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(-x+atan(x))/x^3
Trazado el gráfico de la función/ (-x+atan(x))/x^3

Gráfico de la función y = (-x+atan(x))/x^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -x + atan(x)
f(x) = ------------
             3     
            x
f(x)=−x+atan⁡(x)x3f{\left(x \right)} = \frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}} 
f = (-x + atan(x))/x^3
Gráfico de la función
02468-8-6-4-2-1010-0.50.0
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 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:
−x+atan⁡(x)x3=0\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}} = 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 (-x + atan(x))/x^3.
−0+atan⁡(0)03\frac{- 0 + \operatorname{atan}{\left(0 \right)}}{0^{3}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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+1x2+1x3−3(−x+atan⁡(x))x4=0\frac{-1 + \frac{1}{x^{2} + 1}}{x^{3}} - \frac{3 \left(- x + \operatorname{atan}{\left(x \right)}\right)}{x^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=37695.4423747551x_{1} = 37695.4423747551
x2=31762.8148239848x_{2} = 31762.8148239848
x3=24983.0265404262x_{3} = 24983.0265404262
x4=−18072.8562532625x_{4} = -18072.8562532625
x5=34305.3436017062x_{5} = 34305.3436017062
x6=19051.3391839221x_{6} = 19051.3391839221
x7=15662.345886645x_{7} = 15662.345886645
x8=−20614.8195410024x_{8} = -20614.8195410024
x9=−40954.3706546333x_{9} = -40954.3706546333
x10=−21462.1864505397x_{10} = -21462.1864505397
x11=−27394.1738017033x_{11} = -27394.1738017033
x12=33457.8291397138x_{12} = 33457.8291397138
x13=−24851.8193555671x_{13} = -24851.8193555671
x14=−35021.6422921722x_{14} = -35021.6422921722
x15=12274.1733072055x_{15} = 12274.1733072055
x16=−19767.4735221121x_{16} = -19767.4735221121
x17=−28241.6439581724x_{17} = -28241.6439581724
x18=42780.6862304154x_{18} = 42780.6862304154
x19=−29936.6074421376x_{19} = -29936.6074421376
x20=−30784.0994722808x_{20} = -30784.0994722808
x21=21593.3846176302x_{21} = 21593.3846176302
x22=−24004.389916208x_{22} = -24004.389916208
x23=41085.5943179336x_{23} = 41085.5943179336
x24=16509.5376458429x_{24} = 16509.5376458429
x25=−39259.289744171x_{25} = -39259.289744171
x26=30067.8227785119x_{26} = 30067.8227785119
x27=39390.5125693788x_{27} = 39390.5125693788
x28=36847.9121154636x_{28} = 36847.9121154636
x29=−32479.1013813967x_{29} = -32479.1013813967
x30=18204.0397130023x_{30} = 18204.0397130023
x31=28372.8570679257x_{31} = 28372.8570679257
x32=27525.3856373513x_{32} = 27525.3856373513
x33=17356.7709091699x_{33} = 17356.7709091699
x34=38542.9759312704x_{34} = 38542.9759312704
x35=−16378.3652980817x_{35} = -16378.3652980817
x36=−34174.124093174x_{36} = -34174.124093174
x37=−23156.9735604547x_{37} = -23156.9735604547
x38=13968.1182452817x_{38} = 13968.1182452817
x39=−15531.1805503588x_{39} = -15531.1805503588
x40=40238.0520918502x_{40} = 40238.0520918502
x41=−12989.9682054958x_{41} = -12989.9682054958
x42=−17225.5925849612x_{42} = -17225.5925849612
x43=25830.4694498413x_{43} = 25830.4694498413
x44=−29089.1220673667x_{44} = -29089.1220673667
x45=−40106.8288341669x_{45} = -40106.8288341669
x46=23288.1767401818x_{46} = 23288.1767401818
x47=−14684.0459720068x_{47} = -14684.0459720068
x48=32610.3194185706x_{48} = 32610.3194185706
x49=−33326.6103384257x_{49} = -33326.6103384257
x50=−26546.7123745741x_{50} = -26546.7123745741
x51=14815.2030082688x_{51} = 14815.2030082688
x52=29220.3363398227x_{52} = 29220.3363398227
x53=35152.8624565848x_{53} = 35152.8624565848
x54=−13836.9711337571x_{54} = -13836.9711337571
x55=−38411.7535678211x_{55} = -38411.7535678211
x56=26677.922810015x_{56} = 26677.922810015
x57=−36716.6907745185x_{57} = -36716.6907745185
x58=−22309.5718142799x_{58} = -22309.5718142799
x59=20746.0147149245x_{59} = 20746.0147149245
x60=−25699.2605579154x_{60} = -25699.2605579154
x61=−18920.1512789125x_{61} = -18920.1512789125
x62=−35869.1646158396x_{62} = -35869.1646158396
x63=−41801.9150374336x_{63} = -41801.9150374336
x64=13121.1033142041x_{64} = 13121.1033142041
x65=−37564.2205050519x_{65} = -37564.2205050519
x66=30915.3157846703x_{66} = 30915.3157846703
x67=36000.3853896247x_{67} = 36000.3853896247
x68=−42649.461827958x_{68} = -42649.461827958
x69=19898.6653005008x_{69} = 19898.6653005008
x70=−31631.5976140599x_{70} = -31631.5976140599
x71=41933.1390816712x_{71} = 41933.1390816712
x72=24135.5952065001x_{72} = 24135.5952065001
x73=22440.7726333195x_{73} = 22440.7726333195
x74=−12143.0529026344x_{74} = -12143.0529026344
Signos de extremos en los puntos:
(37695.442374755075, -7.03726999829967e-10)

(31762.81482398482, -9.91152673962698e-10)

(24983.02654042619, -1.60207408398598e-9)

(-18072.85625326246, -3.0613195463161e-9)

(34305.34360170624, -8.49682291355882e-10)

(19051.339183922148, -2.75494652621263e-9)

(15662.345886645018, -4.07608115537644e-9)

(-20614.81954100241, -2.35292363828522e-9)

(-40954.37065463332, -5.96187449881745e-10)

(-21462.186450539717, -2.17080237132013e-9)

(-27394.17380170328, -1.33247381592315e-9)

(33457.829139713795, -8.93272763815866e-10)

(-24851.8193555671, -1.61903476140945e-9)

(-35021.64229217217, -8.15281345135363e-10)

(12274.173307205472, -6.63681814917335e-9)

(-19767.47352211211, -2.55895799826681e-9)

(-28241.643958172423, -1.25370655711109e-9)

(42780.68623041544, -5.4637215795632e-10)

(-29936.60744213758, -1.11576323518388e-9)

(-30784.099472280774, -1.05517606679392e-9)

(21593.384617630203, -2.14450452279427e-9)

(-24004.389916207994, -1.73536260726775e-9)

(41085.59431793356, -5.92385266867228e-10)

(16509.53764584295, -3.66850281768929e-9)

(-39259.289744170994, -6.4878043831623e-10)

(30067.8227785119, -1.10604640353812e-9)

(39390.51256937884, -6.44465119800202e-10)

(36847.91211546358, -7.36471112583106e-10)

(-32479.101381396697, -9.47918471742499e-10)

(18204.03971300229, -3.01735894283166e-9)

(28372.85706792574, -1.24213790730051e-9)

(27525.385637351348, -1.31980081249435e-9)

(17356.77090916988, -3.31911907519973e-9)

(38542.97593127045, -6.73118943310333e-10)

(-16378.365298081748, -3.72749647410585e-9)

(-34174.124093174025, -8.56219774242942e-10)

(-23156.973560454706, -1.86469128439758e-9)

(13968.118245281725, -5.12478151527373e-9)

(-15531.180550358766, -4.1452157438356e-9)

(40238.0520918502, -6.17602647537309e-10)

(-12989.968205495828, -5.92558601529429e-9)

(-17225.59258496116, -3.36986151711332e-9)

(25830.469449841286, -1.49868027442496e-9)

(-29089.122067366698, -1.18172199785001e-9)

(-40106.82883416693, -6.21650577669032e-10)

(23288.17674018178, -1.84374022001946e-9)

(-14684.04597200684, -4.6372666663652e-9)

(32610.31941857058, -9.40305497051386e-10)

(-33326.61033842568, -9.0032071522487e-10)

(-26546.712374574054, -1.41890334782319e-9)

(14815.203008268796, -4.55552819924657e-9)

(29220.33633982265, -1.17113303715945e-9)

(35152.86245658479, -8.09206204108099e-10)

(-13836.9711337571, -5.2223819027556e-9)

(-38411.753567821135, -6.77725727007463e-10)

(26677.92281001498, -1.40498085169395e-9)

(-36716.69077451845, -7.41744536841238e-10)

(-22309.571814279894, -2.00903244998659e-9)

(20746.014714924495, -2.32325967128191e-9)

(-25699.260557915426, -1.51402204783306e-9)

(-18920.151278912526, -2.79328168924977e-9)

(-35869.16461583956, -7.77210137690153e-10)

(-41801.9150374336, -5.72257276955845e-10)

(13121.103314204072, -5.80774176311859e-9)

(-37564.22050505195, -7.08652097474588e-10)

(30915.31578467033, -1.04623816816246e-9)

(36000.38538962473, -7.71554752122418e-10)

(-42649.46182795796, -5.49739437736566e-10)

(19898.665300500757, -2.5253281641932e-9)

(-31631.597614059938, -9.99392713138864e-10)

(41933.13908167115, -5.68681345571628e-10)

(24135.595206500064, -1.71654703584026e-9)

(22440.772633319477, -1.98561017295073e-9)

(-12143.05290263435, -6.78091102495798e-9)


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
Crece 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(−1(x2+1)2+3(1−1x2+1)x2−6(x−atan⁡(x))x3)x2=0\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{3 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} - \frac{6 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4867.53210537788x_{1} = -4867.53210537788
x2=−10318.2039505215x_{2} = -10318.2039505215
x3=7953.48485502851x_{3} = 7953.48485502851
x4=9261.73079916489x_{4} = 9261.73079916489
x5=4683.31340858871x_{5} = 4683.31340858871
x6=6209.27485084669x_{6} = 6209.27485084669
x7=4901.28702868725x_{7} = 4901.28702868725
x8=2722.28930822549x_{8} = 2722.28930822549
x9=10133.9203149188x_{9} = 10133.9203149188
x10=−8137.75760397691x_{10} = -8137.75760397691
x11=1851.91777546029x_{11} = 1851.91777546029
x12=8607.60103964852x_{12} = 8607.60103964852
x13=−8573.83626815949x_{13} = -8573.83626815949
x14=2069.34751479272x_{14} = 2069.34751479272
x15=6645.30962505809x_{15} = 6645.30962505809
x16=4465.34956037736x_{16} = 4465.34956037736
x17=−2470.85333481601x_{17} = -2470.85333481601
x18=−2253.21084726316x_{18} = -2253.21084726316
x19=6863.33216394454x_{19} = 6863.33216394454
x20=−2906.34163889693x_{20} = -2906.34163889693
x21=−7701.68608048424x_{21} = -7701.68608048424
x22=−10100.154265882x_{22} = -10100.154265882
x23=1634.6769971136x_{23} = 1634.6769971136
x24=8389.56064339035x_{24} = 8389.56064339035
x25=3593.62905380545x_{25} = 3593.62905380545
x26=2940.06899128763x_{26} = 2940.06899128763
x27=−4431.59770179171x_{27} = -4431.59770179171
x28=5337.25865222579x_{28} = 5337.25865222579
x29=−2688.56942922662x_{29} = -2688.56942922662
x30=7735.44975075141x_{30} = 7735.44975075141
x31=−8355.79611466611x_{31} = -8355.79611466611
x32=5991.26338522101x_{32} = 5991.26338522101
x33=4029.45777676862x_{33} = 4029.45777676862
x34=3157.89085719709x_{34} = 3157.89085719709
x35=−1601.05829592318x_{35} = -1601.05829592318
x36=9915.87140475585x_{36} = 9915.87140475585
x37=−5303.50140679323x_{37} = -5303.50140679323
x38=−10972.3579735698x_{38} = -10972.3579735698
x39=−4213.64710083852x_{39} = -4213.64710083852
x40=−5957.50358562872x_{40} = -5957.50358562872
x41=−5085.51297240982x_{41} = -5085.51297240982
x42=10788.072291218x_{42} = 10788.072291218
x43=5773.25647992574x_{43} = 5773.25647992574
x44=−3342.00835616088x_{44} = -3342.00835616088
x45=−6829.57005290069x_{45} = -6829.57005290069
x46=8825.64293619637x_{46} = 8825.64293619637
x47=−9664.05772295018x_{47} = -9664.05772295018
x48=10351.9701367697x_{48} = 10351.9701367697
x49=7299.38597330783x_{49} = 7299.38597330783
x50=9479.77657527862x_{50} = 9479.77657527862
x51=−1818.26252555506x_{51} = -1818.26252555506
x52=−7047.59514826828x_{52} = -7047.59514826828
x53=4247.39704550207x_{53} = 4247.39704550207
x54=−6175.51437853999x_{54} = -6175.51437853999
x55=7517.41672670257x_{55} = 7517.41672670257
x56=3375.74634228698x_{56} = 3375.74634228698
x57=−9882.10550224246x_{57} = -9882.10550224246
x58=−2035.66756840836x_{58} = -2035.66756840836
x59=−10536.2544981351x_{59} = -10536.2544981351
x60=10570.0208130553x_{60} = 10570.0208130553
x61=−7483.65339476579x_{61} = -7483.65339476579
x62=−3559.88720742498x_{62} = -3559.88720742498
x63=−5521.49648640676x_{63} = -5521.49648640676
x64=5555.2546872073x_{64} = 5555.2546872073
x65=2286.90825842172x_{65} = 2286.90825842172
x66=9043.68622264216x_{66} = 9043.68622264216
x67=−9227.96540123938x_{67} = -9227.96540123938
x68=5119.26913345252x_{68} = 5119.26913345252
x69=−4649.55990669058x_{69} = -4649.55990669058
x70=11006.1245229238x_{70} = 11006.1245229238
x71=−10754.3058554708x_{71} = -10754.3058554708
x72=6427.29040132553x_{72} = 6427.29040132553
x73=−9009.92101820658x_{73} = -9009.92101820658
x74=8171.52186982164x_{74} = 8171.52186982164
x75=−3995.7100794807x_{75} = -3995.7100794807
x76=−7919.72087446313x_{76} = -7919.72087446313
x77=−3124.15760696891x_{77} = -3124.15760696891
x78=−7265.62301121462x_{78} = -7265.62301121462
x79=9697.8234687662x_{79} = 9697.8234687662
x80=−9446.01099729817x_{80} = -9446.01099729817
x81=3811.53412304893x_{81} = 3811.53412304893
x82=−6393.52932539204x_{82} = -6393.52932539204
x83=−8791.8779400563x_{83} = -8791.8779400563
x84=−6611.5480054393x_{84} = -6611.5480054393
x85=−3777.78908879348x_{85} = -3777.78908879348
x86=7081.35770496613x_{86} = 7081.35770496613
x87=−5739.49743309489x_{87} = -5739.49743309489
x88=2504.56354942868x_{88} = 2504.56354942868
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2(−1(x2+1)2+3(1−1x2+1)x2−6(x−atan⁡(x))x3)x2)=25\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{3 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} - \frac{6 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{2}}\right) = \frac{2}{5}
lim⁡x→0+(2(−1(x2+1)2+3(1−1x2+1)x2−6(x−atan⁡(x))x3)x2)=25\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{3 \left(1 - \frac{1}{x^{2} + 1}\right)}{x^{2}} - \frac{6 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{3}}\right)}{x^{2}}\right) = \frac{2}{5}
- los límites son iguales, es decir omitimos el punto correspondiente

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 verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−x+atan⁡(x)x3)=0\lim_{x \to -\infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(−x+atan⁡(x)x3)=0\lim_{x \to \infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\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 (-x + atan(x))/x^3, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−x+atan⁡(x)xx3)=0\lim_{x \to -\infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x x^{3}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−x+atan⁡(x)xx3)=0\lim_{x \to \infty}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x x^{3}}\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:
−x+atan⁡(x)x3=−x−atan⁡(x)x3\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}} = - \frac{x - \operatorname{atan}{\left(x \right)}}{x^{3}}
- No
−x+atan⁡(x)x3=x−atan⁡(x)x3\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}} = \frac{x - \operatorname{atan}{\left(x \right)}}{x^{3}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (-x+atan(x))/x^3
    © Sr Examen – Калькулятор