Sr Examen

Otras calculadoras

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=−50.2654824574367x_{1} = -50.2654824574367
x2=40.8407044966673x_{2} = 40.8407044966673
x3=62.8318530717959x_{3} = 62.8318530717959
x4=−12.5663706718052x_{4} = -12.5663706718052
x5=−47.1238898038469x_{5} = -47.1238898038469
x6=21.9911485751413x_{6} = 21.9911485751413
x7=−87.9645943005142x_{7} = -87.9645943005142
x8=31.4159265358979x_{8} = 31.4159265358979
x9=−9.42477595605707x_{9} = -9.42477595605707
x10=−65.9734457253857x_{10} = -65.9734457253857
x11=−34.5575191894877x_{11} = -34.5575191894877
x12=87.9645943005142x_{12} = 87.9645943005142
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=56.5486677646163x_{14} = 56.5486677646163
x15=−94.2477796076938x_{15} = -94.2477796076938
x16=97.3893722612836x_{16} = 97.3893722612836
x17=94.2477796076938x_{17} = 94.2477796076938
x18=28.2743338823082x_{18} = 28.2743338823082
x19=−25.1327412287188x_{19} = -25.1327412287188
x20=−329.867228626928x_{20} = -329.867228626928
x21=9.42477595605707x_{21} = 9.42477595605707
x22=−81.6814089933346x_{22} = -81.6814089933346
x23=−75.398223686155x_{23} = -75.398223686155
x24=18.8495559218464x_{24} = 18.8495559218464
x25=−62.8318530717959x_{25} = -62.8318530717959
x26=50.2654824574367x_{26} = 50.2654824574367
x27=−15.7079632737806x_{27} = -15.7079632737806
x28=25.1327412287188x_{28} = 25.1327412287188
x29=−97.3893722612836x_{29} = -97.3893722612836
x30=−43.9822971502571x_{30} = -43.9822971502571
x31=100.530964914873x_{31} = 100.530964914873
x32=−59.6902604182061x_{32} = -59.6902604182061
x33=69.1150383789755x_{33} = 69.1150383789755
x34=47.1238898038469x_{34} = 47.1238898038469
x35=−72.2566310325652x_{35} = -72.2566310325652
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=75.398223686155x_{37} = 75.398223686155
x38=−100.530964914873x_{38} = -100.530964914873
x39=72.2566310325652x_{39} = 72.2566310325652
x40=43.9822971502571x_{40} = 43.9822971502571
x41=697.433569096934x_{41} = 697.433569096934
x42=−84.8230016469244x_{42} = -84.8230016469244
x43=53.4070751110265x_{43} = 53.4070751110265
x44=−91.106186954104x_{44} = -91.106186954104
x45=−18.8495559218464x_{45} = -18.8495559218464
x46=−78.5398163397448x_{46} = -78.5398163397448
x47=15.7079632737806x_{47} = 15.7079632737806
x48=−37.6991118430775x_{48} = -37.6991118430775
x49=59.6902604182061x_{49} = 59.6902604182061
x50=78.5398163397448x_{50} = 78.5398163397448
x51=12.5663706718052x_{51} = 12.5663706718052
x52=81.6814089933346x_{52} = 81.6814089933346
x53=65.9734457253857x_{53} = 65.9734457253857
x54=−21.9911485751413x_{54} = -21.9911485751413
x55=−28.2743338823082x_{55} = -28.2743338823082
x56=−40.8407044966673x_{56} = -40.8407044966673
x57=−31.4159265358979x_{57} = -31.4159265358979
x58=−69.1150383789755x_{58} = -69.1150383789755
x59=37.6991118430775x_{59} = 37.6991118430775
x60=34.5575191894877x_{60} = 34.5575191894877
x61=91.106186954104x_{61} = 91.106186954104
x62=84.8230016469244x_{62} = 84.8230016469244
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=42.4350618814099x_{1} = 42.4350618814099
x2=−83.2642147040886x_{2} = -83.2642147040886
x3=−92.687771772017x_{3} = -92.687771772017
x4=48.7152107175577x_{4} = 48.7152107175577
x5=−95.8290108090195x_{5} = -95.8290108090195
x6=83.2642147040886x_{6} = 83.2642147040886
x7=−64.4181717218392x_{7} = -64.4181717218392
x8=−58.1366632448992x_{8} = -58.1366632448992
x9=102.111554139654x_{9} = 102.111554139654
x10=−70.69997803861x_{10} = -70.69997803861
x11=92.687771772017x_{11} = 92.687771772017
x12=54.9960525574964x_{12} = 54.9960525574964
x13=−42.4350618814099x_{13} = -42.4350618814099
x14=76.9820093304187x_{14} = 76.9820093304187
x15=−54.9960525574964x_{15} = -54.9960525574964
x16=45.57503179559x_{16} = 45.57503179559
x17=−80.1230928148503x_{17} = -80.1230928148503
x18=61.2773745335697x_{18} = 61.2773745335697
x19=−98.9702722883957x_{19} = -98.9702722883957
x20=80.1230928148503x_{20} = 80.1230928148503
x21=−61.2773745335697x_{21} = -61.2773745335697
x22=86.4053708116885x_{22} = 86.4053708116885
x23=64.4181717218392x_{23} = 64.4181717218392
x24=−89.5465575382492x_{24} = -89.5465575382492
x25=73.8409691490209x_{25} = 73.8409691490209
x26=−73.8409691490209x_{26} = -73.8409691490209
x27=−51.855560729152x_{27} = -51.855560729152
x28=−86.4053708116885x_{28} = -86.4053708116885
x29=−48.7152107175577x_{29} = -48.7152107175577
x30=−67.5590428388084x_{30} = -67.5590428388084
x31=67.5590428388084x_{31} = 67.5590428388084
x32=51.855560729152x_{32} = 51.855560729152
x33=−76.9820093304187x_{33} = -76.9820093304187
x34=−45.57503179559x_{34} = -45.57503179559
x35=58.1366632448992x_{35} = 58.1366632448992
x36=89.5465575382492x_{36} = 89.5465575382492
x37=98.9702722883957x_{37} = 98.9702722883957
x38=70.69997803861x_{38} = 70.69997803861
x39=95.8290108090195x_{39} = 95.8290108090195
Signos de extremos en los puntos:
(42.43506188140989, -42.4232840772591 + 3.64406221681416e-19*I)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(95.82901080901948, 95.8237936084657 - 1.23315182610757e-42*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=42.4350618814099x_{1} = 42.4350618814099
x2=−92.687771772017x_{2} = -92.687771772017
x3=48.7152107175577x_{3} = 48.7152107175577
x4=92.687771772017x_{4} = 92.687771772017
x5=54.9960525574964x_{5} = 54.9960525574964
x6=−42.4350618814099x_{6} = -42.4350618814099
x7=−54.9960525574964x_{7} = -54.9960525574964
x8=−80.1230928148503x_{8} = -80.1230928148503
x9=61.2773745335697x_{9} = 61.2773745335697
x10=−98.9702722883957x_{10} = -98.9702722883957
x11=80.1230928148503x_{11} = 80.1230928148503
x12=−61.2773745335697x_{12} = -61.2773745335697
x13=86.4053708116885x_{13} = 86.4053708116885
x14=73.8409691490209x_{14} = 73.8409691490209
x15=−73.8409691490209x_{15} = -73.8409691490209
x16=−86.4053708116885x_{16} = -86.4053708116885
x17=−48.7152107175577x_{17} = -48.7152107175577
x18=−67.5590428388084x_{18} = -67.5590428388084
x19=67.5590428388084x_{19} = 67.5590428388084
x20=98.9702722883957x_{20} = 98.9702722883957
Puntos máximos de la función:
x20=−83.2642147040886x_{20} = -83.2642147040886
x20=−95.8290108090195x_{20} = -95.8290108090195
x20=83.2642147040886x_{20} = 83.2642147040886
x20=−64.4181717218392x_{20} = -64.4181717218392
x20=−58.1366632448992x_{20} = -58.1366632448992
x20=102.111554139654x_{20} = 102.111554139654
x20=−70.69997803861x_{20} = -70.69997803861
x20=76.9820093304187x_{20} = 76.9820093304187
x20=45.57503179559x_{20} = 45.57503179559
x20=64.4181717218392x_{20} = 64.4181717218392
x20=−89.5465575382492x_{20} = -89.5465575382492
x20=−51.855560729152x_{20} = -51.855560729152
x20=51.855560729152x_{20} = 51.855560729152
x20=−76.9820093304187x_{20} = -76.9820093304187
x20=−45.57503179559x_{20} = -45.57503179559
x20=58.1366632448992x_{20} = 58.1366632448992
x20=89.5465575382492x_{20} = 89.5465575382492
x20=70.69997803861x_{20} = 70.69997803861
x20=95.8290108090195x_{20} = 95.8290108090195
Decrece en los intervalos
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Crece en los intervalos
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\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