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Trazado el gráfico de la función/ arctan(1/x)/(x+2)

Gráfico de la función y = arctan(1/x)/(x+2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           /1\
       atan|-|
           \x/
f(x) = -------
        x + 2
f(x)=atan⁡(1x)x+2f{\left(x \right)} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} 
f = atan(1/x)/(x + 2)
Gráfico de la función
02468-8-6-4-2-1010-2020
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−2x_{1} = -2
x2=0x_{2} = 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⁡(1x)x+2=0\frac{\operatorname{atan}{\left(\frac{1}{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(1/x)/(x + 2).
atan⁡(10)2\frac{\operatorname{atan}{\left(\frac{1}{0} \right)}}{2}
Resultado:
f(0)=⟨−π4,π4⟩f{\left(0 \right)} = \left\langle - \frac{\pi}{4}, \frac{\pi}{4}\right\rangle
Punto:
(0, AccumBounds(-pi/4, pi/4))
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
−atan⁡(1x)(x+2)2−1x2(1+1x2)(x+2)=0- \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−28326.9071906733x_{1} = -28326.9071906733
x2=21674.5251522753x_{2} = 21674.5251522753
x3=31847.0269739212x_{3} = 31847.0269739212
x4=36085.3651754688x_{4} = 36085.3651754688
x5=−33413.0308950943x_{5} = -33413.0308950943
x6=42018.9329177414x_{6} = 42018.9329177414
x7=35237.7035205466x_{7} = 35237.7035205466
x8=−13914.1680598466x_{8} = -13914.1680598466
x9=−16458.0478017161x_{9} = -16458.0478017161
x10=−38499.021037117x_{10} = -38499.021037117
x11=−24936.0426238831x_{11} = -24936.0426238831
x12=25913.1837793627x_{12} = 25913.1837793627
x13=−40194.3302854141x_{13} = -40194.3302854141
x14=−14762.1702224834x_{14} = -14762.1702224834
x15=36933.0242421313x_{15} = 36933.0242421313
x16=−30022.3008645925x_{16} = -30022.3008645925
x17=−41889.6312661903x_{17} = -41889.6312661903
x18=42866.5779293747x_{18} = 42866.5779293747
x19=37780.6808915892x_{19} = 37780.6808915892
x20=29303.981040389x_{20} = 29303.981040389
x21=16587.7229190396x_{21} = 16587.7229190396
x22=−31717.6743942306x_{22} = -31717.6743942306
x23=17435.5690134626x_{23} = 17435.5690134626
x24=−30869.9899364173x_{24} = -30869.9899364173
x25=−15610.1277726359x_{25} = -15610.1277726359
x26=15739.8519773568x_{26} = 15739.8519773568
x27=−25783.7700371527x_{27} = -25783.7700371527
x28=25065.4688242842x_{28} = 25065.4688242842
x29=−12217.9913387618x_{29} = -12217.9913387618
x30=12348.0239520496x_{30} = 12348.0239520496
x31=−23240.5602094796x_{31} = -23240.5602094796
x32=−22392.8030712172x_{32} = -22392.8030712172
x33=28456.2900188718x_{33} = 28456.2900188718
x34=−32565.3546042869x_{34} = -32565.3546042869
x35=−19001.634248567x_{35} = -19001.634248567
x36=13196.0450486925x_{36} = 13196.0450486925
x37=−13066.1122911295x_{37} = -13066.1122911295
x38=18283.3935690576x_{38} = 18283.3935690576
x39=−20697.2502454371x_{39} = -20697.2502454371
x40=−26631.4894482298x_{40} = -26631.4894482298
x41=38628.3352801573x_{41} = 38628.3352801573
x42=24217.7463381975x_{42} = 24217.7463381975
x43=30999.349155358x_{43} = 30999.349155358
x44=−39346.6767624349x_{44} = -39346.6767624349
x45=14891.9521692111x_{45} = 14891.9521692111
x46=34390.0390897917x_{46} = 34390.0390897917
x47=39475.9875509635x_{47} = 39475.9875509635
x48=−19849.451180785x_{48} = -19849.451180785
x49=19131.1993334596x_{49} = 19131.1993334596
x50=20826.7633307161x_{50} = 20826.7633307161
x51=−36803.7023616607x_{51} = -36803.7023616607
x52=−17305.9359759458x_{52} = -17305.9359759458
x53=22522.2754805973x_{53} = 22522.2754805973
x54=−35108.372872x_{54} = -35108.372872
x55=−21545.0336007701x_{55} = -21545.0336007701
x56=27608.5937990085x_{56} = 27608.5937990085
x57=32694.7010553391x_{57} = 32694.7010553391
x58=−34260.7035621536x_{58} = -34260.7035621536
x59=30151.6672917348x_{59} = 30151.6672917348
x60=19978.9886071672x_{60} = 19978.9886071672
x61=−35956.0390656664x_{61} = -35956.0390656664
x62=−24088.3063452608x_{62} = -24088.3063452608
x63=26760.8918998626x_{63} = 26760.8918998626
x64=23370.015526887x_{64} = 23370.015526887
x65=40323.6378353104x_{65} = 40323.6378353104
x66=−41041.9817443474x_{66} = -41041.9817443474
x67=41171.2862538706x_{67} = 41171.2862538706
x68=−27479.2016118189x_{68} = -27479.2016118189
x69=−29174.6067691777x_{69} = -29174.6067691777
x70=−18153.7968768281x_{70} = -18153.7968768281
x71=14044.018564154x_{71} = 14044.018564154
x72=33542.3716770708x_{72} = 33542.3716770708
x73=−37651.3629585359x_{73} = -37651.3629585359
Signos de extremos en los puntos:
(-28326.907190673326, 1.24632797891631e-9)

(21674.525152275255, 2.12843672576819e-9)

(31847.026973921213, 9.85904696571643e-10)

(36085.36517546877, 7.67916007881665e-10)

(-33413.03089509429, 8.95765334698308e-10)

(42018.93291774137, 5.66355719448469e-10)

(35237.703520546616, 8.05304557132222e-10)

(-13914.168059846603, 5.16592312791241e-9)

(-16458.04780171614, 3.69229285477188e-9)

(-38499.02103711696, 6.7471938717723e-10)

(-24936.042623883113, 1.60834706395684e-9)

(25913.183779362742, 1.48910366201143e-9)

(-40194.33028541407, 6.19001948967272e-10)

(-14762.170222483426, 4.58942655745189e-9)

(36933.02424213132, 7.33072183377707e-10)

(-30022.300864592482, 1.10953494899044e-9)

(-41889.63126619027, 5.69911815064452e-10)

(42866.57792937472, 5.44179413311006e-10)

(37780.68089158923, 7.00547276479032e-10)

(29303.98104038901, 1.16443998730426e-9)

(16587.72291903956, 3.63390939414846e-9)

(-31717.674394230628, 9.94087728879106e-10)

(17435.569013462577, 3.28910649123316e-9)

(-30869.989936417267, 1.0494340759053e-9)

(-15610.127772635873, 4.10433432669865e-9)

(15739.851977356784, 4.03592912764465e-9)

(-25783.770037152663, 1.50432215043283e-9)

(25065.468824284155, 1.59152580333635e-9)

(-12217.991338761805, 6.69994858512238e-9)

(12348.023952049642, 6.55744619684061e-9)

(-23240.560209479583, 1.85158728724227e-9)

(-22392.803071217215, 1.99444410054978e-9)

(28456.290018871823, 1.23484634452392e-9)

(-32565.35460428687, 9.43007289173052e-10)

(-19001.634248567043, 2.76989817779042e-9)

(13196.045048692522, 5.74178072835781e-9)

(-13066.112291129475, 5.85832833253184e-9)

(18283.393569057633, 2.99115472903318e-9)

(-20697.250245437055, 2.33462250029408e-9)

(-26631.48944822983, 1.41007334595823e-9)

(38628.33528015727, 6.70139982754219e-10)

(24217.7463381975, 1.70489125630029e-9)

(30999.349155358046, 1.04055928711977e-9)

(-39346.676762434894, 6.45960497977612e-10)

(14891.952169211147, 4.50856579523168e-9)

(34390.03908979168, 8.45491809216041e-10)

(39475.98755096351, 6.41670377669519e-10)

(-19849.451180785014, 2.53832223355915e-9)

(19131.199333459597, 2.73193402262787e-9)

(20826.763330716087, 2.30523249325733e-9)

(-36803.70236166067, 7.38313112289218e-10)

(-17305.935975945784, 3.33933513746353e-9)

(22522.275480597287, 1.97122819879297e-9)

(-35108.37287200003, 8.1134083830826e-10)

(-21545.033600770126, 2.15449735440318e-9)

(27608.593799008457, 1.31183727205356e-9)

(32694.7010553391, 9.35445935974442e-10)

(-34260.70356215356, 8.51986667303278e-10)

(30151.667291734844, 1.09988816496254e-9)

(19978.988607167194, 2.50501037096808e-9)

(-35956.03906566644, 7.73535893033487e-10)

(-24088.306345260837, 1.72354857755394e-9)

(26760.891899862592, 1.39626025604068e-9)

(23370.015526887033, 1.83081656259178e-9)

(40323.63783531036, 6.14977248444631e-10)

(-41041.981744347395, 5.93696540305781e-10)

(41171.28625387064, 5.89915806463741e-10)

(-27479.201611818877, 1.3244128597311e-9)

(-29174.606769177743, 1.17495098782929e-9)

(-18153.796876828063, 3.03468001564172e-9)

(14044.01856415396, 5.06938606079722e-9)

(33542.37167707083, 8.88764238480341e-10)

(-37651.36295853592, 7.05442573280625e-10)


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(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2=0\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10155.4590528456x_{1} = 10155.4590528456
x2=2302.78597024282x_{2} = 2302.78597024282
x3=1647.6121150313x_{3} = 1647.6121150313
x4=5793.55185085321x_{4} = 5793.55185085321
x5=−1614.76423318349x_{5} = -1614.76423318349
x6=−4016.55503372661x_{6} = -4016.55503372661
x7=−7942.75750807917x_{7} = -7942.75750807917
x8=−10777.9167556495x_{8} = -10777.9167556495
x9=−7070.36014300167x_{9} = -7070.36014300167
x10=−5543.57396632486x_{10} = -5543.57396632486
x11=−1833.45731857528x_{11} = -1833.45731857528
x12=−6416.04080676295x_{12} = -6416.04080676295
x13=11027.7936852099x_{13} = 11027.7936852099
x14=−2707.18224487169x_{14} = -2707.18224487169
x15=−7724.66068520671x_{15} = -7724.66068520671
x16=9719.28819648547x_{16} = 9719.28819648547
x17=−3361.96514138765x_{17} = -3361.96514138765
x18=−10123.6616831784x_{18} = -10123.6616831784
x19=4484.79808915367x_{19} = 4484.79808915367
x20=−8160.85288071644x_{20} = -8160.85288071644
x21=−4671.03531898808x_{21} = -4671.03531898808
x22=6229.77317138162x_{22} = 6229.77317138162
x23=9283.11454648586x_{23} = 9283.11454648586
x24=2957.5753803023x_{24} = 2957.5753803023
x25=8410.75717085281x_{25} = 8410.75717085281
x26=−9033.22200687244x_{26} = -9033.22200687244
x27=2739.34003587x_{27} = 2739.34003587
x28=−4452.88432742144x_{28} = -4452.88432742144
x29=7102.18691631106x_{29} = 7102.18691631106
x30=3175.78984726872x_{30} = 3175.78984726872
x31=−9251.31164213201x_{31} = -9251.31164213201
x32=−6852.25605773224x_{32} = -6852.25605773224
x33=−4234.72471680378x_{33} = -4234.72471680378
x34=−2270.4661524954x_{34} = -2270.4661524954
x35=−2925.47299918569x_{35} = -2925.47299918569
x36=7538.38258474053x_{36} = 7538.38258474053
x37=3612.17008848393x_{37} = 3612.17008848393
x38=10373.5435412909x_{38} = 10373.5435412909
x39=4266.65367445192x_{39} = 4266.65367445192
x40=9501.20174420201x_{40} = 9501.20174420201
x41=−8815.13139310588x_{41} = -8815.13139310588
x42=6447.87986596005x_{42} = 6447.87986596005
x43=−3143.73181577214x_{43} = -3143.73181577214
x44=−3580.17780168701x_{44} = -3580.17780168701
x45=−8378.94691851695x_{45} = -8378.94691851695
x46=2521.07901482993x_{46} = 2521.07901482993
x47=−10341.7473415644x_{47} = -10341.7473415644
x48=7320.28554933361x_{48} = 7320.28554933361
x49=5139.19521736223x_{49} = 5139.19521736223
x50=−8597.0397249602x_{50} = -8597.0397249602
x51=9937.37395136824x_{51} = 9937.37395136824
x52=−5107.31598521205x_{52} = -5107.31598521205
x53=3393.98712459383x_{53} = 3393.98712459383
x54=−9469.4003676517x_{54} = -9469.4003676517
x55=4048.50171707908x_{55} = 4048.50171707908
x56=−9687.48824590579x_{56} = -9687.48824590579
x57=8192.66534690497x_{57} = 8192.66534690497
x58=1866.06649845331x_{58} = 1866.06649845331
x59=4702.93595515051x_{59} = 4702.93595515051
x60=7974.57237255382x_{60} = 7974.57237255382
x61=8846.93769953506x_{61} = 8846.93769953506
x62=8628.84792943952x_{62} = 8628.84792943952
x63=−6197.92913031119x_{63} = -6197.92913031119
x64=6884.08653869763x_{64} = 6884.08653869763
x65=−5979.81434909453x_{65} = -5979.81434909453
x66=4921.06809983418x_{66} = 4921.06809983418
x67=3830.34101271359x_{67} = 3830.34101271359
x68=−9905.57533377917x_{68} = -9905.57533377917
x69=−4889.17887858538x_{69} = -4889.17887858538
x70=−5761.69610121046x_{70} = -5761.69610121046
x71=6665.98425095459x_{71} = 6665.98425095459
x72=7756.47815347094x_{72} = 7756.47815347094
x73=5575.43662977994x_{73} = 5575.43662977994
x74=−3798.37347953912x_{74} = -3798.37347953912
x75=−10559.8323524182x_{75} = -10559.8323524182
x76=9065.02655079485x_{76} = 9065.02655079485
x77=6011.6639252941x_{77} = 6011.6639252941
x78=−2052.01060774317x_{78} = -2052.01060774317
x79=10591.6274538209x_{79} = 10591.6274538209
x80=−6634.14969172341x_{80} = -6634.14969172341
x81=−5325.44745340199x_{81} = -5325.44745340199
x82=−2488.85074573824x_{82} = -2488.85074573824
x83=2084.45236245165x_{83} = 2084.45236245165
x84=−7506.56228301616x_{84} = -7506.56228301616
x85=5357.3178943352x_{85} = 5357.3178943352
x86=−7288.46215665169x_{86} = -7288.46215665169
x87=10809.710824617x_{87} = 10809.710824617
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=−2x_{1} = -2
x2=0x_{2} = 0

lim⁡x→−2−(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=∞\lim_{x \to -2^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \infty
lim⁡x→−2+(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=−∞\lim_{x \to -2^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = -\infty
- los límites no son iguales, signo
x1=−2x_{1} = -2
- es el punto de flexión
lim⁡x→0−(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=12−π8\lim_{x \to 0^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{1}{2} - \frac{\pi}{8}
lim⁡x→0+(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=π8+12\lim_{x \to 0^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{\pi}{8} + \frac{1}{2}
- los límites no son iguales, signo
x2=0x_{2} = 0
- es el punto de flexión

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=−2x_{1} = -2
x2=0x_{2} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(atan⁡(1x)x+2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{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⁡(1x)x+2)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{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(1/x)/(x + 2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡(1x)x(x+2))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + 2\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(atan⁡(1x)x(x+2))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + 2\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⁡(1x)x+2=−atan⁡(1x)2−x\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} = - \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{2 - x}
- No
atan⁡(1x)x+2=atan⁡(1x)2−x\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{2 - x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор