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-4*cos(x)/65+7*sin(x)/65+(-cos(2*x)-sin(2*x))*exp(-2*x)
Trazado el gráfico de la función/ -4*cos(x)/65+7*sin(x)/65+(-cos(2*x)-sin(2*x))*exp(-2*x)

Gráfico de la función y = -4*cos(x)/65+7*sin(x)/65+(-cos(2*x)-sin(2*x))*exp(-2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -4*cos(x)   7*sin(x)                           -2*x
f(x) = --------- + -------- + (-cos(2*x) - sin(2*x))*e    
           65         65
f(x)=(7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2xf{\left(x \right)} = \left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} 
f = (7*sin(x))/65 + (-4*cos(x))/65 + (-sin(2*x) - cos(2*x))*exp(-2*x)
Gráfico de la función
02468-8-6-4-2-1010-250000000250000000
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:
(7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2x=0\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−5.10508905002088x_{1} = -5.10508905002088
x2=13.0855167286532x_{2} = 13.0855167286532
x3=69.634184493222x_{3} = 69.634184493222
x4=82.2005551075812x_{4} = 82.2005551075812
x5=85.3421477611709x_{5} = 85.3421477611709
x6=91.6253330683505x_{6} = 91.6253330683505
x7=−9.81747704236516x_{7} = -9.81747704236516
x8=47.6430359180934x_{8} = 47.6430359180934
x9=97.9085183755301x_{9} = 97.9085183755301
x10=22.5102946893751x_{10} = 22.5102946893751
x11=35.0766653037342x_{11} = 35.0766653037342
x12=94.7669257219403x_{12} = 94.7669257219403
x13=41.3598506109138x_{13} = 41.3598506109138
x14=1.01586970801377x_{14} = 1.01586970801377
x15=60.2094065324526x_{15} = 60.2094065324526
x16=50.7846285716832x_{16} = 50.7846285716832
x17=38.218257957324x_{17} = 38.218257957324
x18=63.3509991860424x_{18} = 63.3509991860424
x19=19.3687020357853x_{19} = 19.3687020357853
x20=53.926221225273x_{20} = 53.926221225273
x21=−11.3882733692664x_{21} = -11.3882733692664
x22=28.7934799965547x_{22} = 28.7934799965547
x23=75.9173698004016x_{23} = 75.9173698004016
x24=−12.9590696960581x_{24} = -12.9590696960581
x25=6.80234505119303x_{25} = 6.80234505119303
x26=−3.53426221346273x_{26} = -3.53426221346273
x27=79.0589624539914x_{27} = 79.0589624539914
x28=16.2271093821954x_{28} = 16.2271093821954
x29=104.19170368271x_{29} = 104.19170368271
x30=88.4837404147607x_{30} = 88.4837404147607
x31=3.6533028112241x_{31} = 3.6533028112241
x32=101.05011102912x_{32} = 101.05011102912
x33=9.94392404956222x_{33} = 9.94392404956222
x34=25.6518873429649x_{34} = 25.6518873429649
x35=44.5014432645036x_{35} = 44.5014432645036
x36=31.9350726501445x_{36} = 31.9350726501445
x37=−1.96296545103812x_{37} = -1.96296545103812
x38=57.0678138788628x_{38} = 57.0678138788628
x39=66.4925918396322x_{39} = 66.4925918396322
x40=119.899666950659x_{40} = 119.899666950659
x41=72.7757771468118x_{41} = 72.7757771468118
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(x))/65 + (7*sin(x))/65 + (-cos(2*x) - sin(2*x))*exp(-2*x).
(−cos⁡(0⋅2)−sin⁡(0⋅2))e−0+((−1)4cos⁡(0)65+7sin⁡(0)65)\left(- \cos{\left(0 \cdot 2 \right)} - \sin{\left(0 \cdot 2 \right)}\right) e^{- 0} + \left(\frac{\left(-1\right) 4 \cos{\left(0 \right)}}{65} + \frac{7 \sin{\left(0 \right)}}{65}\right)
Resultado:
f(0)=−6965f{\left(0 \right)} = - \frac{69}{65}
Punto:
(0, -69/65)
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
−2(−sin⁡(2x)−cos⁡(2x))e−2x+(2sin⁡(2x)−2cos⁡(2x))e−2x+4sin⁡(x)65+7cos⁡(x)65=0- 2 \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} + \left(2 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{- 2 x} + \frac{4 \sin{\left(x \right)}}{65} + \frac{7 \cos{\left(x \right)}}{65} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−10.9955742875621x_{1} = -10.9955742875621
x2=99.479314702325x_{2} = 99.479314702325
x3=55.4970175520679x_{3} = 55.4970175520679
x4=33.5058689769394x_{4} = 33.5058689769394
x5=61.7802028592475x_{5} = 61.7802028592475
x6=71.2049808200169x_{6} = 71.2049808200169
x7=90.0545367415556x_{7} = 90.0545367415556
x8=−1.57112871387924x_{8} = -1.57112871387924
x9=20.9394983625802x_{9} = 20.9394983625802
x10=−15.707963267949x_{10} = -15.707963267949
x11=102.620907355915x_{11} = 102.620907355915
x12=77.4881661271965x_{12} = 77.4881661271965
x13=5.23232842483375x_{13} = 5.23232842483375
x14=58.6386102056577x_{14} = 58.6386102056577
x15=24.08109101617x_{15} = 24.08109101617
x16=−3.14156751331246x_{16} = -3.14156751331246
x17=14.6563130553954x_{17} = 14.6563130553954
x18=1.74610286808071x_{18} = 1.74610286808071
x19=−14.1371669411541x_{19} = -14.1371669411541
x20=96.3377220487352x_{20} = 96.3377220487352
x21=30.3642763233496x_{21} = 30.3642763233496
x22=52.3554248984781x_{22} = 52.3554248984781
x23=68.0633881664271x_{23} = 68.0633881664271
x24=27.2226836697598x_{24} = 27.2226836697598
x25=−0.0130183622675053x_{25} = -0.0130183622675053
x26=−9.42477796068171x_{26} = -9.42477796068171
x27=86.9129440879658x_{27} = 86.9129440879658
x28=36.6474616305291x_{28} = 36.6474616305291
x29=46.0722395912985x_{29} = 46.0722395912985
x30=8.37312626575237x_{30} = 8.37312626575237
x31=49.2138322448883x_{31} = 49.2138322448883
x32=42.9306469377087x_{32} = 42.9306469377087
x33=83.771351434376x_{33} = 83.771351434376
x34=−7.85398163513373x_{34} = -7.85398163513373
x35=80.6297587807862x_{35} = 80.6297587807862
x36=93.1961293951454x_{36} = 93.1961293951454
x37=11.5147204045792x_{37} = 11.5147204045792
x38=39.7890542841189x_{38} = 39.7890542841189
x39=74.3465734736067x_{39} = 74.3465734736067
x40=17.7979057089904x_{40} = 17.7979057089904
x41=64.9217955128373x_{41} = 64.9217955128373
Signos de extremos en los puntos:
(-10.995574287562112, 3553321280.95474)

