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Trazado el gráfico de la función/ arctg(x)^2*1/x^2

Gráfico de la función y = arctg(x)^2*1/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           2   
       atan (x)
f(x) = --------
           2   
          x
f(x)=atan⁡2(x)x2f{\left(x \right)} = \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}} 
f = atan(x)^2/x^2
Gráfico de la función
02468-8-6-4-2-101002
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:
atan⁡2(x)x2=0\frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}} = 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 atan(x)^2/x^2.
atan⁡2(0)02\frac{\operatorname{atan}^{2}{\left(0 \right)}}{0^{2}}
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
2atan⁡(x)x2(x2+1)−2atan⁡2(x)x3=0\frac{2 \operatorname{atan}{\left(x \right)}}{x^{2} \left(x^{2} + 1\right)} - \frac{2 \operatorname{atan}^{2}{\left(x \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−16315.1152175869x_{1} = -16315.1152175869
x2=−23093.8276177627x_{2} = -23093.8276177627
x3=33394.7591253945x_{3} = 33394.7591253945
x4=27462.2789903763x_{4} = 27462.2789903763
x5=−29026.0265630144x_{5} = -29026.0265630144
x6=−40043.786799176x_{6} = -40043.786799176
x7=−38348.7053355296x_{7} = -38348.7053355296
x8=42717.6529562189x_{8} = 42717.6529562189
x9=−25636.139152875x_{9} = -25636.139152875
x10=37632.3913504509x_{10} = 37632.3913504509
x11=−23941.2527454618x_{11} = -23941.2527454618
x12=−20551.6428217193x_{12} = -20551.6428217193
x13=−41738.8786897091x_{13} = -41738.8786897091
x14=−30721.0147356399x_{14} = -30721.0147356399
x15=−18009.6397917844x_{15} = -18009.6397917844
x16=30004.7336381735x_{16} = 30004.7336381735
x17=34242.2777675229x_{17} = 34242.2777675229
x18=−42586.428151884x_{18} = -42586.428151884
x19=−34111.057629099x_{19} = -34111.057629099
x20=−33263.5396612328x_{20} = -33263.5396612328
x21=41870.1031525007x_{21} = 41870.1031525007
x22=−32416.0262669935x_{22} = -32416.0262669935
x23=35089.8005974058x_{23} = 35089.8005974058
x24=22377.6187368982x_{24} = 22377.6187368982
x25=24072.4593299464x_{25} = 24072.4593299464
x26=19835.4784339352x_{26} = 19835.4784339352
x27=18988.1393061528x_{27} = 18988.1393061528
x28=20682.8397638684x_{28} = 20682.8397638684
x29=−21399.0208242685x_{29} = -21399.0208242685
x30=−18856.9492916941x_{30} = -18856.9492916941
x31=−31568.5178199851x_{31} = -31568.5178199851
x32=−19704.2847279852x_{32} = -19704.2847279852
x33=−26483.598079533x_{33} = -26483.598079533
x34=−24788.6903433097x_{34} = -24788.6903433097
x35=25767.3491704884x_{35} = 25767.3491704884
x36=40175.0105120384x_{36} = 40175.0105120384
x37=−35806.1059703664x_{37} = -35806.1059703664
x38=−36653.6357630571x_{38} = -36653.6357630571
x39=31699.7357670217x_{39} = 31699.7357670217
x40=26614.8095684922x_{40} = 26614.8095684922
x41=28309.7566128221x_{41} = 28309.7566128221
x42=35937.3273150574x_{42} = 35937.3273150574
x43=32547.245002683x_{43} = 32547.245002683
x44=39327.4679790075x_{44} = 39327.4679790075
x45=36784.8576484751x_{45} = 36784.8576484751
x46=−28178.5425745249x_{46} = -28178.5425745249
x47=−40891.3315230118x_{47} = -40891.3315230118
x48=24919.8987337046x_{48} = 24919.8987337046
x49=38479.9281958096x_{49} = 38479.9281958096
x50=23225.0321904926x_{50} = 23225.0321904926
x51=29157.241709707x_{51} = 29157.241709707
x52=17293.5410559894x_{52} = 17293.5410559894
x53=−39196.2446784783x_{53} = -39196.2446784783
x54=−34958.5798336815x_{54} = -34958.5798336815
x55=18140.8255698816x_{55} = 18140.8255698816
x56=−22246.41641428x_{56} = -22246.41641428
x57=−29873.5174772408x_{57} = -29873.5174772408
x58=−27331.0661666544x_{58} = -27331.0661666544
x59=−17162.3601718632x_{59} = -17162.3601718632
x60=16446.2904066561x_{60} = 16446.2904066561
x61=21530.2206192242x_{61} = 21530.2206192242
x62=−37501.1689609137x_{62} = -37501.1689609137
x63=30852.2318270272x_{63} = 30852.2318270272
x64=41022.5556226027x_{64} = 41022.5556226027
Signos de extremos en los puntos:
(-16315.115217586943, 9.26884395294446e-9)

(-23093.827617762727, 4.62619537587167e-9)

(33394.759125394485, 2.21241485104516e-9)

(27462.27899037631, 3.27149656656176e-9)

(-29026.026563014428, 2.9285020110065e-9)

(-40043.786799176, 1.53870605985372e-9)

(-38348.70533552958, 1.67773715094651e-9)

(42717.65295621894, 1.35211005704354e-9)

(-25636.139152874985, 3.7541614317263e-9)

(37632.391350450904, 1.74221371118306e-9)

(-23941.252745461825, 4.30450200285183e-9)

(-20551.64282171927, 5.8414378551297e-9)

(-41738.878689709076, 1.41626639579825e-9)

(-30721.01473563989, 2.61427121581817e-9)

(-18009.639791784448, 7.60674743326579e-9)

(30004.733638173526, 2.7405755146306e-9)

(34242.27776752293, 2.10425473783945e-9)

(-42586.428151884, 1.36045548959822e-9)

(-34111.05762909898, 2.12047508052798e-9)

(-33263.53966123277, 2.22990420626879e-9)

(41870.103152500655, 1.40740304304426e-9)

(-32416.026266993464, 2.34802731296408e-9)

(35089.80059740584, 2.00383609330237e-9)

(22377.618736898214, 4.92705378156517e-9)

(24072.45932994644, 4.25770786495003e-9)

(19835.478433935237, 6.27085149334799e-9)

(18988.139306152807, 6.84298855579961e-9)

(20682.83976386838, 5.76756747011126e-9)

(-21399.020824268537, 5.38798168093402e-9)

(-18856.94929169411, 6.93853148235274e-9)

(-31568.517819985056, 2.47579028383063e-9)

(-19704.284727985178, 6.35463106817894e-9)

(-26483.598079533032, 3.51774940133591e-9)

(-24788.690343309696, 4.01522865494592e-9)

(25767.349170488418, 3.71602655548288e-9)

(40175.01051203836, 1.528670877624e-9)

(-35806.10597036638, 1.92446548022248e-9)

(-36653.63576305709, 1.83649837389837e-9)

(31699.73576702167, 2.45533653346031e-9)

(26614.80956849221, 3.48315059069789e-9)

(28309.75661282207, 3.07856232379491e-9)

(35937.32731505741, 1.91043742278516e-9)

(32547.245002682954, 2.32913305087589e-9)

(39327.467979007466, 1.59526824849965e-9)

(36784.85764847505, 1.8234193625082e-9)

(-28178.542574524876, 3.10729922013013e-9)

(-40891.3315230118, 1.47558328252327e-9)

(24919.89873370463, 3.97305903237599e-9)

(38479.92819580964, 1.66631413320395e-9)

(23225.032190492588, 4.57407515094515e-9)

(29157.24170970699, 2.90220385640576e-9)

(17293.541055989386, 8.24973351161963e-9)

(-39196.24467847833, 1.60596740444435e-9)

(-34958.57983368149, 2.01890727715575e-9)

(18140.825569881647, 7.49713233859708e-9)

(-22246.41641428004, 4.98533989209954e-9)

(-29873.517477240795, 2.76470323040961e-9)

(-27331.066166654437, 3.30298327610778e-9)

(-17162.36017186324, 8.37632482089311e-9)

(16446.290406656062, 9.12158315748907e-9)

(21530.220619224183, 5.32251765375969e-9)

(-37501.168960913725, 1.75442738568204e-9)

(30852.231827027164, 2.59208153851824e-9)

(41022.55562260269, 1.46615825322182e-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(−2xatan⁡(x)−1(x2+1)2−4atan⁡(x)x(x2+1)+3atan⁡2(x)x2)x2=0\frac{2 \left(- \frac{2 x \operatorname{atan}{\left(x \right)} - 1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} + \frac{3 \operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2270.62016178769x_{1} = 2270.62016178769
x2=−6813.43008516762x_{2} = -6813.43008516762
x3=−5069.3490089084x_{3} = -5069.3490089084
x4=7501.28305891524x_{4} = 7501.28305891524
x5=10117.802611372x_{5} = 10117.802611372
x6=5975.11369859257x_{6} = 5975.11369859257
x7=9463.65572815645x_{7} = 9463.65572815645
x8=−8993.79762446118x_{8} = -8993.79762446118
x9=−8121.62867857239x_{9} = -8121.62867857239
x10=−8775.75326915176x_{10} = -8775.75326915176
x11=8373.43343578503x_{11} = 8373.43343578503
x12=−6159.36706474872x_{12} = -6159.36706474872
x13=−10520.1385301813x_{13} = -10520.1385301813
x14=4667.14123445909x_{14} = 4667.14123445909
x15=−6595.40575657108x_{15} = -6595.40575657108
x16=3795.33780638999x_{16} = 3795.33780638999
x17=2488.29614057458x_{17} = 2488.29614057458
x18=−7031.45731558056x_{18} = -7031.45731558056
x19=−2890.10353096108x_{19} = -2890.10353096108
x20=−4851.36387078328x_{20} = -4851.36387078328
x21=−3325.79341436438x_{21} = -3325.79341436438
x22=−3107.93195766563x_{22} = -3107.93195766563
x23=2923.83293459433x_{23} = 2923.83293459433
x24=2706.03895255285x_{24} = 2706.03895255285
x25=8591.47523670622x_{25} = 8591.47523670622
x26=−5505.34000740711x_{26} = -5505.34000740711
x27=−10302.0870594216x_{27} = -10302.0870594216
x28=−2672.31664375281x_{28} = -2672.31664375281
x29=−2019.34935025405x_{29} = -2019.34935025405
x30=−3761.59160337364x_{30} = -3761.59160337364
x31=9245.60880290231x_{31} = 9245.60880290231
x32=6411.14609776215x_{32} = 6411.14609776215
x33=6629.16773822359x_{33} = 6629.16773822359
x34=−7685.55390443464x_{34} = -7685.55390443464
x35=9681.70371780889x_{35} = 9681.70371780889
x36=2053.03379556303x_{36} = 2053.03379556303
x37=−10738.190772382x_{37} = -10738.190772382
x38=6847.19253475579x_{38} = 6847.19253475579
x39=7719.31783892417x_{39} = 7719.31783892417
x40=−2236.91916561067x_{40} = -2236.91916561067
x41=−8557.71025328054x_{41} = -8557.71025328054
x42=−4633.38697872621x_{42} = -4633.38697872621
x43=−10084.0364108156x_{43} = -10084.0364108156
x44=7937.35459990823x_{44} = 7937.35459990823
x45=3141.66696261438x_{45} = 3141.66696261438
x46=5757.10378365535x_{46} = 5757.10378365535
x47=5539.09873443113x_{47} = 5539.09873443113
x48=3359.53291862833x_{48} = 3359.53291862833
x49=8155.39318033922x_{49} = 8155.39318033922
x50=−4415.41959229893x_{50} = -4415.41959229893
x51=−5287.34134664562x_{51} = -5287.34134664562
x52=9027.56302032818x_{52} = 9027.56302032818
x53=−8339.66868361336x_{53} = -8339.66868361336
x54=−9429.88997641172x_{54} = -9429.88997641172
x55=6193.12795404245x_{55} = 6193.12795404245
x56=−7903.59036987145x_{56} = -7903.59036987145
x57=3577.42473969158x_{57} = 3577.42473969158
x58=4231.21411343505x_{58} = 4231.21411343505
x59=−2454.58300226324x_{59} = -2454.58300226324
x60=8809.5184665848x_{60} = 8809.5184665848
x61=4449.17228479232x_{61} = 4449.17228479232
x62=4885.11947885748x_{62} = 4885.11947885748
x63=10990.0104166897x_{63} = 10990.0104166897
x64=−9865.98663962057x_{64} = -9865.98663962057
x65=−5723.34425135156x_{65} = -5723.34425135156
x66=−10956.2437393977x_{66} = -10956.2437393977
x67=−6377.38463377321x_{67} = -6377.38463377321
x68=9899.75270056862x_{68} = 9899.75270056862
x69=−4197.46324132013x_{69} = -4197.46324132013
x70=−7249.48718108111x_{70} = -7249.48718108111
x71=−9211.8432226996x_{71} = -9211.8432226996
x72=7065.22018954741x_{72} = 7065.22018954741
x73=10335.8533907174x_{73} = 10335.8533907174
x74=10553.9049840832x_{74} = 10553.9049840832
x75=10771.9573414176x_{75} = 10771.9573414176
x76=−3979.51980244928x_{76} = -3979.51980244928
x77=−3543.68156694623x_{77} = -3543.68156694623
x78=5321.0991644072x_{78} = 5321.0991644072
x79=−5941.3534497212x_{79} = -5941.3534497212
x80=4013.26853740293x_{80} = 4013.26853740293
x81=−7467.51944665703x_{81} = -7467.51944665703
x82=5103.10579472174x_{82} = 5103.10579472174
x83=−9647.93780615877x_{83} = -9647.93780615877
x84=7283.25044111232x_{84} = 7283.25044111232
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(−2xatan⁡(x)−1(x2+1)2−4atan⁡(x)x(x2+1)+3atan⁡2(x)x2)x2)=−43\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{2 x \operatorname{atan}{\left(x \right)} - 1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} + \frac{3 \operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}\right) = - \frac{4}{3}
lim⁡x→0+(2(−2xatan⁡(x)−1(x2+1)2−4atan⁡(x)x(x2+1)+3atan⁡2(x)x2)x2)=−43\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{2 x \operatorname{atan}{\left(x \right)} - 1}{\left(x^{2} + 1\right)^{2}} - \frac{4 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} + \frac{3 \operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}\right) = - \frac{4}{3}
- 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→−∞(atan⁡2(x)x2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(atan⁡2(x)x2)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\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)^2/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡2(x)xx2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(atan⁡2(x)xx2)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x x^{2}}\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⁡2(x)x2=atan⁡2(x)x2\frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}} = \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}
- Sí
atan⁡2(x)x2=−atan⁡2(x)x2\frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}} = - \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}
- No
es decir, función
es
par
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