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Trazado el gráfico de la función/ x*cos(1/(x+pi))

Gráfico de la función y = x*cos(1/(x+pi))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
            /  1   \
f(x) = x*cos|------|
            \x + pi/
f(x)=xcos⁡(1x+π)f{\left(x \right)} = x \cos{\left(\frac{1}{x + \pi} \right)} 
f = x*cos(1/(x + pi))
Gráfico de la función
02468-8-6-4-2-1010-2020
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−3.14159265358979x_{1} = -3.14159265358979
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:
xcos⁡(1x+π)=0x \cos{\left(\frac{1}{x + \pi} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−π−2πx_{2} = - \pi - \frac{2}{\pi}
x3=−π+2πx_{3} = - \pi + \frac{2}{\pi}
Solución numérica
x1=0x_{1} = 0
x2=−3.77821242595737x_{2} = -3.77821242595737
x3=−2.50497288122221x_{3} = -2.50497288122221
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*cos(1/(x + pi)).
0cos⁡(1π)0 \cos{\left(\frac{1}{\pi} \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
xsin⁡(1x+π)(x+π)2+cos⁡(1x+π)=0\frac{x \sin{\left(\frac{1}{x + \pi} \right)}}{\left(x + \pi\right)^{2}} + \cos{\left(\frac{1}{x + \pi} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.82508094381759x_{1} = -1.82508094381759
Signos de extremos en los puntos:
                                          /          1           \ 
(-1.825080943817593, -1.82508094381759*cos|----------------------|)
                                          \-1.82508094381759 + 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:
Puntos mínimos de la función:
x1=−1.82508094381759x_{1} = -1.82508094381759
La función no tiene puntos máximos
Decrece en los intervalos
[−1.82508094381759,∞)\left[-1.82508094381759, \infty\right)
Crece en los intervalos
(−∞,−1.82508094381759]\left(-\infty, -1.82508094381759\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(2sin⁡(1x+π)+cos⁡(1x+π)x+π)x+π+2sin⁡(1x+π)(x+π)2=0\frac{- \frac{x \left(2 \sin{\left(\frac{1}{x + \pi} \right)} + \frac{\cos{\left(\frac{1}{x + \pi} \right)}}{x + \pi}\right)}{x + \pi} + 2 \sin{\left(\frac{1}{x + \pi} \right)}}{\left(x + \pi\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=26645.9391787033x_{1} = 26645.9391787033
x2=39288.7282737351x_{2} = 39288.7282737351
x3=19954.393431161x_{3} = 19954.393431161
x4=−38057.8502257135x_{4} = -38057.8502257135
x5=−23696.6524302943x_{5} = -23696.6524302943
x6=10486.4652374222x_{6} = 10486.4652374222
x7=−38903.6139021637x_{7} = -38903.6139021637
x8=24967.2334548888x_{8} = 24967.2334548888
x9=−27916.1316966692x_{9} = -27916.1316966692
x10=13423.642035784x_{10} = 13423.642035784
x11=−9434.73438768768x_{11} = -9434.73438768768
x12=−24540.1185862957x_{12} = -24540.1185862957
x13=11157.4958402703x_{13} = 11157.4958402703
x14=−12768.3836519539x_{14} = -12768.3836519539
x15=−20325.5929496416x_{15} = -20325.5929496416
x16=−11932.7709549707x_{16} = -11932.7709549707
x17=−15281.3735269617x_{17} = -15281.3735269617
x18=−37212.1608015591x_{18} = -37212.1608015591
x19=36755.5613612399x_{19} = 36755.5613612399
x20=27486.2421230335x_{20} = 27486.2421230335
x21=37599.7817299683x_{21} = 37599.7817299683
x22=40978.2691164883x_{22} = 40978.2691164883
x23=11884.4360261309x_{23} = 11884.4360261309
x24=15021.5580087522x_{24} = 15021.5580087522
x25=−39749.4475907453x_{25} = -39749.4475907453
x26=−13605.1179844857x_{26} = -13605.1179844857
x27=−34675.5859170572x_{27} = -34675.5859170572
x28=25806.2432025494x_{28} = 25806.2432025494
x29=30852.3093713367x_{29} = 30852.3093713367
x30=15832.9446243121x_{30} = 15832.9446243121
x31=43513.5217475525x_{31} = 43513.5217475525
x32=−32985.0025588812x_{32} = -32985.0025588812
x33=−10265.6907093232x_{33} = -10265.6907093232
x34=14217.5904924769x_{34} = 14217.5904924769
x35=−16120.6654585894x_{35} = -16120.6654585894
x36=19124.236720939x_{36} = 19124.236720939
x37=6.30675194335987x_{37} = 6.30675194335987
x38=17471.4447244252x_{38} = 17471.4447244252
x39=32537.6192625036x_{39} = 32537.6192625036
x40=−26227.7307994329x_{40} = -26227.7307994329
x41=20786.4943013504x_{41} = 20786.4943013504
x42=23291.6655954694x_{42} = 23291.6655954694
x43=35911.5282839706x_{43} = 35911.5282839706
x44=9693.61664396049x_{44} = 9693.61664396049
x45=12643.7282080722x_{45} = 12643.7282080722
x46=24129.0036813952x_{46} = 24129.0036813952
x47=34224.0899582447x_{47} = 34224.0899582447
x48=−36366.5502196385x_{48} = -36366.5502196385
x49=44358.8243250077x_{49} = 44358.8243250077
x50=9928.93120786112x_{50} = 9928.93120786112
x51=−16960.6109489304x_{51} = -16960.6109489304
x52=−18642.1759125762x_{52} = -18642.1759125762
x53=21620.2298931873x_{53} = 21620.2298931873
x54=−19483.6773869411x_{54} = -19483.6773869411
x55=−28760.5904369373x_{55} = -28760.5904369373
x56=41823.2365461285x_{56} = 41823.2365461285
x57=40133.429830421x_{57} = 40133.429830421
x58=29168.4086813793x_{58} = 29168.4086813793
x59=33380.7226809453x_{59} = 33380.7226809453
x60=−32139.8701482325x_{60} = -32139.8701482325
x61=−21167.8817409629x_{61} = -21167.8817409629
x62=31694.8051950957x_{62} = 31694.8051950957
x63=−29605.2043566423x_{63} = -29605.2043566423
x64=−17801.1355963518x_{64} = -17801.1355963518
x65=42668.3234037973x_{65} = 42668.3234037973
x66=−22010.5081175392x_{66} = -22010.5081175392
x67=35067.6985159432x_{67} = 35067.6985159432
x68=−27071.8405560765x_{68} = -27071.8405560765
x69=12620.2673129473x_{69} = 12620.2673129473
x70=−14442.8229306848x_{70} = -14442.8229306848
x71=−22853.4408439251x_{71} = -22853.4408439251
x72=−30449.9622347168x_{72} = -30449.9622347168
x73=45204.2245000187x_{73} = 45204.2245000187
x74=30010.1646721676x_{74} = 30010.1646721676
x75=18296.4176654701x_{75} = 18296.4176654701
x76=38444.1750140489x_{76} = 38444.1750140489
x77=−11098.4641907873x_{77} = -11098.4641907873
x78=−31294.8539077778x_{78} = -31294.8539077778
x79=16649.9871071842x_{79} = 16649.9871071842
x80=22455.3534208511x_{80} = 22455.3534208511
x81=−25383.8177679283x_{81} = -25383.8177679283
x82=28327.0845598711x_{82} = 28327.0845598711
x83=−33830.243481466x_{83} = -33830.243481466
x84=−35521.0234565255x_{84} = -35521.0234565255
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=−3.14159265358979x_{1} = -3.14159265358979

lim⁡x→−3.14159265358979−(−x(2sin⁡(1x+π)+cos⁡(1x+π)x+π)x+π+2sin⁡(1x+π)(x+π)2)=1(1.29006137732798π4sin⁡(13.14159265358979−1π)+0.0392131637166406π6sin⁡(13.14159265358979−1π)+3.81971863420549π2sin⁡(13.14159265358979−1π)+0.322515344331995π3cos⁡(13.14159265358979−1π)+1.59154943091895πcos⁡(13.14159265358979−1π)+0.00326776364305339π5cos⁡(13.14159265358979−1π)−1cos⁡(13.14159265358979−1π)−0.0513299112734217π4cos⁡(13.14159265358979−1π)−1.01321183642338π2cos⁡(13.14159265358979−1π)−2πsin⁡(13.14159265358979−1π)−0.00208032294659171π7sin⁡(13.14159265358979−1π)−3.03963550927013π3sin⁡(13.14159265358979−1π)−0.30797946764053π5sin⁡(13.14159265358979−1π))−40.1070456591576π4−2.70912889238876π6−113.097335529233π2−0.0294098727874805π8−31.0062766802998+0.00104016147329585π9+88.8264396098042π+0.369575361168636π7+84π3+12.7664691389346π5\lim_{x \to -3.14159265358979^-}\left(\frac{- \frac{x \left(2 \sin{\left(\frac{1}{x + \pi} \right)} + \frac{\cos{\left(\frac{1}{x + \pi} \right)}}{x + \pi}\right)}{x + \pi} + 2 \sin{\left(\frac{1}{x + \pi} \right)}}{\left(x + \pi\right)^{2}}\right) = \frac{1 \left(1.29006137732798 \pi^{4} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} + 0.0392131637166406 \pi^{6} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} + 3.81971863420549 \pi^{2} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} + 0.322515344331995 \pi^{3} \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} + 1.59154943091895 \pi \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} + 0.00326776364305339 \pi^{5} \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 1 \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 0.0513299112734217 \pi^{4} \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 1.01321183642338 \pi^{2} \cos{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 2 \pi \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 0.00208032294659171 \pi^{7} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 3.03963550927013 \pi^{3} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)} - 0.30797946764053 \pi^{5} \sin{\left(\frac{1}{3.14159265358979 - 1 \pi} \right)}\right)}{- 40.1070456591576 \pi^{4} - 2.70912889238876 \pi^{6} - 113.097335529233 \pi^{2} - 0.0294098727874805 \pi^{8} - 31.0062766802998 + 0.00104016147329585 \pi^{9} + 88.8264396098042 \pi + 0.369575361168636 \pi^{7} + 84 \pi^{3} + 12.7664691389346 \pi^{5}}
lim⁡x→−3.14159265358979+(−x(2sin⁡(1x+π)+cos⁡(1x+π)x+π)x+π+2sin⁡(1x+π)(x+π)2)=1(−1.29006137732798π4sin⁡(1−3.14159265358979+1π)−0.0392131637166406π6sin⁡(1−3.14159265358979+1π)−3.81971863420549π2sin⁡(1−3.14159265358979+1π)+0.322515344331995π3cos⁡(1−3.14159265358979+1π)+1.59154943091895πcos⁡(1−3.14159265358979+1π)+0.00326776364305339π5cos⁡(1−3.14159265358979+1π)−1cos⁡(1−3.14159265358979+1π)−0.0513299112734217π4cos⁡(1−3.14159265358979+1π)−1.01321183642338π2cos⁡(1−3.14159265358979+1π)+2πsin⁡(1−3.14159265358979+1π)+0.00208032294659171π7sin⁡(1−3.14159265358979+1π)+3.03963550927013π3sin⁡(1−3.14159265358979+1π)+0.30797946764053π5sin⁡(1−3.14159265358979+1π))−40.1070456591576π4−2.70912889238876π6−113.097335529233π2−0.0294098727874805π8−31.0062766802998+0.00104016147329585π9+88.8264396098042π+0.369575361168636π7+84π3+12.7664691389346π5\lim_{x \to -3.14159265358979^+}\left(\frac{- \frac{x \left(2 \sin{\left(\frac{1}{x + \pi} \right)} + \frac{\cos{\left(\frac{1}{x + \pi} \right)}}{x + \pi}\right)}{x + \pi} + 2 \sin{\left(\frac{1}{x + \pi} \right)}}{\left(x + \pi\right)^{2}}\right) = \frac{1 \left(- 1.29006137732798 \pi^{4} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} - 0.0392131637166406 \pi^{6} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} - 3.81971863420549 \pi^{2} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 0.322515344331995 \pi^{3} \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 1.59154943091895 \pi \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 0.00326776364305339 \pi^{5} \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} - 1 \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} - 0.0513299112734217 \pi^{4} \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} - 1.01321183642338 \pi^{2} \cos{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 2 \pi \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 0.00208032294659171 \pi^{7} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 3.03963550927013 \pi^{3} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)} + 0.30797946764053 \pi^{5} \sin{\left(\frac{1}{-3.14159265358979 + 1 \pi} \right)}\right)}{- 40.1070456591576 \pi^{4} - 2.70912889238876 \pi^{6} - 113.097335529233 \pi^{2} - 0.0294098727874805 \pi^{8} - 31.0062766802998 + 0.00104016147329585 \pi^{9} + 88.8264396098042 \pi + 0.369575361168636 \pi^{7} + 84 \pi^{3} + 12.7664691389346 \pi^{5}}
- los límites no son iguales, signo
x1=−3.14159265358979x_{1} = -3.14159265358979
- 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
(−∞,6.30675194335987]\left(-\infty, 6.30675194335987\right]
Convexa en los intervalos
[6.30675194335987,∞)\left[6.30675194335987, \infty\right)
Asíntotas verticales
Hay:
x1=−3.14159265358979x_{1} = -3.14159265358979
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xcos⁡(1x+π))=−∞\lim_{x \to -\infty}\left(x \cos{\left(\frac{1}{x + \pi} \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(xcos⁡(1x+π))=∞\lim_{x \to \infty}\left(x \cos{\left(\frac{1}{x + \pi} \right)}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*cos(1/(x + pi)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞cos⁡(1x+π)=1\lim_{x \to -\infty} \cos{\left(\frac{1}{x + \pi} \right)} = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xy = x
lim⁡x→∞cos⁡(1x+π)=1\lim_{x \to \infty} \cos{\left(\frac{1}{x + \pi} \right)} = 1
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xy = 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:
xcos⁡(1x+π)=−xcos⁡(1π−x)x \cos{\left(\frac{1}{x + \pi} \right)} = - x \cos{\left(\frac{1}{\pi - x} \right)}
- No
xcos⁡(1x+π)=xcos⁡(1π−x)x \cos{\left(\frac{1}{x + \pi} \right)} = x \cos{\left(\frac{1}{\pi - x} \right)}
- No
es decir, función
no es
par ni impar
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