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Trazado el gráfico de la función/ x*log((2*atan(x))/pi)

Gráfico de la función y = x*log((2*atan(x))/pi)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            /2*atan(x)\
f(x) = x*log|---------|
            \    pi   /
f(x)=xlog⁡(2atan⁡(x)π)f{\left(x \right)} = x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} 
f = x*log((2*atan(x))/pi)
Gráfico de la función
02468-8-6-4-2-1010-1.00.0
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:
xlog⁡(2atan⁡(x)π)=0x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=2.37465254083109⋅1027x_{1} = 2.37465254083109 \cdot 10^{27}
x2=9.09267034590915⋅1027x_{2} = 9.09267034590915 \cdot 10^{27}
x3=1.01355623048019⋅1028x_{3} = 1.01355623048019 \cdot 10^{28}
x4=1.01662561671128⋅1027x_{4} = 1.01662561671128 \cdot 10^{27}
x5=1.70510589767168⋅1027x_{5} = 1.70510589767168 \cdot 10^{27}
x6=1.71892810070898⋅1027x_{6} = 1.71892810070898 \cdot 10^{27}
x7=1.59769911871709⋅1027x_{7} = 1.59769911871709 \cdot 10^{27}
x8=4.66751238402882⋅1027x_{8} = 4.66751238402882 \cdot 10^{27}
x9=1.65459899478169⋅1027x_{9} = 1.65459899478169 \cdot 10^{27}
x10=3.36382991775434⋅1027x_{10} = 3.36382991775434 \cdot 10^{27}
x11=2.544270251303⋅1027x_{11} = 2.544270251303 \cdot 10^{27}
x12=2.08670666140796⋅1027x_{12} = 2.08670666140796 \cdot 10^{27}
x13=1.34694290111058⋅1027x_{13} = 1.34694290111058 \cdot 10^{27}
x14=1.10940265077509⋅1027x_{14} = 1.10940265077509 \cdot 10^{27}
x15=2.70392527894452⋅1027x_{15} = 2.70392527894452 \cdot 10^{27}
x16=2.9529898466685⋅1027x_{16} = 2.9529898466685 \cdot 10^{27}
x17=9.44299400086797⋅1026x_{17} = 9.44299400086797 \cdot 10^{26}
x18=2.02983961199207⋅1027x_{18} = 2.02983961199207 \cdot 10^{27}
x19=2.19828498256076⋅1027x_{19} = 2.19828498256076 \cdot 10^{27}
x20=1.61808573256392⋅1027x_{20} = 1.61808573256392 \cdot 10^{27}
x21=1.40693046903675⋅1027x_{21} = 1.40693046903675 \cdot 10^{27}
x22=2.57391599385481⋅1027x_{22} = 2.57391599385481 \cdot 10^{27}
x23=3.78355930912823⋅1027x_{23} = 3.78355930912823 \cdot 10^{27}
x24=2.94808529900454⋅1027x_{24} = 2.94808529900454 \cdot 10^{27}
x25=2.71772035896347⋅1027x_{25} = 2.71772035896347 \cdot 10^{27}
x26=1.2658989739658⋅1027x_{26} = 1.2658989739658 \cdot 10^{27}
x27=1.01192348319571⋅1027x_{27} = 1.01192348319571 \cdot 10^{27}
x28=3.192554125383⋅1027x_{28} = 3.192554125383 \cdot 10^{27}
x29=1.27867043651213⋅1027x_{29} = 1.27867043651213 \cdot 10^{27}
x30=9.22072667088938⋅1026x_{30} = 9.22072667088938 \cdot 10^{26}
x31=9.708174068544⋅1026x_{31} = 9.708174068544 \cdot 10^{26}
x32=2.30031512047749⋅1027x_{32} = 2.30031512047749 \cdot 10^{27}
x33=1.73098845697979⋅1027x_{33} = 1.73098845697979 \cdot 10^{27}
x34=1.23665996986943⋅1027x_{34} = 1.23665996986943 \cdot 10^{27}
x35=1.61020041210548⋅1028x_{35} = 1.61020041210548 \cdot 10^{28}
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*log((2*atan(x))/pi).
0log⁡(2atan⁡(0)π)0 \log{\left(\frac{2 \operatorname{atan}{\left(0 \right)}}{\pi} \right)}
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
x(x2+1)atan⁡(x)+log⁡(2atan⁡(x)π)=0\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=448904.467395942x_{1} = 448904.467395942
x2=489341.039648711x_{2} = 489341.039648711
x3=469122.587942295x_{3} = 469122.587942295
x4=438795.545945876x_{4} = 438795.545945876
x5=428686.725982567x_{5} = 428686.725982567
x6=418578.015038876x_{6} = 418578.015038876
x7=499450.376862916x_{7} = 499450.376862916
x8=398360.954263746x_{8} = 398360.954263746
x9=479231.775075814x_{9} = 479231.775075814
x10=459013.483478323x_{10} = 459013.483478323
x11=509559.782308744x_{11} = 509559.782308744
x12=408469.421412152x_{12} = 408469.421412152
Signos de extremos en los puntos:
                                         /3.14158819829887\ 
(448904.46739594196, 448904.467395942*log|----------------|)
                                         \       pi       / 

                                        /3.1415885664607\ 
(489341.0396487106, 489341.039648711*log|---------------|)
                                        \       pi      / 

                                         /3.14158839031181\ 
(469122.58794229454, 469122.587942295*log|----------------|)
                                         \       pi       / 

                                        /3.14158809565839\ 
(438795.5459458755, 438795.545945876*log|----------------|)
                                        \       pi       / 

                                         /3.14158798817825\ 
(428686.72598256654, 428686.725982567*log|----------------|)
                                         \       pi       / 

                                         /3.14158787550798\ 
(418578.01503887575, 418578.015038876*log|----------------|)
                                         \       pi       / 

                                         /3.14158864918797\ 
(499450.37686291616, 499450.376862916*log|----------------|)
                                         \       pi       / 

                                        /3.14158763301742\ 
(398360.9542637464, 398360.954263746*log|----------------|)
                                        \       pi       / 

                                         /3.14158848024382\ 
(479231.77507581434, 479231.775075814*log|----------------|)
                                         \       pi       / 

                                        /3.14158829641931\ 
(459013.4834783227, 459013.483478323*log|----------------|)
                                        \       pi       / 

                                        /3.14158872863325\ 
(509559.7823087445, 509559.782308744*log|----------------|)
                                        \       pi       / 

                                         /3.14158775726244\ 
(408469.42141215154, 408469.421412152*log|----------------|)
                                         \       pi       /


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
Puntos máximos de la función:
x12=479231.775075814x_{12} = 479231.775075814
Decrece en los intervalos
(−∞,479231.775075814]\left(-\infty, 479231.775075814\right]
Crece en los intervalos
[479231.775075814,∞)\left[479231.775075814, \infty\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
−x(2x+1atan⁡(x))x2+1+2(x2+1)atan⁡(x)=0\frac{- \frac{x \left(2 x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{x^{2} + 1} + 2}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=23230.024004552x_{1} = 23230.024004552
x2=7167.2354620335x_{2} = 7167.2354620335
x3=36785.7126288471x_{3} = 36785.7126288471
x4=−15478.8197012543x_{4} = -15478.8197012543
x5=34243.649370143x_{5} = 34243.649370143
x6=18149.0816677683x_{6} = 18149.0816677683
x7=−29875.9909139865x_{7} = -29875.9909139865
x8=42717.5488462566x_{8} = 42717.5488462566
x9=−11250.8616253629x_{9} = -11250.8616253629
x10=9693.10193650276x_{10} = 9693.10193650276
x11=−33265.1318463999x_{11} = -33265.1318463999
x12=−18018.0057264033x_{12} = -18018.0057264033
x13=−23098.8841994312x_{13} = -23098.8841994312
x14=8007.80313807833x_{14} = 8007.80313807833
x15=−28181.5398429727x_{15} = -28181.5398429727
x16=25771.2167319068x_{16} = 25771.2167319068
x17=35938.3460880304x_{17} = 35938.3460880304
x18=38480.4779619115x_{18} = 38480.4779619115
x19=26618.3524715985x_{19} = 26618.3524715985
x20=20689.2500969357x_{20} = 20689.2500969357
x21=−13786.8329402522x_{21} = -13786.8329402522
x22=−12095.8351865752x_{22} = -12095.8351865752
x23=39327.8753275521x_{23} = 39327.8753275521
x24=31701.7102656076x_{24} = 31701.7102656076
x25=41022.6963438243x_{25} = 41022.6963438243
x26=11381.6615466408x_{26} = 11381.6615466408
x27=29159.9289611697x_{27} = 29159.9289611697
x28=−38349.2776146624x_{28} = -38349.2776146624
x29=37633.0901689145x_{29} = 37633.0901689145
x30=41870.1189041515x_{30} = 41870.1189041515
x31=35090.9913670332x_{31} = 35090.9913670332
x32=17302.542491929x_{32} = 17302.542491929
x33=−21404.9948702379x_{33} = -21404.9948702379
x34=−30723.2488528237x_{34} = -30723.2488528237
x35=−34112.4579108891x_{35} = -34112.4579108891
x36=−22251.9132074708x_{36} = -22251.9132074708
x37=13072.0980035978x_{37} = 13072.0980035978
x38=40175.2816444387x_{38} = 40175.2816444387
x39=24077.048555564x_{39} = 24077.048555564
x40=−26487.189557406x_{40} = -26487.189557406
x41=−17171.4834086717x_{41} = -17171.4834086717
x42=30007.1692999306x_{42} = 30007.1692999306
x43=−2.33112237041442x_{43} = -2.33112237041442
x44=−20558.1360436213x_{44} = -20558.1360436213
x45=−24792.9616156613x_{45} = -24792.9616156613
x46=−39196.6735475476x_{46} = -39196.6735475476
x47=−42586.3421831168x_{47} = -42586.3421831168
x48=−40044.0785238142x_{48} = -40044.0785238142
x49=12226.698693816x_{49} = 12226.698693816
x50=8849.91095105935x_{50} = 8849.91095105935
x51=−31570.5259971979x_{51} = -31570.5259971979
x52=−32417.8207922022x_{52} = -32417.8207922022
x53=−37501.8913551979x_{53} = -37501.8913551979
x54=32549.0076519436x_{54} = 32549.0076519436
x55=−35807.1506836181x_{55} = -35807.1506836181
x56=−7037.26362170559x_{56} = -7037.26362170559
x57=27465.5182312612x_{57} = 27465.5182312612
x58=21536.1185810074x_{58} = 21536.1185810074
x59=−12941.1839661986x_{59} = -12941.1839661986
x60=22383.0454455925x_{60} = 22383.0454455925
x61=13917.7877997787x_{61} = 13917.7877997787
x62=−10406.3656388881x_{62} = -10406.3656388881
x63=−41738.9133485496x_{63} = -41738.9133485496
x64=−16325.0810229119x_{64} = -16325.0810229119
x65=30854.430305527x_{65} = 30854.430305527
x66=−25640.058714171x_{66} = -25640.058714171
x67=−19711.3448284427x_{67} = -19711.3448284427
x68=−8719.44409149685x_{68} = -8719.44409149685
x69=19842.4478851941x_{69} = 19842.4478851941
x70=15609.835771182x_{70} = 15609.835771182
x71=−23945.9020047737x_{71} = -23945.9020047737
x72=−7877.53782198138x_{72} = -7877.53782198138
x73=18995.7213280503x_{73} = 18995.7213280503
x74=−40891.4919669806x_{74} = -40891.4919669806
x75=−14632.7258559082x_{75} = -14632.7258559082
x76=28312.7112143158x_{76} = 28312.7112143158
x77=14763.7141682514x_{77} = 14763.7141682514
x78=2.33112237041442x_{78} = 2.33112237041442
x79=−27334.3508848309x_{79} = -27334.3508848309
x80=−18864.6308666567x_{80} = -18864.6308666567
x81=16456.1203792091x_{81} = 16456.1203792091
x82=24924.1142055314x_{82} = 24924.1142055314
x83=10537.0839889211x_{83} = 10537.0839889211
x84=−29028.753923803x_{84} = -29028.753923803
x85=−9562.49064635093x_{85} = -9562.49064635093
x86=−36654.5154591195x_{86} = -36654.5154591195
x87=33396.3210963241x_{87} = 33396.3210963241
x88=−34959.7978615038x_{88} = -34959.7978615038

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
[41870.1189041515,∞)\left[41870.1189041515, \infty\right)
Convexa en los intervalos
(−∞,7167.2354620335]\left(-\infty, 7167.2354620335\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xlog⁡(2atan⁡(x)π))=−∞i\lim_{x \to -\infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \infty i
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(xlog⁡(2atan⁡(x)π))=−2π\lim_{x \to \infty}\left(x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}\right) = - \frac{2}{\pi}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=−2πy = - \frac{2}{\pi}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*log((2*atan(x))/pi), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞log⁡(2atan⁡(x)π)=iπ\lim_{x \to -\infty} \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} = i \pi
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=iπxy = i \pi x
lim⁡x→∞log⁡(2atan⁡(x)π)=0\lim_{x \to \infty} \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \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:
xlog⁡(2atan⁡(x)π)=−xlog⁡(−2atan⁡(x)π)x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} = - x \log{\left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}
- No
xlog⁡(2atan⁡(x)π)=xlog⁡(−2atan⁡(x)π)x \log{\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)} = x \log{\left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\pi} \right)}
- No
es decir, función
no es
par ni impar
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