Sr Examen

Otras calculadoras

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

Gráfico de la función y = (1+x^2+2*x+(-2*sin(x)+cos(x))*exp(x))*exp(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /     2                               x\  x
f(x) = \1 + x  + 2*x + (-2*sin(x) + cos(x))*e /*e
f(x)=((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)exf{\left(x \right)} = \left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} 
f = (2*x + x^2 + 1 + (-2*sin(x) + cos(x))*exp(x))*exp(x)
Gráfico de la función
02468201012141618-500000000000000000500000000000000000
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:
((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex=0\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−65.7042784040527x_{1} = -65.7042784040527
x2=−57.862646187602x_{2} = -57.862646187602
x3=−53.9671706290045x_{3} = -53.9671706290045
x4=−63.7388815603694x_{4} = -63.7388815603694
x5=−36.98833015896x_{5} = -36.98833015896
x6=3.29434699426817x_{6} = 3.29434699426817
x7=−85.4672817459601x_{7} = -85.4672817459601
x8=−44.095029077399x_{8} = -44.095029077399
x9=−52.0283094760939x_{9} = -52.0283094760939
x10=−101.357042977005x_{10} = -101.357042977005
x11=−50.0968145541527x_{11} = -50.0968145541527
x12=−73.5898302946287x_{12} = -73.5898302946287
x13=−113.297937218974x_{13} = -113.297937218974
x14=−97.3805776811866x_{14} = -97.3805776811866
x15=−111.30676502525x_{15} = -111.30676502525
x16=−77.5437076550386x_{16} = -77.5437076550386
x17=−42.480814213486x_{17} = -42.480814213486
x18=−81.5031793443065x_{18} = -81.5031793443065
x19=−89.4352613345747x_{19} = -89.4352613345747
x20=6.77764951408605x_{20} = 6.77764951408605
x21=−59.8176028297081x_{21} = -59.8176028297081
x22=−93.4065201895675x_{22} = -93.4065201895675
x23=−67.6723516792106x_{23} = -67.6723516792106
x24=−40.6193242741437x_{24} = -40.6193242741437
x25=−115.289461427472x_{25} = -115.289461427472
x26=−35.2437105789005x_{26} = -35.2437105789005
x27=−121.265947133446x_{27} = -121.265947133446
x28=−79.5228101394918x_{28} = -79.5228101394918
x29=1.17559091199161x_{29} = 1.17559091199161
x30=−75.5659993047837x_{30} = -75.5659993047837
x31=−95.3932253723146x_{31} = -95.3932253723146
x32=−44.3632645762336x_{32} = -44.3632645762336
x33=−87.450827396631x_{33} = -87.450827396631
x34=9.88572836927688x_{34} = 9.88572836927688
x35=13.0302112843657x_{35} = 13.0302112843657
x36=−105.335594954962x_{36} = -105.335594954962
x37=−71.615366862757x_{37} = -71.615366862757
x38=−119.273484753832x_{38} = -119.273484753832
x39=−107.325568409955x_{39} = -107.325568409955
x40=−48.1741343685241x_{40} = -48.1741343685241
x41=−55.9122546875788x_{41} = -55.9122546875788
x42=−117.281316969027x_{42} = -117.281316969027
x43=−83.4847029913442x_{43} = -83.4847029913442
x44=−69.6428002602809x_{44} = -69.6428002602809
x45=−46.2621345566293x_{45} = -46.2621345566293
x46=−103.34607591362x_{46} = -103.34607591362
x47=−109.315967289782x_{47} = -109.315967289782
x48=−99.3685308617713x_{48} = -99.3685308617713
x49=−38.7852769640357x_{49} = -38.7852769640357
x50=−61.7765159711448x_{50} = -61.7765159711448
x51=−91.420513277015x_{51} = -91.420513277015
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 + x^2 + 2*x + (-2*sin(x) + cos(x))*exp(x))*exp(x).
((0⋅2+(02+1))+(−2sin⁡(0)+cos⁡(0))e0)e0\left(\left(0 \cdot 2 + \left(0^{2} + 1\right)\right) + \left(- 2 \sin{\left(0 \right)} + \cos{\left(0 \right)}\right) e^{0}\right) e^{0}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
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
((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex+(2x+(−2sin⁡(x)+cos⁡(x))ex+(−sin⁡(x)−2cos⁡(x))ex+2)ex=0\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} + \left(2 x + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} + \left(- \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{x} + 2\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−93.4135149755732x_{1} = -93.4135149755732
x2=−52.0625314136456x_{2} = -52.0625314136456
x3=−65.7215707857911x_{3} = -65.7215707857911
x4=−38.8866033852926x_{4} = -38.8866033852926
x5=−61.7970469090044x_{5} = -61.7970469090044
x6=−40.7021606138828x_{6} = -40.7021606138828
x7=−105.34083441056x_{7} = -105.34083441056
x8=−109.320766981846x_{8} = -109.320766981846
x9=−67.6883070820612x_{9} = -67.6883070820612
x10=−69.6575690448132x_{10} = -69.6575690448132
x11=12.5665178864624x_{11} = 12.5665178864624
x12=−59.8401098081453x_{12} = -59.8401098081453
x13=2.88302862300745x_{13} = 2.88302862300745
x14=−113.302350381263x_{14} = -113.302350381263
x15=−81.5129914542486x_{15} = -81.5129914542486
x16=−87.4590521969804x_{16} = -87.4590521969804
x17=−101.362785703358x_{17} = -101.362785703358
x18=6.30797146170616x_{18} = 6.30797146170616
x19=−119.277400270678x_{19} = -119.277400270678
x20=0.677247254969318x_{20} = 0.677247254969318
x21=−48.218087356325x_{21} = -48.218087356325
x22=−57.8874330100957x_{22} = -57.8874330100957
x23=−103.351558330607x_{23} = -103.351558330607
x24=−63.7576880804052x_{24} = -63.7576880804052
x25=−79.5332552098772x_{25} = -79.5332552098772
x26=−35.4100272197758x_{26} = -35.4100272197758
x27=−99.3745529423273x_{27} = -99.3745529423273
x28=−1.69742009717348x_{28} = -1.69742009717348
x29=−91.4278853676496x_{29} = -91.4278853676496
x30=−117.285388562094x_{30} = -117.285388562094
x31=−83.4939382270426x_{31} = -83.4939382270426
x32=−42.5499668447831x_{32} = -42.5499668447831
x33=−44.4219620932777x_{33} = -44.4219620932777
x34=−2.97924417035108x_{34} = -2.97924417035108
x35=−111.311365356229x_{35} = -111.311365356229
x36=−75.577910111863x_{36} = -75.577910111863
x37=−95.3998711831007x_{37} = -95.3998711831007
x38=−121.269715394308x_{38} = -121.269715394308
x39=−55.93969182993x_{39} = -55.93969182993
x40=−107.330580740598x_{40} = -107.330580740598
x41=−115.293698637214x_{41} = -115.293698637214
x42=−85.4759897300653x_{42} = -85.4759897300653
x43=−71.6290775076557x_{43} = -71.6290775076557
x44=9.4226838207314x_{44} = 9.4226838207314
x45=−77.5548493298981x_{45} = -77.5548493298981
x46=−73.6025932625681x_{46} = -73.6025932625681
x47=−53.9977149340079x_{47} = -53.9977149340079
x48=−97.3869000709892x_{48} = -97.3869000709892
x49=−37.1157134040609x_{49} = -37.1157134040609
x50=−46.3126397955165x_{50} = -46.3126397955165
x51=−50.1354367467416x_{51} = -50.1354367467416
x52=15.7079538468433x_{52} = 15.7079538468433
x53=−89.4430422232197x_{53} = -89.4430422232197
Signos de extremos en los puntos:
(-93.41351497557318, 2.30407647152512e-37)

