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Trazado el gráfico de la función/ xsinx+(e^(-x)(-e)^x)/(e^(-x)+e^x)

Gráfico de la función y = xsinx+(e^(-x)(-e)^x)/(e^(-x)+e^x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   -x     x
                  E  *(-E) 
f(x) = x*sin(x) + ---------
                    -x    x
                   E   + E
f(x)=xsin⁡(x)+e−x(−e)xex+e−xf{\left(x \right)} = x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}} 
f = x*sin(x) + (E^(-x)*(-E)^x)/(E^x + E^(-x))
Gráfico de la función
02468-8-6-4-2-1010-5.45-5.43
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:
xsin⁡(x)+e−x(−e)xex+e−x=0x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−25.1327412287188x_{2} = -25.1327412287188
x3=−91.106186954104x_{3} = -91.106186954104
x4=9.42477595605707x_{4} = 9.42477595605707
x5=−72.2566310325652x_{5} = -72.2566310325652
x6=−15.7079632737806x_{6} = -15.7079632737806
x7=18.8495559218464x_{7} = 18.8495559218464
x8=34.5575191894877x_{8} = 34.5575191894877
x9=72.2566310325652x_{9} = 72.2566310325652
x10=−56.5486677646163x_{10} = -56.5486677646163
x11=37.6991118430775x_{11} = 37.6991118430775
x12=69.1150383789755x_{12} = 69.1150383789755
x13=87.9645943005142x_{13} = 87.9645943005142
x14=−87.9645943005142x_{14} = -87.9645943005142
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=43.9822971502571x_{16} = 43.9822971502571
x17=50.2654824574367x_{17} = 50.2654824574367
x18=−69.1150383789755x_{18} = -69.1150383789755
x19=−18.8495559218464x_{19} = -18.8495559218464
x20=12.5663706718052x_{20} = 12.5663706718052
x21=56.5486677646163x_{21} = 56.5486677646163
x22=59.6902604182061x_{22} = 59.6902604182061
x23=75.398223686155x_{23} = 75.398223686155
x24=−59.6902604182061x_{24} = -59.6902604182061
x25=−329.867228626928x_{25} = -329.867228626928
x26=−34.5575191894877x_{26} = -34.5575191894877
x27=−12.5663706718052x_{27} = -12.5663706718052
x28=40.8407044966673x_{28} = 40.8407044966673
x29=81.6814089933346x_{29} = 81.6814089933346
x30=53.4070751110265x_{30} = 53.4070751110265
x31=−78.5398163397448x_{31} = -78.5398163397448
x32=−81.6814089933346x_{32} = -81.6814089933346
x33=−65.9734457253857x_{33} = -65.9734457253857
x34=−94.2477796076938x_{34} = -94.2477796076938
x35=91.106186954104x_{35} = 91.106186954104
x36=−97.3893722612836x_{36} = -97.3893722612836
x37=−100.530964914873x_{37} = -100.530964914873
x38=−43.9822971502571x_{38} = -43.9822971502571
x39=−31.4159265358979x_{39} = -31.4159265358979
x40=31.4159265358979x_{40} = 31.4159265358979
x41=100.530964914873x_{41} = 100.530964914873
x42=−53.4070751110265x_{42} = -53.4070751110265
x43=−84.8230016469244x_{43} = -84.8230016469244
x44=−75.398223686155x_{44} = -75.398223686155
x45=84.8230016469244x_{45} = 84.8230016469244
x46=97.3893722612836x_{46} = 97.3893722612836
x47=62.8318530717959x_{47} = 62.8318530717959
x48=−21.9911485751413x_{48} = -21.9911485751413
x49=−40.8407044966673x_{49} = -40.8407044966673
x50=25.1327412287188x_{50} = 25.1327412287188
x51=21.9911485751413x_{51} = 21.9911485751413
x52=94.2477796076938x_{52} = 94.2477796076938
x53=−28.2743338823082x_{53} = -28.2743338823082
x54=697.433569096934x_{54} = 697.433569096934
x55=−50.2654824574367x_{55} = -50.2654824574367
x56=15.7079632737806x_{56} = 15.7079632737806
x57=−47.1238898038469x_{57} = -47.1238898038469
x58=78.5398163397448x_{58} = 78.5398163397448
x59=47.1238898038469x_{59} = 47.1238898038469
x60=−9.42477595605707x_{60} = -9.42477595605707
x61=28.2743338823082x_{61} = 28.2743338823082
x62=−37.6991118430775x_{62} = -37.6991118430775
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*sin(x) + (E^(-x)*(-E)^x)/(E^(-x) + E^x).
0sin⁡(0)+e−0(−e)0e−0+e00 \sin{\left(0 \right)} + \frac{e^{- 0} \left(- e\right)^{0}}{e^{- 0} + e^{0}}
Resultado:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
Punto:
(0, 1/2)
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
xcos⁡(x)+(−e)x(−ex+e−x)e−x(ex+e−x)2+sin⁡(x)+−(−e)xe−x+(−e)x(1+iπ)e−xex+e−x=0x \cos{\left(x \right)} + \frac{\left(- e\right)^{x} \left(- e^{x} + e^{- x}\right) e^{- x}}{\left(e^{x} + e^{- x}\right)^{2}} + \sin{\left(x \right)} + \frac{- \left(- e\right)^{x} e^{- x} + \left(- e\right)^{x} \left(1 + i \pi\right) e^{- x}}{e^{x} + e^{- x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−64.4181717218392x_{1} = -64.4181717218392
x2=76.9820093304187x_{2} = 76.9820093304187
x3=−86.4053708116885x_{3} = -86.4053708116885
x4=80.1230928148503x_{4} = 80.1230928148503
x5=−42.4350618814099x_{5} = -42.4350618814099
x6=−95.8290108090195x_{6} = -95.8290108090195
x7=−61.2773745335697x_{7} = -61.2773745335697
x8=89.5465575382492x_{8} = 89.5465575382492
x9=−73.8409691490209x_{9} = -73.8409691490209
x10=98.9702722883957x_{10} = 98.9702722883957
x11=−45.57503179559x_{11} = -45.57503179559
x12=73.8409691490209x_{12} = 73.8409691490209
x13=51.855560729152x_{13} = 51.855560729152
x14=54.9960525574964x_{14} = 54.9960525574964
x15=83.2642147040886x_{15} = 83.2642147040886
x16=92.687771772017x_{16} = 92.687771772017
x17=−83.2642147040886x_{17} = -83.2642147040886
x18=−67.5590428388084x_{18} = -67.5590428388084
x19=−92.687771772017x_{19} = -92.687771772017
x20=−76.9820093304187x_{20} = -76.9820093304187
x21=−89.5465575382492x_{21} = -89.5465575382492
x22=−48.7152107175577x_{22} = -48.7152107175577
x23=102.111554139654x_{23} = 102.111554139654
x24=42.4350618814099x_{24} = 42.4350618814099
x25=−70.69997803861x_{25} = -70.69997803861
x26=45.57503179559x_{26} = 45.57503179559
x27=64.4181717218392x_{27} = 64.4181717218392
x28=−51.855560729152x_{28} = -51.855560729152
x29=−54.9960525574964x_{29} = -54.9960525574964
x30=48.7152107175577x_{30} = 48.7152107175577
x31=95.8290108090195x_{31} = 95.8290108090195
x32=70.69997803861x_{32} = 70.69997803861
x33=−80.1230928148503x_{33} = -80.1230928148503
x34=67.5590428388084x_{34} = 67.5590428388084
x35=−58.1366632448992x_{35} = -58.1366632448992
x36=86.4053708116885x_{36} = 86.4053708116885
x37=61.2773745335697x_{37} = 61.2773745335697
x38=58.1366632448992x_{38} = 58.1366632448992
x39=−98.9702722883957x_{39} = -98.9702722883957
Signos de extremos en los puntos:
(-64.41817172183916, 64.4104113393753 - 1.02101539286205e-28*I)

