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Trazado el gráfico de la función/ -exp(-x*cos(x)+sin(x))

Gráfico de la función y = -exp(-x*cos(x)+sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         -x*cos(x) + sin(x)
f(x) = -e
f(x)=−e−xcos⁡(x)+sin⁡(x)f{\left(x \right)} = - e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} 
f = -exp((-x)*cos(x) + sin(x))
Gráfico de la función
02468-8-6-4-2-1010-2000010000
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:
−e−xcos⁡(x)+sin⁡(x)=0- e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 -exp((-x)*cos(x) + sin(x)).
−e−0cos⁡(0)+sin⁡(0)- e^{- 0 \cos{\left(0 \right)} + \sin{\left(0 \right)}}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
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
−xe−xcos⁡(x)+sin⁡(x)sin⁡(x)=0- x e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−15.707963267949x_{1} = -15.707963267949
x2=0x_{2} = 0
x3=−91.1998699041465x_{3} = -91.1998699041465
x4=81.6814089933346x_{4} = 81.6814089933346
x5=−72.2566310325652x_{5} = -72.2566310325652
x6=−34.9294771386704x_{6} = -34.9294771386704
x7=50.2654824574367x_{7} = 50.2654824574367
x8=69.5606259232977x_{8} = 69.5606259232977
x9=−40.8417340899375x_{9} = -40.8417340899375
x10=25.1327412287183x_{10} = 25.1327412287183
x11=−65.9734457253857x_{11} = -65.9734457253857
x12=−53.4070751110265x_{12} = -53.4070751110265
x13=−59.6902604182061x_{13} = -59.6902604182061
x14=74.5739798656666x_{14} = 74.5739798656666
x15=−46.5524448088427x_{15} = -46.5524448088427
x16=18.8495559215388x_{16} = 18.8495559215388
x17=80.5118670544782x_{17} = 80.5118670544782
x18=63.6359781723457x_{18} = 63.6359781723457
x19=12.5663706143592x_{19} = 12.5663706143592
x20=56.5486677646163x_{20} = 56.5486677646163
x21=−21.9911485751286x_{21} = -21.9911485751286
x22=6.28318530717959x_{22} = 6.28318530717959
x23=−78.5398163397448x_{23} = -78.5398163397448
x24=37.6991118430775x_{24} = 37.6991118430775
x25=−96.2124947519558x_{25} = -96.2124947519558
x26=21.9911485751286x_{26} = 21.9911485751286
x27=−97.3893722612836x_{27} = -97.3893722612836
x28=100.530964914873x_{28} = 100.530964914873
x29=−47.1238898038469x_{29} = -47.1238898038469
x30=28.2743338823081x_{30} = 28.2743338823081
x31=94.2477796076938x_{31} = 94.2477796076938
x32=−40.8407044966673x_{32} = -40.8407044966673
x33=−34.5575191894877x_{33} = -34.5575191894877
x34=−28.2743338823081x_{34} = -28.2743338823081
x35=−85.4121130834002x_{35} = -85.4121130834002
x36=−90.260314866971x_{36} = -90.260314866971
x37=−34.9842404193317x_{37} = -34.9842404193317
x38=−79.4606661258204x_{38} = -79.4606661258204
x39=43.9822971502571x_{39} = 43.9822971502571
x40=75.398223686155x_{40} = 75.398223686155
x41=62.8318530717959x_{41} = 62.8318530717959
x42=−84.3813701398482x_{42} = -84.3813701398482
x43=−3.14159265358979x_{43} = -3.14159265358979
x44=68.7396225684591x_{44} = 68.7396225684591
x45=87.9645943005142x_{45} = 87.9645943005142
x46=−9.42477796076938x_{46} = -9.42477796076938
x47=31.4159265358979x_{47} = 31.4159265358979
x48=31.2527380956438x_{48} = 31.2527380956438
Signos de extremos en los puntos:
(-15.707963267948966, -1.50701727539007e-7)

(0, -1)

(-91.19986990414654, -4.04303150065854e-40)

(81.68140899333463, -3.35903709639911e-36)

(-72.25663103256524, -4.16240046723054e-32)

(-34.92947713867037, -1.06047627266875e-14)

