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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
02468-8-6-4-2-10100.00.5
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.
(−1)atan⁡(0)03\frac{\left(-1\right) \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=−28241.6439581724x_{1} = -28241.6439581724
x2=41933.1390816712x_{2} = 41933.1390816712
x3=−24851.8193555671x_{3} = -24851.8193555671
x4=−33326.6103384257x_{4} = -33326.6103384257
x5=38542.9759312704x_{5} = 38542.9759312704
x6=36847.9121154636x_{6} = 36847.9121154636
x7=−40106.8288341669x_{7} = -40106.8288341669
x8=−25699.2605579154x_{8} = -25699.2605579154
x9=40238.0520918502x_{9} = 40238.0520918502
x10=30915.3157846703x_{10} = 30915.3157846703
x11=21593.3846176302x_{11} = 21593.3846176302
x12=−42649.461827958x_{12} = -42649.461827958
x13=13968.1182452817x_{13} = 13968.1182452817
x14=−24004.389916208x_{14} = -24004.389916208
x15=−12989.9682054958x_{15} = -12989.9682054958
x16=−36716.6907745185x_{16} = -36716.6907745185
x17=−31631.5976140599x_{17} = -31631.5976140599
x18=42780.6862304154x_{18} = 42780.6862304154
x19=−34174.124093174x_{19} = -34174.124093174
x20=28372.8570679257x_{20} = 28372.8570679257
x21=41085.5943179336x_{21} = 41085.5943179336
x22=24983.0265404262x_{22} = 24983.0265404262
x23=−12143.0529026344x_{23} = -12143.0529026344
x24=−18920.1512789125x_{24} = -18920.1512789125
x25=−18072.8562532625x_{25} = -18072.8562532625
x26=−29936.6074421376x_{26} = -29936.6074421376
x27=19898.6653005008x_{27} = 19898.6653005008
x28=−40954.3706546333x_{28} = -40954.3706546333
x29=13121.1033142041x_{29} = 13121.1033142041
x30=−23156.9735604547x_{30} = -23156.9735604547
x31=−32479.1013813967x_{31} = -32479.1013813967
x32=−17225.5925849612x_{32} = -17225.5925849612
x33=12274.1733072055x_{33} = 12274.1733072055
x34=−38411.7535678211x_{34} = -38411.7535678211
x35=−21462.1864505397x_{35} = -21462.1864505397
x36=−27394.1738017033x_{36} = -27394.1738017033
x37=32610.3194185706x_{37} = 32610.3194185706
x38=−30784.0994722808x_{38} = -30784.0994722808
x39=34305.3436017062x_{39} = 34305.3436017062
x40=24135.5952065001x_{40} = 24135.5952065001
x41=19051.3391839221x_{41} = 19051.3391839221
x42=35152.8624565848x_{42} = 35152.8624565848
x43=−29089.1220673667x_{43} = -29089.1220673667
x44=20746.0147149245x_{44} = 20746.0147149245
x45=−19767.4735221121x_{45} = -19767.4735221121
x46=25830.4694498413x_{46} = 25830.4694498413
x47=31762.8148239848x_{47} = 31762.8148239848
x48=−26546.7123745741x_{48} = -26546.7123745741
x49=−22309.5718142799x_{49} = -22309.5718142799
x50=29220.3363398227x_{50} = 29220.3363398227
x51=−37564.2205050519x_{51} = -37564.2205050519
x52=−14684.0459720068x_{52} = -14684.0459720068
x53=36000.3853896247x_{53} = 36000.3853896247
x54=30067.8227785119x_{54} = 30067.8227785119
x55=23288.1767401818x_{55} = 23288.1767401818
x56=27525.3856373513x_{56} = 27525.3856373513
x57=−35869.1646158396x_{57} = -35869.1646158396
x58=−16378.3652980817x_{58} = -16378.3652980817
x59=39390.5125693788x_{59} = 39390.5125693788
x60=15662.345886645x_{60} = 15662.345886645
x61=37695.4423747551x_{61} = 37695.4423747551
x62=16509.5376458429x_{62} = 16509.5376458429
x63=26677.922810015x_{63} = 26677.922810015
x64=−35021.6422921722x_{64} = -35021.6422921722
x65=22440.7726333195x_{65} = 22440.7726333195
x66=−15531.1805503588x_{66} = -15531.1805503588
x67=−13836.9711337571x_{67} = -13836.9711337571
x68=33457.8291397138x_{68} = 33457.8291397138
x69=17356.7709091699x_{69} = 17356.7709091699
x70=14815.2030082688x_{70} = 14815.2030082688
x71=−20614.8195410024x_{71} = -20614.8195410024
x72=−41801.9150374336x_{72} = -41801.9150374336
x73=−39259.289744171x_{73} = -39259.289744171
x74=18204.0397130023x_{74} = 18204.0397130023
Signos de extremos en los puntos:
(-28241.643958172423, 1.25370655711109e-9)

(41933.13908167115, 5.68681345571628e-10)

(-24851.8193555671, 1.61903476140945e-9)

(-33326.61033842568, 9.0032071522487e-10)

(38542.97593127045, 6.73118943310333e-10)

(36847.91211546358, 7.36471112583106e-10)

(-40106.82883416693, 6.21650577669032e-10)

(-25699.260557915426, 1.51402204783306e-9)

(40238.0520918502, 6.17602647537309e-10)

(30915.31578467033, 1.04623816816246e-9)

(21593.384617630203, 2.14450452279427e-9)

(-42649.46182795796, 5.49739437736566e-10)

(13968.118245281725, 5.12478151527373e-9)

(-24004.389916207994, 1.73536260726775e-9)

(-12989.968205495828, 5.92558601529429e-9)

(-36716.69077451845, 7.41744536841238e-10)

(-31631.597614059938, 9.99392713138864e-10)

(42780.68623041544, 5.4637215795632e-10)

(-34174.124093174025, 8.56219774242942e-10)

(28372.85706792574, 1.24213790730051e-9)

(41085.59431793356, 5.92385266867228e-10)

(24983.02654042619, 1.60207408398598e-9)

(-12143.05290263435, 6.78091102495798e-9)

(-18920.151278912526, 2.79328168924977e-9)

(-18072.85625326246, 3.0613195463161e-9)

(-29936.60744213758, 1.11576323518388e-9)

(19898.665300500757, 2.5253281641932e-9)

(-40954.37065463332, 5.96187449881745e-10)

(13121.103314204072, 5.80774176311859e-9)

(-23156.973560454706, 1.86469128439758e-9)

(-32479.101381396697, 9.47918471742499e-10)

(-17225.59258496116, 3.36986151711332e-9)

(12274.173307205472, 6.63681814917335e-9)

(-38411.753567821135, 6.77725727007463e-10)

(-21462.186450539717, 2.17080237132013e-9)

(-27394.17380170328, 1.33247381592315e-9)

(32610.31941857058, 9.40305497051386e-10)

(-30784.099472280774, 1.05517606679392e-9)

(34305.34360170624, 8.49682291355882e-10)

(24135.595206500064, 1.71654703584026e-9)

(19051.339183922148, 2.75494652621263e-9)

(35152.86245658479, 8.09206204108099e-10)

(-29089.122067366698, 1.18172199785001e-9)

(20746.014714924495, 2.32325967128191e-9)

(-19767.47352211211, 2.55895799826681e-9)

(25830.469449841286, 1.49868027442496e-9)

(31762.81482398482, 9.91152673962698e-10)

(-26546.712374574054, 1.41890334782319e-9)

(-22309.571814279894, 2.00903244998659e-9)

(29220.33633982265, 1.17113303715945e-9)

(-37564.22050505195, 7.08652097474588e-10)

(-14684.04597200684, 4.6372666663652e-9)

(36000.38538962473, 7.71554752122418e-10)

(30067.8227785119, 1.10604640353812e-9)

(23288.17674018178, 1.84374022001946e-9)

(27525.385637351348, 1.31980081249435e-9)

(-35869.16461583956, 7.77210137690153e-10)

(-16378.365298081748, 3.72749647410585e-9)

(39390.51256937884, 6.44465119800202e-10)

(15662.345886645018, 4.07608115537644e-9)

(37695.442374755075, 7.03726999829967e-10)

(16509.53764584295, 3.66850281768929e-9)

(26677.92281001498, 1.40498085169395e-9)

(-35021.64229217217, 8.15281345135363e-10)

(22440.772633319477, 1.98561017295073e-9)

(-15531.180550358766, 4.1452157438356e-9)

(-13836.9711337571, 5.2223819027556e-9)

(33457.829139713795, 8.93272763815866e-10)

(17356.77090916988, 3.31911907519973e-9)

(14815.203008268796, 4.55552819924657e-9)

(-20614.81954100241, 2.35292363828522e-9)

(-41801.9150374336, 5.72257276955845e-10)

(-39259.289744170994, 6.4878043831623e-10)

(18204.03971300229, 3.01735894283166e-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=7299.38597330783x_{1} = 7299.38597330783
x2=−4649.55990669058x_{2} = -4649.55990669058
x3=4465.34956037736x_{3} = 4465.34956037736
x4=11006.1245229238x_{4} = 11006.1245229238
x5=−7919.72087446313x_{5} = -7919.72087446313
x6=8825.64293619637x_{6} = 8825.64293619637
x7=6863.33216394454x_{7} = 6863.33216394454
x8=−5739.49743309489x_{8} = -5739.49743309489
x9=3375.74634228698x_{9} = 3375.74634228698
x10=5337.25865222579x_{10} = 5337.25865222579
x11=5119.26913345252x_{11} = 5119.26913345252
x12=10133.9203149188x_{12} = 10133.9203149188
x13=2069.34751479272x_{13} = 2069.34751479272
x14=−10536.2544981351x_{14} = -10536.2544981351
x15=2286.90825842172x_{15} = 2286.90825842172
x16=6427.29040132553x_{16} = 6427.29040132553
x17=−9227.96540123938x_{17} = -9227.96540123938
x18=−8355.79611466611x_{18} = -8355.79611466611
x19=−9882.10550224246x_{19} = -9882.10550224246
x20=−5521.49648640676x_{20} = -5521.49648640676
x21=−7265.62301121462x_{21} = -7265.62301121462
x22=−6611.5480054393x_{22} = -6611.5480054393
x23=−2906.34163889693x_{23} = -2906.34163889693
x24=−9446.01099729817x_{24} = -9446.01099729817
x25=−8573.83626815949x_{25} = -8573.83626815949
x26=4683.31340858871x_{26} = 4683.31340858871
x27=1851.91777546029x_{27} = 1851.91777546029
x28=−3559.88720742498x_{28} = -3559.88720742498
x29=−5303.50140679323x_{29} = -5303.50140679323
x30=−5085.51297240982x_{30} = -5085.51297240982
x31=−10972.3579735698x_{31} = -10972.3579735698
x32=9043.68622264216x_{32} = 9043.68622264216
x33=−9009.92101820658x_{33} = -9009.92101820658
x34=2504.56354942868x_{34} = 2504.56354942868
x35=−2253.21084726316x_{35} = -2253.21084726316
x36=9479.77657527862x_{36} = 9479.77657527862
x37=10351.9701367697x_{37} = 10351.9701367697
x38=2722.28930822549x_{38} = 2722.28930822549
x39=6645.30962505809x_{39} = 6645.30962505809
x40=−9664.05772295018x_{40} = -9664.05772295018
x41=−10100.154265882x_{41} = -10100.154265882
x42=3811.53412304893x_{42} = 3811.53412304893
x43=4247.39704550207x_{43} = 4247.39704550207
x44=7953.48485502851x_{44} = 7953.48485502851
x45=−3124.15760696891x_{45} = -3124.15760696891
x46=−2688.56942922662x_{46} = -2688.56942922662
x47=7517.41672670257x_{47} = 7517.41672670257
x48=9697.8234687662x_{48} = 9697.8234687662
x49=4901.28702868725x_{49} = 4901.28702868725
x50=3157.89085719709x_{50} = 3157.89085719709
x51=−1818.26252555506x_{51} = -1818.26252555506
x52=9915.87140475585x_{52} = 9915.87140475585
x53=−4213.64710083852x_{53} = -4213.64710083852
x54=8389.56064339035x_{54} = 8389.56064339035
x55=−4431.59770179171x_{55} = -4431.59770179171
x56=5773.25647992574x_{56} = 5773.25647992574
x57=10788.072291218x_{57} = 10788.072291218
x58=10570.0208130553x_{58} = 10570.0208130553
x59=−3777.78908879348x_{59} = -3777.78908879348
x60=7735.44975075141x_{60} = 7735.44975075141
x61=5555.2546872073x_{61} = 5555.2546872073
x62=−6175.51437853999x_{62} = -6175.51437853999
x63=5991.26338522101x_{63} = 5991.26338522101
x64=8607.60103964852x_{64} = 8607.60103964852
x65=6209.27485084669x_{65} = 6209.27485084669
x66=−10754.3058554708x_{66} = -10754.3058554708
x67=−7701.68608048424x_{67} = -7701.68608048424
x68=−4867.53210537788x_{68} = -4867.53210537788
x69=3593.62905380545x_{69} = 3593.62905380545
x70=−8791.8779400563x_{70} = -8791.8779400563
x71=−6393.52932539204x_{71} = -6393.52932539204
x72=−2035.66756840836x_{72} = -2035.66756840836
x73=−8137.75760397691x_{73} = -8137.75760397691
x74=2940.06899128763x_{74} = 2940.06899128763
x75=9261.73079916489x_{75} = 9261.73079916489
x76=−7047.59514826828x_{76} = -7047.59514826828
x77=−7483.65339476579x_{77} = -7483.65339476579
x78=−2470.85333481601x_{78} = -2470.85333481601
x79=1634.6769971136x_{79} = 1634.6769971136
x80=−10318.2039505215x_{80} = -10318.2039505215
x81=8171.52186982164x_{81} = 8171.52186982164
x82=7081.35770496613x_{82} = 7081.35770496613
x83=−3342.00835616088x_{83} = -3342.00835616088
x84=−1601.05829592318x_{84} = -1601.05829592318
x85=4029.45777676862x_{85} = 4029.45777676862
x86=−6829.57005290069x_{86} = -6829.57005290069
x87=−3995.7100794807x_{87} = -3995.7100794807
x88=−5957.50358562872x_{88} = -5957.50358562872
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
    © Sr Examen – Калькулятор