(76.98200933041872, 76.9755151282637 + 2.08499057569284e-35*I)

(-86.40537081168854, -86.3995847156108 - 2.85195995242999e-38*I)

(80.12309281485025, -80.1168531456592 + 6.01847913110505e-36*I)

(-42.43506188140989, -42.4232840772591 - 3.64406221681416e-19*I)

(-95.82901080901948, 95.8237936084657 + 1.23315182610757e-42*I)

(-61.277374533569656, -61.2692165444766 + 1.86795364076796e-27*I)

(89.54655753824919, 89.5409743728852 - 1.27573952608076e-39*I)

(-73.8409691490209, -73.8341987715416 + 4.08964550506982e-33*I)

(98.9702722883957, -98.9652206531187 + 9.71486370052824e-45*I)

(-45.57503179559002, 45.5640648360268 + 1.56616053871094e-20*I)

(73.8409691490209, -73.8341987715416 - 4.08964550506982e-33*I)

(51.85556072915197, 51.8459212502015 - 1.32203699404896e-23*I)

(54.99605255749639, -54.9869632496976 + 1.61797721607199e-26*I)

(83.26421470408864, 83.2582103729533 - 5.09153602460438e-37*I)

(92.687771772017, -92.6823777880592 + 4.63244820243205e-41*I)

(-83.26421470408864, 83.2582103729533 + 5.09153602460438e-37*I)

(-67.5590428388084, -67.5516431209725 + 4.48710347494403e-30*I)

(-92.687771772017, -92.6823777880592 - 4.63244820243205e-41*I)

(-76.98200933041872, 76.9755151282637 - 2.08499057569284e-35*I)

(-89.54655753824919, 89.5409743728852 + 1.27573952608076e-39*I)

(-48.715210717557724, -48.7049502253679 - 5.43696449694713e-22*I)

(102.11155413965392, 102.106657886316 + 1.54605283051173e-45*I)

(42.43506188140989, -42.4232840772591 + 3.64406221681416e-19*I)

(-70.69997803861, 70.6929069615931 - 1.59723596255949e-31*I)

(45.57503179559002, 45.5640648360268 - 1.56616053871094e-20*I)

(64.41817172183916, 64.4104113393753 + 1.02101539286205e-28*I)

(-51.85556072915197, 51.8459212502015 + 1.32203699404896e-23*I)

(-54.99605255749639, -54.9869632496976 - 1.61797721607199e-26*I)

(48.715210717557724, -48.7049502253679 + 5.43696449694713e-22*I)

(95.82901080901948, 95.8237936084657 - 1.23315182610757e-42*I)

(70.69997803861, 70.6929069615931 + 1.59723596255949e-31*I)

(-80.12309281485025, -80.1168531456592 - 6.01847913110505e-36*I)

(67.5590428388084, -67.5516431209725 - 4.48710347494403e-30*I)

(-58.13666324489916, 58.1280647280857 - 2.34933057997456e-26*I)

(86.40537081168854, -86.3995847156108 + 2.85195995242999e-38*I)

(61.277374533569656, -61.2692165444766 - 1.86795364076796e-27*I)

(58.13666324489916, 58.1280647280857 + 2.34933057997456e-26*I)

(-98.9702722883957, -98.9652206531187 - 9.71486370052824e-45*I)


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=−86.4053708116885x_{1} = -86.4053708116885
x2=80.1230928148503x_{2} = 80.1230928148503
x3=−42.4350618814099x_{3} = -42.4350618814099
x4=−61.2773745335697x_{4} = -61.2773745335697
x5=−73.8409691490209x_{5} = -73.8409691490209
x6=98.9702722883957x_{6} = 98.9702722883957
x7=73.8409691490209x_{7} = 73.8409691490209
x8=54.9960525574964x_{8} = 54.9960525574964
x9=92.687771772017x_{9} = 92.687771772017
x10=−67.5590428388084x_{10} = -67.5590428388084
x11=−92.687771772017x_{11} = -92.687771772017
x12=−48.7152107175577x_{12} = -48.7152107175577
x13=42.4350618814099x_{13} = 42.4350618814099
x14=−54.9960525574964x_{14} = -54.9960525574964
x15=48.7152107175577x_{15} = 48.7152107175577
x16=−80.1230928148503x_{16} = -80.1230928148503
x17=67.5590428388084x_{17} = 67.5590428388084
x18=86.4053708116885x_{18} = 86.4053708116885
x19=61.2773745335697x_{19} = 61.2773745335697
x20=−98.9702722883957x_{20} = -98.9702722883957
Puntos máximos de la función:
x20=−64.4181717218392x_{20} = -64.4181717218392
x20=76.9820093304187x_{20} = 76.9820093304187
x20=−95.8290108090195x_{20} = -95.8290108090195
x20=89.5465575382492x_{20} = 89.5465575382492
x20=−45.57503179559x_{20} = -45.57503179559
x20=51.855560729152x_{20} = 51.855560729152
x20=83.2642147040886x_{20} = 83.2642147040886
x20=−83.2642147040886x_{20} = -83.2642147040886
x20=−76.9820093304187x_{20} = -76.9820093304187
x20=−89.5465575382492x_{20} = -89.5465575382492
x20=102.111554139654x_{20} = 102.111554139654
x20=−70.69997803861x_{20} = -70.69997803861
x20=45.57503179559x_{20} = 45.57503179559
x20=64.4181717218392x_{20} = 64.4181717218392
x20=−51.855560729152x_{20} = -51.855560729152
x20=95.8290108090195x_{20} = 95.8290108090195
x20=70.69997803861x_{20} = 70.69997803861
x20=−58.1366632448992x_{20} = -58.1366632448992
x20=58.1366632448992x_{20} = 58.1366632448992
Decrece en los intervalos
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Crece en los intervalos
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\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
−xsin⁡(x)+2(−e)x(ex−e−x)2e−x(ex+e−x)3+(−e)x(ex−e−x)e−x(ex+e−x)2−(−e)x(1+iπ)(ex−e−x)e−x(ex+e−x)2−iπ(−e)x(ex−e−x)e−x(ex+e−x)2−(−e)xe−xex+e−x+(−e)x(−1−2iπ+(1+iπ)2)e−xex+e−x+2cos⁡(x)=0- x \sin{\left(x \right)} + \frac{2 \left(- e\right)^{x} \left(e^{x} - e^{- x}\right)^{2} e^{- x}}{\left(e^{x} + e^{- x}\right)^{3}} + \frac{\left(- e\right)^{x} \left(e^{x} - e^{- x}\right) e^{- x}}{\left(e^{x} + e^{- x}\right)^{2}} - \frac{\left(- e\right)^{x} \left(1 + i \pi\right) \left(e^{x} - e^{- x}\right) e^{- x}}{\left(e^{x} + e^{- x}\right)^{2}} - \frac{i \pi \left(- e\right)^{x} \left(e^{x} - e^{- x}\right) e^{- x}}{\left(e^{x} + e^{- x}\right)^{2}} - \frac{\left(- e\right)^{x} e^{- x}}{e^{x} + e^{- x}} + \frac{\left(- e\right)^{x} \left(-1 - 2 i \pi + \left(1 + i \pi\right)^{2}\right) e^{- x}}{e^{x} + e^{- x}} + 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−72.2842925036825x_{1} = -72.2842925036825
x2=−56.5839987378634x_{2} = -56.5839987378634
x3=62.863657228703x_{3} = 62.863657228703
x4=78.5652673845995x_{4} = 78.5652673845995
x5=−84.8465692433091x_{5} = -84.8465692433091
x6=−59.7237354324305x_{6} = -59.7237354324305
x7=47.1662676027767x_{7} = 47.1662676027767
x8=94.2689923093066x_{8} = 94.2689923093066
x9=−78.5652673845995x_{9} = -78.5652673845995
x10=100.550852725424x_{10} = 100.550852725424
x11=−62.863657228703x_{11} = -62.863657228703
x12=−69.1439554764926x_{12} = -69.1439554764926
x13=−47.1662676027767x_{13} = -47.1662676027767
x14=−128.820822990274x_{14} = -128.820822990274
x15=−53.4444796697636x_{15} = -53.4444796697636
x16=69.1439554764926x_{16} = 69.1439554764926
x17=−87.9873209346887x_{17} = -87.9873209346887
x18=−100.550852725424x_{18} = -100.550852725424
x19=−81.7058821480364x_{19} = -81.7058821480364
x20=91.1281305511393x_{20} = 91.1281305511393
x21=72.2842925036825x_{21} = 72.2842925036825
x22=87.9873209346887x_{22} = 87.9873209346887
x23=−94.2689923093066x_{23} = -94.2689923093066
x24=−91.1281305511393x_{24} = -91.1281305511393
x25=75.4247339745236x_{25} = 75.4247339745236
x26=−97.4099011706723x_{26} = -97.4099011706723
x27=−66.0037377708277x_{27} = -66.0037377708277
x28=44.0276918992479x_{28} = 44.0276918992479
x29=59.7237354324305x_{29} = 59.7237354324305
x30=84.8465692433091x_{30} = 84.8465692433091
x31=66.0037377708277x_{31} = 66.0037377708277
x32=−75.4247339745236x_{32} = -75.4247339745236
x33=−44.0276918992479x_{33} = -44.0276918992479
x34=53.4444796697636x_{34} = 53.4444796697636
x35=−50.3052188363296x_{35} = -50.3052188363296
x36=81.7058821480364x_{36} = 81.7058821480364
x37=50.3052188363296x_{37} = 50.3052188363296
x38=97.4099011706723x_{38} = 97.4099011706723
x39=56.5839987378634x_{39} = 56.5839987378634

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
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Convexa en los intervalos
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(xsin⁡(x)+e−x(−e)xex+e−x)\lim_{x \to -\infty}\left(x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(xsin⁡(x)+e−x(−e)xex+e−x)\lim_{x \to \infty}\left(x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(x) + (E^(-x)*(-E)^x)/(E^(-x) + E^x), dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(xsin⁡(x)+e−x(−e)xex+e−xx)\lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}}}{x}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(xsin⁡(x)+e−x(−e)xex+e−xx)\lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}}}{x}\right)
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:
xsin⁡(x)+e−x(−e)xex+e−x=xsin⁡(x)+(−e)−xexex+e−xx \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}} = x \sin{\left(x \right)} + \frac{\left(- e\right)^{- x} e^{x}}{e^{x} + e^{- x}}
- No
xsin⁡(x)+e−x(−e)xex+e−x=−xsin⁡(x)−(−e)−xexex+e−xx \sin{\left(x \right)} + \frac{e^{- x} \left(- e\right)^{x}}{e^{x} + e^{- x}} = - x \sin{\left(x \right)} - \frac{\left(- e\right)^{- x} e^{x}}{e^{x} + e^{- x}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор