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Trazado el gráfico de la función/ -cos(x)-sin(x)-3*cos(3*x)*exp(2*x)/160-exp(2*x)*sin(3*x)/160

Gráfico de la función y = -cos(x)-sin(x)-3*cos(3*x)*exp(2*x)/160-exp(2*x)*sin(3*x)/160

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                      2*x    2*x         
                          3*cos(3*x)*e      e   *sin(3*x)
f(x) = -cos(x) - sin(x) - --------------- - -------------
                                160              160
f(x)=−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))f{\left(x \right)} = - \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) 
f = -exp(2*x)*sin(3*x)/160 - exp(2*x)*(3*cos(3*x))/160 - sin(x) - cos(x)
Gráfico de la función
02468-8-6-4-2-1010-50000005000000
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:
−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))=0- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−88.7499924639117x_{1} = -88.7499924639117
x2=−54.1924732744239x_{2} = -54.1924732744239
x3=14.2444171259466x_{3} = 14.2444171259466
x4=−25.9181393921158x_{4} = -25.9181393921158
x5=−7.06858346151519x_{5} = -7.06858346151519
x6=−38.484510006475x_{6} = -38.484510006475
x7=−63.6172512351933x_{7} = -63.6172512351933
x8=5.86676772293971x_{8} = 5.86676772293971
x9=−16.4933614313464x_{9} = -16.4933614313464
x10=−98.174770424681x_{10} = -98.174770424681
x11=−85.6083998103219x_{11} = -85.6083998103219
x12=−91.8915851175014x_{12} = -91.8915851175014
x13=3.78462892030548x_{13} = 3.78462892030548
x14=10.0556269646276x_{14} = 10.0556269646276
x15=−22.776546738526x_{15} = -22.776546738526
x16=−107.59954838545x_{16} = -107.59954838545
x17=−3.92698596443542x_{17} = -3.92698596443542
x18=−79.3252145031423x_{18} = -79.3252145031423
x19=12.1500220233191x_{19} = 12.1500220233191
x20=−13.3517687777566x_{20} = -13.3517687777566
x21=−73.0420291959627x_{21} = -73.0420291959627
x22=16.3388122283463x_{22} = 16.3388122283463
x23=7.96122999911698x_{23} = 7.96122999911698
x24=−10.2101761241499x_{24} = -10.2101761241499
x25=−82.4668071567321x_{25} = -82.4668071567321
x26=−95.0331777710912x_{26} = -95.0331777710912
x27=−41.6261026600648x_{27} = -41.6261026600648
x28=−76.1836218495525x_{28} = -76.1836218495525
x29=−44.7676953136546x_{29} = -44.7676953136546
x30=−66.7588438887831x_{30} = -66.7588438887831
x31=−60.4756585816035x_{31} = -60.4756585816035
x32=−0.782796382530714x_{32} = -0.782796382530714
x33=−35.3429173528852x_{33} = -35.3429173528852
x34=−19.6349540849362x_{34} = -19.6349540849362
x35=−47.9092879672443x_{35} = -47.9092879672443
x36=−32.2013246992954x_{36} = -32.2013246992954
x37=−51.0508806208341x_{37} = -51.0508806208341
x38=−69.9004365423729x_{38} = -69.9004365423729
x39=−57.3340659280137x_{39} = -57.3340659280137
x40=−29.0597320457056x_{40} = -29.0597320457056
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 -cos(x) - sin(x) - (3*cos(3*x))*exp(2*x)/160 - exp(2*x)*sin(3*x)/160.
((−cos⁡(0)−sin⁡(0))−e0⋅2⋅3cos⁡(0⋅3)160)−e0⋅2sin⁡(0⋅3)160\left(\left(- \cos{\left(0 \right)} - \sin{\left(0 \right)}\right) - \frac{e^{0 \cdot 2} \cdot 3 \cos{\left(0 \cdot 3 \right)}}{160}\right) - \frac{e^{0 \cdot 2} \sin{\left(0 \cdot 3 \right)}}{160}
Resultado:
f(0)=−163160f{\left(0 \right)} = - \frac{163}{160}
Punto:
(0, -163/160)
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
7e2xsin⁡(3x)160−9e2xcos⁡(3x)160+sin⁡(x)−cos⁡(x)=0\frac{7 e^{2 x} \sin{\left(3 x \right)}}{160} - \frac{9 e^{2 x} \cos{\left(3 x \right)}}{160} + \sin{\left(x \right)} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−52.621676947629x_{1} = -52.621676947629
x2=−33.7721210260903x_{2} = -33.7721210260903
x3=−55.7632696012188x_{3} = -55.7632696012188
x4=−96.6039740978861x_{4} = -96.6039740978861
x5=−11.7809724509646x_{5} = -11.7809724509646
x6=9.72802902430936x_{6} = 9.72802902430936
x7=−14.9225651045515x_{7} = -14.9225651045515
x8=1.56827245191667x_{8} = 1.56827245191667
x9=0.634564451570739x_{9} = 0.634564451570739
x10=−71.4712328691678x_{10} = -71.4712328691678
x11=−99.7455667514759x_{11} = -99.7455667514759
x12=−58.9048622548086x_{12} = -58.9048622548086
x13=−18.0641577581413x_{13} = -18.0641577581413
x14=−2.35664317595788x_{14} = -2.35664317595788
x15=16.0112143205971x_{15} = 16.0112143205971
x16=−77.7544181763474x_{16} = -77.7544181763474
x17=−36.9137136796801x_{17} = -36.9137136796801
x18=−62.0464549083984x_{18} = -62.0464549083984
x19=−21.2057504117311x_{19} = -21.2057504117311
x20=−68.329640215578x_{20} = -68.329640215578
x21=−24.3473430653209x_{21} = -24.3473430653209
x22=−90.3207887907066x_{22} = -90.3207887907066
x23=−74.6128255227576x_{23} = -74.6128255227576
x24=7.63363474044993x_{24} = 7.63363474044993
x25=−46.3384916404494x_{25} = -46.3384916404494
x26=−27.4889357189107x_{26} = -27.4889357189107
x27=−30.6305283725005x_{27} = -30.6305283725005
x28=−84.037603483527x_{28} = -84.037603483527
x29=−5.49778798256954x_{29} = -5.49778798256954
x30=−49.4800842940392x_{30} = -49.4800842940392
x31=−40.0553063332699x_{31} = -40.0553063332699
x32=−65.1880475619882x_{32} = -65.1880475619882
x33=−43.1968989868597x_{33} = -43.1968989868597
x34=−8.63937979893832x_{34} = -8.63937979893832
x35=13.9168192182067x_{35} = 13.9168192182067
x36=−80.8960108299372x_{36} = -80.8960108299372
x37=5.5391367293406x_{37} = 5.5391367293406
x38=−93.4623814442964x_{38} = -93.4623814442964
x39=2.33534033389595x_{39} = 2.33534033389595
x40=−228.550865548657x_{40} = -228.550865548657
x41=−87.1791961371168x_{41} = -87.1791961371168
Signos de extremos en los puntos:
(-52.621676947629034, 1.41421356237309)

(-33.772121026090275, 1.41421356237309)

(-55.76326960121883, -1.41421356237309)

(-96.60397409788614, 1.41421356237309)

(-11.780972450964649, -1.41421356237258)

(9.728029024309363, 4631144.73551137)

(-14.922565104551524, 1.41421356237309)

(1.5682724519166662, -0.855355031870809)

(0.6345644515707395, -1.39737302604807)

(-71.47123286916779, 1.41421356237309)

(-99.74556675147593, -1.41421356237309)

(-58.90486225480862, 1.41421356237309)

(-18.06415775814131, -1.41421356237309)

(-2.3566431759578816, 1.414134303054)

(16.011214320597073, 1327986474937.03)

(-77.75441817634739, 1.41421356237309)

(-36.91371367968007, -1.41421356237309)

(-62.04645490839842, -1.41421356237309)

(-21.205750411731103, 1.4142135623731)

(-68.329640215578, -1.41421356237309)

(-24.3473430653209, -1.4142135623731)

(-90.32078879070656, 1.41421356237309)

(-74.61282552275759, -1.41421356237309)

(7.633634740449935, 70228.3360603857)

(-46.33849164044945, 1.41421356237309)

(-27.488935718910692, 1.41421356237309)

(-30.630528372500486, -1.41421356237309)

(-84.03760348352696, 1.41421356237309)

(-5.497787982569544, -1.41421341409523)

(-49.480084294039244, -1.41421356237309)

(-40.05530633326986, 1.41421356237309)

(-65.18804756198821, 1.41421356237309)

(-43.19689898685966, -1.41421356237309)

(-8.639379798938322, 1.41421356209619)

(13.916819218206733, 20138410076.4658)

(-80.89601082993718, -1.41421356237309)

(5.539136729340605, 1064.94559994152)

(-93.46238144429635, -1.4142135623731)

(2.335340333895947, -1.97225252817596)

(-228.55086554865747, 1.41421356237309)

(-87.17919613711676, -1.41421356237309)


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=−55.7632696012188x_{1} = -55.7632696012188
x2=−11.7809724509646x_{2} = -11.7809724509646
x3=0.634564451570739x_{3} = 0.634564451570739
x4=−99.7455667514759x_{4} = -99.7455667514759
x5=−18.0641577581413x_{5} = -18.0641577581413
x6=−36.9137136796801x_{6} = -36.9137136796801
x7=−62.0464549083984x_{7} = -62.0464549083984
x8=−68.329640215578x_{8} = -68.329640215578
x9=−24.3473430653209x_{9} = -24.3473430653209
x10=−74.6128255227576x_{10} = -74.6128255227576
x11=−30.6305283725005x_{11} = -30.6305283725005
x12=−5.49778798256954x_{12} = -5.49778798256954
x13=−49.4800842940392x_{13} = -49.4800842940392
x14=−43.1968989868597x_{14} = -43.1968989868597
x15=−80.8960108299372x_{15} = -80.8960108299372
x16=−93.4623814442964x_{16} = -93.4623814442964
x17=2.33534033389595x_{17} = 2.33534033389595
x18=−87.1791961371168x_{18} = -87.1791961371168
Puntos máximos de la función:
x18=−52.621676947629x_{18} = -52.621676947629
x18=−33.7721210260903x_{18} = -33.7721210260903
x18=−96.6039740978861x_{18} = -96.6039740978861
x18=9.72802902430936x_{18} = 9.72802902430936
x18=−14.9225651045515x_{18} = -14.9225651045515
x18=1.56827245191667x_{18} = 1.56827245191667
x18=−71.4712328691678x_{18} = -71.4712328691678
x18=−58.9048622548086x_{18} = -58.9048622548086
x18=−2.35664317595788x_{18} = -2.35664317595788
x18=16.0112143205971x_{18} = 16.0112143205971
x18=−77.7544181763474x_{18} = -77.7544181763474
x18=−21.2057504117311x_{18} = -21.2057504117311
x18=−90.3207887907066x_{18} = -90.3207887907066
x18=7.63363474044993x_{18} = 7.63363474044993
x18=−46.3384916404494x_{18} = -46.3384916404494
x18=−27.4889357189107x_{18} = -27.4889357189107
x18=−84.037603483527x_{18} = -84.037603483527
x18=−40.0553063332699x_{18} = -40.0553063332699
x18=−65.1880475619882x_{18} = -65.1880475619882
x18=−8.63937979893832x_{18} = -8.63937979893832
x18=13.9168192182067x_{18} = 13.9168192182067
x18=5.5391367293406x_{18} = 5.5391367293406
x18=−228.550865548657x_{18} = -228.550865548657
Decrece en los intervalos
[2.33534033389595,∞)\left[2.33534033389595, \infty\right)
Crece en los intervalos
(−∞,−99.7455667514759]\left(-\infty, -99.7455667514759\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
41e2xsin⁡(3x)160+3e2xcos⁡(3x)160+sin⁡(x)+cos⁡(x)=0\frac{41 e^{2 x} \sin{\left(3 x \right)}}{160} + \frac{3 e^{2 x} \cos{\left(3 x \right)}}{160} + \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−88.7499924639117x_{1} = -88.7499924639117
x2=−54.1924732744239x_{2} = -54.1924732744239
x3=2.06141267219052x_{3} = 2.06141267219052
x4=−25.9181393921158x_{4} = -25.9181393921158
x5=8.35323352571835x_{5} = 8.35323352571835
x6=−38.484510006475x_{6} = -38.484510006475
x7=−63.6172512351933x_{7} = -63.6172512351933
x8=−16.4933614313464x_{8} = -16.4933614313464
x9=−98.174770424681x_{9} = -98.174770424681
x10=10.4476286583664x_{10} = 10.4476286583664
x11=−85.6083998103219x_{11} = -85.6083998103219
x12=−7.06858337089674x_{12} = -7.06858337089674
x13=−91.8915851175014x_{13} = -91.8915851175014
x14=−22.776546738526x_{14} = -22.776546738526
x15=−107.59954838545x_{15} = -107.59954838545
x16=−79.3252145031423x_{16} = -79.3252145031423
x17=13.5892213104575x_{17} = 13.5892213104575
x18=4.16487348164952x_{18} = 4.16487348164952
x19=−73.0420291959627x_{19} = -73.0420291959627
x20=−0.752402287345429x_{20} = -0.752402287345429
x21=−82.4668071567321x_{21} = -82.4668071567321
x22=−95.0331777710912x_{22} = -95.0331777710912
x23=6.25883381906193x_{23} = 6.25883381906193
x24=−41.6261026600648x_{24} = -41.6261026600648
x25=−76.1836218495525x_{25} = -76.1836218495525
x26=−44.7676953136546x_{26} = -44.7676953136546
x27=−13.3517687777563x_{27} = -13.3517687777563
x28=−66.7588438887831x_{28} = -66.7588438887831
x29=−60.4756585816035x_{29} = -60.4756585816035
x30=−35.3429173528852x_{30} = -35.3429173528852
x31=−10.2101761239807x_{31} = -10.2101761239807
x32=−19.6349540849362x_{32} = -19.6349540849362
x33=−47.9092879672443x_{33} = -47.9092879672443
x34=−32.2013246992954x_{34} = -32.2013246992954
x35=15.6836164128479x_{35} = 15.6836164128479
x36=−3.92693742596262x_{36} = -3.92693742596262
x37=−51.0508806208341x_{37} = -51.0508806208341
x38=−69.9004365423729x_{38} = -69.9004365423729
x39=−57.3340659280137x_{39} = -57.3340659280137
x40=−29.0597320457056x_{40} = -29.0597320457056

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
[10.4476286583664,∞)\left[10.4476286583664, \infty\right)
Convexa en los intervalos
(−∞,−107.59954838545]\left(-\infty, -107.59954838545\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x))))=⟨−2,2⟩\lim_{x \to -\infty}\left(- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x))))y = \lim_{x \to \infty}\left(- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -cos(x) - sin(x) - (3*cos(3*x))*exp(2*x)/160 - exp(2*x)*sin(3*x)/160, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))x)=0\lim_{x \to -\infty}\left(\frac{- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))x)y = x \lim_{x \to \infty}\left(\frac{- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \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:
−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))=sin⁡(x)−cos⁡(x)+e−2xsin⁡(3x)160−3e−2xcos⁡(3x)160- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) = \sin{\left(x \right)} - \cos{\left(x \right)} + \frac{e^{- 2 x} \sin{\left(3 x \right)}}{160} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{160}
- No
−e2xsin⁡(3x)160+(−e2x3cos⁡(3x)160+(−sin⁡(x)−cos⁡(x)))=−sin⁡(x)+cos⁡(x)−e−2xsin⁡(3x)160+3e−2xcos⁡(3x)160- \frac{e^{2 x} \sin{\left(3 x \right)}}{160} + \left(- \frac{e^{2 x} 3 \cos{\left(3 x \right)}}{160} + \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) = - \sin{\left(x \right)} + \cos{\left(x \right)} - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{160} + \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{160}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор