Sr Examen

Otras calculadoras

acos(exp(x*(1-x)))
Trazado el gráfico de la función/ acos(exp(x*(1-x)))

Gráfico de la función y = acos(exp(x*(1-x)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           / x*(1 - x)\
f(x) = acos\e         /
f(x)=acos⁡(ex(1−x))f{\left(x \right)} = \operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} 
f = acos(exp(x*(1 - x)))
Gráfico de la función
02468-8-6-4-2-101002
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:
acos⁡(ex(1−x))=0\operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=1x_{2} = 1
Solución numérica
x1=0x_{1} = 0
x2=1x_{2} = 1
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 acos(exp(x*(1 - x))).
acos⁡(e0(1−0))\operatorname{acos}{\left(e^{0 \left(1 - 0\right)} \right)}
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
−(1−2x)ex(1−x)1−e2x(1−x)=0- \frac{\left(1 - 2 x\right) e^{x \left(1 - x\right)}}{\sqrt{1 - e^{2 x \left(1 - x\right)}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−14.6644965281302x_{1} = -14.6644965281302
x2=36.5333611543142x_{2} = 36.5333611543142
x3=96.3561054822232x_{3} = 96.3561054822232
x4=−72.135311107138x_{4} = -72.135311107138
x5=98.2551162704909x_{5} = 98.2551162704909
x6=64.4092717290461x_{6} = 64.4092717290461
x7=66.4044367940282x_{7} = 66.4044367940282
x8=−90.1084290662785x_{8} = -90.1084290662785
x9=30.5900037685939x_{9} = 30.5900037685939
x10=−12.7663556907184x_{10} = -12.7663556907184
x11=44.481805306273x_{11} = 44.481805306273
x12=−56.1735407689525x_{12} = -56.1735407689525
x13=−100.004969508435x_{13} = -100.004969508435
x14=−98.0050769414502x_{14} = -98.0050769414502
x15=−92.1060868900058x_{15} = -92.1060868900058
x16=62.4144188276356x_{16} = 62.4144188276356
x17=86.3684653455661x_{17} = 86.3684653455661
x18=−86.1134376887439x_{18} = -86.1134376887439
x19=9.48164106753931x_{19} = 9.48164106753931
x20=13.0899263689562x_{20} = 13.0899263689562
x21=80.3773664031498x_{21} = 80.3773664031498
x22=90.3631913509983x_{22} = 90.3631913509983
x23=20.7590676347421x_{23} = 20.7590676347421
x24=−10.9043062831487x_{24} = -10.9043062831487
x25=42.4928561998145x_{25} = 42.4928561998145
x26=−94.1038440239097x_{26} = -94.1038440239097
x27=54.4388221101575x_{27} = 54.4388221101575
x28=−18.5244356600151x_{28} = -18.5244356600151
x29=−96.1016930096477x_{29} = -96.1016930096477
x30=−80.121882215662x_{30} = -80.121882215662
x31=−78.1249833637931x_{31} = -78.1249833637931
x32=76.3840821424105x_{32} = 76.3840821424105
x33=40.5050104922868x_{33} = 40.5050104922868
x34=34.5500301132873x_{34} = 34.5500301132873
x35=72.3915448860942x_{35} = 72.3915448860942
x36=84.3712909199937x_{36} = 84.3712909199937
x37=88.365768348138x_{37} = 88.365768348138
x38=70.3955964118507x_{38} = 70.3955964118507
x39=−84.1161195200159x_{39} = -84.1161195200159
x40=24.6747374826298x_{40} = 24.6747374826298
x41=94.358366768573x_{41} = 94.358366768573
x42=26.642187727879x_{42} = 26.642187727879
x43=−66.147497673365x_{43} = -66.147497673365
x44=−38.2542965417263x_{44} = -38.2542965417263
x45=−48.2020724431024x_{45} = -48.2020724431024
x46=100.255013050999x_{46} = 100.255013050999
x47=−68.1431988999997x_{47} = -68.1431988999997
x48=−28.342738753825x_{48} = -28.342738753825
x49=14.973184916168x_{49} = 14.973184916168
x50=48.4624637640073x_{50} = 48.4624637640073
x51=82.3742544880409x_{51} = 82.3742544880409
x52=92.3607265271323x_{52} = 92.3607265271323
x53=−64.1520622640929x_{53} = -64.1520622640929
x54=78.3806380820164x_{54} = 78.3806380820164
x55=22.713127614282x_{55} = 22.713127614282
x56=56.4320673764301x_{56} = 56.4320673764301
x57=−82.1189311443725x_{57} = -82.1189311443725
x58=46.4717143780255x_{58} = 46.4717143780255
x59=32.5687739425934x_{59} = 32.5687739425934
x60=58.4257786220294x_{60} = 58.4257786220294
x61=−42.2304778452627x_{61} = -42.2304778452627
x62=−52.1867245973796x_{62} = -52.1867245973796
x63=−62.1569180872444x_{63} = -62.1569180872444
x64=−16.5863055170952x_{64} = -16.5863055170952
x65=−30.3204690788784x_{65} = -30.3204690788784
x66=50.453952871065x_{66} = 50.453952871065
x67=−46.2107311674476x_{67} = -46.2107311674476
x68=−32.3009063046151x_{68} = -32.3009063046151
x69=−54.1798917899715x_{69} = -54.1798917899715
x70=−74.1316841006271x_{70} = -74.1316841006271
x71=68.3998864860113x_{71} = 68.3998864860113
x72=7.8403540900179x_{72} = 7.8403540900179
x73=−76.1282463272436x_{73} = -76.1282463272436
x74=−20.4742841415853x_{74} = -20.4742841415853
x75=52.4460965208043x_{75} = 52.4460965208043
x76=−22.4328254071079x_{76} = -22.4328254071079
x77=−88.1108768750308x_{77} = -88.1108768750308
x78=−7.39969072860318x_{78} = -7.39969072860318
x79=−44.2201635391922x_{79} = -44.2201635391922
x80=−50.1940960821807x_{80} = -50.1940960821807
x81=16.884523551043x_{81} = 16.884523551043
x82=−60.162093900683x_{82} = -60.162093900683
x83=−40.2418035171732x_{83} = -40.2418035171732
x84=−34.283586749193x_{84} = -34.283586749193
x85=74.387712565726x_{85} = 74.387712565726
x86=11.2500538346575x_{86} = 11.2500538346575
x87=−36.268146617465x_{87} = -36.268146617465
x88=−9.1008195104702x_{88} = -9.1008195104702
x89=−24.3979901158107x_{89} = -24.3979901158107
x90=−58.1676223726593x_{90} = -58.1676223726593
x91=6.44491475275573x_{91} = 6.44491475275573
x92=−26.3683155327555x_{92} = -26.3683155327555
x93=38.51844159893x_{93} = 38.51844159893
x94=18.8149982347286x_{94} = 18.8149982347286
x95=−70.1391434001417x_{95} = -70.1391434001417
x96=−5.89271805573471x_{96} = -5.89271805573471
x97=60.4199092923593x_{97} = 60.4199092923593
x98=28.6142463819744x_{98} = 28.6142463819744
Signos de extremos en los puntos:
(-14.664496528130183, 1.5707963267949)

(36.53336115431421, 1.5707963267949)

(96.35610548222323, 1.5707963267949)

(-72.13531110713797, 1.5707963267949)

(98.25511627049089, 1.5707963267949)

(64.40927172904611, 1.5707963267949)

(66.40443679402819, 1.5707963267949)

(-90.10842906627853, 1.5707963267949)

(30.590003768593895, 1.5707963267949)

(-12.766355690718424, 1.5707963267949)

(44.48180530627295, 1.5707963267949)

(-56.17354076895255, 1.5707963267949)

(-100.00496950843483, 1.5707963267949)

(-98.00507694145021, 1.5707963267949)

(-92.1060868900058, 1.5707963267949)

(62.4144188276356, 1.5707963267949)

(86.3684653455661, 1.5707963267949)

(-86.11343768874391, 1.5707963267949)

(9.481641067539305, 1.5707963267949)

(13.089926368956169, 1.5707963267949)

(80.37736640314978, 1.5707963267949)

(90.36319135099829, 1.5707963267949)

(20.759067634742106, 1.5707963267949)

(-10.904306283148685, 1.5707963267949)

(42.49285619981447, 1.5707963267949)

(-94.1038440239097, 1.5707963267949)

(54.43882211015752, 1.5707963267949)

(-18.52443566001512, 1.5707963267949)

(-96.10169300964768, 1.5707963267949)

(-80.121882215662, 1.5707963267949)

(-78.1249833637931, 1.5707963267949)

(76.3840821424105, 1.5707963267949)

(40.50501049228682, 1.5707963267949)

(34.55003011328726, 1.5707963267949)

(72.39154488609422, 1.5707963267949)

(84.37129091999368, 1.5707963267949)

(88.36576834813798, 1.5707963267949)

(70.39559641185075, 1.5707963267949)

(-84.11611952001589, 1.5707963267949)

(24.67473748262979, 1.5707963267949)

(94.35836676857302, 1.5707963267949)

(26.64218772787897, 1.5707963267949)

(-66.14749767336497, 1.5707963267949)

(-38.254296541726276, 1.5707963267949)

(-48.20207244310242, 1.5707963267949)

(100.25501305099921, 1.5707963267949)

(-68.14319889999966, 1.5707963267949)

(-28.34273875382503, 1.5707963267949)

(14.973184916167991, 1.5707963267949)

(48.462463764007275, 1.5707963267949)

(82.37425448804089, 1.5707963267949)

(92.36072652713227, 1.5707963267949)

(-64.15206226409286, 1.5707963267949)

(78.38063808201643, 1.5707963267949)

(22.713127614281984, 1.5707963267949)

(56.432067376430055, 1.5707963267949)

(-82.11893114437245, 1.5707963267949)

(46.47171437802554, 1.5707963267949)

(32.56877394259343, 1.5707963267949)

(58.42577862202938, 1.5707963267949)

(-42.23047784526274, 1.5707963267949)

(-52.18672459737958, 1.5707963267949)

(-62.15691808724441, 1.5707963267949)

(-16.586305517095177, 1.5707963267949)

(-30.3204690788784, 1.5707963267949)

(50.45395287106497, 1.5707963267949)

(-46.21073116744759, 1.5707963267949)

(-32.300906304615054, 1.5707963267949)

(-54.17989178997149, 1.5707963267949)

(-74.13168410062706, 1.5707963267949)

(68.3998864860113, 1.5707963267949)

(7.840354090017897, 1.5707963267949)

(-76.12824632724364, 1.5707963267949)

(-20.474284141585258, 1.5707963267949)

(52.4460965208043, 1.5707963267949)

(-22.432825407107853, 1.5707963267949)

(-88.11087687503078, 1.5707963267949)

(-7.399690728603181, 1.5707963267949)

(-44.22016353919216, 1.5707963267949)

(-50.1940960821807, 1.5707963267949)

(16.88452355104299, 1.5707963267949)

(-60.16209390068298, 1.5707963267949)

(-40.241803517173196, 1.5707963267949)

(-34.28358674919297, 1.5707963267949)

(74.387712565726, 1.5707963267949)

(11.250053834657459, 1.5707963267949)

(-36.26814661746504, 1.5707963267949)

(-9.100819510470203, 1.5707963267949)

(-24.397990115810703, 1.5707963267949)

(-58.16762237265933, 1.5707963267949)

(6.444914752755732, 1.5707963267949)

(-26.36831553275548, 1.5707963267949)

(38.518441598930046, 1.5707963267949)

(18.8149982347286, 1.5707963267949)

(-70.13914340014168, 1.5707963267949)

(-5.892718055734707, 1.5707963267949)

(60.41990929235934, 1.5707963267949)

(28.614246381974446, 1.5707963267949)


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
No cambia el valor 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
(−(2x−1)2+2−(2x−1)2e−2x(x−1)1−e2x(1−x))e−x(x−1)1−e2x(1−x)=0\frac{\left(- \left(2 x - 1\right)^{2} + 2 - \frac{\left(2 x - 1\right)^{2} e^{- 2 x \left(x - 1\right)}}{1 - e^{2 x \left(1 - x\right)}}\right) e^{- x \left(x - 1\right)}}{\sqrt{1 - e^{2 x \left(1 - x\right)}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−82.1189398655108x_{1} = -82.1189398655108
x2=−16.5873386663431x_{2} = -16.5873386663431
x3=−84.1161276367699x_{3} = -84.1161276367699
x4=−20.4748330197422x_{4} = -20.4748330197422
x5=−34.2837046931748x_{5} = -34.2837046931748
x6=26.6424780144044x_{6} = 26.6424780144044
x7=14.9749824857761x_{7} = 14.9749824857761
x8=−76.1282572618827x_{8} = -76.1282572618827
x9=68.3999027689785x_{9} = 68.3999027689785
x10=−46.2107796288234x_{10} = -46.2107796288234
x11=48.4625101018812x_{11} = 48.4625101018812
x12=−40.2418766927414x_{12} = -40.2418766927414
x13=−32.301047151143x_{13} = -32.301047151143
x14=−98.0050743238786x_{14} = -98.0050743238786
x15=−30.3206391428518x_{15} = -30.3206391428518
x16=84.3712995506549x_{16} = 84.3712995506549
x17=80.3773763965944x_{17} = 80.3773763965944
x18=−68.143214120251x_{18} = -68.143214120251
x19=24.6751058559075x_{19} = 24.6751058559075
x20=−44.2202187951075x_{20} = -44.2202187951075
x21=82.3742637667605x_{21} = 82.3742637667605
x22=11.2547368089636x_{22} = 11.2547368089636
x23=9.49024588225921x_{23} = 9.49024588225921
x24=−94.1038498325156x_{24} = -94.1038498325156
x25=−5.92402617076406x_{25} = -5.92402617076406
x26=78.3806488651423x_{26} = 78.3806488651423
x27=70.395611336994x_{27} = 70.395611336994
x28=−72.1353239493314x_{28} = -72.1353239493314
x29=66.4044546045305x_{29} = 66.4044546045305
x30=−90.1084356631546x_{30} = -90.1084356631546
x31=−92.1060931321846x_{31} = -92.1060931321846
x32=−86.1134452607208x_{32} = -86.1134452607208
x33=100.255010790106x_{33} = 100.255010790106
x34=62.4144403171831x_{34} = 62.4144403171831
x35=−50.1941339587386x_{35} = -50.1941339587386
x36=72.3915586002076x_{36} = 72.3915586002076
x37=52.4461329625053x_{37} = 52.4461329625053
x38=−18.5251765300086x_{38} = -18.5251765300086
x39=−26.3685733915089x_{39} = -26.3685733915089
x40=56.4320965487307x_{40} = 56.4320965487307
x41=−10.9080521426508x_{41} = -10.9080521426508
x42=6.48915870510635x_{42} = 6.48915870510635
x43=96.3561109158819x_{43} = 96.3561109158819
x44=38.5185349670573x_{44} = 38.5185349670573
x45=−14.6659967073356x_{45} = -14.6659967073356
x46=20.7597001055415x_{46} = 20.7597001055415
x47=−7.41308381939881x_{47} = -7.41308381939881
x48=−80.1218916025257x_{48} = -80.1218916025257
x49=76.3840938006122x_{49} = 76.3840938006122
x50=13.0927264960382x_{50} = 13.0927264960382
x51=36.5334709380707x_{51} = 36.5334709380707
x52=−100.004974408232x_{52} = -100.004974408232
x53=−22.4332430971178x_{53} = -22.4332430971178
x54=−24.3983151872144x_{54} = -24.3983151872144
x55=32.5689301221016x_{55} = 32.5689301221016
x56=46.4717670280435x_{56} = 46.4717670280435
x57=−48.2021151786712x_{57} = -48.2021151786712
x58=42.4929253680612x_{58} = 42.4929253680612
x59=−9.10750078550253x_{59} = -9.10750078550253
x60=−60.1621159709617x_{60} = -60.1621159709617
x61=98.2551145950588x_{61} = 98.2551145950588
x62=30.5901931787706x_{62} = 30.5901931787706
x63=18.8158611350303x_{63} = 18.8158611350303
x64=−66.1475143051055x_{64} = -66.1475143051055
x65=50.4539938657847x_{65} = 50.4539938657847
x66=−74.1316959379875x_{66} = -74.1316959379875
x67=86.3684733862224x_{67} = 86.3684733862224
x68=−70.1391573641318x_{68} = -70.1391573641318
x69=−56.1735678489804x_{69} = -56.1735678489804
x70=7.85828925030317x_{70} = 7.85828925030317
x71=−58.1676467773744x_{71} = -58.1676467773744
x72=−78.124993485196x_{72} = -78.124993485196
x73=22.7136045921971x_{73} = 22.7136045921971
x74=28.6144791191814x_{74} = 28.6144791191814
x75=−88.1108839441568x_{75} = -88.1108839441568
x76=34.5501603867115x_{76} = 34.5501603867115
x77=−12.7686486633886x_{77} = -12.7686486633886
x78=−36.268246358677x_{78} = -36.268246358677
x79=−64.1520804872676x_{79} = -64.1520804872676
x80=74.3877251963013x_{80} = 74.3877251963013
x81=−54.179921950687x_{81} = -54.179921950687
x82=−38.254381635506x_{82} = -38.254381635506
x83=64.4092912641406x_{83} = 64.4092912641406
x84=16.8857424273909x_{84} = 16.8857424273909
x85=94.3583729112083x_{85} = 94.3583729112083
x86=90.3631983715661x_{86} = 90.3631983715661
x87=40.5050905560828x_{87} = 40.5050905560828
x88=−52.1867583238569x_{88} = -52.1867583238569
x89=−62.1569381113902x_{89} = -62.1569381113902
x90=60.4199330057959x_{90} = 60.4199330057959
x91=−28.3429466774556x_{91} = -28.3429466774556
x92=−96.101698521238x_{92} = -96.101698521238
x93=44.4818654670654x_{93} = 44.4818654670654
x94=92.360733133595x_{94} = 92.360733133595
x95=58.4258048768246x_{95} = 58.4258048768246
x96=54.4388546482833x_{96} = 54.4388546482833
x97=88.3657758562432x_{97} = 88.3657758562432
x98=−42.23054122548x_{98} = -42.23054122548

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→−∞acos⁡(ex(1−x))=π2\lim_{x \to -\infty} \operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} = \frac{\pi}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=π2y = \frac{\pi}{2}
lim⁡x→∞acos⁡(ex(1−x))=π2\lim_{x \to \infty} \operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} = \frac{\pi}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=π2y = \frac{\pi}{2}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función acos(exp(x*(1 - x))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(acos⁡(ex(1−x))x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(acos⁡(ex(1−x))x)=0\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)}}{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:
acos⁡(ex(1−x))=acos⁡(e−x(x+1))\operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} = \operatorname{acos}{\left(e^{- x \left(x + 1\right)} \right)}
- No
acos⁡(ex(1−x))=−acos⁡(e−x(x+1))\operatorname{acos}{\left(e^{x \left(1 - x\right)} \right)} = - \operatorname{acos}{\left(e^{- x \left(x + 1\right)} \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = acos(exp(x*(1-x)))
    © Sr Examen – Калькулятор