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

Gráfico de la función y = (2*x*cos(2*x)+3*x*sin(2*x))*exp(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                      3*x
f(x) = (2*x*cos(2*x) + 3*x*sin(2*x))*e
f(x)=(2xcos⁡(2x)+3xsin⁡(2x))e3xf{\left(x \right)} = \left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x} 
f = ((2*x)*cos(2*x) + (3*x)*sin(2*x))*exp(3*x)
Gráfico de la función
02468-8-6-4-2-1010-500000000000000500000000000000
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:
(2xcos⁡(2x)+3xsin⁡(2x))e3x=0\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−ilog⁡(−e−iatan⁡(125)4)x_{2} = - i \log{\left(- e^{- \frac{i \operatorname{atan}{\left(\frac{12}{5} \right)}}{4}} \right)}
x3=−atan⁡(125)4x_{3} = - \frac{\operatorname{atan}{\left(\frac{12}{5} \right)}}{4}
Solución numérica
x1=−30.1391315108768x_{1} = -30.1391315108768
x2=−17.5727608965176x_{2} = -17.5727608965176
x3=−78.8338176415186x_{3} = -78.8338176415186
x4=−72.550632334339x_{4} = -72.550632334339
x5=−96.1125772362625x_{5} = -96.1125772362625
x6=−67.8382433539543x_{6} = -67.8382433539543
x7=−50.5594837592105x_{7} = -50.5594837592105
x8=−16.0019645697228x_{8} = -16.0019645697228
x9=−64.6966507003645x_{9} = -64.6966507003645
x10=−89.8293919290829x_{10} = -89.8293919290829
x11=−92.9709845826727x_{11} = -92.9709845826727
x12=−94.5417809094676x_{12} = -94.5417809094676
x13=−86.6877992754931x_{13} = -86.6877992754931
x14=−37.9931131448513x_{14} = -37.9931131448513
x15=−75.6922249879288x_{15} = -75.6922249879288
x16=−52.1302800860054x_{16} = -52.1302800860054
x17=−70.9798360075441x_{17} = -70.9798360075441
x18=−26.997538857287x_{18} = -26.997538857287
x19=−22.2851498769023x_{19} = -22.2851498769023
x20=−58.413465393185x_{20} = -58.413465393185
x21=−6.57718660895337x_{21} = -6.57718660895337
x22=−45.8470947788258x_{22} = -45.8470947788258
x23=7.5599803322007x_{23} = 7.5599803322007
x24=−59.9842617199799x_{24} = -59.9842617199799
x25=−100.824966216647x_{25} = -100.824966216647
x26=−9.71877926254316x_{26} = -9.71877926254316
x27=−34.8515204912615x_{27} = -34.8515204912615
x28=−14.4311682429279x_{28} = -14.4311682429279
x29=−53.7010764128003x_{29} = -53.7010764128003
x30=−88.258595602288x_{30} = -88.258595602288
x31=−48.9886874324156x_{31} = -48.9886874324156
x32=−44.2762984520309x_{32} = -44.2762984520309
x33=1.27679502502111x_{33} = 1.27679502502111
x34=0x_{34} = 0
x35=−97.6833735630574x_{35} = -97.6833735630574
x36=−1.86479762856868x_{36} = -1.86479762856868
x37=−31.7099278376717x_{37} = -31.7099278376717
x38=−23.8559462036972x_{38} = -23.8559462036972
x39=−20.7143535501074x_{39} = -20.7143535501074
x40=−12.860371916133x_{40} = -12.860371916133
x41=−8.14798293574827x_{41} = -8.14798293574827
x42=−74.1214286611339x_{42} = -74.1214286611339
x43=5.9891840054058x_{43} = 5.9891840054058
x44=−39.75x_{44} = -39.75
x45=−42.705502125236x_{45} = -42.705502125236
x46=−80.4046139683135x_{46} = -80.4046139683135
x47=−36.4223168180564x_{47} = -36.4223168180564
x48=−66.2674470271594x_{48} = -66.2674470271594
x49=−231.601271460591x_{49} = -231.601271460591
x50=9.1307766589956x_{50} = 9.1307766589956
x51=−28.5683351840819x_{51} = -28.5683351840819
x52=−56.8426690663901x_{52} = -56.8426690663901
x53=−81.9754102951084x_{53} = -81.9754102951084
x54=−5.00639028215847x_{54} = -5.00639028215847
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 ((2*x)*cos(2*x) + (3*x)*sin(2*x))*exp(3*x).
(0⋅2cos⁡(0⋅2)+0⋅3sin⁡(0⋅2))e0⋅3\left(0 \cdot 2 \cos{\left(0 \cdot 2 \right)} + 0 \cdot 3 \sin{\left(0 \cdot 2 \right)}\right) e^{0 \cdot 3}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
3(2xcos⁡(2x)+3xsin⁡(2x))e3x+(−4xsin⁡(2x)+6xcos⁡(2x)+3sin⁡(2x)+2cos⁡(2x))e3x=03 \left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x} + \left(- 4 x \sin{\left(2 x \right)} + 6 x \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{3 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.5825925919772x_{1} = -22.5825925919772
x2=−91.6950305747138x_{2} = -91.6950305747138
x3=−86.9826872780801x_{3} = -86.9826872780801
x4=−68.1333774980042x_{4} = -68.1333774980042
x5=−2.19782517345074x_{5} = -2.19782517345074
x6=7.27622445141199x_{6} = 7.27622445141199
x7=−36.7184263018089x_{7} = -36.7184263018089
x8=−105.832084927443x_{8} = -105.832084927443
x9=−97.9781618220212x_{9} = -97.9781618220212
x10=−96.4073783480111x_{10} = -96.4073783480111
x11=−55.75x_{11} = -55.75
x12=−11.5903481257722x_{12} = -11.5903481257722
x13=−28.8650228606344x_{13} = -28.8650228606344
x14=−33.5770322473965x_{14} = -33.5770322473965
x15=−69.7041482116849x_{15} = -69.7041482116849
x16=−46.1427715202489x_{16} = -46.1427715202489
x17=−10.020637334688x_{17} = -10.020637334688
x18=−60.2795440288829x_{18} = -60.2795440288829
x19=5.70813221590859x_{19} = 5.70813221590859
x20=−6.88274978547193x_{20} = -6.88274978547193
x21=−88.5534678353211x_{21} = -88.5534678353211
x22=−3.75143040881351x_{22} = -3.75143040881351
x23=−52.4257551472372x_{23} = -52.4257551472372
x24=−64.9918397980093x_{24} = -64.9918397980093
x25=−61.8503077012464x_{25} = -61.8503077012464
x26=−66.5626079973583x_{26} = -66.5626079973583
x27=−17.871122719003x_{27} = -17.871122719003
x28=−74.4164668604379x_{28} = -74.4164668604379
x29=−53.9965084187389x_{29} = -53.9965084187389
x30=−30.435679498307x_{30} = -30.435679498307
x31=−47.7135124234798x_{31} = -47.7135124234798
x32=−50.8550045474444x_{32} = -50.8550045474444
x33=−90.1242489436633x_{33} = -90.1242489436633
x34=−72.8456929632852x_{34} = -72.8456929632852
x35=−25.7237612161477x_{35} = -25.7237612161477
x36=−80.6995712055717x_{36} = -80.6995712055717
x37=1.0428975219455x_{37} = 1.0428975219455
x38=−58.7087821107136x_{38} = -58.7087821107136
x39=−38.2891356226561x_{39} = -38.2891356226561
x40=−14.7304744932575x_{40} = -14.7304744932575
x41=−83.8411279411402x_{41} = -83.8411279411402
x42=−24.1531629870512x_{42} = -24.1531629870512
x43=−8.45134053350092x_{43} = -8.45134053350092
x44=−75.98724168771x_{44} = -75.98724168771
x45=−16.3007524805305x_{45} = -16.3007524805305
x46=−21.0120563392475x_{46} = -21.0120563392475
x47=−82.2703492295073x_{47} = -82.2703492295073
x48=−43.0013019256564x_{48} = -43.0013019256564
x49=−94.8365953007974x_{49} = -94.8365953007974
x50=−44.5720345434575x_{50} = -44.5720345434575
x51=−32.0063499441685x_{51} = -32.0063499441685
x52=4.14197062912355x_{52} = 4.14197062912355
x53=−0.118279356560721x_{53} = -0.118279356560721
x54=−39.8598518400666x_{54} = -39.8598518400666
Signos de extremos en los puntos:
(-22.582592591977225, 1.72491739457128e-28)

(-91.69503057471384, 6.25939213081828e-118)

(-86.98268727808012, -8.19054028565859e-112)

(-68.13337749800418, -2.32280428417938e-87)

(-2.1978251734507377, -0.00670448291291365)

(7.276224451411985, 42594872645.1624)

(-36.718426301808925, -1.06860818414622e-46)

(-105.83208492744305, -2.75247531407714e-136)

(-97.97816182202118, 4.35569811524792e-126)

(-96.40737834801105, -4.77093204884725e-124)

(-55.75, -3.79932597305764e-71)

(-11.590348125772172, -1.87477086550131e-14)

(-28.86502286063435, 1.43589018319356e-36)

(-33.57703224739653, -1.21088624734345e-42)

(-69.70414821168492, 2.13474967074143e-89)

(-46.142771520248886, -7.05757535736511e-59)

(-10.020637334688034, 1.8041295749359e-12)

(-60.27954402888286, 3.51272625290329e-77)

(5.70813221590859, -300052483.294721)

(-6.882749785471929, 1.53485060827482e-8)

(-88.55346783532114, 7.49067320346596e-114)

(-3.7514304088135058, 0.000103444814131522)

(-52.425755147237204, -5.22204011678467e-67)

(-64.9918397980093, -2.74561876327491e-83)

(-61.850307701246365, -3.23781413016059e-79)

(-66.56260799735826, 2.52608167078262e-85)

(-17.871122719003, -1.88286821351864e-22)

(-74.41646686043794, -1.65220605562237e-95)

(-53.99650841873891, 4.83166714147487e-69)

(-30.43567949830701, -1.36009686456514e-38)

(-47.713512423479806, 6.55585316908051e-61)

(-50.8550045474444, 5.63888649296277e-65)

(-90.12424894366329, -6.84845299013871e-116)

(-72.84569296328516, 1.80037693417602e-93)

(-25.723761216147697, 1.58565085911933e-32)

(-80.69957120557174, -1.16683304779487e-103)

(1.0428975219454952, 38.7334370951176)

(-58.70878211071359, -3.80839247816075e-75)

(-38.28913562265615, 1.00102857705261e-48)

(-14.730474493257542, -1.92304073899592e-18)

(-83.84112794114019, -9.78285765525164e-108)

(-24.153162987051186, -1.65732692958366e-30)

(-8.451340533500924, -1.69352449049274e-10)

(-75.98724168770995, 1.51555407382563e-97)

(-16.30075248053046, 1.91174259567256e-20)

(-21.012056339247465, -1.78657972459598e-26)

(-82.27034922950726, 1.06860303057626e-105)

(-43.00130192565645, -8.15006951914614e-55)

(-94.83659530079738, 5.22436345162697e-122)

(-44.57203454345748, 7.5888901835825e-57)

(-32.00634994416847, 1.28487301469379e-40)

(4.141970629123551, 1954009.7270685)

(-0.11827935656072107, -0.102956828898738)

(-39.859851840066575, -9.36144636810416e-51)


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=−86.9826872780801x_{1} = -86.9826872780801
x2=−68.1333774980042x_{2} = -68.1333774980042
x3=−2.19782517345074x_{3} = -2.19782517345074
x4=−36.7184263018089x_{4} = -36.7184263018089
x5=−105.832084927443x_{5} = -105.832084927443
x6=−96.4073783480111x_{6} = -96.4073783480111
x7=−11.5903481257722x_{7} = -11.5903481257722
x8=−33.5770322473965x_{8} = -33.5770322473965
x9=−46.1427715202489x_{9} = -46.1427715202489
x10=5.70813221590859x_{10} = 5.70813221590859
x11=−52.4257551472372x_{11} = -52.4257551472372
x12=−64.9918397980093x_{12} = -64.9918397980093
x13=−61.8503077012464x_{13} = -61.8503077012464
x14=−17.871122719003x_{14} = -17.871122719003
x15=−74.4164668604379x_{15} = -74.4164668604379
x16=−30.435679498307x_{16} = -30.435679498307
x17=−90.1242489436633x_{17} = -90.1242489436633
x18=−80.6995712055717x_{18} = -80.6995712055717
x19=−58.7087821107136x_{19} = -58.7087821107136
x20=−14.7304744932575x_{20} = -14.7304744932575
x21=−83.8411279411402x_{21} = -83.8411279411402
x22=−24.1531629870512x_{22} = -24.1531629870512
x23=−8.45134053350092x_{23} = -8.45134053350092
x24=−21.0120563392475x_{24} = -21.0120563392475
x25=−43.0013019256564x_{25} = -43.0013019256564
x26=−0.118279356560721x_{26} = -0.118279356560721
x27=−39.8598518400666x_{27} = -39.8598518400666
Puntos máximos de la función:
x27=−22.5825925919772x_{27} = -22.5825925919772
x27=−91.6950305747138x_{27} = -91.6950305747138
x27=7.27622445141199x_{27} = 7.27622445141199
x27=−97.9781618220212x_{27} = -97.9781618220212
x27=−28.8650228606344x_{27} = -28.8650228606344
x27=−69.7041482116849x_{27} = -69.7041482116849
x27=−10.020637334688x_{27} = -10.020637334688
x27=−60.2795440288829x_{27} = -60.2795440288829
x27=−6.88274978547193x_{27} = -6.88274978547193
x27=−88.5534678353211x_{27} = -88.5534678353211
x27=−3.75143040881351x_{27} = -3.75143040881351
x27=−66.5626079973583x_{27} = -66.5626079973583
x27=−53.9965084187389x_{27} = -53.9965084187389
x27=−47.7135124234798x_{27} = -47.7135124234798
x27=−50.8550045474444x_{27} = -50.8550045474444
x27=−72.8456929632852x_{27} = -72.8456929632852
x27=−25.7237612161477x_{27} = -25.7237612161477
x27=1.0428975219455x_{27} = 1.0428975219455
x27=−38.2891356226561x_{27} = -38.2891356226561
x27=−75.98724168771x_{27} = -75.98724168771
x27=−16.3007524805305x_{27} = -16.3007524805305
x27=−82.2703492295073x_{27} = -82.2703492295073
x27=−94.8365953007974x_{27} = -94.8365953007974
x27=−44.5720345434575x_{27} = -44.5720345434575
x27=−32.0063499441685x_{27} = -32.0063499441685
x27=4.14197062912355x_{27} = 4.14197062912355
Decrece en los intervalos
[5.70813221590859,∞)\left[5.70813221590859, \infty\right)
Crece en los intervalos
(−∞,−105.832084927443]\left(-\infty, -105.832084927443\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
(9x(3sin⁡(2x)+2cos⁡(2x))−36xsin⁡(2x)+28xcos⁡(2x)+10sin⁡(2x)+24cos⁡(2x))e3x=0\left(9 x \left(3 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) - 36 x \sin{\left(2 x \right)} + 28 x \cos{\left(2 x \right)} + 10 \sin{\left(2 x \right)} + 24 \cos{\left(2 x \right)}\right) e^{3 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−18.1694506152937x_{1} = -18.1694506152937
x2=−32.3027619567363x_{2} = -32.3027619567363
x3=−96.7021783914176x_{3} = -96.7021783914176
x4=−0.316649795177296x_{4} = -0.316649795177296
x5=−68.4285094849262x_{5} = -68.4285094849262
x6=−88.8483387998132x_{6} = -88.8483387998132
x7=−16.5994992026264x_{7} = -16.5994992026264
x8=−37.0145281750185x_{8} = -37.0145281750185
x9=−19.7395395489895x_{9} = -19.7395395489895
x10=−5.62415769536112x_{10} = -5.62415769536112
x11=−30.7322162885513x_{11} = -30.7322162885513
x12=−26.020764045764x_{12} = -26.020764045764
x13=−2.52675038860602x_{13} = -2.52675038860602
x14=−46.438443494014x_{14} = -46.438443494014
x15=−90.4191047340094x_{15} = -90.4191047340094
x16=−27.5912119882652x_{16} = -27.5912119882652
x17=−4.0661734412533x_{17} = -4.0661734412533
x18=−41.7264365901015x_{18} = -41.7264365901015
x19=6.99260503010282x_{19} = 6.99260503010282
x20=−85.7068102914697x_{20} = -85.7068102914697
x21=−44.8677655177043x_{21} = -44.8677655177043
x22=−80.9945269120609x_{22} = -80.9945269120609
x23=−48.0091292885884x_{23} = -48.0091292885884
x24=−71.5700021396748x_{24} = -71.5700021396748
x25=−91.9898717113413x_{25} = -91.9898717113413
x26=−63.7162891255144x_{26} = -63.7162891255144
x27=−49.579822151541x_{27} = -49.579822151541
x28=−76.2822566583722x_{28} = -76.2822566583722
x29=−98.2729490469616x_{29} = -98.2729490469616
x30=−69.9992546829513x_{30} = -69.9992546829513
x31=−74.7115032558754x_{31} = -74.7115032558754
x32=−77.8530118031971x_{32} = -77.8530118031971
x33=5.42728278477148x_{33} = 5.42728278477148
x34=−112.18554520386x_{34} = -112.18554520386
x35=−59.0040959095527x_{35} = -59.0040959095527
x36=−55.862652265148x_{36} = -55.862652265148
x37=−84.1360478434666x_{37} = -84.1360478434666
x38=−33.8733311843724x_{38} = -33.8733311843724
x39=−11.8910353909642x_{39} = -11.8910353909642
x40=−52.7212265336866x_{40} = -52.7212265336866
x41=−10.3223785344063x_{41} = -10.3223785344063
x42=−62.1455547304287x_{42} = -62.1455547304287
x43=−82.5652866917158x_{43} = -82.5652866917158
x44=−38.5851511165807x_{44} = -38.5851511165807
x45=−8.7545283830898x_{45} = -8.7545283830898
x46=2.31477730881784x_{46} = 2.31477730881784
x47=−66.8577667058875x_{47} = -66.8577667058875
x48=−60.5748235716208x_{48} = -60.5748235716208
x49=−24.4503616986276x_{49} = -24.4503616986276
x50=3.86587685540702x_{50} = 3.86587685540702
x51=−99.8437205139716x_{51} = -99.8437205139716
x52=−15.0297296921925x_{52} = -15.0297296921925
x53=−40.1557877769703x_{53} = -40.1557877769703
x54=−22.880014521131x_{54} = -22.880014521131
x55=−54.2919369643226x_{55} = -54.2919369643226