(99.47931470232501, -0.124034734589208)

(55.4970175520679, -0.124034734589208)

(33.505868976939354, 0.124034734589208)

(61.78020285924749, -0.124034734589208)

(71.20498082001687, 0.124034734589208)

(90.05453674155564, 0.124034734589208)

(-1.5711287138792445, 23.0330105546489)

(20.939498362580178, 0.124034734589208)

(-15.707963267948966, -44031505860632)

(102.62090735591481, 0.124034734589208)

(77.48816612719645, 0.124034734589208)

(5.232328424833752, -0.12399565147073)

(58.6386102056577, 0.124034734589208)

(24.08109101616997, -0.124034734589208)

(-3.141567513312459, -535.430118416911)

(14.656313055395422, 0.124034734589463)

(1.7461028680807114, 0.155809962189241)

(-14.137166941154074, 1902773895292.05)

(96.33772204873522, 0.124034734589208)

(30.364276323349557, -0.124034734589208)

(52.35542489847811, 0.124034734589208)

(68.06338816642707, -0.124034734589208)

(27.222683669759764, 0.124034734589208)

(-0.013018362267505296, -1.06224543011657)

(-9.424777960681713, -153552935.333908)

(86.91294408796584, -0.124034734589208)

(36.64746163052914, -0.124034734589208)

(46.072239591298526, 0.124034734589208)

(8.373126265752369, 0.124034807647768)

(49.213832244888316, -0.124034734589208)

(42.93064693770873, -0.124034734589208)

(83.77135143437604, 0.124034734589208)

(-7.853981635133727, 6635623.89164883)

(80.62975878078625, -0.124034734589208)

(93.19612939514542, -0.124034734589208)

(11.51472040457922, -0.124034734452776)

(39.78905428411894, 0.124034734589208)

(74.34657347360667, -0.124034734589208)

(17.797905708990395, -0.124034734589208)

(64.92179551283728, 0.124034734589208)


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=99.479314702325x_{1} = 99.479314702325
x2=55.4970175520679x_{2} = 55.4970175520679
x3=61.7802028592475x_{3} = 61.7802028592475
x4=−15.707963267949x_{4} = -15.707963267949
x5=5.23232842483375x_{5} = 5.23232842483375
x6=24.08109101617x_{6} = 24.08109101617
x7=−3.14156751331246x_{7} = -3.14156751331246
x8=30.3642763233496x_{8} = 30.3642763233496
x9=68.0633881664271x_{9} = 68.0633881664271
x10=−0.0130183622675053x_{10} = -0.0130183622675053
x11=−9.42477796068171x_{11} = -9.42477796068171
x12=86.9129440879658x_{12} = 86.9129440879658
x13=36.6474616305291x_{13} = 36.6474616305291
x14=49.2138322448883x_{14} = 49.2138322448883
x15=42.9306469377087x_{15} = 42.9306469377087
x16=80.6297587807862x_{16} = 80.6297587807862
x17=93.1961293951454x_{17} = 93.1961293951454
x18=11.5147204045792x_{18} = 11.5147204045792
x19=74.3465734736067x_{19} = 74.3465734736067
x20=17.7979057089904x_{20} = 17.7979057089904
Puntos máximos de la función:
x20=−10.9955742875621x_{20} = -10.9955742875621
x20=33.5058689769394x_{20} = 33.5058689769394
x20=71.2049808200169x_{20} = 71.2049808200169
x20=90.0545367415556x_{20} = 90.0545367415556
x20=−1.57112871387924x_{20} = -1.57112871387924
x20=20.9394983625802x_{20} = 20.9394983625802
x20=102.620907355915x_{20} = 102.620907355915
x20=77.4881661271965x_{20} = 77.4881661271965
x20=58.6386102056577x_{20} = 58.6386102056577
x20=14.6563130553954x_{20} = 14.6563130553954
x20=1.74610286808071x_{20} = 1.74610286808071
x20=−14.1371669411541x_{20} = -14.1371669411541
x20=96.3377220487352x_{20} = 96.3377220487352
x20=52.3554248984781x_{20} = 52.3554248984781
x20=27.2226836697598x_{20} = 27.2226836697598
x20=46.0722395912985x_{20} = 46.0722395912985
x20=8.37312626575237x_{20} = 8.37312626575237
x20=83.771351434376x_{20} = 83.771351434376
x20=−7.85398163513373x_{20} = -7.85398163513373
x20=39.7890542841189x_{20} = 39.7890542841189
x20=64.9217955128373x_{20} = 64.9217955128373
Decrece en los intervalos
[99.479314702325,∞)\left[99.479314702325, \infty\right)
Crece en los intervalos
(−∞,−15.707963267949]\left(-\infty, -15.707963267949\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
−8(sin⁡(2x)−cos⁡(2x))e−2x−7sin⁡(x)65+4cos⁡(x)65=0- 8 \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} - \frac{7 \sin{\left(x \right)}}{65} + \frac{4 \cos{\left(x \right)}}{65} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−7.46128255407314x_{1} = -7.46128255407314
x2=69.634184493222x_{2} = 69.634184493222
x3=82.2005551075812x_{3} = 82.2005551075812
x4=−5.8904862201934x_{4} = -5.8904862201934
x5=85.3421477611709x_{5} = 85.3421477611709
x6=91.6253330683505x_{6} = 91.6253330683505
x7=47.6430359180934x_{7} = 47.6430359180934
x8=97.9085183755301x_{8} = 97.9085183755301
x9=22.5102946893751x_{9} = 22.5102946893751
x10=0.394201857251154x_{10} = 0.394201857251154
x11=35.0766653037342x_{11} = 35.0766653037342
x12=94.7669257219403x_{12} = 94.7669257219403
x13=41.3598506109138x_{13} = 41.3598506109138
x14=60.2094065324526x_{14} = 60.2094065324526
x15=9.94392412763926x_{15} = 9.94392412763926
x16=50.7846285716832x_{16} = 50.7846285716832
x17=38.218257957324x_{17} = 38.218257957324
x18=63.3509991860424x_{18} = 63.3509991860424
x19=19.3687020357853x_{19} = 19.3687020357853
x20=3.67719184177229x_{20} = 3.67719184177229
x21=53.926221225273x_{21} = 53.926221225273
x22=28.7934799965547x_{22} = 28.7934799965547
x23=75.9173698004016x_{23} = 75.9173698004016
x24=16.2271093821957x_{24} = 16.2271093821957
x25=6.80230324662182x_{25} = 6.80230324662182
x26=13.0855167285074x_{26} = 13.0855167285074
x27=79.0589624539914x_{27} = 79.0589624539914
x28=104.19170368271x_{28} = 104.19170368271
x29=88.4837404147607x_{29} = 88.4837404147607
x30=101.05011102912x_{30} = 101.05011102912
x31=25.6518873429649x_{31} = 25.6518873429649
x32=44.5014432645036x_{32} = 44.5014432645036
x33=31.9350726501445x_{33} = 31.9350726501445
x34=−12.1736715326604x_{34} = -12.1736715326604
x35=57.0678138788628x_{35} = 57.0678138788628
x36=66.4925918396322x_{36} = 66.4925918396322
x37=119.899666950659x_{37} = 119.899666950659
x38=72.7757771468118x_{38} = 72.7757771468118
x39=−13.7444678594554x_{39} = -13.7444678594554
x40=−1.17861208151654x_{40} = -1.17861208151654
x41=−9.03207887908053x_{41} = -9.03207887908053

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
[104.19170368271,∞)\left[104.19170368271, \infty\right)
Convexa en los intervalos
(−∞,−13.7444678594554]\left(-\infty, -13.7444678594554\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2x)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2x)=⟨−1165,1165⟩\lim_{x \to \infty}\left(\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x}\right) = \left\langle - \frac{11}{65}, \frac{11}{65}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1165,1165⟩y = \left\langle - \frac{11}{65}, \frac{11}{65}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-4*cos(x))/65 + (7*sin(x))/65 + (-cos(2*x) - sin(2*x))*exp(-2*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→−∞((7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2xx)y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x}}{x}\right)
lim⁡x→∞((7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2xx)=0\lim_{x \to \infty}\left(\frac{\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
(7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2x=(sin⁡(2x)−cos⁡(2x))e2x−7sin⁡(x)65+(−1)4cos⁡(x)65\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} = \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{2 x} - \frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}
- No
(7sin⁡(x)65+(−1)4cos⁡(x)65)+(−sin⁡(2x)−cos⁡(2x))e−2x=−(sin⁡(2x)−cos⁡(2x))e2x+7sin⁡(x)65−(−1)4cos⁡(x)65\left(\frac{7 \sin{\left(x \right)}}{65} + \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}\right) + \left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- 2 x} = - \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{2 x} + \frac{7 \sin{\left(x \right)}}{65} - \frac{\left(-1\right) 4 \cos{\left(x \right)}}{65}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = -4*cos(x)/65+7*sin(x)/65+(-cos(2*x)-sin(2*x))*exp(-2*x)
    © Sr Examen – Калькулятор