(50.26548245743669, -1.47903461596179e-22)

(69.56062592329772, -8.45545260626203e-28)

(-40.84173408993747, -1.83280728040168e-18)

(25.132741228718345, -1.21615567094093e-11)

(-65.97344572538566, -2.22893071715432e-29)

(-53.40707511102649, -6.39148810034623e-24)

(-59.69026041820607, -1.19357379977897e-26)

(74.57397986566656, -4.86137607527994e-23)

(-46.55244480884272, -5.75158728239587e-18)

(18.84955592153876, -6.51241213607991e-9)

(80.51186705447822, -8.77942288424734e-15)

(63.63597817234566, -1.38045601340376e-19)

(12.566370614359172, -3.487342356209e-6)

(56.548667764616276, -2.76201244352236e-25)

(-21.991148575128552, -2.81426845748556e-10)

(6.283185307179586, -0.00186744273170799)

(-78.53981633974483, -7.77304449898755e-35)

(37.69911184307752, -4.24115118301608e-17)

(-96.21249475195584, -3.64447893646061e-17)

(21.991148575128552, -3553321280.84704)

(-97.3893722612836, -5.06212693294953e-43)

(100.53096491487338, -2.18754339521324e-44)

(-47.1238898038469, -3.42258854412123e-21)

(28.274333882308138, -1902773895292.16)

(94.2477796076938, -1.17141123423499e-41)

(-40.840704496667314, -1.83276760567157e-18)

(-34.55751918948773, -9.8143175935323e-16)

(-28.274333882308138, -5.25548517600645e-13)

(-85.41211308340019, -2.51221442358378e-31)

(-90.26031486697102, -4.81411641652805e-27)

(-34.984240419331734, -2.23213156812239e-14)

(-79.46066612582041, -2.90162212518714e-21)

(43.982297150257104, -7.92010695079813e-20)

(75.39822368615503, -1.79873633571987e-33)

(62.83185307179586, -5.15790006254285e-28)

(-84.3813701398482, -4.8309026479408e-34)

(-3.141592653589793, -0.0432139182637723)

(68.73962256845907, -1.1658553642877e-28)

(87.96459430051421, -6.27280941120808e-39)

(-9.42477796076938, -8.06995175703046e-5)

(31.41592653589793, -2.2711010683241e-14)

(31.25273809564381, -3.44251713837872e-14)


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=21.9911485751286x_{1} = 21.9911485751286
x2=28.2743338823081x_{2} = 28.2743338823081
Puntos máximos de la función:
x2=−15.707963267949x_{2} = -15.707963267949
x2=81.6814089933346x_{2} = 81.6814089933346
x2=−72.2566310325652x_{2} = -72.2566310325652
x2=50.2654824574367x_{2} = 50.2654824574367
x2=25.1327412287183x_{2} = 25.1327412287183
x2=−65.9734457253857x_{2} = -65.9734457253857
x2=−53.4070751110265x_{2} = -53.4070751110265
x2=−59.6902604182061x_{2} = -59.6902604182061
x2=18.8495559215388x_{2} = 18.8495559215388
x2=12.5663706143592x_{2} = 12.5663706143592
x2=56.5486677646163x_{2} = 56.5486677646163
x2=−21.9911485751286x_{2} = -21.9911485751286
x2=6.28318530717959x_{2} = 6.28318530717959
x2=−78.5398163397448x_{2} = -78.5398163397448
x2=37.6991118430775x_{2} = 37.6991118430775
x2=−97.3893722612836x_{2} = -97.3893722612836
x2=100.530964914873x_{2} = 100.530964914873
x2=−47.1238898038469x_{2} = -47.1238898038469
x2=94.2477796076938x_{2} = 94.2477796076938
x2=−40.8407044966673x_{2} = -40.8407044966673
x2=−34.5575191894877x_{2} = -34.5575191894877
x2=−28.2743338823081x_{2} = -28.2743338823081
x2=43.9822971502571x_{2} = 43.9822971502571
x2=75.398223686155x_{2} = 75.398223686155
x2=62.8318530717959x_{2} = 62.8318530717959
x2=−3.14159265358979x_{2} = -3.14159265358979
x2=87.9645943005142x_{2} = 87.9645943005142
x2=−9.42477796076938x_{2} = -9.42477796076938
x2=31.4159265358979x_{2} = 31.4159265358979
Decrece en los intervalos
[28.2743338823081,∞)\left[28.2743338823081, \infty\right)
Crece en los intervalos
(−∞,21.9911485751286]\left(-\infty, 21.9911485751286\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
−(x2sin⁡2(x)+xcos⁡(x)+sin⁡(x))e−xcos⁡(x)+sin⁡(x)=0- \left(x^{2} \sin^{2}{\left(x \right)} + x \cos{\left(x \right)} + \sin{\left(x \right)}\right) e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−59.6402640011949x_{1} = -59.6402640011949
x2=−1.23267117887385x_{2} = -1.23267117887385
x3=74.5705047152689x_{3} = 74.5705047152689
x4=−66.0202928956355x_{4} = -66.0202928956355
x5=−5.89130279784936x_{5} = -5.89130279784936
x6=0x_{6} = 0
x7=15.9589298351923x_{7} = 15.9589298351923
x8=−91.2497626387973x_{8} = -91.2497626387973
x9=22.2035816252474x_{9} = 22.2035816252474
x10=93.4553424019354x_{10} = 93.4553424019354
x11=−34.6022901645322x_{11} = -34.6022901645322
x12=−84.3697900389548x_{12} = -84.3697900389548
x13=−96.2118616154068x_{13} = -96.2118616154068
x14=−40.8367181903417x_{14} = -40.8367181903417
x15=69.5754429784409x_{15} = 69.5754429784409
x16=−40.8091574400446x_{16} = -40.8091574400446
x17=43.9722451925316x_{17} = 43.9722451925316
x18=87.8028770529953x_{18} = 87.8028770529953
x19=−53.9641122518576x_{19} = -53.9641122518576
x20=−47.2295815828054x_{20} = -47.2295815828054
x21=−90.2579525230567x_{21} = -90.2579525230567
x22=63.6407614766509x_{22} = 63.6407614766509
x23=−85.4187404353139x_{23} = -85.4187404353139
x24=−72.2927352502085x_{24} = -72.2927352502085
x25=100.559990867994x_{25} = 100.559990867994
x26=−78.4659866505624x_{26} = -78.4659866505624
x27=−46.5374386723423x_{27} = -46.5374386723423
x28=82.1659717866707x_{28} = 82.1659717866707
x29=75.4683544003024x_{29} = 75.4683544003024
x30=50.2604655230831x_{30} = 50.2604655230831
x31=63.3618002772679x_{31} = 63.3618002772679
x32=−79.4630444602951x_{32} = -79.4630444602951
x33=37.2989271761257x_{33} = 37.2989271761257
x34=37.8221426519431x_{34} = 37.8221426519431
x35=−97.3237257578261x_{35} = -97.3237257578261
x36=68.7197821653715x_{36} = 68.7197821653715

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
(−∞,−5.89130279784936]∪[−1.23267117887385,∞)\left(-\infty, -5.89130279784936\right] \cup \left[-1.23267117887385, \infty\right)
Convexa en los intervalos
(−∞,−1.23267117887385]∪[22.2035816252474,∞)\left(-\infty, -1.23267117887385\right] \cup \left[22.2035816252474, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(−e−xcos⁡(x)+sin⁡(x))y = \lim_{x \to -\infty}\left(- e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−e−xcos⁡(x)+sin⁡(x))y = \lim_{x \to \infty}\left(- e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -exp((-x)*cos(x) + sin(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(−e−xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to -\infty}\left(- \frac{e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−e−xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to \infty}\left(- \frac{e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}}}{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:
−e−xcos⁡(x)+sin⁡(x)=−excos⁡(x)−sin⁡(x)- e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} = - e^{x \cos{\left(x \right)} - \sin{\left(x \right)}}
- No
−e−xcos⁡(x)+sin⁡(x)=excos⁡(x)−sin⁡(x)- e^{- x \cos{\left(x \right)} + \sin{\left(x \right)}} = e^{x \cos{\left(x \right)} - \sin{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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