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
[5.42728278477148,∞)\left[5.42728278477148, \infty\right)
Convexa en los intervalos
(−∞,−99.8437205139716]\left(-\infty, -99.8437205139716\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((2xcos⁡(2x)+3xsin⁡(2x))e3x)=0\lim_{x \to -\infty}\left(\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((2xcos⁡(2x)+3xsin⁡(2x))e3x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((2*x)*cos(2*x) + (3*x)*sin(2*x))*exp(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((2xcos⁡(2x)+3xsin⁡(2x))e3xx)=0\lim_{x \to -\infty}\left(\frac{\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((2xcos⁡(2x)+3xsin⁡(2x))e3xx)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\frac{\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x}}{x}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
(2xcos⁡(2x)+3xsin⁡(2x))e3x=(3xsin⁡(2x)−2xcos⁡(2x))e−3x\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x} = \left(3 x \sin{\left(2 x \right)} - 2 x \cos{\left(2 x \right)}\right) e^{- 3 x}
- No
(2xcos⁡(2x)+3xsin⁡(2x))e3x=−(3xsin⁡(2x)−2xcos⁡(2x))e−3x\left(2 x \cos{\left(2 x \right)} + 3 x \sin{\left(2 x \right)}\right) e^{3 x} = - \left(3 x \sin{\left(2 x \right)} - 2 x \cos{\left(2 x \right)}\right) e^{- 3 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор