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exp(2*x-x^2)
Trazado el gráfico de la función/ exp(2*x-x^2)

Gráfico de la función y = exp(2*x-x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               2
        2*x - x 
f(x) = e
f(x)=e−x2+2xf{\left(x \right)} = e^{- x^{2} + 2 x} 
f = exp(-x^2 + 2*x)
Gráfico de la función
02468-8-6-4-2-10100.05.0
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:
e−x2+2x=0e^{- x^{2} + 2 x} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 exp(2*x - x^2).
e0⋅2−02e^{0 \cdot 2 - 0^{2}}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
(2−2x)e−x2+2x=0\left(2 - 2 x\right) e^{- x^{2} + 2 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=86.3691593312175x_{1} = 86.3691593312175
x2=14.9992809268135x_{2} = 14.9992809268135
x3=78.3814824148947x_{3} = 78.3814824148947
x4=28.6208538623602x_{4} = 28.6208538623602
x5=96.3566621914633x_{5} = 96.3566621914633
x6=58.4273097814622x_{6} = 58.4273097814622
x7=34.554507149414x_{7} = 34.554507149414
x8=−92.1055171244878x_{8} = -92.1055171244878
x9=13.125084358924x_{9} = 13.125084358924
x10=−14.6430797877549x_{10} = -14.6430797877549
x11=−72.1343857669297x_{11} = -72.1343857669297
x12=48.4647034274775x_{12} = 48.4647034274775
x13=20.7719971377255x_{13} = 20.7719971377255
x14=54.4405897196054x_{14} = 54.4405897196054
x15=94.358947085372x_{15} = 94.358947085372
x16=−40.2388692669542x_{16} = -40.2388692669542
x17=−26.3615719551025x_{17} = -26.3615719551025
x18=6.66015534698603x_{18} = 6.66015534698603
x19=22.7238244881469x_{19} = 22.7238244881469
x20=72.392536498399x_{20} = 72.392536498399
x21=−58.1662052407992x_{21} = -58.1662052407992
x22=−78.1241933853779x_{22} = -78.1241933853779
x23=−76.1274147285399x_{23} = -76.1274147285399
x24=84.372018487179x_{24} = 84.372018487179
x25=−24.3901348160467x_{25} = -24.3901348160467
x26=100.255036987721x_{26} = 100.255036987721
x27=−44.2177273205401x_{27} = -44.2177273205401
x28=−38.2510541282376x_{28} = -38.2510541282376
x29=26.6498514574821x_{29} = 26.6498514574821
x30=16.9046192283974x_{30} = 16.9046192283974
x31=−18.5109382379574x_{31} = -18.5109382379574
x32=92.3613326305303x_{32} = 92.3613326305303
x33=−88.1102546346649x_{33} = -88.1102546346649
x34=66.4056178257571x_{34} = 66.4056178257571
x35=−64.1508948593432x_{35} = -64.1508948593432
x36=−12.7381124092375x_{36} = -12.7381124092375
x37=−46.2084977857701x_{37} = -46.2084977857701
x38=32.5738300131898x_{38} = 32.5738300131898
x39=−74.1308075062542x_{39} = -74.1308075062542
x40=88.3664310248785x_{40} = 88.3664310248785
x41=82.375018132477x_{41} = 82.375018132477
x42=62.4157579453486x_{42} = 62.4157579453486
x43=−86.1127864643158x_{43} = -86.1127864643158