(-52.0625314136456, 6.39343972624134e-20)

(-65.72157078579109, 1.20110939427611e-25)

(-38.88660338529258, 1.85666723330323e-14)

(-61.79704690900439, 5.3659731984164e-24)

(-40.70216061388277, 3.31821504233813e-15)

(-105.34083441056026, 1.94073173390061e-42)

(-109.32076698184598, 3.90856730846474e-44)

(-67.68830708206119, 1.78419012120052e-26)

(-69.65756904481317, 2.63924061365338e-27)

(12.566517886462412, 82279095595.3478)

(-59.84010980814534, 3.55727518601997e-23)

(2.8830286230074504, -202.519543113505)

(-113.30235038126258, 7.83776312909701e-46)

(-81.51299145424862, 2.57685234663256e-32)

(-87.45905219698041, 7.77386830240892e-35)

(-101.36278570335809, 9.59062506726681e-41)

(6.307971461706159, 315612.507291013)

(-119.27740027067756, 2.20946327407096e-48)

(0.6772472549693177, 3.70091126944644)

(-48.21808735632502, 2.55486240594015e-18)

(-57.88743301009572, 2.34336503915629e-22)

(-103.35155833060679, 1.36514077767275e-41)

(-63.75768808040524, 8.04862624344384e-25)

(-79.53325520987718, 1.77522470555565e-31)

(-35.41002721977578, 4.95439474937152e-13)

(-99.37455294232734, 6.72892300611602e-40)

(-1.6974200971734845, 0.151404169344921)

(-91.42788536764964, 1.60686223129953e-36)

(-117.28538856209434, 1.56550059270147e-47)

(-83.49393822704256, 3.73153423063277e-33)

(-42.549966844783086, 5.72697856040409e-16)

(-44.421962093277685, 9.62068234384569e-17)

(-2.9792441703510804, 0.197412114144383)

(-111.31136535622898, 5.53769168851023e-45)

(-75.57791011186302, 8.35893797444063e-30)

(-95.3998711831007, 3.29841406899876e-38)

(-121.26971539430798, 3.11561903698671e-49)

(-55.93969182993003, 1.53275576033734e-21)

(-107.33058074059751, 2.75573477437007e-43)

(-115.29369863721371, 1.10822376178835e-46)

(-85.47598973006525, 5.39157812314283e-34)

(-71.62907750765571, 3.88914345170604e-28)

(9.422683820731399, -152207938.972468)

(-77.55484932989813, 1.21983840758805e-30)

(-73.60259326256813, 5.71089478551578e-29)

(-53.997714934007874, 9.94499704535209e-21)

(-97.38690007098918, 4.71457680860106e-39)

(-37.11571340406087, 9.9138620796487e-14)

(-46.31263979551651, 1.58166615656916e-17)

(-50.13543674674165, 4.0667372566538e-19)

(15.707953846843301, -44029653495906.4)

(-89.44304222321975, 1.11868461008945e-35)


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=2.88302862300745x_{1} = 2.88302862300745
x2=−1.69742009717348x_{2} = -1.69742009717348
x3=9.4226838207314x_{3} = 9.4226838207314
x4=15.7079538468433x_{4} = 15.7079538468433
Puntos máximos de la función:
x4=12.5665178864624x_{4} = 12.5665178864624
x4=6.30797146170616x_{4} = 6.30797146170616
x4=0.677247254969318x_{4} = 0.677247254969318
x4=−2.97924417035108x_{4} = -2.97924417035108
Decrece en los intervalos
[15.7079538468433,∞)\left[15.7079538468433, \infty\right)
Crece en los intervalos
(−∞,−1.69742009717348]\left(-\infty, -1.69742009717348\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
(x2+6x−4(sin⁡(x)+2cos⁡(x))ex−3(2sin⁡(x)−cos⁡(x))ex+7)ex=0\left(x^{2} + 6 x - 4 \left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{x} - 3 \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x} + 7\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2.46436440950957x_{1} = 2.46436440950957
x2=−40.7941628641187x_{2} = -40.7941628641187
x3=−65.7396188614987x_{3} = -65.7396188614987
x4=−93.4206982622687x_{4} = -93.4206982622687
x5=−73.6158294137195x_{5} = -73.6158294137195
x6=−101.368668013017x_{6} = -101.368668013017
x7=−41.4947390630672x_{7} = -41.4947390630672
x8=−95.4066913476332x_{8} = -95.4066913476332
x9=−113.306857078541x_{9} = -113.306857078541
x10=−46.3672208724377x_{10} = -46.3672208724377
x11=12.1028351149199x_{11} = 12.1028351149199
x12=−111.316065313131x_{12} = -111.316065313131
x13=−50.1767141989494x_{13} = -50.1767141989494
x14=−121.273557249601x_{14} = -121.273557249601
x15=−52.0989460197047x_{15} = -52.0989460197047
x16=−109.325672932625x_{16} = -109.325672932625
x17=−39.0008140386269x_{17} = -39.0008140386269
x18=−54.03009245017x_{18} = -54.03009245017
x19=−71.6433164421341x_{19} = -71.6433164421341
x20=−97.3933840527324x_{20} = -97.3933840527324
x21=−35.6073726027789x_{21} = -35.6073726027789
x22=−44.4858568772657x_{22} = -44.4858568772657
x23=−69.672930203154x_{23} = -69.672930203154
x24=−87.4675183643145x_{24} = -87.4675183643145
x25=−115.298023778539x_{25} = -115.298023778539
x26=−107.33570656639x_{26} = -107.33570656639
x27=−81.5231196989186x_{27} = -81.5231196989186
x28=−55.9686782648593x_{28} = -55.9686782648593
x29=−89.4510448712257x_{29} = -89.4510448712257
x30=−67.704929844768x_{30} = -67.704929844768
x31=−57.9135416592293x_{31} = -57.9135416592293
x32=−59.8637540821341x_{32} = -59.8637540821341
x33=−77.5663751001662x_{33} = -77.5663751001662
x34=−63.7773551203297x_{34} = -63.7773551203297
x35=−42.6259139055306x_{35} = -42.6259139055306
x36=0.186401959420659x_{36} = 0.186401959420659
x37=−117.289542922953x_{37} = -117.289542922953
x38=−103.357170818505x_{38} = -103.357170818505
x39=−105.346195271843x_{39} = -105.346195271843
x40=−61.8185637723201x_{40} = -61.8185637723201
x41=5.83935990103028x_{41} = 5.83935990103028
x42=−37.2622641517238x_{42} = -37.2622641517238
x43=15.244308568669x_{43} = 15.244308568669
x44=8.9595093226765x_{44} = 8.9595093226765
x45=−4.44833009887346x_{45} = -4.44833009887346
x46=−85.4849610815635x_{46} = -85.4849610815635
x47=−91.4354616849822x_{47} = -91.4354616849822
x48=−2.15514583450327x_{48} = -2.15514583450327
x49=−75.5902464456082x_{49} = -75.5902464456082
x50=−48.2653023401513x_{50} = -48.2653023401513
x51=−119.281393788779x_{51} = -119.281393788779
x52=−99.3807250722054x_{52} = -99.3807250722054
x53=−79.544048180243x_{53} = -79.544048180243
x54=−83.5034616017267x_{54} = -83.5034616017267

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
[15.244308568669,∞)\left[15.244308568669, \infty\right)
Convexa en los intervalos
(−∞,−2.15514583450327]\left(-\infty, -2.15514583450327\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex)=0\lim_{x \to -\infty}\left(\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex)y = \lim_{x \to \infty}\left(\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 + x^2 + 2*x + (-2*sin(x) + cos(x))*exp(x))*exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)exx)=0\lim_{x \to -\infty}\left(\frac{\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x}}{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→∞(((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)exx)y = x \lim_{x \to \infty}\left(\frac{\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x}}{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:
((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex=(x2−2x+(2sin⁡(x)+cos⁡(x))e−x+1)e−x\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} = \left(x^{2} - 2 x + \left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x} + 1\right) e^{- x}
- No
((2x+(x2+1))+(−2sin⁡(x)+cos⁡(x))ex)ex=−(x2−2x+(2sin⁡(x)+cos⁡(x))e−x+1)e−x\left(\left(2 x + \left(x^{2} + 1\right)\right) + \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{x} = - \left(x^{2} - 2 x + \left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x} + 1\right) e^{- x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор