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-4*cos(2*x)/5+22*sin(2*x)/5+(-1-x)*exp(-x)-4*x*cos(2*x)-3*x*sin(2*x)
Trazado el gráfico de la función/ -4*cos(2*x)/5+22*sin(2*x)/5+(-1-x)*exp(-x)-4*x*cos(2*x)-3*x*sin(2*x)

Gráfico de la función y = -4*cos(2*x)/5+22*sin(2*x)/5+(-1-x)*exp(-x)-4*x*cos(2*x)-3*x*sin(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -4*cos(2*x)   22*sin(2*x)             -x                              
f(x) = ----------- + ----------- + (-1 - x)*e   - 4*x*cos(2*x) - 3*x*sin(2*x)
            5             5
f(x)=−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))f{\left(x \right)} = - 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right) 
f = -3*x*sin(2*x) - 4*x*cos(2*x) + (-x - 1)*exp(-x) + (22*sin(2*x))/5 + (-4*cos(2*x))/5
Gráfico de la función
02468-8-6-4-2-1010-200000200000
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:
−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))=0- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=19.9362413049626x_{1} = 19.9362413049626
x2=0.494898601335865x_{2} = 0.494898601335865
x3=26.224405674641x_{3} = 26.224405674641
x4=48.2226750686665x_{4} = 48.2226750686665
x5=98.4924432834008x_{5} = 98.4924432834008
x6=10.4923786605713x_{6} = 10.4923786605713
x7=2.48141440535073x_{7} = 2.48141440535073
x8=21.5085603730973x_{8} = 21.5085603730973
x9=71.7873804268691x_{9} = 71.7873804268691
x10=92.208978909833x_{10} = 92.208978909833
x11=46.6515947066983x_{11} = 46.6515947066983
x12=24.6526065251789x_{12} = 24.6526065251789
x13=100.063303878727x_{13} = 100.063303878727
x14=70.2164580393816x_{14} = 70.2164580393816
x15=5.74485152931299x_{15} = 5.74485152931299
x16=85.9254735319816x_{16} = 85.9254735319816
x17=49.793737363806x_{17} = 49.793737363806
x18=54.5068315856654x_{18} = 54.5068315856654
x19=43.5093718813221x_{19} = 43.5093718813221
x20=32.5106208925014x_{20} = 32.5106208925014
x21=68.6455298486732x_{21} = 68.6455298486732
x22=76.500116363812x_{22} = 76.500116363812
x23=84.3545897016882x_{23} = 84.3545897016882
x24=57.6488298947873x_{24} = 57.6488298947873
x25=−1.68318234537615x_{25} = -1.68318234537615
x26=79.6419173955904x_{26} = 79.6419173955904
x27=13.6433522888925x_{27} = 13.6433522888925
x28=63.9327061525031x_{28} = 63.9327061525031
x29=35.6533234063887x_{29} = 35.6533234063887
x30=65.5036543772557x_{30} = 65.5036543772557
x31=90.638106744116x_{31} = 90.638106744116
x32=78.0710189868474x_{32} = 78.0710189868474
x33=62.361750225907x_{33} = 62.361750225907
x34=18.363655832466x_{34} = 18.363655832466
x35=87.4963542059256x_{35} = 87.4963542059256
x36=34.0819980328772x_{36} = 34.0819980328772
x37=12.0684955852973x_{37} = 12.0684955852973
x38=93.7798485240398x_{38} = 93.7798485240398
x39=56.0778364627472x_{39} = 56.0778364627472
x40=40.3670499796827x_{40} = 40.3670499796827
x41=4.14565895890117x_{41} = 4.14565895890117
x42=41.9382247202911x_{42} = 41.9382247202911
x43=27.7960897974582x_{43} = 27.7960897974582
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 (-4*cos(2*x))/5 + (22*sin(2*x))/5 + (-1 - x)*exp(-x) - 4*x*cos(2*x) - 3*x*sin(2*x).
(((−1−0)e−0+((−1)4cos⁡(0⋅2)5+22sin⁡(0⋅2)5))−0⋅4cos⁡(0⋅2))−0⋅3sin⁡(0⋅2)\left(\left(\left(-1 - 0\right) e^{- 0} + \left(\frac{\left(-1\right) 4 \cos{\left(0 \cdot 2 \right)}}{5} + \frac{22 \sin{\left(0 \cdot 2 \right)}}{5}\right)\right) - 0 \cdot 4 \cos{\left(0 \cdot 2 \right)}\right) - 0 \cdot 3 \sin{\left(0 \cdot 2 \right)}
Resultado:
f(0)=−95f{\left(0 \right)} = - \frac{9}{5}
Punto:
(0, -9/5)
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
8xsin⁡(2x)−6xcos⁡(2x)−(−x−1)e−x−7sin⁡(2x)5+24cos⁡(2x)5−e−x=08 x \sin{\left(2 x \right)} - 6 x \cos{\left(2 x \right)} - \left(- x - 1\right) e^{- x} - \frac{7 \sin{\left(2 x \right)}}{5} + \frac{24 \cos{\left(2 x \right)}}{5} - e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=82.0013213504849x_{1} = 82.0013213504849
x2=97.7095813485123x_{2} = 97.7095813485123
x3=42.7297078266193x_{3} = 42.7297078266193
x4=30.1618409233784x_{4} = 30.1618409233784
x5=96.1387597304536x_{5} = 96.1387597304536
x6=67.8637692055864x_{6} = 67.8637692055864
x7=86.7138106903959x_{7} = 86.7138106903959
x8=36.4459048063813x_{8} = 36.4459048063813
x9=88.2846380786111x_{9} = 88.2846380786111
x10=39.5878310801846x_{10} = 39.5878310801846
x11=61.5803555516686x_{11} = 61.5803555516686
x12=102.422041529783x_{12} = 102.422041529783
x13=66.2929198749139x_{13} = 66.2929198749139
x14=5.00195544130954x_{14} = 5.00195544130954
x15=52.1551311077314x_{15} = 52.1551311077314
x16=9.7304545664145x_{16} = 9.7304545664145
x17=50.5842440614957x_{17} = 50.5842440614957
x18=60.0094946161975x_{18} = 60.0094946161975
x19=16.020111993054x_{19} = 16.020111993054
x20=80.4304889440277x_{20} = 80.4304889440277
x21=23.8773066554791x_{21} = 23.8773066554791
x22=83.5721523940922x_{22} = 83.5721523940922
x23=−1.04531244075747x_{23} = -1.04531244075747
x24=64.7220679553212x_{24} = 64.7220679553212
x25=45.871545313038x_{25} = 45.871545313038
x26=14.4482416526545x_{26} = 14.4482416526545
x27=31.7328899347876x_{27} = 31.7328899347876
x28=53.7260128090016x_{28} = 53.7260128090016
x29=6.58061170058247x_{29} = 6.58061170058247
x30=3.41184475352026x_{30} = 3.41184475352026
x31=1.79722028457545x_{31} = 1.79722028457545
x32=94.5679372686185x_{32} = 94.5679372686185
x33=17.5917859387702x_{33} = 17.5917859387702
x34=58.4386301832932x_{34} = 58.4386301832932
x35=8.15641807249072x_{35} = 8.15641807249072
x36=28.59076374588x_{36} = 28.59076374588
x37=75.7179826948227x_{37} = 75.7179826948227
x38=89.8554643759487x_{38} = 89.8554643759487
x39=44.3006309504545x_{39} = 44.3006309504545
x40=22.3060521347805x_{40} = 22.3060521347805
x41=38.0168749099347x_{41} = 38.0168749099347
x42=20.7347267942285x_{42} = 20.7347267942285
x43=74.1471439479031x_{43} = 74.1471439479031
x44=72.5763033545058x_{44} = 72.5763033545058
Signos de extremos en los puntos:
(82.00132135048491, -408.018555016927)

(97.70958134851229, -486.557926268775)

(42.72970782661926, 211.671572090084)

(30.16184092337843, 148.84196321498)

(96.13875973045359, 478.703982571177)

(67.86376920558638, 337.33329810323)

(86.71381069039586, 431.580349382289)

(36.445904806381265, 180.256572045683)

(88.28463807861112, -439.434284424917)

(39.5878310801846, 195.964034928584)

(61.58035555166859, 305.917714076825)

(102.4220415297831, 510.119764370742)

(66.29291987491392, -329.479395972791)

(5.001955441309536, 23.1790590108791)

(52.155131107731364, 258.794493970511)

(9.730454566414501, -46.7573749584005)

(50.58424406149571, -250.940648409015)

(60.009494616197514, -298.063829352476)

(16.02011199305398, -78.1629712162921)

(80.43048894402772, 400.16462750139)

(23.877306655479092, 117.428059742964)

(83.57215239409223, 415.872484576662)

(-1.0453124407574717, -8.09116397932016)

(64.72206795532122, 321.625497725504)

(45.87154531303799, 227.379168133157)

(14.448241652654461, 70.3105896305165)

(31.73288993478761, -156.695565873901)

(53.726012809001595, -266.64834754869)

(6.580611700582473, -31.0715280641441)

(3.4118447535202554, -15.5361810809344)

(1.7972202845754457, 7.15385460972602)

(94.56793726861854, -470.850040139209)

(17.591785938770194, 86.0156347557838)

(58.4386301832932, 290.2099498739)

(8.156418072490723, 38.9050074363988)

(28.59076374587995, -140.988402792998)

(75.71798269482274, -376.602858498971)

(89.85546437594871, 447.288221103806)

(44.30063095045447, -219.525363541552)

(22.306052134780472, -109.574765235038)

(38.01687490993474, -188.110293039044)

(20.734726794228454, 101.721576868008)

(74.14714394790312, 368.748940493723)

(72.57630335450584, -360.895025258062)


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=82.0013213504849x_{1} = 82.0013213504849
x2=97.7095813485123x_{2} = 97.7095813485123
x3=88.2846380786111x_{3} = 88.2846380786111
x4=66.2929198749139x_{4} = 66.2929198749139
x5=9.7304545664145x_{5} = 9.7304545664145
x6=50.5842440614957x_{6} = 50.5842440614957
x7=60.0094946161975x_{7} = 60.0094946161975
x8=16.020111993054x_{8} = 16.020111993054
x9=−1.04531244075747x_{9} = -1.04531244075747
x10=31.7328899347876x_{10} = 31.7328899347876
x11=53.7260128090016x_{11} = 53.7260128090016
x12=6.58061170058247x_{12} = 6.58061170058247
x13=3.41184475352026x_{13} = 3.41184475352026
x14=94.5679372686185x_{14} = 94.5679372686185
x15=28.59076374588x_{15} = 28.59076374588
x16=75.7179826948227x_{16} = 75.7179826948227
x17=44.3006309504545x_{17} = 44.3006309504545
x18=22.3060521347805x_{18} = 22.3060521347805
x19=38.0168749099347x_{19} = 38.0168749099347
x20=72.5763033545058x_{20} = 72.5763033545058
Puntos máximos de la función:
x20=42.7297078266193x_{20} = 42.7297078266193
x20=30.1618409233784x_{20} = 30.1618409233784
x20=96.1387597304536x_{20} = 96.1387597304536
x20=67.8637692055864x_{20} = 67.8637692055864
x20=86.7138106903959x_{20} = 86.7138106903959
x20=36.4459048063813x_{20} = 36.4459048063813
x20=39.5878310801846x_{20} = 39.5878310801846
x20=61.5803555516686x_{20} = 61.5803555516686
x20=102.422041529783x_{20} = 102.422041529783
x20=5.00195544130954x_{20} = 5.00195544130954
x20=52.1551311077314x_{20} = 52.1551311077314
x20=80.4304889440277x_{20} = 80.4304889440277
x20=23.8773066554791x_{20} = 23.8773066554791
x20=83.5721523940922x_{20} = 83.5721523940922
x20=64.7220679553212x_{20} = 64.7220679553212
x20=45.871545313038x_{20} = 45.871545313038
x20=14.4482416526545x_{20} = 14.4482416526545
x20=1.79722028457545x_{20} = 1.79722028457545
x20=17.5917859387702x_{20} = 17.5917859387702
x20=58.4386301832932x_{20} = 58.4386301832932
x20=8.15641807249072x_{20} = 8.15641807249072
x20=89.8554643759487x_{20} = 89.8554643759487
x20=20.7347267942285x_{20} = 20.7347267942285
x20=74.1471439479031x_{20} = 74.1471439479031
Decrece en los intervalos
[97.7095813485123,∞)\left[97.7095813485123, \infty\right)
Crece en los intervalos
(−∞,−1.04531244075747]\left(-\infty, -1.04531244075747\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
12xsin⁡(2x)+16xcos⁡(2x)−(x+1)e−x−8sin⁡(2x)5−44cos⁡(2x)5+2e−x=012 x \sin{\left(2 x \right)} + 16 x \cos{\left(2 x \right)} - \left(x + 1\right) e^{- x} - \frac{8 \sin{\left(2 x \right)}}{5} - \frac{44 \cos{\left(2 x \right)}}{5} + 2 e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2.82581551034232x_{1} = -2.82581551034232
x2=78.0774561027055x_{2} = 78.0774561027055
x3=68.6528559059379x_{3} = 68.6528559059379
x4=48.2331291107184x_{4} = 48.2331291107184
x5=76.5066863459246x_{5} = 76.5066863459246
x6=98.497540371216x_{6} = 98.497540371216
x7=40.3795581450707x_{7} = 40.3795581450707
x8=84.3605450712926x_{8} = 84.3605450712926
x9=46.6624037638282x_{9} = 46.6624037638282
x10=54.516071700286x_{10} = 54.516071700286
x11=93.7852028305173x_{11} = 93.7852028305173
x12=18.3914662745253x_{12} = 18.3914662745253
x13=70.2236192729809x_{13} = 70.2236192729809
x14=43.5209685819974x_{14} = 43.5209685819974
x15=49.8038589707576x_{15} = 49.8038589707576
x16=67.0820940916461x_{16} = 67.0820940916461
x17=0.361689969968521x_{17} = 0.361689969968521
x18=27.8143339353724x_{18} = 27.8143339353724
x19=4.27423624729829x_{19} = 4.27423624729829
x20=100.068320632357x_{20} = 100.068320632357
x21=12.1112610958542x_{21} = 12.1112610958542
x22=71.7943840902932x_{22} = 71.7943840902932
x23=5.83798336763077x_{23} = 5.83798336763077
x24=13.6810482618941x_{24} = 13.6810482618941
x25=63.9405755819083x_{25} = 63.9405755819083
x26=87.50209476751x_{26} = 87.50209476751
x27=62.3698191459425x_{27} = 62.3698191459425
x28=85.9313195249547x_{28} = 85.9313195249547
x29=34.0968391830336x_{29} = 34.0968391830336
x30=32.526187938843x_{30} = 32.526187938843
x31=41.9502599196877x_{31} = 41.9502599196877
x32=35.6675033483689x_{32} = 35.6675033483689
x33=10.5417849768304x_{33} = 10.5417849768304
x34=24.6732132941872x_{34} = 24.6732132941872
x35=−2.46272674827621x_{35} = -2.46272674827621
x36=19.9618164608789x_{36} = 19.9618164608789
x37=57.6575629694789x_{37} = 57.6575629694789
x38=56.0868159050835x_{38} = 56.0868159050835
x39=79.6482269127449x_{39} = 79.6482269127449
x40=26.2437592753781x_{40} = 26.2437592753781
x41=21.532232932456x_{41} = 21.532232932456
x42=92.214424823652x_{42} = 92.214424823652
x43=65.511333942476x_{43} = 65.511333942476
x44=90.6436474545281x_{44} = 90.6436474545281

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
[100.068320632357,∞)\left[100.068320632357, \infty\right)
Convexa en los intervalos
(−∞,−2.46272674827621]\left(-\infty, -2.46272674827621\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5))))=∞\lim_{x \to -\infty}\left(- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right)\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5))))y = \lim_{x \to \infty}\left(- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right)\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-4*cos(2*x))/5 + (22*sin(2*x))/5 + (-1 - x)*exp(-x) - 4*x*cos(2*x) - 3*x*sin(2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))x)=−∞\lim_{x \to -\infty}\left(\frac{- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right)}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))x)y = x \lim_{x \to \infty}\left(\frac{- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\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:
−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))=−3xsin⁡(2x)+4xcos⁡(2x)+(x−1)ex−22sin⁡(2x)5+(−1)4cos⁡(2x)5- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right) = - 3 x \sin{\left(2 x \right)} + 4 x \cos{\left(2 x \right)} + \left(x - 1\right) e^{x} - \frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}
- No
−3xsin⁡(2x)+(−4xcos⁡(2x)+((−x−1)e−x+(22sin⁡(2x)5+(−1)4cos⁡(2x)5)))=3xsin⁡(2x)−4xcos⁡(2x)−(x−1)ex+22sin⁡(2x)5−(−1)4cos⁡(2x)5- 3 x \sin{\left(2 x \right)} + \left(- 4 x \cos{\left(2 x \right)} + \left(\left(- x - 1\right) e^{- x} + \left(\frac{22 \sin{\left(2 x \right)}}{5} + \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}\right)\right)\right) = 3 x \sin{\left(2 x \right)} - 4 x \cos{\left(2 x \right)} - \left(x - 1\right) e^{x} + \frac{22 \sin{\left(2 x \right)}}{5} - \frac{\left(-1\right) 4 \cos{\left(2 x \right)}}{5}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = -4*cos(2*x)/5+22*sin(2*x)/5+(-1-x)*exp(-x)-4*x*cos(2*x)-3*x*sin(2*x)
    © Sr Examen – Калькулятор