x44=−5.74301612101296x_{44} = -5.74301612101296
x45=−28.3368875035311x_{45} = -28.3368875035311
x46=−94.1032983570039x_{46} = -94.1032983570039
x47=18.8309297171705x_{47} = 18.8309297171705
x48=−7.3115816730434x_{48} = -7.3115816730434
x49=−36.2645449807319x_{49} = -36.2645449807319
x50=−52.1849682625092x_{50} = -52.1849682625092
x51=−90.1078338924361x_{51} = -90.1078338924361
x52=−70.1381651356283x_{52} = -70.1381651356283
x53=−80.1211308116339x_{53} = -80.1211308116339
x54=56.4337104086693x_{54} = 56.4337104086693
x55=−82.1182155582792x_{55} = -82.1182155582792
x56=64.4105280865885x_{56} = 64.4105280865885
x57=−16.5695229622524x_{57} = -16.5695229622524
x58=−54.1782609089954x_{58} = -54.1782609089954
x59=−20.4631983204608x_{59} = -20.4631983204608
x60=7.96154504225972x_{60} = 7.96154504225972
x61=−68.1421630415863x_{61} = -68.1421630415863
x62=24.6837303756218x_{62} = 24.6837303756218
x63=76.3849716980399x_{63} = 76.3849716980399
x64=42.495785122687x_{64} = 42.495785122687
x65=52.4480034075534x_{65} = 52.4480034075534
x66=40.5082410274504x_{66} = 40.5082410274504
x67=70.3966457620089x_{67} = 70.3966457620089
x68=−96.1011702662716x_{68} = -96.1011702662716
x69=−30.3153447286813x_{69} = -30.3153447286813
x70=−100.004956913936x_{70} = -100.004956913936
x71=−10.8654506014968x_{71} = -10.8654506014968
x72=9.55624995284133x_{72} = 9.55624995284133
x73=−9.04433982996113x_{73} = -9.04433982996113
x74=30.5957583046983x_{74} = 30.5957583046983
x75=−60.1607682345028x_{75} = -60.1607682345028
x76=−62.1556753178058x_{76} = -62.1556753178058
x77=38.5220226949901x_{77} = 38.5220226949901
x78=68.400998762587x_{78} = 68.400998762587
x79=−42.2278099160914x_{79} = -42.2278099160914
x80=−56.1720223810675x_{80} = -56.1720223810675
x81=11.2997190058202x_{81} = 11.2997190058202
x82=−32.296381804162x_{82} = -32.296381804162
x83=−98.005050700024x_{83} = -98.005050700024
x84=98.3544708350782x_{84} = 98.3544708350782
x85=90.3638247874991x_{85} = 90.3638247874991
x86=−66.1463989860564x_{86} = -66.1463989860564
x87=50.4560161379821x_{87} = 50.4560161379821
x88=80.3781688739502x_{88} = 80.3781688739502
x89=−22.4235611844925x_{89} = -22.4235611844925
x90=36.5373529142981x_{90} = 36.5373529142981
x91=−48.2000176225044x_{91} = -48.2000176225044
x92=−84.1154372504332x_{92} = -84.1154372504332
x93=74.3886510740335x_{93} = 74.3886510740335
x94=44.4844728781857x_{94} = 44.4844728781857
x95=−34.2795629354021x_{95} = -34.2795629354021
x96=60.4213396148336x_{96} = 60.4213396148336
x97=−50.1921992657937x_{97} = -50.1921992657937
x98=46.4741540144396x_{98} = 46.4741540144396
Signos de extremos en los puntos:
(86.36915933121747, 2.19999350339782e-3165)

(14.999280926813473, 2.09565559302052e-85)

(78.38148241489469, 8.41491857024383e-2601)

(28.62085386236018, 1.27646902925939e-331)

(96.35666219146327, 2.76076533517936e-3949)

(58.42730978146217, 1.50071915735733e-1432)

(34.55450714941399, 2.88394870733637e-489)

(-92.10551712448779, 4.93107049926044e-3765)

(13.125084358924044, 3.8488532261596e-64)

(-14.643079787754933, 1.44494984454117e-106)

(-72.13438576692971, 3.54943884469736e-2323)

(48.4647034274775, 1.03060858789133e-978)

(20.771997137725517, 4.51585706397205e-170)

(54.44058971960544, 1.36190674118343e-1240)

(94.358947085372, 1.47931048865774e-3785)

(-40.23886926695423, 7.14237465404434e-739)

(-26.361571955102548, 1.98285032736697e-325)

(6.6601553469860315, 3.31624377683807e-14)

(22.72382448814687, 3.02041328683687e-205)

(72.39253649839898, 7.60745139429253e-2214)

(-58.166205240799194, 1.3357577777206e-1520)

(-78.12419338537786, 2.97691927111302e-2719)

(-76.1274147285399, 9.41004361378032e-2584)

(84.37201848717905, 5.00746161235727e-3019)

(-24.390134816046736, 2.90049108207911e-280)

(100.25503698772079, 9.01820658402517e-4279)

(-44.217727320540064, 2.86537302960703e-888)

(-38.25105412823759, 2.19060340917184e-669)

(26.64985145748212, 5.07566121646344e-286)

(16.904619228397404, 3.77140431345026e-110)

(-18.51093823795736, 1.28380640455433e-165)

(92.36133263053033, 2.65876592457153e-3625)

(-88.1102546346649, 7.23192087191618e-3449)

(66.4056178257571, 3.69972914878973e-1858)

(-64.15089485934324, 1.02554626780828e-1843)

(-12.738112409237461, 2.93364317966072e-82)

(-46.20849778577015, 3.52579812533857e-968)

(32.573830013189806, 3.04226883578918e-433)

(-74.13080750625416, 9.97828747176261e-2452)

(88.36643102487854, 3.24239822987928e-3315)

(82.37501813247704, 3.82344324594221e-2876)

(62.415757945348595, 2.0939627483844e-1638)

(-86.11278646431583, 5.38114367666287e-3296)

(-5.743016121012964, 4.87166345986923e-20)

(-28.33688750353106, 4.54422131417275e-374)

(-94.10329835700394, 2.50149086293248e-3928)

(18.830929717170516, 2.25847149427632e-138)

(-7.311581673043402, 2.70516492847735e-30)

(-36.26454498073191, 2.25353130419225e-603)

(-52.18496826250919, 9.35798702166659e-1229)

(-90.10783389243608, 3.26044494353328e-3605)

(-70.13816513562826, 4.23547523132339e-2198)

(-80.12113081163389, 3.15924039939545e-2858)

(56.433710408669306, 7.80566283555038e-1335)

(-82.11821555827918, 1.12470568666248e-3000)

(64.41052808658847, 1.5196805489091e-1746)

(-16.569522962252368, 2.35906496696053e-134)

(-54.17826090899544, 1.45792763158548e-1322)

(-20.463198320460812, 2.33529087565485e-200)

(7.9615450422597185, 2.43799889413567e-21)

(-68.14216304158627, 1.69544117354065e-2076)

(24.683730375621803, 6.76443124823753e-244)

(76.3849716980399, 2.42553373002255e-2468)

(42.495785122686975, 4.1932102554903e-748)

(52.448003407553365, 7.97093913434912e-1150)

(40.50824102745043, 3.49587592584824e-678)

(70.39664576200886, 8.27775276032343e-2092)

(-96.10117026627161, 4.25752808934263e-4095)

(-30.315344728681275, 3.49185858154282e-426)

(-100.00495691393613, 5.77319625326306e-4431)

(-10.865450601496772, 1.95184034871264e-61)

(9.556249952841327, 4.36365374055561e-32)

(-9.04433982996113, 4.15778080665637e-44)

(30.595758304698325, 1.07618858108817e-380)

(-60.16076823450285, 7.85543053683815e-1625)

(-62.15567531780577, 1.54969202491514e-1732)

(38.522022694990106, 9.7756029712129e-612)

(68.40099876258702, 3.0215026775798e-1973)

(-42.22780991609136, 7.81109664461969e-812)

(-56.17202238106753, 7.61933358820865e-1420)

(11.299719005820185, 2.30412888023818e-46)

(-32.29638180416196, 8.99774125888357e-482)

(-98.005050700024, 3.01824869641447e-4257)

(98.35447083507825, 1.72849516886926e-4116)

(90.36382478749914, 1.60307210276243e-3468)

(-66.14639898605638, 2.27667327527876e-1958)

(50.45601613798206, 1.56492284011286e-1062)

(80.37816887395016, 9.79337114815848e-2737)

(-22.42356118449253, 1.42195625644996e-238)

(36.5373529142981, 9.16841017071105e-549)

(-48.200017622504404, 1.45528052393883e-1051)

(-84.11543725043325, 1.343185762218e-3146)

(74.38865107403352, 2.34532983556296e-2339)

(44.48447287818566, 1.6870473225035e-821)

(-34.27956293540211, 7.77547381294577e-541)

(60.42133961483356, 9.67873326312275e-1534)

(-50.19219926579367, 2.0149011930674e-1138)

(46.4741540144396, 2.27670717977593e-898)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
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
2(2(x−1)2−1)ex(2−x)=02 \left(2 \left(x - 1\right)^{2} - 1\right) e^{x \left(2 - x\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−52.1850010470255x_{1} = -52.1850010470255
x2=−44.2177807645343x_{2} = -44.2177807645343
x3=16.9059564908195x_{3} = 16.9059564908195
x4=−70.1381788072998x_{4} = -70.1381788072998
x5=−56.1720487570141x_{5} = -56.1720487570141
x6=−42.2278711235207x_{6} = -42.2278711235207
x7=−40.2389398145583x_{7} = -40.2389398145583
x8=−90.1078404144279x_{8} = -90.1078404144279
x9=86.3691675149948x_{9} = 86.3691675149948
x10=−78.1242033161843x_{10} = -78.1242033161843
x11=26.6501590282445x_{11} = 26.6501590282445
x12=62.4157799638918x_{12} = 62.4157799638918
x13=−88.1102615833855x_{13} = -88.1102615833855
x14=−34.2796759394231x_{14} = -34.2796759394231
x15=80.3781790571961x_{15} = 80.3781790571961
x16=−16.5704704273981x_{16} = -16.5704704273981
x17=−68.1421779340921x_{17} = -68.1421779340921
x18=−20.4637097805595x_{18} = -20.4637097805595
x19=20.7726787189143x_{19} = 20.7726787189143
x20=88.3664386613479x_{20} = 88.3664386613479
x21=64.4105480873281x_{21} = 64.4105480873281
x22=−38.2511360134471x_{22} = -38.2511360134471
x23=−9.0500655359739x_{23} = -9.0500655359739
x24=42.4958568200158x_{24} = 42.4958568200158
x25=18.8318671133789x_{25} = 18.8318671133789
x26=24.6841225372817x_{26} = 24.6841225372817
x27=36.5374673937235x_{27} = 36.5374673937235
x28=30.5959574669433x_{28} = 30.5959574669433
x29=46.4742084194675x_{29} = 46.4742084194675
x30=15.0012777744454x_{30} = 15.0012777744454
x31=90.3638319253342x_{31} = 90.3638319253342
x32=100.255037700499x_{32} = 100.255037700499
x33=−84.115445225953x_{33} = -84.115445225953
x34=54.4406231792732x_{34} = 54.4406231792732
x35=74.3886639644834x_{35} = 74.3886639644834
x36=−96.1011751469827x_{36} = -96.1011751469827
x37=92.3613393047312x_{37} = 92.3613393047312
x38=58.4273367278067x_{38} = 58.4273367278067
x39=68.401015410542x_{39} = 68.401015410542
x40=−66.1464152490962x_{40} = -66.1464152490962
x41=−26.3618159522672x_{41} = -26.3618159522672
x42=−46.2085447242652x_{42} = -46.2085447242652
x43=48.4647512445571x_{43} = 48.4647512445571
x44=−7.32269391999105x_{44} = -7.32269391999105
x45=84.3720272748887x_{45} = 84.3720272748887
x46=−28.3370849881563x_{46} = -28.3370849881563
x47=52.448040921684x_{47} = 52.448040921684
x48=−48.2000590688826x_{48} = -48.2000590688826
x49=−10.8687398765478x_{49} = -10.8687398765478
x50=82.375027583386x_{50} = 82.375027583386
x51=−14.6444405159595x_{51} = -14.6444405159595
x52=−5.76791523640854x_{52} = -5.76791523640854
x53=32.5739937230882x_{53} = 32.5739937230882
x54=−62.1556948702322x_{54} = -62.1556948702322
x55=−24.3904410852112x_{55} = -24.3904410852112
x56=78.3814934082712x_{56} = 78.3814934082712
x57=13.1282450910777x_{57} = 13.1282450910777
x58=−76.127425451889x_{58} = -76.127425451889
x59=60.4213639317561x_{59} = 60.4213639317561
x60=−18.5116236618385x_{60} = -18.5116236618385
x61=38.5221198429226x_{61} = 38.5221198429226
x62=−36.2646407635773x_{62} = -36.2646407635773
x63=1.70710678118655x_{63} = 1.70710678118655
x64=−64.1509126662648x_{64} = -64.1509126662648
x65=−60.1607897680498x_{65} = -60.1607897680498
x66=7.98362634704309x_{66} = 7.98362634704309
x67=−80.121140025594x_{67} = -80.121140025594
x68=−50.1922360437359x_{68} = -50.1922360437359
x69=50.4560583881637x_{69} = 50.4560583881637
x70=96.3566674788483x_{70} = 96.3566674788483
x71=−30.3155067841679x_{71} = -30.3155067841679
x72=44.4845351370641x_{72} = 44.4845351370641
x73=70.3966610120181x_{73} = 70.3966610120181
x74=34.5546433285019x_{74} = 34.5546433285019
x75=72.3925505026189x_{75} = 72.3925505026189
x76=−22.4239527325761x_{76} = -22.4239527325761
x77=94.3589534992624x_{77} = 94.3589534992624
x78=−82.1182241228293x_{78} = -82.1182241228293
x79=76.3849835896893x_{79} = 76.3849835896893
x80=66.4056360477541x_{80} = 66.4056360477541
x81=−86.1127939015948x_{81} = -86.1127939015948
x82=−100.004952760382x_{82} = -100.004952760382
x83=9.56645267084672x_{83} = 9.56645267084672
x84=40.5083241676641x_{84} = 40.5083241676641
x85=−98.0050513301156x_{85} = -98.0050513301156
x86=22.7243351041127x_{86} = 22.7243351041127
x87=−58.1662290321839x_{87} = -58.1662290321839
x88=−54.1782902574728x_{88} = -54.1782902574728
x89=−12.7401632108074x_{89} = -12.7401632108074
x90=56.4337403772313x_{90} = 56.4337403772313
x91=98.3544751989217x_{91} = 98.3544751989217
x92=0.292893218813452x_{92} = 0.292893218813452
x93=11.3051175196805x_{93} = 11.3051175196805
x94=−92.1055231124176x_{94} = -92.1055231124176
x95=−72.1343983475769x_{95} = -72.1343983475769
x96=28.6210994527089x_{96} = 28.6210994527089
x97=−32.2965164060315x_{97} = -32.2965164060315
x98=−74.1308191088924x_{98} = -74.1308191088924
x99=−94.1033035166246x_{99} = -94.1033035166246

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
(−∞,0.292893218813452]∪[1.70710678118655,∞)\left(-\infty, 0.292893218813452\right] \cup \left[1.70710678118655, \infty\right)
Convexa en los intervalos
[0.292893218813452,1.70710678118655]\left[0.292893218813452, 1.70710678118655\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞e−x2+2x=0\lim_{x \to -\infty} e^{- x^{2} + 2 x} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞e−x2+2x=0\lim_{x \to \infty} e^{- x^{2} + 2 x} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función exp(2*x - x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(e−x2+2xx)=0\lim_{x \to -\infty}\left(\frac{e^{- x^{2} + 2 x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(e−x2+2xx)=0\lim_{x \to \infty}\left(\frac{e^{- x^{2} + 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:
e−x2+2x=e−x2−2xe^{- x^{2} + 2 x} = e^{- x^{2} - 2 x}
- No
e−x2+2x=−e−x2−2xe^{- x^{2} + 2 x} = - e^{- x^{2} - 2 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = exp(2*x-x^2)
    © Sr Examen – Калькулятор