Sr Examen

Otras calculadoras

exp(4*x-x^2)
Trazado el gráfico de la función/ exp(4*x-x^2)

Gráfico de la función y = exp(4*x-x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               2
        4*x - x 
f(x) = e
f(x)=e−x2+4xf{\left(x \right)} = e^{- x^{2} + 4 x} 
f = exp(-x^2 + 4*x)
Gráfico de la función
02468-8-6-4-2-10100100
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+4x=0e^{- x^{2} + 4 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(4*x - x^2).
e0⋅4−02e^{0 \cdot 4 - 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
(4−2x)e−x2+4x=0\left(4 - 2 x\right) e^{- x^{2} + 4 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=76.3867866673358x_{1} = 76.3867866673358
x2=15.0574223701178x_{2} = 15.0574223701178
x3=−20.4425007185933x_{3} = -20.4425007185933
x4=−72.1325725141081x_{4} = -72.1325725141081
x5=94.3601270814273x_{5} = 94.3601270814273
x6=72.3945619595659x_{6} = 72.3945619595659
x7=−56.1690638483732x_{7} = -56.1690638483732
x8=100.255089257012x_{8} = 100.255089257012
x9=46.4791976169932x_{9} = 46.4791976169932
x10=−4.0660448195699x_{10} = -4.0660448195699
x11=−42.2226547762937x_{11} = -42.2226547762937
x12=78.3832042269373x_{12} = 78.3832042269373
x13=−52.1815527450525x_{13} = -52.1815527450525
x14=50.4602702583442x_{14} = 50.4602702583442
x15=60.4242736475066x_{15} = 60.4242736475066
x16=58.4304534331944x_{16} = 58.4304534331944
x17=−22.4061660554474x_{17} = -22.4061660554474
x18=−48.1960306485386x_{18} = -48.1960306485386
x19=−40.2332088562321x_{19} = -40.2332088562321
x20=56.4370869389189x_{20} = 56.4370869389189
x21=−80.1196554417412x_{21} = -80.1196554417412
x22=−24.3753139220651x_{22} = -24.3753139220651
x23=54.4442259550818x_{23} = 54.4442259550818
x24=22.7467570641594x_{24} = 22.7467570641594
x25=−86.1115061900845x_{25} = -86.1115061900845
x26=74.3905669642191x_{26} = 74.3905669642191
x27=−84.1140964763775x_{27} = -84.1140964763775
x28=−70.1362492686553x_{28} = -70.1362492686553
x29=−78.1226429944108x_{29} = -78.1226429944108
x30=−14.6040954166054x_{30} = -14.6040954166054
x31=−64.1486129602884x_{31} = -64.1486129602884
x32=11.4146972457383x_{32} = 11.4146972457383
x33=88.3677793923608x_{33} = 88.3677793923608
x34=−62.1532478546075x_{34} = -62.1532478546075
x35=28.6348104846935x_{35} = 28.6348104846935
x36=−90.1066629479608x_{36} = -90.1066629479608
x37=86.3705719720027x_{37} = 86.3705719720027
x38=−74.129088844583x_{38} = -74.129088844583
x39=82.3765739117452x_{39} = 82.3765739117452
x40=−44.2130128196551x_{40} = -44.2130128196551
x41=−36.2576235616253x_{41} = -36.2576235616253
x42=38.5294784207713x_{42} = 38.5294784207713
x43=68.4032735361933x_{43} = 68.4032735361933
x44=−50.1885145543938x_{44} = -50.1885145543938
x45=98.3555539933023x_{45} = 98.3555539933023
x46=−94.1022227575014x_{46} = -94.1022227575014
x47=−26.3487957849917x_{47} = -26.3487957849917
x48=−82.1168098950627x_{48} = -82.1168098950627
x49=66.408034863895x_{49} = 66.408034863895
x50=44.4899960942614x_{50} = 44.4899960942614
x51=−60.1581808370582x_{51} = -60.1581808370582
x52=8.25999679126562x_{52} = 8.25999679126562
x53=34.5638726903793x_{53} = 34.5638726903793
x54=−66.1442499509518x_{54} = -66.1442499509518
x55=92.3625649357664x_{55} = 92.3625649357664
x56=−54.1750861644869x_{56} = -54.1750861644869
x57=9.73387960889725x_{57} = 9.73387960889725
x58=48.4693271688763x_{58} = 48.4693271688763
x59=−5.49944089361232x_{59} = -5.49944089361232
x60=−30.3055705254499x_{60} = -30.3055705254499
x61=84.3735001099732x_{61} = 84.3735001099732
x62=−12.6873692419514x_{62} = -12.6873692419514
x63=40.5149532926537x_{63} = 40.5149532926537
x64=−96.0051017677228x_{64} = -96.0051017677228
x65=−76.125783449603x_{65} = -76.125783449603
x66=20.7999073667175x_{66} = 20.7999073667175
x67=52.4519304520445x_{67} = 52.4519304520445
x68=−98.0050074899868x_{68} = -98.0050074899868
x69=2x_{69} = 2
x70=−68.1401356046363x_{70} = -68.1401356046363
x71=−34.2718473606153x_{71} = -34.2718473606153
x72=−8.94672674319965x_{72} = -8.94672674319965
x73=96.3577924641229x_{73} = 96.3577924641229
x74=16.9488147933141x_{74} = 16.9488147933141
x75=32.5844367675871x_{75} = 32.5844367675871
x76=26.6661073409247x_{76} = 26.6661073409247
x77=−32.2877276681532x_{77} = -32.2877276681532
x78=−16.5386580960287x_{78} = -16.5386580960287
x79=70.3987904608748x_{79} = 70.3987904608748
x80=−28.3257619544685x_{80} = -28.3257619544685
x81=64.4131011494136x_{81} = 64.4131011494136
x82=−18.4859087535338x_{82} = -18.4859087535338
x83=−92.104395712328x_{83} = -92.104395712328
x84=−88.1090308486605x_{84} = -88.1090308486605
x85=−38.244810563854x_{85} = -38.244810563854
x86=18.8656081194959x_{86} = 18.8656081194959
x87=90.3651131734546x_{87} = 90.3651131734546
x88=36.5456823438052x_{88} = 36.5456823438052
x89=−7.16348321175021x_{89} = -7.16348321175021
x90=30.6078690007847x_{90} = 30.6078690007847
x91=13.204731998154x_{91} = 13.204731998154
x92=80.3798045164051x_{92} = 80.3798045164051
x93=24.7028991452193x_{93} = 24.7028991452193
x94=−46.2041698444887x_{94} = -46.2041698444887
x95=42.5018595731864x_{95} = 42.5018595731864
x96=−10.7968091698462x_{96} = -10.7968091698462
x97=−100.004913250976x_{97} = -100.004913250976
x98=−58.1634415856681x_{98} = -58.1634415856681
x99=62.4185026243398x_{99} = 62.4185026243398
Signos de extremos en los puntos:
(76.3867866673358, 4.11797970393336e-2402)

(15.057422370117813, 4.91569347340168e-73)

(-20.442500718593337, 9.95138531024423e-218)

(-72.13257251410812, 1.02751085447705e-2385)

(94.36012708142727, 1.08277247644004e-3703)

(72.3945619595659, 4.33265947542034e-2151)

(-56.169063848373206, 1.74181473196989e-1468)

(100.25508925701156, 1.07418242220151e-4191)

(46.47919761699318, 3.38376548208837e-858)

(-4.0660448195699015, 5.70804533463009e-15)

(-42.22265477629371, 2.58303069031435e-848)

(78.3832042269373, 7.80022500368688e-2533)

(-52.181552745052464, 6.37733698347579e-1274)

(50.46027025834418, 6.93385717432854e-1019)

(60.42427364750664, 2.08083964022229e-1481)

(58.430453433194394, 5.90927908562814e-1382)

(-22.406166055447407, 1.10896525006072e-257)

(-48.19603064853855, 2.95652257944216e-1093)

(-40.23320885623211, 1.28961869611762e-773)

(56.437086938918874, 5.62935682488956e-1286)

(-80.1196554417412, 1.02917480739733e-2927)

(-24.37531392206507, 4.14096355590178e-301)

(54.44422595508178, 1.79890818004307e-1193)

(22.746757064159397, 6.37828095107243e-186)

(-86.1115061900845, 1.07706559825572e-3370)

(74.39056696421908, 7.29289918493022e-2275)

(-84.11409647637747, 1.46785537212356e-3219)

(-70.13624926865535, 6.69436765248258e-2259)

(-78.1226429944108, 5.29484531172697e-2787)

(-14.604095416605448, 1.00909750255223e-118)

(-64.14861296028836, 2.63819514368756e-1899)

(11.414697245738271, 1.74913995328833e-37)

(88.36777939236077, 1.45821625731569e-3238)

(-62.15324785460746, 2.17660384472811e-1786)

(28.634810484693503, 4.3951163042085e-307)

(-90.1066629479608, 2.18920417596409e-3683)

(86.37057197200268, 1.81216537265271e-3090)

(-74.12908884458302, 5.29056828620092e-2516)

(82.37657391174523, 1.05650571249707e-2804)

(-44.21301281965513, 1.73540747698719e-926)

(-36.2576235616253, 1.21311129207606e-634)

(38.52947842077126, 1.63458681024575e-578)

(68.40327353619331, 5.77265242976012e-1914)

(-50.18851455439384, 7.49719180767543e-1182)

(98.35555399330232, 3.77244941540845e-4031)

(-94.1022227575014, 5.63503846583692e-4010)

(-26.348795784991665, 5.18303039091582e-348)

(-82.11680989506266, 6.71066972237406e-3072)

(66.40803486389501, 1.29461371433712e-1800)

(44.489996094261436, 4.59220235079293e-783)

(-60.15818083705822, 6.02402703628695e-1677)

(8.259996791265625, 5.22681267561149e-16)

(34.56387269037934, 1.61729190849048e-459)

(-66.14424995095182, 1.07268087455588e-2015)

(92.36256493576639, 3.56446115329e-3545)

(-54.175086164486885, 1.81972742123995e-1369)

(9.73387960889725, 5.76458704601128e-25)

(48.46932716887627, 8.36339635670837e-937)

(-5.4994408936123165, 2.05000743931597e-23)

(-30.305570525449934, 3.06049007536586e-452)

(84.37350010997318, 7.55465245519232e-2946)

(-12.687369241951417, 1.12638490590186e-92)

(40.5149532926537, 3.19180842282743e-643)

(-96.0051017677228, 2.18117840491067e-4170)

(-76.12578344960298, 9.13813960066049e-2650)

(20.799907366717463, 1.74441130540746e-152)

(52.45193045204447, 1.92833382605477e-1104)

(-98.00500748998677, 2.27720931318018e-4342)

(2, 54.5981500331442)

(-68.14013560463634, 1.46309039201704e-2135)

(-34.27184736061533, 2.28551653802721e-570)

(-8.94672674319965, 4.95805413571058e-51)

(96.3577924641229, 1.10338830142306e-3865)

(16.948814793314096, 4.86032448732091e-96)

(32.584436767587086, 3.12423755823658e-405)

(26.66610734092469, 3.1996073636404e-263)

(-32.28772766815322, 1.44418289137864e-509)

(-16.53865809602872, 3.00641927707013e-148)

(70.39879046087482, 8.63468611178062e-2031)

(-28.325761954468536, 2.17500049589626e-398)

(64.41310114941363, 9.73957192023436e-1691)

(-18.485908753533757, 2.99051129307274e-181)

(-92.10439571232797, 6.06412450341211e-3845)

(-88.10903084866051, 2.65119987133656e-3525)

(-38.24481056385404, 2.15967546197596e-702)

(18.865608119495942, 1.59463378001323e-122)

(90.36511317345456, 3.93629172033413e-3390)

(36.54568234380517, 2.80758442615998e-517)

(-7.163483211750211, 1.86088779458037e-35)

(30.607869000784714, 2.02375390143655e-354)

(13.204731998153951, 1.63393428476891e-53)

(80.37980451640513, 4.95643625361034e-2667)

(24.702899145219273, 7.80563680241777e-223)

(-46.204169844488675, 3.91096185195431e-1008)

(42.50185957318639, 2.09043511364748e-711)

(-10.796809169846151, 4.1480219419717e-70)

(-100.00491325097592, 7.98147161986681e-4518)

(-58.16344158566805, 5.59278660715047e-1571)

(62.41850262433981, 2.45795065385434e-1584)


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
Puntos máximos de la función:
x99=2x_{99} = 2
Decrece en los intervalos
(−∞,2]\left(-\infty, 2\right]
Crece en los intervalos
[2,∞)\left[2, \infty\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
2(2(x−2)2−1)ex(4−x)=02 \left(2 \left(x - 2\right)^{2} - 1\right) e^{x \left(4 - x\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=36.5458070473973x_{1} = 36.5458070473973
x2=−12.6890284388678x_{2} = -12.6890284388678
x3=56.4371185890045x_{3} = 56.4371185890045
x4=92.3625718232063x_{4} = 92.3625718232063
x5=−94.1022286542423x_{5} = -94.1022286542423
x6=72.3945765690017x_{6} = 72.3945765690017
x7=84.3735092208767x_{7} = 84.3735092208767
x8=−96.0051010366872x_{8} = -96.0051010366872
x9=80.3798150944542x_{9} = 80.3798150944542
x10=46.4792557727931x_{10} = 46.4792557727931
x11=−82.116818158387x_{11} = -82.116818158387
x12=−90.1066692202883x_{12} = -90.1066692202883
x13=76.3867990449735x_{13} = 76.3867990449735
x14=52.4519702405016x_{14} = 52.4519702405016
x15=34.564021793184x_{15} = 34.564021793184
x16=−28.3259405618292x_{16} = -28.3259405618292
x17=20.8007034086588x_{17} = 20.8007034086588
x18=−98.005004913554x_{18} = -98.005004913554
x19=−76.1257937665358x_{19} = -76.1257937665358
x20=78.3832156577194x_{20} = 78.3832156577194
x21=98.3555601161029x_{21} = 98.3555601161029
x22=−24.3755867085268x_{22} = -24.3755867085268
x23=−26.3490148940033x_{23} = -26.3490148940033
x24=38.5295837616735x_{24} = 38.5295837616735
x25=50.4603151773295x_{25} = 50.4603151773295
x26=62.4185257540807x_{26} = 62.4185257540807
x27=−22.4065114935024x_{27} = -22.4065114935024
x28=−7.17133632670237x_{28} = -7.17133632670237
x29=26.6664538034799x_{29} = 26.6664538034799
x30=22.7473450052957x_{30} = 22.7473450052957
x31=42.5019367102417x_{31} = 42.5019367102417
x32=−52.1815837482288x_{32} = -52.1815837482288
x33=11.422011464692x_{33} = 11.422011464692
x34=−86.1115133706781x_{34} = -86.1115133706781
x35=−54.1751139738584x_{35} = -54.1751139738584
x36=−36.2577120334991x_{36} = -36.2577120334991
x37=−74.1290999960711x_{37} = -74.1290999960711
x38=−46.2042139232102x_{38} = -46.2042139232102
x39=74.3905803961827x_{39} = 74.3905803961827
x40=−44.2130628704228x_{40} = -44.2130628704228
x41=8.29464172933834x_{41} = 8.29464172933834
x42=−68.1401498693331x_{42} = -68.1401498693331
x43=60.4242992341965x_{43} = 60.4242992341965
x44=13.2088180591481x_{44} = 13.2088180591481
x45=−8.95101589105128x_{45} = -8.95101589105128
x46=40.5150430741865x_{46} = 40.5150430741865
x47=94.3601335054317x_{47} = 94.3601335054317
x48=−70.1362623797001x_{48} = -70.1362623797001
x49=−58.1634642106518x_{49} = -58.1634642106518
x50=64.4131221267642x_{50} = 64.4131221267642
x51=−4.13719667299394x_{51} = -4.13719667299394
x52=−5.51573811786742x_{52} = -5.51573811786742
x53=32.5846170503136x_{53} = 32.5846170503136
x54=48.4693781384692x_{54} = 48.4693781384692
x55=−60.1582013482095x_{55} = -60.1582013482095
x56=24.703345361781x_{56} = 24.703345361781
x57=−64.1486299719532x_{57} = -64.1486299719532
x58=−62.1532665071025x_{58} = -62.1532665071025
x59=70.3988063894397x_{59} = 70.3988063894397
x60=−48.1960696678598x_{60} = -48.1960696678598
x61=96.3577984827687x_{61} = 96.3577984827687
x62=−30.3057180079325x_{62} = -30.3057180079325
x63=−42.222711926854x_{63} = -42.222711926854
x64=−34.2719512808812x_{64} = -34.2719512808812
x65=54.4442613648032x_{65} = 54.4442613648032
x66=−66.1442655086309x_{66} = -66.1442655086309
x67=90.3651205447553x_{67} = 90.3651205447553
x68=9.7485998775194x_{68} = 9.7485998775194
x69=−100.004889927025x_{69} = -100.004889927025
x70=68.4032909474903x_{70} = 68.4032909474903
x71=15.0599139899326x_{71} = 15.0599139899326
x72=28.6350847534115x_{72} = 28.6350847534115
x73=88.367787292426x_{73} = 88.367787292426
x74=−4.10591360282363x_{74} = -4.10591360282363
x75=−78.1226525581753x_{75} = -78.1226525581753
x76=−40.2332745131468x_{76} = -40.2332745131468
x77=86.37058045082x_{77} = 86.37058045082
x78=−56.1690888876657x_{78} = -56.1690888876657
x79=30.6080897634358x_{79} = 30.6080897634358
x80=−18.4864987711344x_{80} = -18.4864987711344
x81=−88.1090375563286x_{81} = -88.1090375563286
x82=−32.2878508435882x_{82} = -32.2878508435882
x83=−50.1885492592296x_{83} = -50.1885492592296
x84=−38.2448864993236x_{84} = -38.2448864993236
x85=−14.6052248328518x_{85} = -14.6052248328518
x86=16.9504384793318x_{86} = 16.9504384793318
x87=18.8667220714146x_{87} = 18.8667220714146
x88=44.4900628508112x_{88} = 44.4900628508112
x89=−80.1196643230321x_{89} = -80.1196643230321
x90=−72.1325845925649x_{90} = -72.1325845925649
x91=66.4080539476779x_{91} = 66.4080539476779
x92=58.4304818371702x_{92} = 58.4304818371702
x93=−16.5394605224035x_{93} = -16.5394605224035
x94=100.255088428374x_{94} = 100.255088428374
x95=−84.1141041757895x_{95} = -84.1141041757895
x96=−92.1044015172776x_{96} = -92.1044015172776
x97=82.3765837198604x_{97} = 82.3765837198604
x98=−10.7993835180225x_{98} = -10.7993835180225
x99=−20.4429469192179x_{99} = -20.4429469192179

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:
No tiene corvaduras en todo el eje numérico
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞e−x2+4x=0\lim_{x \to -\infty} e^{- x^{2} + 4 x} = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞e−x2+4x=0\lim_{x \to \infty} e^{- x^{2} + 4 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(4*x - x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(e−x2+4xx)=0\lim_{x \to -\infty}\left(\frac{e^{- x^{2} + 4 x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(e−x2+4xx)=0\lim_{x \to \infty}\left(\frac{e^{- x^{2} + 4 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+4x=e−x2−4xe^{- x^{2} + 4 x} = e^{- x^{2} - 4 x}
- No
e−x2+4x=−e−x2−4xe^{- x^{2} + 4 x} = - e^{- x^{2} - 4 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = exp(4*x-x^2)
    © Sr Examen – Калькулятор