5465771541859
x6=18.8046828073616x_{6} = 18.8046828073616
x7=13.0705877948477x_{7} = 13.0705877948477
x8=64.4081949332316x_{8} = 64.4081949332316
x9=82.3735809218471x_{9} = 82.3735809218471
x10=62.413275811043x_{10} = 62.413275811043
x11=60.4186937391442x_{11} = 60.4186937391442
x12=28.6093725411008x_{12} = 28.6093725411008
x13=84.3706475718115x_{13} = 84.3706475718115
x14=44.4796434307671x_{14} = 44.4796434307671
x15=68.3989259386441x_{15} = 68.3989259386441
x16=36.5302456893077x_{16} = 36.5302456893077
x17=86.3678502219319x_{17} = 86.3678502219319
x18=6.38136158385154x_{18} = 6.38136158385154
x19=42.4905019047743x_{19} = 42.4905019047743
x20=100.254990033728x_{20} = 100.254990033728
x21=7.79482083692762x_{21} = 7.79482083692762
x22=40.5024368583271x_{22} = 40.5024368583271
x23=96.3556072865186x_{23} = 96.3556072865186
x24=52.4445087552595x_{24} = 52.4445087552595
x25=48.4606220913302x_{25} = 48.4606220913302
x26=80.3766604342783x_{26} = 80.3766604342783
x27=11.2250538592056x_{27} = 11.2250538592056
x28=14.957829305469x_{28} = 14.957829305469
x29=72.3906826994777x_{29} = 72.3906826994777
x30=58.4244833860834x_{30} = 58.4244833860834
x31=66.4034206162255x_{31} = 66.4034206162255
x32=26.6366405657216x_{32} = 26.6366405657216
x33=92.360185787405x_{33} = 92.360185787405
x34=56.4306843403131x_{34} = 56.4306843403131
x35=94.3578479042189x_{35} = 94.3578479042189
x36=24.6683677501596x_{36} = 24.6683677501596
x37=54.4373420142764x_{37} = 54.4373420142764
x38=22.7057388790964x_{38} = 22.7057388790964
x39=70.3946870403354x_{39} = 70.3946870403354
x40=78.3798973095204x_{40} = 78.3798973095204
x41=90.3626273903747x_{41} = 90.3626273903747
x42=98.2550919407595x_{42} = 98.2550919407595
x43=−98x_{43} = -98
x44=88.3651796240614x_{44} = 88.3651796240614
x45=32.5649255248198x_{45} = 32.5649255248198
x46=30.5856878970632x_{46} = 30.5856878970632
x47=76.3833039219478x_{47} = 76.3833039219478
x48=74.3868939778675x_{48} = 74.3868939778675
x49=16.8720556120598x_{49} = 16.8720556120598
x50=46.4697222327844x_{50} = 46.4697222327844
x51=9.44836611993232x_{51} = 9.44836611993232
Signos de extremos en los puntos:
(20.75039631869186, -1.69169431587095e-183)

                                                              /                         2\ 
                                                              \(4.60517018598809 + pi*I) / 
(-100, -1.13548386531474e-4345*(209.210340371976 + 2*pi*I)*e                            )

(50.45224522030347, -3.00016562651398e-1099)

(38.51561644487376, -6.1903093245304e-639)

(34.54657715418586, -2.44262327283898e-513)

(18.804682807361562, -2.46984199609007e-150)

(13.070587794847711, -7.58986843348118e-72)

(64.40819493323156, -1.48825717005403e-1794)

(82.37358092184706, -7.30178802095274e-2939)

(62.413275811043, -8.59881473611415e-1685)

(60.41869373914422, -1.65570273323801e-1578)

(28.609372541100775, -4.60511174588636e-351)

(84.37064757181149, -2.1683995620047e-3083)

(44.47964343076712, -1.95939386840372e-853)

(68.39892593864406, -1.65232427959918e-2024)

(36.53024568930773, -2.14055133518378e-574)

(86.36785022193187, -2.1517449650608e-3231)

(6.381361583851538, -9.09055077086583e-17)

(42.49050190477434, -1.87072450797457e-778)

(100.25499003372838, -2.39837261797915e-4356)

(7.794820836927622, -4.09380741414915e-25)

(40.50243685832706, -5.91601497156799e-707)

(96.3556072865186, -1.50439746250718e-4023)

(52.44450875525948, -3.81483535841022e-1188)

(48.46062209133024, -7.84049116964963e-1014)

(80.37666043427832, -8.21464013419875e-2798)

(11.225053859205614, -1.03103861404899e-52)

(14.957829305468962, -1.6791486237881e-94)

(72.39068269947767, -2.27143303500089e-2268)

(58.42448338608341, -1.06205157769996e-1475)

(66.40342061622555, -8.58718444322707e-1908)

(26.63664056572165, -6.11230061548531e-304)

(92.36018578740499, -2.93120822309661e-3696)

(56.43068434031313, -2.2685558725968e-1376)

(94.35784790421889, -3.63152625506616e-3858)

(24.668367750159582, -2.64606933577122e-260)

(54.43734201427636, -1.61286499040111e-1280)

(22.705738879096383, -3.72133047542398e-220)

(70.39468704033544, -1.06056922055151e-2144)

(78.37989730952043, -3.08702696157713e-2660)

(90.36262739037467, -7.91011379955707e-3538)

(98.25509194075947, -5.17603915761359e-4184)

                                                             /                         2\ 
                                                             \(4.58496747867057 + pi*I) / 
(-98, -1.10807645148737e-4173*(205.169934957341 + 2*pi*I)*e                            )

(88.3651796240614, -7.13594835229694e-3383)

(32.564925524819785, -9.18323715795255e-456)

(30.58568789706321, -1.13531333978164e-401)

(76.38330392194779, -3.87438423587964e-2526)

(74.38689397786752, -1.623630994812e-2395)

(16.87205561205983, -1.14815034007864e-120)

(46.469722232784385, -6.80420762342333e-932)

(9.448366119932317, -4.01311817195712e-37)


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
Crece en todo el eje numérico
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(4(1−1x)(x−log⁡(x)x)−2(x−log⁡(x))(2(x−log⁡(x)x)2−1−log⁡(x)x2+1x2)+2(2(−x+log⁡(x))(x−log⁡(x)x)+1−1x)x+2(−x+log⁡(x))x2−1x2)e−x2+log⁡(x)2x=0\frac{2 \left(4 \left(1 - \frac{1}{x}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) - 2 \left(x - \log{\left(x \right)}\right) \left(2 \left(x - \frac{\log{\left(x \right)}}{x}\right)^{2} - 1 - \frac{\log{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right) + \frac{2 \left(2 \left(- x + \log{\left(x \right)}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) + 1 - \frac{1}{x}\right)}{x} + \frac{2 \left(- x + \log{\left(x \right)}\right)}{x^{2}} - \frac{1}{x^{2}}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=66.4034380905146x_{1} = 66.4034380905146
x2=9.45644063594105x_{2} = 9.45644063594105
x3=100.254991210947x_{3} = 100.254991210947
x4=36.5303521115337x_{4} = 36.5303521115337
x5=72.390696173179x_{5} = 72.390696173179
x6=80.3766702675784x_{6} = 80.3766702675784
x7=58.4245090886833x_{7} = 58.4245090886833
x8=98.2550911296364x_{8} = 98.2550911296364
x9=94.3578537703565x_{9} = 94.3578537703565
x10=84.370656069936x_{10} = 84.370656069936
x11=48.4606672925143x_{11} = 48.4606672925143
x12=34.5467032675926x_{12} = 34.5467032675926
x13=13.073228075059x_{13} = 13.073228075059
x14=54.4373738266384x_{14} = 54.4373738266384
x15=6.42306719092416x_{15} = 6.42306719092416
x16=50.4522852417599x_{16} = 50.4522852417599
x17=74.3869063921747x_{17} = 74.3869063921747
x18=64.4082140898631x_{18} = 64.4082140898631
x19=18.805504387007x_{19} = 18.805504387007
x20=7.81166989276969x_{20} = 7.81166989276969
x21=76.3833153850303x_{21} = 76.3833153850303
x22=56.4307128810347x_{22} = 56.4307128810347
x23=44.479702010772x_{23} = 44.479702010772
x24=38.515707067307x_{24} = 38.515707067307
x25=90.3626342883669x_{25} = 90.3626342883669
x26=−98x_{26} = -98
x27=26.6369196858441x_{27} = 26.6369196858441
x28=96.3556130056482x_{28} = 96.3556130056482
x29=32.5650764942418x_{29} = 32.5650764942418
x30=40.5025146575044x_{30} = 40.5025146575044
x31=42.4905691885884x_{31} = 42.4905691885884
x32=20.7510002042727x_{32} = 20.7510002042727
x33=22.7061954709362x_{33} = 22.7061954709362
x34=11.22945614599x_{34} = 11.22945614599
x35=92.360192205913x_{35} = 92.360192205913
x36=62.4132968730228x_{36} = 62.4132968730228
x37=60.4187169676738x_{37} = 60.4187169676738
x38=68.3989419219101x_{38} = 68.3989419219101
x39=78.3799079161728x_{39} = 78.3799079161728
x40=46.4697735468833x_{40} = 46.4697735468833
x41=88.3651870184483x_{41} = 88.3651870184483
x42=30.5858706942994x_{42} = 30.5858706942994
x43=86.3678581436757x_{43} = 86.3678581436757
x44=82.3735900556088x_{44} = 82.3735900556088
x45=14.9595299490862x_{45} = 14.9595299490862
x46=70.3947016974629x_{46} = 70.3947016974629
x47=16.8732125600477x_{47} = 16.8732125600477
x48=−100x_{48} = -100
x49=24.6687212030401x_{49} = 24.6687212030401
x50=52.4445443587715x_{50} = 52.4445443587715
x51=28.6095967578267x_{51} = 28.6095967578267
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(4(1−1x)(x−log⁡(x)x)−2(x−log⁡(x))(2(x−log⁡(x)x)2−1−log⁡(x)x2+1x2)+2(2(−x+log⁡(x))(x−log⁡(x)x)+1−1x)x+2(−x+log⁡(x))x2−1x2)e−x2+log⁡(x)2x)=∞\lim_{x \to 0^-}\left(\frac{2 \left(4 \left(1 - \frac{1}{x}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) - 2 \left(x - \log{\left(x \right)}\right) \left(2 \left(x - \frac{\log{\left(x \right)}}{x}\right)^{2} - 1 - \frac{\log{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right) + \frac{2 \left(2 \left(- x + \log{\left(x \right)}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) + 1 - \frac{1}{x}\right)}{x} + \frac{2 \left(- x + \log{\left(x \right)}\right)}{x^{2}} - \frac{1}{x^{2}}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x}\right) = \infty
lim⁡x→0+(2(4(1−1x)(x−log⁡(x)x)−2(x−log⁡(x))(2(x−log⁡(x)x)2−1−log⁡(x)x2+1x2)+2(2(−x+log⁡(x))(x−log⁡(x)x)+1−1x)x+2(−x+log⁡(x))x2−1x2)e−x2+log⁡(x)2x)=−∞\lim_{x \to 0^+}\left(\frac{2 \left(4 \left(1 - \frac{1}{x}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) - 2 \left(x - \log{\left(x \right)}\right) \left(2 \left(x - \frac{\log{\left(x \right)}}{x}\right)^{2} - 1 - \frac{\log{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}\right) + \frac{2 \left(2 \left(- x + \log{\left(x \right)}\right) \left(x - \frac{\log{\left(x \right)}}{x}\right) + 1 - \frac{1}{x}\right)}{x} + \frac{2 \left(- x + \log{\left(x \right)}\right)}{x^{2}} - \frac{1}{x^{2}}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{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:
No tiene corvaduras 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→−∞(((−2)x+2log⁡(x))e−x2+log⁡(x)2x)=0\lim_{x \to -\infty}\left(\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(((−2)x+2log⁡(x))e−x2+log⁡(x)2x)=0\lim_{x \to \infty}\left(\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{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 ((x*(-2) + 2*log(x))*exp(-x^2 + log(x)^2))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−2)x+2log⁡(x))e−x2+log⁡(x)2x2)=0\lim_{x \to -\infty}\left(\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(((−2)x+2log⁡(x))e−x2+log⁡(x)2x2)=0\lim_{x \to \infty}\left(\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x^{2}}\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:
((−2)x+2log⁡(x))e−x2+log⁡(x)2x=−(2x+2log⁡(−x))e−x2+log⁡(−x)2x\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} = - \frac{\left(2 x + 2 \log{\left(- x \right)}\right) e^{- x^{2} + \log{\left(- x \right)}^{2}}}{x}
- No
((−2)x+2log⁡(x))e−x2+log⁡(x)2x=(2x+2log⁡(−x))e−x2+log⁡(−x)2x\frac{\left(\left(-2\right) x + 2 \log{\left(x \right)}\right) e^{- x^{2} + \log{\left(x \right)}^{2}}}{x} = \frac{\left(2 x + 2 \log{\left(- x \right)}\right) e^{- x^{2} + \log{\left(- x \right)}^{2}}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор