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Trazado el gráfico de la función/ atan(3*x)^2/(1+9*x^2)

Gráfico de la función y = atan(3*x)^2/(1+9*x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           2     
       atan (3*x)
f(x) = ----------
               2 
        1 + 9*x
f(x)=atan⁡2(3x)9x2+1f{\left(x \right)} = \frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1} 
f = atan(3*x)^2/(9*x^2 + 1)
Gráfico de la función
02468-8-6-4-2-10100.00.5
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(3x)9x2+1=0\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=0x_{1} = 0
x2=466968.951506058x_{2} = 466968.951506058
x3=507412.49431553x_{3} = 507412.49431553
x4=497301.600186387x_{4} = 497301.600186387
x5=−465404.173303906x_{5} = -465404.173303906
x6=−475515.049251489x_{6} = -475515.049251489
x7=−485625.93136021x_{7} = -485625.93136021
x8=487190.711440954x_{8} = 487190.711440954
x9=−495736.819251574x_{9} = -495736.819251574
x10=−505847.712577467x_{10} = -505847.712577467
x11=477079.828422966x_{11} = 477079.828422966
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(3*x)^2/(1 + 9*x^2).
atan⁡2(0⋅3)9⋅02+1\frac{\operatorname{atan}^{2}{\left(0 \cdot 3 \right)}}{9 \cdot 0^{2} + 1}
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
−18xatan⁡2(3x)(9x2+1)2+6atan⁡(3x)(9x2+1)2=0- \frac{18 x \operatorname{atan}^{2}{\left(3 x \right)}}{\left(9 x^{2} + 1\right)^{2}} + \frac{6 \operatorname{atan}{\left(3 x \right)}}{\left(9 x^{2} + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−25453.3897677548x_{1} = -25453.3897677548
x2=−16130.4206813256x_{2} = -16130.4206813256
x3=40841.0607305034x_{3} = 40841.0607305034
x4=14566.645056127x_{4} = 14566.645056127
x5=9482.13453755639x_{5} = 9482.13453755639
x6=−41557.4194475537x_{6} = -41557.4194475537
x7=33212.7888402883x_{7} = 33212.7888402883
x8=23889.4907884857x_{8} = 23889.4907884857
x9=37450.7092635504x_{9} = 37450.7092635504
x10=−29691.2445869544x_{10} = -29691.2445869544
x11=−15282.9195645598x_{11} = -15282.9195645598
x12=−27996.0965725767x_{12} = -27996.0965725767
x13=−39862.2411907947x_{13} = -39862.2411907947
x14=26432.1838705652x_{14} = 26432.1838705652
x15=−12740.4998376586x_{15} = -12740.4998376586
x16=42536.2403963004x_{16} = 42536.2403963004
x17=22194.3774317304x_{17} = 22194.3774317304
x18=38298.2960378627x_{18} = 38298.2960378627
x19=15414.1359312968x_{19} = 15414.1359312968
x20=31517.6283055379x_{20} = 31517.6283055379
x21=−33929.1412071627x_{21} = -33929.1412071627
x22=17956.6734820545x_{22} = 17956.6734820545
x23=−30538.8211274058x_{23} = -30538.8211274058
x24=−39014.653052031x_{24} = -39014.653052031
x25=−37319.478979684x_{25} = -37319.478979684
x26=41688.6502691788x_{26} = 41688.6502691788
x27=−19520.5156124822x_{27} = -19520.5156124822
x28=28974.898092493x_{28} = 28974.898092493
x29=−21215.6020344714x_{29} = -21215.6020344714
x30=34907.9540572246x_{30} = 34907.9540572246
x31=35755.5382122508x_{31} = 35755.5382122508
x32=30670.0500381674x_{32} = 30670.0500381674
x33=21346.8264711072x_{33} = 21346.8264711072
x34=7787.62982828489x_{34} = 7787.62982828489
x35=−34776.7241948771x_{35} = -34776.7241948771
x36=−32233.9786046248x_{36} = -32233.9786046248
x37=−36471.8931491507x_{37} = -36471.8931491507
x38=−22063.1523419379x_{38} = -22063.1523419379
x39=32365.2079413765x_{39} = 32365.2079413765
x40=25584.6168473202x_{40} = 25584.6168473202
x41=−33081.5593147672x_{41} = -33081.5593147672
x42=34060.3709074307x_{42} = 34060.3709074307
x43=−14435.4307255607x_{43} = -14435.4307255607
x44=12871.7087953663x_{44} = 12871.7087953663
x45=−18672.9808002007x_{45} = -18672.9808002007
x46=29822.4732567211x_{46} = 29822.4732567211
x47=−11893.0645713475x_{47} = -11893.0645713475
x48=−13587.9564959512x_{48} = -13587.9564959512
x49=17109.1517713068x_{49} = 17109.1517713068
x50=−17825.4526639404x_{50} = -17825.4526639404
x51=−20368.0562591866x_{51} = -20368.0562591866
x52=11176.856602686x_{52} = 11176.856602686
x53=−22910.7066742105x_{53} = -22910.7066742105
x54=−26300.9564102858x_{54} = -26300.9564102858
x55=10329.4752210354x_{55} = 10329.4752210354
x56=39993.4718178721x_{56} = 39993.4718178721
x57=−28843.6696853761x_{57} = -28843.6696853761
x58=20499.2799588917x_{58} = 20499.2799588917
x59=−38167.0656319192x_{59} = -38167.0656319192
x60=−7656.46446138043x_{60} = -7656.46446138043
x61=18804.2027113685x_{61} = 18804.2027113685
x62=−40709.8300031049x_{62} = -40709.8300031049
x63=0x_{63} = 0
x64=19651.7384765184x_{64} = 19651.7384765184
x65=39145.883572156x_{65} = 39145.883572156
x66=−24605.8257360461x_{66} = -24605.8257360461
x67=24737.0523946641x_{67} = 24737.0523946641
x68=−9350.94656395263x_{68} = -9350.94656395263
x69=−8503.66844061373x_{69} = -8503.66844061373
x70=−10198.2799749787x_{70} = -10198.2799749787
x71=36603.1233022994x_{71} = 36603.1233022994
x72=16261.6387692744x_{72} = 16261.6387692744
x73=23041.9323456187x_{73} = 23041.9323456187
x74=−35624.3081992955x_{74} = -35624.3081992955
x75=−16977.9322153433x_{75} = -16977.9322153433
x76=28127.3246927144x_{76} = 28127.3246927144
x77=−23758.2645969239x_{77} = -23758.2645969239
x78=−42405.0094860591x_{78} = -42405.0094860591
x79=−31386.3991731197x_{79} = -31386.3991731197
x80=27279.753223048x_{80} = 27279.753223048
x81=13719.1683935771x_{81} = 13719.1683935771
x82=−27148.5254173193x_{82} = -27148.5254173193
x83=8634.84682209105x_{83} = 8634.84682209105
x84=12024.2699308479x_{84} = 12024.2699308479
x85=−11045.6557119668x_{85} = -11045.6557119668
Signos de extremos en los puntos:
(-25453.389767754827, 4.23154289069959e-10)

(-16130.420681325562, 1.05364528878586e-9)

(40841.060730503355, 1.64360973620241e-10)

(14566.64505612695, 1.29200892830183e-9)

(9482.13453755639, 3.04905725165556e-9)

(-41557.419447553664, 1.58743395577554e-10)

(33212.78884028826, 2.48531251177041e-10)

(23889.490788485706, 4.80369852242893e-10)

(37450.709263550416, 1.95466451487248e-10)

(-29691.24458695437, 3.10981269046097e-10)

(-15282.919564559794, 1.1737416859579e-9)

(-27996.09657257674, 3.497805810516e-10)

(-39862.241190794666, 1.72531816782174e-10)

(26432.183870565226, 3.92395644092528e-10)

(-12740.499837658586, 1.68892297509467e-9)

(42536.24039630045, 1.51521655885333e-10)

(22194.377431730372, 5.56548485339069e-10)

(38298.296037862696, 1.86910426360432e-10)

(15414.135931296803, 1.15384352525886e-9)

(31517.62830553795, 2.75984274890955e-10)

(-33929.14120716271, 2.38147527898935e-10)

(17956.6734820545, 8.50227618562664e-10)

(-30538.821127405798, 2.93958939883183e-10)

(-39014.65305203102, 1.80109666392998e-10)

(-37319.47897968401, 1.96843537773707e-10)

(41688.65026917884, 1.57745563695807e-10)

(-19520.51561248225, 7.19457645825396e-10)

(28974.898092493, 3.26548015491439e-10)

(-21215.602034471383, 6.0908490437413e-10)

(34907.954057224626, 2.249796169199e-10)

(35755.53821225075, 2.14439828003555e-10)

(30670.050038167443, 2.91448796098933e-10)

(21346.82647110723, 6.01619591580903e-10)

(7787.629828284889, 4.52025541769381e-9)

(-34776.724194877075, 2.2668072985971e-10)

(-32233.9786046248, 2.63854010515731e-10)

(-36471.89314915066, 2.06098854086382e-10)

(-22063.15234193787, 5.63188480102419e-10)

(32365.20794137646, 2.61718695088448e-10)

(25584.61684732025, 4.18824622154535e-10)

(-33081.5593147672, 2.50506920852841e-10)

(34060.37090743066, 2.36315978263408e-10)

(-14435.430725560725, 1.31560338380833e-9)

(12871.708795366323, 1.65466664580886e-9)

(-18672.980800200665, 7.86248938534151e-10)

(29822.473256721143, 3.08250470641355e-10)

(-11893.06457134755, 1.93818037748311e-9)

(-13587.956495951206, 1.48482538515329e-9)

(17109.15177130684, 9.36546862960578e-10)

(-17825.452663940443, 8.6279132465024e-10)

(-20368.056259186636, 6.60828893142411e-10)

(11176.85660268599, 2.19452935611824e-9)

(-22910.70667421052, 5.22290623488181e-10)

(-26300.956410285813, 3.96321062368272e-10)

(10329.475221035374, 2.56934772294664e-9)

(39993.471817872116, 1.71401422694693e-10)

(-28843.669685376106, 3.29526106780598e-10)

(20499.27995889174, 6.52395624457449e-10)

(-38167.06563191924, 1.88197942806359e-10)

(-7656.464461380434, 4.67645357845298e-9)

(18804.202711368536, 7.75313942157111e-10)

(-40709.830003104886, 1.65422332126805e-10)

(0, 0)

(19651.738476518363, 7.09881589730627e-10)

(39145.883572155995, 1.78904117429925e-10)

(-24605.82573604609, 4.52807764996633e-10)

(24737.052394664144, 4.48016382955832e-10)

(-9350.94656395263, 3.135208168526e-9)

(-8503.66844061373, 3.7910798424386e-9)

(-10198.279974978675, 2.63587802349588e-9)

(36603.12330229937, 2.04623693393831e-10)

(16261.63876927437, 1.03671000682342e-9)

(23041.93234561868, 5.16358643085914e-10)

(-35624.30819929549, 2.16022601467523e-10)

(-16977.93221534326, 9.51079448500519e-10)

(28127.324692714363, 3.46524414446513e-10)

(-23758.264596923887, 4.85691000942603e-10)

(-42405.009486059076, 1.52460931376032e-10)

(-31386.399173119684, 2.78296910014492e-10)

(27279.753223047952, 3.68391510353174e-10)

(13719.168393577056, 1.45655951774159e-9)

(-27148.525417319343, 3.71961477219723e-10)

(8634.84682209105, 3.67677134219538e-9)

(12024.269930847862, 1.89611415753716e-9)

(-11045.655711966832, 2.24697143936919e-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:
Puntos mínimos de la función:
x1=0x_{1} = 0
La función no tiene puntos máximos
Decrece en los intervalos
[0,∞)\left[0, \infty\right)
Crece en los intervalos
(−∞,0]\left(-\infty, 0\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
18(−12xatan⁡(3x)9x2+1+(36x29x2+1−1)atan⁡2(3x)−6xatan⁡(3x)−19x2+1)(9x2+1)2=0\frac{18 \left(- \frac{12 x \operatorname{atan}{\left(3 x \right)}}{9 x^{2} + 1} + \left(\frac{36 x^{2}}{9 x^{2} + 1} - 1\right) \operatorname{atan}^{2}{\left(3 x \right)} - \frac{6 x \operatorname{atan}{\left(3 x \right)} - 1}{9 x^{2} + 1}\right)}{\left(9 x^{2} + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−5458.23899662611x_{1} = -5458.23899662611
x2=3529.62187442465x_{2} = 3529.62187442465
x3=−8511.06253348204x_{3} = -8511.06253348204
x4=9635.14981430586x_{4} = 9635.14981430586
x5=10289.3445659585x_{5} = 10289.3445659585
x6=6146.1689946175x_{6} = 6146.1689946175
x7=−2187.86501687924x_{7} = -2187.86501687924
x8=−6984.63316783656x_{8} = -6984.63316783656
x9=7018.4007711312x_{9} = 7018.4007711312
x10=−3713.88817942979x_{10} = -3713.88817942979
x11=−10255.5760567888x_{11} = -10255.5760567888
x12=−6330.45842937166x_{12} = -6330.45842937166
x13=−3277.83621827431x_{13} = -3277.83621827431
x14=−2841.81200158736x_{14} = -2841.81200158736
x15=9417.08539227737x_{15} = 9417.08539227737
x16=−4804.08925687195x_{16} = -4804.08925687195
x17=−4367.99971209797x_{17} = -4367.99971209797
x18=−6548.51587016711x_{18} = -6548.51587016711
x19=−1969.9228368665x_{19} = -1969.9228368665
x20=−8729.12566057696x_{20} = -8729.12566057696
x21=1567.88497543018x_{21} = 1567.88497543018
x22=−4586.04325820123x_{22} = -4586.04325820123
x23=3311.59775381421x_{23} = 3311.59775381421
x24=0.13421371891138x_{24} = 0.13421371891138
x25=5928.1133327414x_{25} = 5928.1133327414
x26=3747.65144687779x_{26} = 3747.65144687779
x27=8544.83068907972x_{27} = 8544.83068907972
x28=2221.61667216732x_{28} = 2221.61667216732
x29=7454.52100630592x_{29} = 7454.52100630592
x30=6364.22566410655x_{30} = 6364.22566410655
x31=−10037.5109424752x_{31} = -10037.5109424752
x32=−7856.8755357295x_{32} = -7856.8755357295
x33=8980.95739992478x_{33} = 8980.95739992478
x34=−3931.9216348001x_{34} = -3931.9216348001
x35=−1316.36324699077x_{35} = -1316.36324699077
x36=1350.08245110134x_{36} = 1350.08245110134
x37=−9165.25293530557x_{37} = -9165.25293530557
x38=5710.05879511816x_{38} = 5710.05879511816
x39=−8074.93743863882x_{39} = -8074.93743863882
x40=−7202.69287721068x_{40} = -7202.69287721068
x41=5055.90339956845x_{41} = 5055.90339956845
x42=2875.57093413496x_{42} = 2875.57093413496
x43=−9383.31703520695x_{43} = -9383.31703520695
x44=3965.68555818632x_{44} = 3965.68555818632
x45=1132.38898664745x_{45} = 1132.38898664745
x46=−10473.6413853316x_{46} = -10473.6413853316
x47=−6766.57414250379x_{47} = -6766.57414250379
x48=9199.02124737738x_{48} = 9199.02124737738
x49=9853.21449495342x_{49} = 9853.21449495342
x50=10943.5412334737x_{50} = 10943.5412334737
x51=1785.75518103508x_{51} = 1785.75518103508
x52=−5240.18737322031x_{52} = -5240.18737322031
x53=10071.2794173231x_{53} = 10071.2794173231
x54=7890.64349393817x_{54} = 7890.64349393817
x55=7672.58199275606x_{55} = 7672.58199275606
x56=8762.89387227463x_{56} = 8762.89387227463
x57=−8947.1891361972x_{57} = -8947.1891361972
x58=−1534.15218839159x_{58} = -1534.15218839159
x59=−2405.83100356537x_{59} = -2405.83100356537
x60=−9601.3814152807x_{60} = -9601.3814152807
x61=7236.46058150029x_{61} = 7236.46058150029
x62=10725.4754865203x_{62} = 10725.4754865203
x63=4619.80861469893x_{63} = 4619.80861469893
x64=4401.76466237667x_{64} = 4401.76466237667
x65=−1752.01367265696x_{65} = -1752.01367265696
x66=4183.72348845104x_{66} = 4183.72348845104
x67=−9819.44605674893x_{67} = -9819.44605674893
x68=8108.70546802287x_{68} = 8108.70546802287
x69=−5894.34641528893x_{69} = -5894.34641528893
x70=−7420.75320984139x_{70} = -7420.75320984139
x71=−10909.7726334431x_{71} = -10909.7726334431
x72=−10691.7069149233x_{72} = -10691.7069149233
x73=8326.76787759361x_{73} = 8326.76787759361
x74=10507.4099266935x_{74} = 10507.4099266935
x75=2003.67027635031x_{75} = 2003.67027635031
x76=−4149.9590107776x_{76} = -4149.9590107776
x77=2439.58576234042x_{77} = 2439.58576234042
x78=5273.95365673431x_{78} = 5273.95365673431
x79=−3495.85939103282x_{79} = -3495.85939103282
x80=−5022.13738482951x_{80} = -5022.13738482951
x81=−6112.40190998043x_{81} = -6112.40190998043
x82=−7638.81411193874x_{82} = -7638.81411193874
x83=−2623.81472177863x_{83} = -2623.81472177863
x84=6582.28324016457x_{84} = 6582.28324016457
x85=4837.8549650783x_{85} = 4837.8549650783
x86=2657.57183124886x_{86} = 2657.57183124886
x87=−1098.69265146447x_{87} = -1098.69265146447
x88=5492.00551711356x_{88} = 5492.00551711356
x89=−8292.99978260187x_{89} = -8292.99978260187
x90=3093.58026051747x_{90} = 3093.58026051747
x91=−3059.81988564699x_{91} = -3059.81988564699
x92=−5676.29206463263x_{92} = -5676.29206463263
x93=6800.34163482113x_{93} = 6800.34163482113

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
(−∞,0.13421371891138]\left(-\infty, 0.13421371891138\right]
Convexa en los intervalos
[0.13421371891138,∞)\left[0.13421371891138, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(atan⁡2(3x)9x2+1)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(atan⁡2(3x)9x2+1)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1}\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(3*x)^2/(1 + 9*x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡2(3x)x(9x2+1))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{x \left(9 x^{2} + 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(atan⁡2(3x)x(9x2+1))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{x \left(9 x^{2} + 1\right)}\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(3x)9x2+1=atan⁡2(3x)9x2+1\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1} = \frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1}
- Sí
atan⁡2(3x)9x2+1=−atan⁡2(3x)9x2+1\frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1} = - \frac{\operatorname{atan}^{2}{\left(3 x \right)}}{9 x^{2} + 1}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор