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(-1-40*cos(2*x)*exp(x/2)/17+10*exp(x/2)*sin(2*x)/17)*exp(-x/2)
Trazado el gráfico de la función/ (-1-40*cos(2*x)*exp(x/2)/17+10*exp(x/2)*sin(2*x)/17)*exp(-x/2)

Gráfico de la función y = (-1-40*cos(2*x)*exp(x/2)/17+10*exp(x/2)*sin(2*x)/17)*exp(-x/2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /                  x       x         \     
       |                  -       -         |  -x 
       |                  2       2         |  ---
       |     40*cos(2*x)*e    10*e *sin(2*x)|   2 
f(x) = |-1 - -------------- + --------------|*e   
       \           17               17      /
f(x)=(10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2f{\left(x \right)} = \left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}} 
f = (((10*exp(x/2))*sin(2*x))/17 - exp(x/2)*(40*cos(2*x))/17 - 1)*exp((-x)/2)
Gráfico de la función
02468-8-6-4-2-1010-200200
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:
(10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2=0\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=39.932817001267x_{1} = 39.932817001267
x2=153.030152530939x_{2} = 153.030152530939
x3=10.0890153891536x_{3} = 10.0890153891536
x4=44.6452059821328x_{4} = 44.6452059821328
x5=77.631928844784x_{5} = 77.631928844784
x6=−1.3963457481719x_{6} = -1.3963457481719
x7=33.649631684358x_{7} = 33.649631684358
x8=8.5139702712992x_{8} = 8.5139702712992
x9=2.16359181153048x_{9} = 2.16359181153048
x10=98.0522810931176x_{10} = 98.0522810931176
x11=11.6578767758034x_{11} = 11.6578767758034
x12=66.6363545572197x_{12} = 66.6363545572197
x13=24.2248526017884x_{13} = 24.2248526017884
x14=90.1982994591431x_{14} = 90.1982994591431
x15=22.6540598896833x_{15} = 22.6540598896833
x16=6.95246938121567x_{16} = 6.95246938121567
x17=76.0611325179891x_{17} = 76.0611325179891
x18=30.5080389920203x_{18} = 30.5080389920203
x19=52.4991876160648x_{19} = 52.4991876160648
x20=19.512476696388x_{20} = 19.512476696388
x21=91.769095785938x_{21} = 91.769095785938
x22=88.6275031323482x_{22} = 88.6275031323482
x23=60.3531692500401x_{23} = 60.3531692500401
x24=41.5036133287017x_{24} = 41.5036133287017
x25=3.83482221260289x_{25} = 3.83482221260289
x26=47.7867986356896x_{26} = 47.7867986356896
x27=74.4903361911942x_{27} = 74.4903361911942
x28=63.4947619036299x_{28} = 63.4947619036299
x29=55.6407802696552x_{29} = 55.6407802696552
x30=25.7956505766598x_{30} = 25.7956505766598
x31=79.2027251715788x_{31} = 79.2027251715788
x32=16.3709295500805x_{32} = 16.3709295500805
x33=82.3443178251686x_{33} = 82.3443178251686
x34=17.9416422317013x_{34} = 17.9416422317013
x35=0.80272569852102x_{35} = 0.80272569852102
x36=99.6230774199125x_{36} = 99.6230774199125
x37=54.0699839428609x_{37} = 54.0699839428609
x38=32.078835390035x_{38} = 32.078835390035
x39=85.4859104787584x_{39} = 85.4859104787584
x40=69.7779472108095x_{40} = 69.7779472108095
x41=68.2071508840146x_{41} = 68.2071508840146
x42=83.9151141519635x_{42} = 83.9151141519635
x43=46.216002308867x_{43} = 46.216002308867
x44=38.3620206758753x_{44} = 38.3620206758753
x45=96.4814847663227x_{45} = 96.4814847663227
x46=61.923965576835x_{46} = 61.923965576835
x47=−1.75612164764682x_{47} = -1.75612164764682
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 (-1 - (40*cos(2*x))*exp(x/2)/17 + ((10*exp(x/2))*sin(2*x))/17)*exp((-x)/2).
((−e02⋅40cos⁡(0⋅2)17−1)+10e02sin⁡(0⋅2)17)e(−1)02\left(\left(- \frac{e^{\frac{0}{2}} \cdot 40 \cos{\left(0 \cdot 2 \right)}}{17} - 1\right) + \frac{10 e^{\frac{0}{2}} \sin{\left(0 \cdot 2 \right)}}{17}\right) e^{\frac{\left(-1\right) 0}{2}}
Resultado:
f(0)=−5717f{\left(0 \right)} = - \frac{57}{17}
Punto:
(0, -57/17)
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
−(10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x22+5e(−1)x2ex2sin⁡(2x)=0- \frac{\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}}}{2} + 5 e^{\frac{\left(-1\right) x}{2}} e^{\frac{x}{2}} \sin{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=21.8686583243532x_{1} = 21.8686583243532
x2=72.1341417010018x_{2} = 72.1341417010018
x3=81.5589196617712x_{3} = 81.5589196617712
x4=−1.57879569949206x_{4} = -1.57879569949206
x5=6.15832529660957x_{5} = 6.15832529660957
x6=40.7182151650297x_{6} = 40.7182151650297
x7=92.5544939493355x_{7} = 92.5544939493355
x8=87.8421049689508x_{8} = 87.8421049689508
x9=4.59507969182983x_{9} = 4.59507969182983
x10=37.5766225111572x_{10} = 37.5766225111572
x11=28.151844511022x_{11} = 28.151844511022
x12=26.5810483110729x_{12} = 26.5810483110729
x13=43.8598078186783x_{13} = 43.8598078186783
x14=20.2978649328565x_{14} = 20.2978649328565
x15=70.5633453742069x_{15} = 70.5633453742069
x16=50.1429931258726x_{16} = 50.1429931258726
x17=100.40847558331x_{17} = 100.40847558331
x18=86.2713086421559x_{18} = 86.2713086421559
x19=67.4217527206171x_{19} = 67.4217527206171
x20=73.7049380277967x_{20} = 73.7049380277967
x21=14.0147242621816x_{21} = 14.0147242621816
x22=18.7270621681194x_{22} = 18.7270621681194
x23=59.5677710866426x_{23} = 59.5677710866426
x24=−3.60092572697893x_{24} = -3.60092572697893
x25=7.73257131940171x_{25} = 7.73257131940171
x26=57.9969747598478x_{26} = 57.9969747598478
x27=56.4261784330528x_{27} = 56.4261784330528
x28=78.4173270081814x_{28} = 78.4173270081814
x29=42.2890114919326x_{29} = 42.2890114919326
x30=62.7093637402324x_{30} = 62.7093637402324
x31=29.7226408956507x_{31} = 29.7226408956507
x32=−0.178972890023548x_{32} = -0.178972890023548
x33=34.4350298562077x_{33} = 34.4350298562077
x34=51.7137894526685x_{34} = 51.7137894526685
x35=12.4437789529758x_{35} = 12.4437789529758
x36=84.700512315361x_{36} = 84.700512315361
x37=65.8509563938222x_{37} = 65.8509563938222
x38=94.1252902761304x_{38} = 94.1252902761304
x39=36.0058261855018x_{39} = 36.0058261855018
x40=64.2801600670273x_{40} = 64.2801600670273
x41=79.9881233349763x_{41} = 79.9881233349763
x42=95.6960866029253x_{42} = 95.6960866029253
x43=48.5721967990798x_{43} = 48.5721967990798
x44=−4.29438235525874x_{44} = -4.29438235525874
x45=15.5854526649682x_{45} = 15.5854526649682
Signos de extremos en los puntos:
(21.868658324353184, -2.42537408569055)

(72.13414170100181, -2.42535625036333)

(81.5589196617712, -2.42535625036333)

(-1.5787956994920558, 0.159980628969631)

(6.158325296609574, -2.4713267456605)

(40.7182151650297, -2.42535625180263)

(92.55449394933547, 2.42535625036333)

(87.84210496895078, -2.42535625036333)

(4.595079691829826, 2.32472029353113)

(37.57662251115725, -2.42535625728706)

(28.151844511021974, -2.42535702109753)

(26.58104831107288, 2.42535455992743)

(43.859807818678256, -2.42535625066253)

(20.297864932856466, 2.42531713252463)

(70.56334537420692, 2.42535625036333)

(50.14299312587259, -2.42535625037626)

(100.40847558330995, -2.42535625036333)

(86.27130864215589, 2.42535625036333)

(67.42175272061712, 2.42535625036333)

(73.70493802779671, 2.42535625036333)

(14.014724262181575, 2.42445104658303)

(18.727062168119353, -2.42544204687666)

(59.56777108664264, -2.42535625036345)

(-3.600925726978934, -7.9479283735532)

(7.732571319401707, 2.40441461421923)

(57.99697475984775, 2.42535625036307)

(56.42617843305282, -2.42535625036389)

(78.4173270081814, -2.42535625036333)

(42.2890114919326, 2.4253562497071)

(62.70936374023243, -2.42535625036335)

(29.722640895650713, 2.42535589895623)

(-0.1789728900235484, -3.50350953431561)

(34.43502985620772, -2.42535628366977)

(51.71378945266846, 2.42535625035743)

(12.44377895297577, -2.4273416897035)

(84.70051231536098, -2.42535625036333)

(65.85095639382223, -2.42535625036334)

(94.12529027613037, -2.42535625036333)

(36.00582618550184, 2.42535623517765)

(64.28016006702732, 2.42535625036332)

(79.9881233349763, 2.42535625036333)

(95.69608660292526, 2.42535625036333)

(48.572196799079826, 2.42535625033497)

(-4.294382355258739, -7.41976202031498)

(15.585452664968184, -2.42576897428859)


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=21.8686583243532x_{1} = 21.8686583243532
x2=72.1341417010018x_{2} = 72.1341417010018
x3=81.5589196617712x_{3} = 81.5589196617712
x4=6.15832529660957x_{4} = 6.15832529660957
x5=40.7182151650297x_{5} = 40.7182151650297
x6=87.8421049689508x_{6} = 87.8421049689508
x7=37.5766225111572x_{7} = 37.5766225111572
x8=28.151844511022x_{8} = 28.151844511022
x9=43.8598078186783x_{9} = 43.8598078186783
x10=50.1429931258726x_{10} = 50.1429931258726
x11=100.40847558331x_{11} = 100.40847558331
x12=18.7270621681194x_{12} = 18.7270621681194
x13=59.5677710866426x_{13} = 59.5677710866426
x14=−3.60092572697893x_{14} = -3.60092572697893
x15=56.4261784330528x_{15} = 56.4261784330528
x16=78.4173270081814x_{16} = 78.4173270081814
x17=62.7093637402324x_{17} = 62.7093637402324
x18=−0.178972890023548x_{18} = -0.178972890023548
x19=34.4350298562077x_{19} = 34.4350298562077
x20=12.4437789529758x_{20} = 12.4437789529758
x21=84.700512315361x_{21} = 84.700512315361
x22=65.8509563938222x_{22} = 65.8509563938222
x23=94.1252902761304x_{23} = 94.1252902761304
x24=15.5854526649682x_{24} = 15.5854526649682
Puntos máximos de la función:
x24=−1.57879569949206x_{24} = -1.57879569949206
x24=92.5544939493355x_{24} = 92.5544939493355
x24=4.59507969182983x_{24} = 4.59507969182983
x24=26.5810483110729x_{24} = 26.5810483110729
x24=20.2978649328565x_{24} = 20.2978649328565
x24=70.5633453742069x_{24} = 70.5633453742069
x24=86.2713086421559x_{24} = 86.2713086421559
x24=67.4217527206171x_{24} = 67.4217527206171
x24=73.7049380277967x_{24} = 73.7049380277967
x24=14.0147242621816x_{24} = 14.0147242621816
x24=7.73257131940171x_{24} = 7.73257131940171
x24=57.9969747598478x_{24} = 57.9969747598478
x24=42.2890114919326x_{24} = 42.2890114919326
x24=29.7226408956507x_{24} = 29.7226408956507
x24=51.7137894526685x_{24} = 51.7137894526685
x24=36.0058261855018x_{24} = 36.0058261855018
x24=64.2801600670273x_{24} = 64.2801600670273
x24=79.9881233349763x_{24} = 79.9881233349763
x24=95.6960866029253x_{24} = 95.6960866029253
x24=48.5721967990798x_{24} = 48.5721967990798
x24=−4.29438235525874x_{24} = -4.29438235525874
Decrece en los intervalos
[100.40847558331,∞)\left[100.40847558331, \infty\right)
Crece en los intervalos
(−∞,−3.60092572697893]\left(-\infty, -3.60092572697893\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
−(−10ex2sin⁡(2x)+40ex2cos⁡(2x)+17)e−x268−5sin⁡(2x)2+10cos⁡(2x)=0- \frac{\left(- 10 e^{\frac{x}{2}} \sin{\left(2 x \right)} + 40 e^{\frac{x}{2}} \cos{\left(2 x \right)} + 17\right) e^{- \frac{x}{2}}}{68} - \frac{5 \sin{\left(2 x \right)}}{2} + 10 \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=153.030152530939x_{1} = 153.030152530939
x2=33.6496316951624x_{2} = 33.6496316951624
x3=−2.5242679966991x_{3} = -2.5242679966991
x4=22.6540572517923x_{4} = 22.6540572517923
x5=25.7956500282956x_{5} = 25.7956500282956
x6=77.631928844784x_{6} = 77.631928844784
x7=3.80257681447479x_{7} = 3.80257681447479
x8=47.7867986356804x_{8} = 47.7867986356804
x9=16.3708685090298x_{9} = 16.3708685090298
x10=61.923965576835x_{10} = 61.923965576835
x11=−5.87402821228329x_{11} = -5.87402821228329
x12=98.0522810931176x_{12} = 98.0522810931176
x13=11.6585210036587x_{13} = 11.6585210036587
x14=60.3531692500401x_{14} = 60.3531692500401
x15=38.3620206748513x_{15} = 38.3620206748513
x16=90.1982994591431x_{16} = 90.1982994591431
x17=66.6363545572197x_{17} = 66.6363545572197
x18=28.9372427074366x_{18} = 28.9372427074366
x19=63.4947619036299x_{19} = 63.4947619036299
x20=17.9416700637349x_{20} = 17.9416700637349
x21=76.0611325179891x_{21} = 76.0611325179891
x22=44.6452059820885x_{22} = 44.6452059820885
x23=8.51707269403793x_{23} = 8.51707269403793
x24=91.769095785938x_{24} = 91.769095785938
x25=−3.95580794955927x_{25} = -3.95580794955927
x26=88.6275031323482x_{26} = 88.6275031323482
x27=2.23791361513602x_{27} = 2.23791361513602
x28=74.4903361911942x_{28} = 74.4903361911942
x29=79.2027251715788x_{29} = 79.2027251715788
x30=55.6407802696554x_{30} = 55.6407802696554
x31=46.2160023088872x_{31} = 46.2160023088872
x32=39.9328170017339x_{32} = 39.9328170017339
x33=30.5080390439944x_{33} = 30.5080390439944
x34=82.3443178251686x_{34} = 82.3443178251686
x35=−6.76141068997644x_{35} = -6.76141068997644
x36=52.4991876160657x_{36} = 52.4991876160657
x37=19.5124640069296x_{37} = 19.5124640069296
x38=0.65361555931973x_{38} = 0.65361555931973
x39=85.4859104787584x_{39} = 85.4859104787584
x40=−0.887797761422613x_{40} = -0.887797761422613
x41=24.2248538045055x_{41} = 24.2248538045055
x42=41.5036133284888x_{42} = 41.5036133284888
x43=99.6230774199125x_{43} = 99.6230774199125
x44=10.0876036967842x_{44} = 10.0876036967842
x45=54.0699839428605x_{45} = 54.0699839428605
x46=68.2071508840146x_{46} = 68.2071508840146
x47=69.7779472108095x_{47} = 69.7779472108095
x48=83.9151141519635x_{48} = 83.9151141519635
x49=32.078835366338x_{49} = 32.078835366338
x50=96.4814847663227x_{50} = 96.4814847663227

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
[153.030152530939,∞)\left[153.030152530939, \infty\right)
Convexa en los intervalos
(−∞,−6.76141068997644]\left(-\infty, -6.76141068997644\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2)=−∞\lim_{x \to -\infty}\left(\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}}\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→∞((10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2)y = \lim_{x \to \infty}\left(\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-1 - (40*cos(2*x))*exp(x/2)/17 + ((10*exp(x/2))*sin(2*x))/17)*exp((-x)/2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2x)=∞\lim_{x \to -\infty}\left(\frac{\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}}}{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→∞((10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2x)y = x \lim_{x \to \infty}\left(\frac{\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}}}{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:
(10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2=(−1−10e−x2sin⁡(2x)17−40e−x2cos⁡(2x)17)ex2\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}} = \left(-1 - \frac{10 e^{- \frac{x}{2}} \sin{\left(2 x \right)}}{17} - \frac{40 e^{- \frac{x}{2}} \cos{\left(2 x \right)}}{17}\right) e^{\frac{x}{2}}
- No
(10ex2sin⁡(2x)17+(−ex2⋅40cos⁡(2x)17−1))e(−1)x2=−(−1−10e−x2sin⁡(2x)17−40e−x2cos⁡(2x)17)ex2\left(\frac{10 e^{\frac{x}{2}} \sin{\left(2 x \right)}}{17} + \left(- \frac{e^{\frac{x}{2}} \cdot 40 \cos{\left(2 x \right)}}{17} - 1\right)\right) e^{\frac{\left(-1\right) x}{2}} = - \left(-1 - \frac{10 e^{- \frac{x}{2}} \sin{\left(2 x \right)}}{17} - \frac{40 e^{- \frac{x}{2}} \cos{\left(2 x \right)}}{17}\right) e^{\frac{x}{2}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (-1-40*cos(2*x)*exp(x/2)/17+10*exp(x/2)*sin(2*x)/17)*exp(-x/2)
    © Sr Examen – Калькулятор