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

Gráfico de la función y = atan(x)/x^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       atan(x)
f(x) = -------
           3  
          x
f(x)=atan⁡(x)x3f{\left(x \right)} = \frac{\operatorname{atan}{\left(x \right)}}{x^{3}} 
f = atan(x)/x^3
Gráfico de la función
02468-8-6-4-2-10100500
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⁡(x)x3=0\frac{\operatorname{atan}{\left(x \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=14565.9942371401x_{1} = 14565.9942371401
x2=17956.1504709903x_{2} = 17956.1504709903
x3=26431.8331102249x_{3} = 26431.8331102249
x4=−17824.9256677942x_{4} = -17824.9256677942
x5=−13587.2563241785x_{5} = -13587.2563241785
x6=19651.2622347272x_{6} = 19651.2622347272
x7=−28843.3489877175x_{7} = -28843.3489877175
x8=−23757.8731740251x_{8} = -23757.8731740251
x9=39145.64880854x_{9} = 39145.64880854
x10=−33081.2805883304x_{10} = -33081.2805883304
x11=−14434.7737264451x_{11} = -14434.7737264451
x12=16261.0588083246x_{12} = 16261.0588083246
x13=−38166.8247390554x_{13} = -38166.8247390554
x14=−22062.7298639823x_{14} = -22062.7298639823
x15=−33928.8695904438x_{15} = -33928.8695904438
x16=−40709.6044205561x_{16} = -40709.6044205561
x17=13718.475235644x_{17} = 13718.475235644
x18=−27995.765917035x_{18} = -27995.765917035
x19=29822.1633304622x_{19} = 29822.1633304622
x20=−35624.0497668238x_{20} = -35624.0497668238
x21=40840.8358817364x_{21} = 40840.8358817364
x22=27279.4136476329x_{22} = 27279.4136476329
x23=−39862.0107257058x_{23} = -39862.0107257058
x24=−32233.692386317x_{24} = -32233.692386317
x25=−20367.5973756642x_{25} = -20367.5973756642
x26=−16129.8358168267x_{26} = -16129.8358168267
x27=−29690.933265034x_{27} = -29690.933265034
x28=−26300.6038613592x_{28} = -26300.6038613592
x29=−18672.478648118x_{29} = -18672.478648118
x30=24736.6768986951x_{30} = 24736.6768986951
x31=36602.8719109676x_{31} = 36602.8719109676
x32=12023.4731167592x_{32} = 12023.4731167592
x33=−42404.793073229x_{33} = -42404.793073229
x34=39993.2421186136x_{34} = 39993.2421186136
x35=−37319.2325109126x_{35} = -37319.2325109126
x36=−31386.1050491301x_{36} = -31386.1050491301
x37=17108.6017548148x_{37} = 17108.6017548148
x38=31517.3354276164x_{38} = 31517.3354276164
x39=−21215.1621053738x_{39} = -21215.1621053738
x40=34907.6902075057x_{40} = 34907.6902075057
x41=25584.2541403093x_{41} = 25584.2541403093
x42=−36471.6408402545x_{42} = -36471.6408402545
x43=34060.100353754x_{43} = 34060.100353754
x44=12870.9674117666x_{44} = 12870.9674117666
x45=−16977.3777901713x_{45} = -16977.3777901713
x46=−15282.3007264236x_{46} = -15282.3007264236
x47=−34776.4593342323x_{47} = -34776.4593342323
x48=−11044.7836724659x_{48} = -11044.7836724659
x49=18803.7041789897x_{49} = 18803.7041789897
x50=−11892.2584778519x_{50} = -11892.2584778519
x51=32364.9229032012x_{51} = 32364.9229032012
x52=15413.5225797734x_{52} = 15413.5225797734
x53=20498.8241003272x_{53} = 20498.8241003272
x54=11175.9954104209x_{54} = 11175.9954104209
x55=−12739.7504258502x_{55} = -12739.7504258502
x56=−39014.4174883898x_{56} = -39014.4174883898
x57=38298.055981561x_{57} = 38298.055981561
x58=22193.9575193407x_{58} = 22193.9575193407
x59=−22910.3003160847x_{59} = -22910.3003160847
x60=35755.2807423288x_{60} = 35755.2807423288
x61=28974.578875601x_{61} = 28974.578875601
x62=37450.4636704392x_{62} = 37450.4636704392
x63=−19520.0360683423x_{63} = -19520.0360683423
x64=23041.5283616412x_{64} = 23041.5283616412
x65=30669.7488771186x_{65} = 30669.7488771186
x66=−27148.1841658046x_{66} = -27148.1841658046
x67=41688.4300703035x_{67} = 41688.4300703035
x68=−25453.0251479577x_{68} = -25453.0251479577
x69=42536.0246589077x_{69} = 42536.0246589077
x70=28126.9956110713x_{70} = 28126.9956110713
x71=−41557.1985449747x_{71} = -41557.1985449747
x72=33212.5112331461x_{72} = 33212.5112331461
x73=23889.1015689627x_{73} = 23889.1015689627
x74=21346.3893231038x_{74} = 21346.3893231038
x75=−30538.51864867x_{75} = -30538.51864867
x76=−24605.4481896938x_{76} = -24605.4481896938
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)/x^3.
atan⁡(0)03\frac{\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
1x3(x2+1)−3atan⁡(x)x4=0\frac{1}{x^{3} \left(x^{2} + 1\right)} - \frac{3 \operatorname{atan}{\left(x \right)}}{x^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=7236.35716104477x_{1} = 7236.35716104477
x2=7672.48461667795x_{2} = 7672.48461667795
x3=1785.29502719907x_{3} = 1785.29502719907
x4=−7856.7805079803x_{4} = -7856.7805079803
x5=5491.86794127918x_{5} = 5491.86794127918
x6=5709.9266730675x_{6} = 5709.9266730675
x7=−10255.5037217841x_{7} = -10255.5037217841
x8=−1533.60532068274x_{8} = -1533.60532068274
x9=9853.13914060572x_{9} = 9853.13914060572
x10=−8074.84504579217x_{10} = -8074.84504579217
x11=−7638.71629462345x_{11} = -7638.71629462345
x12=−7202.58895891985x_{12} = -7202.58895891985
x13=−5894.21857884875x_{13} = -5894.21857884875
x14=3965.49209127075x_{14} = 3965.49209127075
x15=2439.26010325451x_{15} = 2439.26010325451
x16=−6112.27879598334x_{16} = -6112.27879598334
x17=8108.61346894407x_{17} = 8108.61346894407
x18=2875.29844141364x_{18} = 2875.29844141364
x19=3093.32834085196x_{19} = 3093.32834085196
x20=10943.4735346239x_{20} = 10943.4735346239
x21=−3277.59942069308x_{21} = -3277.59942069308
x22=9635.07271631375x_{22} = 9635.07271631375
x23=4183.54063178357x_{23} = 4183.54063178357
x24=10289.2724725614x_{24} = 10289.2724725614
x25=3311.36352231564x_{25} = 3311.36352231564
x26=−8510.97499518894x_{26} = -8510.97499518894
x27=10725.406383577x_{27} = 10725.406383577
x28=−5676.15912916437x_{28} = -5676.15912916437
x29=6146.04657865205x_{29} = 6146.04657865205
x30=−9819.37043840689x_{30} = -9819.37043840689
x31=−6766.46331179812x_{31} = -6766.46331179812
x32=4619.64383422034x_{32} = 4619.64383422034
x33=−10909.7047215818x_{33} = -10909.7047215818
x34=−9383.23782133527x_{34} = -9383.23782133527
x35=−2187.49806367198x_{35} = -2187.49806367198
x36=10507.3393601875x_{36} = 10507.3393601875
x37=−4367.82496072467x_{37} = -4367.82496072467
x38=8762.80891061845x_{38} = 8762.80891061845
x39=−5240.04291424018x_{39} = -5240.04291424018
x40=−2623.51387781694x_{40} = -2623.51387781694
x41=8544.74350435163x_{41} = 8544.74350435163
x42=3747.44606380726x_{42} = 3747.44606380726
x43=−4149.77459307597x_{43} = -4149.77459307597
x44=1567.35153092681x_{44} = 1567.35153092681
x45=8980.87455080771x_{45} = 8980.87455080771
x46=−1969.51063344567x_{46} = -1969.51063344567
x47=2003.26576637706x_{47} = 2003.26576637706
x48=5927.98624872058x_{48} = 5927.98624872058
x49=−8292.90988250754x_{49} = -8292.90988250754
x50=−10473.5705873586x_{50} = -10473.5705873586
x51=3529.40301266065x_{51} = 3529.40301266065
x52=4401.59131343521x_{52} = 4401.59131343521
x53=−1751.54354846741x_{53} = -1751.54354846741
x54=2657.275113602x_{54} = 2657.275113602
x55=5055.75344464919x_{55} = 5055.75344464919
x56=5273.81015770891x_{56} = 5273.81015770891
x57=−2405.50036947356x_{57} = -2405.50036947356
x58=9417.00646804832x_{58} = 9417.00646804832
x59=−2841.53603153169x_{59} = -2841.53603153169
x60=−10691.6375900281x_{60} = -10691.6375900281
x61=−4803.93108820379x_{61} = -4803.93108820379
x62=9198.94040830908x_{62} = 9198.94040830908
x63=−6984.52590455775x_{63} = -6984.52590455775
x64=10071.2057295056x_{64} = 10071.2057295056
x65=−6548.4012266516x_{65} = -6548.4012266516
x66=−4585.87721090832x_{66} = -4585.87721090832
x67=6364.10758552026x_{67} = 6364.10758552026
x68=−9165.17179234581x_{68} = -9165.17179234581
x69=7454.42069899149x_{69} = 7454.42069899149
x70=6582.16920220582x_{70} = 6582.16920220582
x71=2221.25583331424x_{71} = 2221.25583331424
x72=−7420.65243427039x_{72} = -7420.65243427039
x73=−10037.4370022285x_{73} = -10037.4370022285
x74=6800.23137017706x_{74} = 6800.23137017706
x75=−3495.63829008982x_{75} = -3495.63829008982
x76=−5021.98638138004x_{76} = -5021.98638138004
x77=−8729.04036318698x_{77} = -8729.04036318698
x78=−5458.10053864951x_{78} = -5458.10053864951
x79=−3059.56499592463x_{79} = -3059.56499592463
x80=−8947.1059678621x_{80} = -8947.1059678621
x81=7018.29403815943x_{81} = 7018.29403815943
x82=8326.67835035446x_{82} = 8326.67835035446
x83=−3931.72641976461x_{83} = -3931.72641976461
x84=−3713.68082542498x_{84} = -3713.68082542498
x85=−9601.30404091578x_{85} = -9601.30404091578
x86=7890.54888267919x_{86} = 7890.54888267919
x87=4837.69794635806x_{87} = 4837.69794635806
x88=−6330.33970145248x_{88} = -6330.33970145248
Signos de extremos en los puntos:
(7236.357161044765, 4.14497281294608e-12)

(7672.484616677953, 3.47756904322625e-12)

(1785.2950271990674, 2.7595292267938e-10)

(-7856.780507980298, 3.23855162026853e-12)

(5491.867941279178, 9.48220354976981e-12)

(5709.926673067501, 8.4368409146065e-12)

(-10255.503721784135, 1.45620301960021e-12)

(-1533.60532068274, 4.35310094708241e-10)

(9853.139140605721, 1.64198040059422e-12)

(-8074.845045792167, 2.98320463219618e-12)

(-7638.7162946234475, 3.52389159333441e-12)

(-7202.588958919849, 4.20354398603347e-12)

(-5894.218578848751, 7.6699804396491e-12)

(3965.4920912707485, 2.51859818262929e-11)

(2439.2601032545067, 1.08201225272451e-10)

(-6112.278795983339, 6.87804757216005e-12)

(8108.613468944074, 2.94608981868099e-12)

(2875.29844141364, 6.6065529857031e-11)

(3093.3283408519605, 5.30581895022233e-11)

(10943.47353462391, 1.19847515681457e-12)

(-3277.599420693085, 4.46033924641438e-11)

(9635.072716313749, 1.75600679027795e-12)

(4183.540631783572, 2.14497290356031e-11)

(10289.27247256137, 1.44191281411785e-12)

(3311.363522315636, 4.32529560149683e-11)

(-8510.974995188935, 2.54770680385954e-12)

(10725.406383577012, 1.27307159639954e-12)

(-5676.159129164371, 8.58830537874198e-12)

(6146.046578652049, 6.76530448711292e-12)

(-9819.370438406886, 1.65897861131416e-12)

(-6766.463311798117, 5.06983494489466e-12)

(4619.643834220337, 1.59306884116444e-11)

(-10909.704721581813, 1.2096383436691e-12)

(-9383.237821335268, 1.90121940910476e-12)

(-2187.4980636719824, 1.50020440270723e-10)

(10507.339360187512, 1.35398946704501e-12)

(-4367.82496072467, 1.88478100210337e-11)

(8762.808910618449, 2.33430876481498e-12)

(-5240.042914240175, 1.09159761878483e-11)

(-2623.5138778169407, 8.69689809102294e-11)

(8544.743504351633, 2.51762143305962e-12)

(3747.44606380726, 2.98428241496043e-11)

(-4149.774593075971, 2.19775726475667e-11)

(1567.351530926813, 4.07797166781175e-10)

(8980.874550807712, 2.16836905341858e-12)

(-1969.5106334456698, 2.05543848750264e-10)

(2003.2657663770606, 1.9532873246313e-10)

(5927.986248720577, 7.53965844131424e-12)

(-8292.909882507542, 2.7540111760754e-12)

(-10473.570587358558, 1.36712802563038e-12)

(3529.403012660651, 3.57221654370015e-11)

(4401.591313435207, 1.84173826289931e-11)

(-1751.5435484674142, 2.92212762453538e-10)

(2657.275113602, 8.36962996075909e-11)

(5055.753444649192, 1.21536724603538e-11)

(5273.810157708911, 1.070764518226e-11)

(-2405.500369473562, 1.12820661351227e-10)

(9417.00646804832, 1.88084025033226e-12)

(-2841.536031531689, 6.84483627174075e-11)

(-10691.637590028142, 1.28517221905977e-12)

(-4803.931088203787, 1.41668081825709e-11)

(9198.940408309081, 2.01779203752738e-12)

(-6984.525904557752, 4.60966636245985e-12)

(10071.205729505597, 1.5376164647985e-12)

(-6548.401226651605, 5.59334761106735e-12)

(-4585.8772109083175, 1.62851707682617e-11)

(6364.107585520264, 6.09345899183828e-12)

(-9165.17179234581, 2.04017715460926e-12)

(7454.420698991493, 3.79176216607348e-12)

(6582.16920220582, 5.50770592963753e-12)

(2221.255833314239, 1.43284642641616e-10)

(-7420.652434270391, 3.84376072549042e-12)

(-10037.437002228511, 1.55318731011956e-12)

(6800.231370177061, 4.99468552735776e-12)

(-3495.6382900898216, 3.6767263992254e-11)

(-5021.986381380039, 1.24004723284782e-11)

(-8729.040363186985, 2.36150406340518e-12)

(-5458.100538649511, 9.65927711161091e-12)

(-3059.5649959246252, 5.4834066067361e-11)

(-8947.105967862104, 2.19301312568368e-12)

(7018.2940381594335, 4.54345051878643e-12)

(8326.678350354463, 2.72064141906119e-12)

(-3931.726419764614, 2.58404265617627e-11)

(-3713.6808254249777, 3.06642040058813e-11)

(-9601.304040915778, 1.77459973482031e-12)

(7890.548882679185, 3.19715132297066e-12)

(4837.6979463580565, 1.38722362419844e-11)

(-6330.339701452477, 6.19148965399804e-12)


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
Decrece 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)2−3x2(x2+1)+6atan⁡(x)x3)x2=0\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{6 \operatorname{atan}{\left(x \right)}}{x^{3}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1865.28415090324x_{1} = 1865.28415090324
x2=2604.56780510086x_{2} = 2604.56780510086
x3=3713.53345404081x_{3} = 3713.53345404081
x4=3990.77839429747x_{4} = 3990.77839429747
x5=−4623.37406052588x_{5} = -4623.37406052588
x6=1588.06565602442x_{6} = 1588.06565602442
x7=−1019.37381476972x_{7} = -1019.37381476972
x8=2142.5111769995x_{8} = 2142.5111769995
x9=3159.04706579843x_{9} = 3159.04706579843
x10=4452.85513116147x_{10} = 4452.85513116147
x11=−3329.5631890265x_{11} = -3329.5631890265
x12=−1850.97313095597x_{12} = -1850.97313095597
x13=3343.87524463244x_{13} = 3343.87524463244
x14=479.56544758983x_{14} = 479.56544758983
x15=−3514.39196963933x_{15} = -3514.39196963933
x16=2881.80628804693x_{16} = 2881.80628804693
x17=2050.10143307417x_{17} = 2050.10143307417
x18=−4346.12737901472x_{18} = -4346.12737901472
x19=−2682.66852998539x_{19} = -2682.66852998539
x20=4175.60884400991x_{20} = 4175.60884400991
x21=4268.0241966798x_{21} = 4268.0241966798
x22=2512.15561831596x_{22} = 2512.15561831596
x23=−2590.25605281979x_{23} = -2590.25605281979
x24=3898.36330935314x_{24} = 3898.36330935314
x25=2419.74381294856x_{25} = 2419.74381294856
x26=−1481.35201606092x_{26} = -1481.35201606092
x27=−2959.90772518943x_{27} = -2959.90772518943
x28=1772.87684142981x_{28} = 1772.87684142981
x29=−3884.05113011252x_{29} = -3884.05113011252
x30=4637.68634076976x_{30} = 4637.68634076976
x31=−1666.15994267541x_{31} = -1666.15994267541
x32=−1296.55158436641x_{32} = -1296.55158436641
x33=2789.39317172384x_{33} = 2789.39317172384
x34=−1388.95067771544x_{34} = -1388.95067771544
x35=4360.43962734659x_{35} = 4360.43962734659
x36=−4530.95843341628x_{36} = -4530.95843341628
x37=−2867.49439451636x_{37} = -2867.49439451636
x38=1033.68144969185x_{38} = 1033.68144969185
x39=−649.89976351342x_{39} = -649.89976351342
x40=−926.990929003824x_{40} = -926.990929003824
x41=1218.46425095381x_{41} = 1218.46425095381
x42=−3699.22130992045x_{42} = -3699.22130992045
x43=4545.27070366937x_{43} = 4545.27070366937
x44=2974.21965726039x_{44} = 2974.21965726039
x45=−3791.63616440615x_{45} = -3791.63616440615
x46=−4253.71196039196x_{46} = -4253.71196039196
x47=3436.28958415218x_{47} = 3436.28958415218
x48=−465.275311793613x_{48} = -465.275311793613
x49=756.553819434797x_{49} = 756.553819434797
x50=−4068.88136492396x_{50} = -4068.88136492396
x51=−3052.32128937886x_{51} = -3052.32128937886
x52=−1204.15524833194x_{52} = -1204.15524833194
x53=−3144.73506654531x_{53} = -3144.73506654531
x54=−2313.02087666514x_{54} = -2313.02087666514
x55=−3606.80657518857x_{55} = -3606.80657518857
x56=1495.66220162481x_{56} = 1495.66220162481
x57=−1111.76235034269x_{57} = -1111.76235034269
x58=848.920826643994x_{58} = 848.920826643994
x59=941.29756084813x_{59} = 941.29756084813
x60=−834.615540367377x_{60} = -834.615540367377
x61=−2405.43218297667x_{61} = -2405.43218297667
x62=−3237.14903848913x_{62} = -3237.14903848913
x63=664.200504727886x_{63} = 664.200504727886
x64=3805.94832672617x_{64} = 3805.94832672617
x65=1680.47061389097x_{65} = 1680.47061389097
x66=−4438.54287152777x_{66} = -4438.54287152777
x67=1310.86106515399x_{67} = 1310.86106515399
x68=2327.33243427439x_{68} = 2327.33243427439
x69=4083.19357462386x_{69} = 4083.19357462386
x70=−1943.38123132634x_{70} = -1943.38123132634
x71=2234.92153502719x_{71} = 2234.92153502719
x72=−1573.75520649554x_{72} = -1573.75520649554
x73=1403.26054557315x_{73} = 1403.26054557315
x74=571.867322865663x_{74} = 571.867322865663
x75=3621.11869969799x_{75} = 3621.11869969799
x76=3066.63325655884x_{76} = 3066.63325655884
x77=1957.69238982779x_{77} = 1957.69238982779
x78=2696.98033421243x_{78} = 2696.98033421243
x79=−742.250396105731x_{79} = -742.250396105731
x80=−557.57063924874x_{80} = -557.57063924874
x81=−2220.61005893796x_{81} = -2220.61005893796
x82=−2497.84392381731x_{82} = -2497.84392381731
x83=−3421.9775037179x_{83} = -3421.9775037179
x84=−2775.08132062707x_{84} = -2775.08132062707
x85=−1758.56598224199x_{85} = -1758.56598224199
x86=−4161.29662057471x_{86} = -4161.29662057471
x87=−2035.79015431674x_{87} = -2035.79015431674
x88=3251.46106711963x_{88} = 3251.46106711963
x89=3528.70407297678x_{89} = 3528.70407297678
x90=−2128.19979319987x_{90} = -2128.19979319987
x91=−3976.46619929822x_{91} = -3976.46619929822
x92=1126.07075276109x_{92} = 1126.07075276109
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)2−3x2(x2+1)+6atan⁡(x)x3)x2)=∞\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{6 \operatorname{atan}{\left(x \right)}}{x^{3}}\right)}{x^{2}}\right) = \infty
lim⁡x→0+(2(−1(x2+1)2−3x2(x2+1)+6atan⁡(x)x3)x2)=∞\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{1}{\left(x^{2} + 1\right)^{2}} - \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{6 \operatorname{atan}{\left(x \right)}}{x^{3}}\right)}{x^{2}}\right) = \infty
- 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⁡(x)x3)=0\lim_{x \to -\infty}\left(\frac{\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→∞(atan⁡(x)x3)=0\lim_{x \to \infty}\left(\frac{\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 atan(x)/x^3, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡(x)xx3)=0\lim_{x \to -\infty}\left(\frac{\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→∞(atan⁡(x)xx3)=0\lim_{x \to \infty}\left(\frac{\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:
atan⁡(x)x3=atan⁡(x)x3\frac{\operatorname{atan}{\left(x \right)}}{x^{3}} = \frac{\operatorname{atan}{\left(x \right)}}{x^{3}}
- No
atan⁡(x)x3=−atan⁡(x)x3\frac{\operatorname{atan}{\left(x \right)}}{x^{3}} = - \frac{\operatorname{atan}{\left(x \right)}}{x^{3}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор