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Trazado el gráfico de la función/ arctg(x)/(e^x)

Gráfico de la función y = arctg(x)/(e^x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       atan(x)
f(x) = -------
           x  
          E
f(x)=atan⁡(x)exf{\left(x \right)} = \frac{\operatorname{atan}{\left(x \right)}}{e^{x}} 
f = atan(x)/E^x
Gráfico de la función
02468-8-6-4-2-1010-5000050000
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:
atan⁡(x)ex=0\frac{\operatorname{atan}{\left(x \right)}}{e^{x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=59.1877984220795x_{1} = 59.1877984220795
x2=63.1868495906405x_{2} = 63.1868495906405
x3=103.183268758525x_{3} = 103.183268758525
x4=37.2055911357874x_{4} = 37.2055911357874
x5=77.1848273092174x_{5} = 77.1848273092174
x6=51.1907097452623x_{6} = 51.1907097452623
x7=39.2017366988819x_{7} = 39.2017366988819
x8=69.1857948834502x_{8} = 69.1857948834502
x9=73.1852623919652x_{9} = 73.1852623919652
x10=61.1872937221209x_{10} = 61.1872937221209
x11=113.182965546205x_{11} = 113.182965546205
x12=99.1834205539013x_{12} = 99.1834205539013
x13=33.2179728685119x_{13} = 33.2179728685119
x14=83.1843097779544x_{14} = 83.1843097779544
x15=81.1844670973992x_{15} = 81.1844670973992
x16=31.2285767745976x_{16} = 31.2285767745976
x17=45.1945690353946x_{17} = 45.1945690353946
x18=75.1850342574319x_{18} = 75.1850342574319
x19=107.183135873218x_{19} = 107.183135873218
x20=29.2453850657538x_{20} = 29.2453850657538
x21=47.1930392314136x_{21} = 47.1930392314136
x22=115.182915313439x_{22} = 115.182915313439
x23=117.182867938493x_{23} = 117.182867938493
x24=53.189809510092x_{24} = 53.189809510092
x25=55.1890395124405x_{25} = 55.1890395124405
x26=49.1917721248293x_{26} = 49.1917721248293
x27=111.183018872223x_{27} = 111.183018872223
x28=93.1836922512221x_{28} = 93.1836922512221
x29=109.183075551776x_{29} = 109.183075551776
x30=97.1835046902428x_{30} = 97.1835046902428
x31=57.1883754267393x_{31} = 57.1883754267393
x32=65.186456603753x_{32} = 65.186456603753
x33=79.1846389840386x_{33} = 79.1846389840386
x34=67.186107119178x_{34} = 67.186107119178
x35=35.2107611900178x_{35} = 35.2107611900178
x36=41.1987745768459x_{36} = 41.1987745768459
x37=101.183342076106x_{37} = 101.183342076106
x38=91.1837970221995x_{38} = 91.1837970221995
x39=89.1839101691383x_{39} = 89.1839101691383
x40=119.182823207979x_{40} = 119.182823207979
x41=71.1855147373009x_{41} = 71.1855147373009
x42=43.1964422923481x_{42} = 43.1964422923481
x43=87.18403261512x_{43} = 87.18403261512
x44=85.1841654148423x_{44} = 85.1841654148423
x45=105.183200156706x_{45} = 105.183200156706
x46=121.182780928117x_{46} = 121.182780928117
x47=95.1835950454233x_{47} = 95.1835950454233
x48=0x_{48} = 0
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 atan(x)/E^x.
atan⁡(0)e0\frac{\operatorname{atan}{\left(0 \right)}}{e^{0}}
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
−e−xatan⁡(x)+e−xx2+1=0- e^{- x} \operatorname{atan}{\left(x \right)} + \frac{e^{- x}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=85.1842346063611x_{1} = 85.1842346063611
x2=109.183104778778x_{2} = 109.183104778778
x3=119.182844946427x_{3} = 119.182844946427
x4=43.1974831119063x_{4} = 43.1974831119063
x5=99.1834611670237x_{5} = 99.1834611670237
x6=31.2351226666798x_{6} = 31.2351226666798
x7=39.2034159974866x_{7} = 39.2034159974866
x8=49.1923507345192x_{8} = 49.1923507345192
x9=69.1859426201322x_{9} = 69.1859426201322
x10=111.183046352654x_{10} = 111.183046352654
x11=51.1911976041242x_{11} = 51.1911976041242
x12=55.1893964070374x_{12} = 55.1893964070374
x13=33.2222867338296x_{13} = 33.2222867338296
x14=73.1853822604484x_{14} = 73.1853822604484
x15=53.1902249534802x_{15} = 53.1902249534802
x16=59.1880678461169x_{16} = 59.1880678461169
x17=105.183233349247x_{17} = 105.183233349247
x18=95.1836418842928x_{18} = 95.1836418842928
x19=0.747211955161568x_{19} = 0.747211955161568
x20=61.1875301292092x_{20} = 61.1875301292092
x21=75.1851428155636x_{21} = 75.1851428155636
x22=29.2562563384761x_{22} = 29.2562563384761
x23=37.2078062507627x_{23} = 37.2078062507627
x24=57.1886844308322x_{24} = 57.1886844308322
x25=63.1870582237151x_{25} = 63.1870582237151
x26=107.183166998612x_{26} = 107.183166998612
x27=117.182890949296x_{27} = 117.182890949296
x28=101.183379988955x_{28} = 101.183379988955
x29=113.182991416959x_{29} = 113.182991416959
x30=79.1847288825771x_{30} = 79.1847288825771
x31=115.182939698215x_{31} = 115.182939698215
x32=93.1837426872226x_{32} = 93.1837426872226
x33=45.1954125148498x_{33} = 45.1954125148498
x34=35.2137841865788x_{34} = 35.2137841865788
x35=67.1862721161642x_{35} = 67.1862721161642
x36=81.1845492662939x_{36} = 81.1845492662939
x37=89.1839689919496x_{37} = 89.1839689919496
x38=77.1849259468806x_{38} = 77.1849259468806
x39=71.1856475607887x_{39} = 71.1856475607887
x40=87.1840963402224x_{40} = 87.1840963402224
x41=65.1866416947261x_{41} = 65.1866416947261
x42=41.2000823161778x_{42} = 41.2000823161778
x43=103.183304206226x_{43} = 103.183304206226
x44=121.182801486579x_{44} = 121.182801486579
x45=47.1937332876317x_{45} = 47.1937332876317
x46=83.1843850841488x_{46} = 83.1843850841488
x47=97.1835482676259x_{47} = 97.1835482676259
x48=91.1838514355446x_{48} = 91.1838514355446
Signos de extremos en los puntos:
(85.18423460636112, 1.57695437100074e-37)

(109.18310477877782, 5.96980781732506e-48)

(119.1828449464274, 2.71232680163858e-52)

(43.19748311190632, 2.68685651682442e-19)

(99.18346116702368, 1.31369195546394e-43)

(31.235122666679775, 4.18734272870461e-14)

(39.2034159974866, 1.45607533060368e-17)

(49.192350734519174, 6.70651816500195e-22)

(69.18594262013221, 1.39647240863825e-30)

(111.18304635265427, 8.08058069428484e-49)

(51.1911976041242, 9.09140760125111e-23)

(55.189396407037385, 1.66967263109774e-24)

(33.22228673382964, 5.74729812743755e-15)

(73.18538226044839, 2.5604606491747e-32)

(53.19022495348018, 1.23216819702701e-23)

(59.18806784611694, 3.06459115345288e-26)

(105.18323334924699, 3.25825880344179e-46)

(95.18364188429283, 7.16927243245884e-42)

(0.7472119551615676, 0.303970708865008)

(61.18753012920924, 4.15117785655748e-27)

(75.18514281556361, 3.46684531364579e-33)

(29.25625633847611, 3.02509006665195e-13)

(37.20780625076267, 1.07024139185157e-16)

(57.18868443083221, 2.26218781435376e-25)

(63.18705822371507, 5.62252963853095e-28)

(107.18316699861187, 4.41036737746924e-47)

(117.1828909492963, 2.00387762921248e-51)

(101.18337998895493, 1.77826006606962e-44)

(113.1829914169586, 1.09375903513552e-49)

(79.18472888257706, 6.35511714007681e-35)

(115.18293969821491, 1.48046382513151e-50)

(93.18374268722258, 5.2961162811258e-41)

(45.195412514849764, 3.6462100835775e-20)

(35.21378418657885, 7.85319792249962e-16)

(67.1862721161642, 1.03123629659521e-29)

(81.18454926629394, 8.60397789339621e-36)

(89.18396899194956, 2.89003544325537e-39)

(77.18492594688057, 4.69392069394134e-34)

(71.18564756078865, 1.89097070863853e-31)

(87.18409634022241, 2.13483938416633e-38)

(65.18664169472613, 7.61481256317454e-29)

(41.20008231617777, 1.97874462061717e-18)

(103.18330420622623, 2.4070910178928e-45)

(121.18280148657853, 3.67122001460912e-53)

(47.193733287631716, 4.94589088973195e-21)

(83.18438508414884, 1.16483423430511e-36)

(97.18354826762585, 9.70480791709028e-43)

(91.18385143554461, 3.91231421182066e-40)


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:
x48=0.747211955161568x_{48} = 0.747211955161568
Decrece en los intervalos
(−∞,0.747211955161568]\left(-\infty, 0.747211955161568\right]
Crece en los intervalos
[0.747211955161568,∞)\left[0.747211955161568, \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
(−2x(x2+1)2+atan⁡(x)−2x2+1)e−x=0\left(- \frac{2 x}{\left(x^{2} + 1\right)^{2}} + \operatorname{atan}{\left(x \right)} - \frac{2}{x^{2} + 1}\right) e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31.2431732085877x_{1} = 31.2431732085877
x2=121.182822613408x_{2} = 121.182822613408
x3=41.2015462963447x_{3} = 41.2015462963447
x4=59.1883553043907x_{4} = 59.1883553043907
x5=57.1890151493499x_{5} = 57.1890151493499
x6=113.183018060677x_{6} = 113.183018060677
x7=63.1872796182893x_{7} = 63.1872796182893
x8=87.1841626515966x_{8} = 87.1841626515966
x9=85.1843066861695x_{9} = 85.1843066861695
x10=39.205316262185x_{10} = 39.205316262185
x11=105.183267620794x_{11} = 105.183267620794
x12=73.1855081737694x_{12} = 73.1855081737694
x13=1.4336519078806x_{13} = 1.4336519078806
x14=69.1860983671262x_{14} = 69.1860983671262
x15=89.1840301376831x_{15} = 89.1840301376831
x16=49.1929803162908x_{16} = 49.1929803162908
x17=119.182867297293x_{17} = 119.182867297293
x18=51.1917258531251x_{18} = 51.1917258531251
x19=71.1857873247724x_{19} = 71.1857873247724
x20=37.2103463599711x_{20} = 37.2103463599711
x21=115.182964797136x_{21} = 115.182964797136
x22=111.183074671075x_{22} = 111.183074671075
x23=29.2702561411876x_{23} = 29.2702561411876
x24=101.183419191102x_{24} = 101.183419191102
x25=53.1906728523036x_{25} = 53.1906728523036
x26=55.1897797041079x_{26} = 55.1897797041079
x27=75.185256663858x_{27} = 75.185256663858
x28=91.1839079413109x_{28} = 91.1839079413109
x29=35.2173097203437x_{29} = 35.2173097203437
x30=79.1848228863875x_{30} = 79.1848228863875
x31=99.1835031939156x_{31} = 99.1835031939156
x32=43.1986379476504x_{32} = 43.1986379476504
x33=109.183134915806x_{33} = 109.183134915806
x34=117.182914620932x_{34} = 117.182914620932
x35=77.1850292350287x_{35} = 77.1850292350287
x36=103.18334083221x_{36} = 103.18334083221
x37=33.2274309599796x_{37} = 33.2274309599796
x38=95.1836904346593x_{38} = 95.1836904346593
x39=61.1877816464692x_{39} = 61.1877816464692
x40=93.1837950131725x_{40} = 93.1837950131725
x41=81.1846350733562x_{41} = 81.1846350733562
x42=65.1868376420401x_{42} = 65.1868376420401
x43=47.1944927090152x_{43} = 47.1944927090152
x44=45.1963413875172x_{44} = 45.1963413875172
x45=97.1835933987185x_{45} = 97.1835933987185
x46=77.9110897064623x_{46} = 77.9110897064623
x47=83.1844636263869x_{47} = 83.1844636263869
x48=107.183199114012x_{48} = 107.183199114012
x49=67.1864464101894x_{49} = 67.1864464101894

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
[1.4336519078806,∞)\left[1.4336519078806, \infty\right)
Convexa en los intervalos
(−∞,1.4336519078806]\left(-\infty, 1.4336519078806\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(atan⁡(x)ex)=−∞\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x \right)}}{e^{x}}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(atan⁡(x)ex)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(x \right)}}{e^{x}}\right) = 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 atan(x)/E^x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(e−xatan⁡(x)x)=∞\lim_{x \to -\infty}\left(\frac{e^{- x} \operatorname{atan}{\left(x \right)}}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(e−xatan⁡(x)x)=0\lim_{x \to \infty}\left(\frac{e^{- x} \operatorname{atan}{\left(x \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:
atan⁡(x)ex=−exatan⁡(x)\frac{\operatorname{atan}{\left(x \right)}}{e^{x}} = - e^{x} \operatorname{atan}{\left(x \right)}
- No
atan⁡(x)ex=exatan⁡(x)\frac{\operatorname{atan}{\left(x \right)}}{e^{x}} = e^{x} \operatorname{atan}{\left(x \right)}
- No
es decir, función
no es
par ni impar
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