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  • Gráfico de la función y =:
  • 5-x 5-x
  • (1-x^3)/x^2 (1-x^3)/x^2
  • x/(x^2-5) x/(x^2-5)
  • 3*x-x^3 3*x-x^3

  • Expresiones idénticas

  • acot(dos *x)/(x- uno)
  • arcoco tangente de gente de (2 multiplicar por x) dividir por (x menos 1)
  • arcoco tangente de gente de (dos multiplicar por x) dividir por (x menos uno)
  • acot(2x)/(x-1)
  • acot2x/x-1
  • acot(2*x) dividir por (x-1)

  • Expresiones semejantes

  • acot(2*x)/(x+1)
  • arccot(2*x)/(x-1)

  • Expresiones con funciones

  • Arcocotangente arccot
  • acot(x)-5/x^2
  • acot(3/x+4)
  • acot(x)-x-1
Trazado el gráfico de la función/ acot(2*x)/(x-1)

Gráfico de la función y = acot(2*x)/(x-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acot(2*x)
f(x) = ---------
         x - 1
f(x)=acot⁡(2x)x−1f{\left(x \right)} = \frac{\operatorname{acot}{\left(2 x \right)}}{x - 1} 
f = acot(2*x)/(x - 1)
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=1x_{1} = 1
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:
acot⁡(2x)x−1=0\frac{\operatorname{acot}{\left(2 x \right)}}{x - 1} = 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 acot(2*x)/(x - 1).
acot⁡(0⋅2)−1\frac{\operatorname{acot}{\left(0 \cdot 2 \right)}}{-1}
Resultado:
f(0)=−π2f{\left(0 \right)} = - \frac{\pi}{2}
Punto:
(0, -pi/2)
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
−2(x−1)(4x2+1)−acot⁡(2x)(x−1)2=0- \frac{2}{\left(x - 1\right) \left(4 x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(2 x \right)}}{\left(x - 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10023.3059886798x_{1} = 10023.3059886798
x2=−16248.99951668x_{2} = -16248.99951668
x3=11718.8349580713x_{3} = 11718.8349580713
x4=−28115.9864417731x_{4} = -28115.9864417731
x5=13414.2811900563x_{5} = 13414.2811900563
x6=−38287.431577539x_{6} = -38287.431577539
x7=−31506.4819421501x_{7} = -31506.4819421501
x8=−18791.980335574x_{8} = -18791.980335574
x9=26976.7635661688x_{9} = 26976.7635661688
x10=39691.0813598296x_{10} = 39691.0813598296
x11=28672.0191082501x_{11} = 28672.0191082501
x12=−12858.2578651732x_{12} = -12858.2578651732
x13=17652.6936745703x_{13} = 17652.6936745703
x14=31214.8927402376x_{14} = 31214.8927402376
x15=−37439.8151451701x_{15} = -37439.8151451701
x16=10871.0832955839x_{16} = 10871.0832955839
x17=25281.5016345177x_{17} = 25281.5016345177
x18=41386.3118402928x_{18} = 41386.3118402928
x19=−29811.2363998511x_{19} = -29811.2363998511
x20=−10315.0664908658x_{20} = -10315.0664908658
x21=15957.3540863923x_{21} = 15957.3540863923
x22=−15401.3271925303x_{22} = -15401.3271925303
x23=−11162.8154801548x_{23} = -11162.8154801548
x24=18500.3543120234x_{24} = 18500.3543120234
x25=−21334.9217457585x_{25} = -21334.9217457585
x26=−26420.7312291888x_{26} = -26420.7312291888
x27=34605.3768635164x_{27} = 34605.3768635164
x28=−32354.1032719495x_{28} = -32354.1032719495
x29=33757.7571094617x_{29} = 33757.7571094617
x30=38843.4653933966x_{30} = 38843.4653933966
x31=23586.2319179501x_{31} = 23586.2319179501
x32=21890.9525830866x_{32} = 21890.9525830866
x33=12566.5662777284x_{33} = 12566.5662777284
x34=−25573.1013303947x_{34} = -25573.1013303947
x35=30367.2693415002x_{35} = 30367.2693415002
x36=32062.515115973x_{36} = 32062.515115973
x37=40538.6968319284x_{37} = 40538.6968319284
x38=27824.3920621207x_{38} = 27824.3920621207
x39=42233.9264130482x_{39} = 42233.9264130482
x40=14261.9826775785x_{40} = 14261.9826775785
x41=16805.0272459016x_{41} = 16805.0272459016
x42=−20487.2780980658x_{42} = -20487.2780980658
x43=36300.6141788641x_{43} = 36300.6141788641
x44=−12010.5446730764x_{44} = -12010.5446730764
x45=26129.1334776658x_{45} = 26129.1334776658
x46=−42525.5065046146x_{46} = -42525.5065046146
x47=−39135.0474861716x_{47} = -39135.0474861716
x48=−17096.6652306606x_{48} = -17096.6652306606
x49=15109.673041667x_{49} = 15109.673041667
x50=−19639.6311091475x_{50} = -19639.6311091475
x51=−36592.1981530122x_{51} = -36592.1981530122
x52=0.299468985102263x_{52} = 0.299468985102263
x53=−39982.6629040901x_{53} = -39982.6629040901
x54=24433.8678518666x_{54} = 24433.8678518666
x55=−14553.6471327275x_{55} = -14553.6471327275
x56=−40830.2778615979x_{56} = -40830.2778615979
x57=−33201.7237379436x_{57} = -33201.7237379436
x58=−27268.359552755x_{58} = -27268.359552755
x59=−34049.343403984x_{59} = -34049.343403984
x60=37148.2318410501x_{60} = 37148.2318410501
x61=−13705.9579412476x_{61} = -13705.9579412476
x62=−28963.6120206595x_{62} = -28963.6120206595
x63=20195.661172063x_{63} = 20195.661172063
x64=−24725.4696966835x_{64} = -24725.4696966835
x65=19348.0099308358x_{65} = 19348.0099308358
x66=32910.1365484833x_{66} = 32910.1365484833
x67=−35744.5805616267x_{67} = -35744.5805616267
x68=29519.6448307279x_{68} = 29519.6448307279
x69=21043.3085720763x_{69} = 21043.3085720763
x70=−23877.8361460348x_{70} = -23877.8361460348
x71=−22182.5624290172x_{71} = -22182.5624290172
x72=−17944.3252518746x_{72} = -17944.3252518746
x73=−41677.892386554x_{73} = -41677.892386554
x74=−30658.8596777143x_{74} = -30658.8596777143
x75=−23030.2004700538x_{75} = -23030.2004700538
x76=35452.9958690241x_{76} = 35452.9958690241
x77=−34896.9623277814x_{77} = -34896.9623277814
x78=−9467.2926099517x_{78} = -9467.2926099517
x79=37995.8488992771x_{79} = 37995.8488992771
x80=22738.5935890233x_{80} = 22738.5935890233
Signos de extremos en los puntos:
(10023.30598867983, 4.97727179998685e-9)

(-16248.999516679962, 1.89360776601023e-9)

(11718.834958071282, 3.64114680508727e-9)

(-28115.986441773104, 6.32481617412191e-10)

(13414.2811900563, 2.77886577091334e-9)

(-38287.431577539006, 3.41072099113022e-10)

(-31506.481942150105, 5.03681959987282e-10)

(-18791.980335574022, 1.41579963358164e-9)

(26976.763566168815, 6.87078584294831e-10)

(39691.0813598296, 3.17391348115915e-10)

(28672.019108250137, 6.08231063639072e-10)

(-12858.257865173153, 3.02393163075937e-9)

(17652.693674570288, 1.60462161135569e-9)

(31214.892740237614, 5.13168776007676e-10)

(-37439.81514517007, 3.56690065091881e-10)

(10871.083295583927, 4.23120735206181e-9)

(25281.501634517706, 7.82314628168441e-10)

(41386.311840292765, 2.91922148402099e-10)

(-29811.236399851103, 5.62594470647764e-10)

(-10315.06649086584, 4.69876623793991e-9)

(15957.354086392263, 1.96370143326721e-9)

(-15401.327192530276, 2.10778112918953e-9)

(-11162.815480154843, 4.01220955995748e-9)

(18500.35431202337, 1.46094339039383e-9)

(-21334.92174575853, 1.09841774463209e-9)

(-26420.73122918883, 7.1624878901441e-10)

(34605.37686351635, 4.17537405235419e-10)

(-32354.103271949505, 4.77636873128713e-10)

(33757.7571094617, 4.38768762856835e-10)

(38843.465393396575, 3.3139446354388e-10)

(23586.23191795006, 8.98817022075509e-10)

(21890.95258308659, 1.04342328253237e-9)

(12566.56627772839, 3.16644036216381e-9)

(-25573.101330394697, 7.64515373854258e-10)

(30367.269341500167, 5.42216617057071e-10)

(32062.515115973005, 4.86394186318491e-10)

(40538.696831928406, 3.04257400376883e-10)

(27824.39206212073, 6.45853837877993e-10)

(42233.92641304821, 2.80322120504772e-10)

(14261.98267757852, 2.45833263849057e-9)

(16805.02724590162, 1.77058755052948e-9)

(-20487.278098065755, 1.19118792177403e-9)

(36300.61417886415, 3.79449532632734e-10)

(-12010.544673076394, 3.46583944632635e-9)

(26129.133477665782, 7.32380227407848e-10)

(-42525.506504614576, 2.76478141728713e-10)

(-39135.04748617162, 3.26457895680577e-10)

(-17096.665230660605, 1.71049491297e-9)

(15109.673041667047, 2.19022440802021e-9)

(-19639.631109147533, 1.296227524837e-9)

(-36592.19815301222, 3.7340586744676e-10)

(0.29946898510226344, -1.47196637230352)

(-39982.66290409008, 3.12763245855605e-10)

(24433.86785186661, 8.37535718469471e-10)

(-14553.647132727501, 2.36045911551622e-9)

(-40830.27786159789, 2.99912589764952e-10)

(-33201.723737943634, 4.53560954733933e-10)

(-27268.35955275504, 6.72412927209326e-10)

(-34049.343403984, 4.31260583883113e-10)

(37148.23184105011, 3.62330923326709e-10)

(-13705.957941247598, 2.66145746157632e-9)

(-28963.612020659526, 5.96004539352621e-10)

(20195.66117206302, 1.22595733811094e-9)

(-24725.46969668346, 8.17830568060985e-10)

(19348.009930835764, 1.33573359189659e-9)

(32910.13654848328, 4.6166168504241e-10)

(-35744.580561626666, 3.91324864896379e-10)

(29519.644830727913, 5.73802532598462e-10)

(21043.30857207633, 1.12917848775663e-9)

(-23877.83614603476, 8.7692384847109e-10)

(-22182.56242901715, 1.01607789110859e-9)

(-17944.325251874623, 1.55271424577534e-9)

(-41677.89238655404, 2.87837972075299e-10)

(-30658.859677714296, 5.31916967142086e-10)

(-23030.200470053816, 9.42661370342373e-10)

(35452.99586902411, 3.97810621184273e-10)

(-34896.96232778141, 4.10565365367217e-10)

(-9467.292609951697, 5.57792322985131e-9)

(37995.84889927713, 3.46345166149953e-10)

(22738.593589023312, 9.67078893909473e-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
Puntos máximos de la función:
x80=0.299468985102263x_{80} = 0.299468985102263
Decrece en los intervalos
(−∞,0.299468985102263]\left(-\infty, 0.299468985102263\right]
Crece en los intervalos
[0.299468985102263,∞)\left[0.299468985102263, \infty\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
2(8x(4x2+1)2+2(x−1)(4x2+1)+acot⁡(2x)(x−1)2)x−1=0\frac{2 \left(\frac{8 x}{\left(4 x^{2} + 1\right)^{2}} + \frac{2}{\left(x - 1\right) \left(4 x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(2 x \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4180.0807038557x_{1} = -4180.0807038557
x2=2579.04331201249x_{2} = 2579.04331201249
x3=4541.96924066321x_{3} = 4541.96924066321
x4=7158.95004982661x_{4} = 7158.95004982661
x5=9121.6331447341x_{5} = 9121.6331447341
x6=−5706.6737887799x_{6} = -5706.6737887799
x7=5632.39375709389x_{7} = 5632.39375709389
x8=−3961.98977326656x_{8} = -3961.98977326656
x9=−7233.22469496698x_{9} = -7233.22469496698
x10=7595.1041560603x_{10} = 7595.1041560603
x11=−4834.34255044352x_{11} = -4834.34255044352
x12=6504.71540956074x_{12} = 6504.71540956074
x13=5850.47532705702x_{13} = 5850.47532705702
x14=6940.87234485317x_{14} = 6940.87234485317
x15=−2217.12330500479x_{15} = -2217.12330500479
x16=1924.59554434664x_{16} = 1924.59554434664
x17=−5488.59244862096x_{17} = -5488.59244862096
x18=−10504.3477714382x_{18} = -10504.3477714382
x19=−3525.80047619857x_{19} = -3525.80047619857
x20=10430.0777759907x_{20} = 10430.0777759907
x21=−2871.48982329053x_{21} = -2871.48982329053
x22=3669.59745505432x_{22} = 3669.59745505432
x23=0.0584278431012344x_{23} = 0.0584278431012344
x24=−10286.2742630911x_{24} = -10286.2742630911
x25=3233.39266844105x_{25} = 3233.39266844105
x26=9993.93024849973x_{26} = 9993.93024849973
x27=9557.78206955215x_{27} = 9557.78206955215
x28=−8977.82990801102x_{28} = -8977.82990801102
x29=4760.05688473916x_{29} = 4760.05688473916
x30=−9850.12681560958x_{30} = -9850.12681560958
x31=8467.40810407493x_{31} = 8467.40810407493
x32=6068.55606953298x_{32} = 6068.55606953298
x33=−5052.42695344654x_{33} = -5052.42695344654
x34=−2653.37580051623x_{34} = -2653.37580051623
x35=−5924.7543261568x_{35} = -5924.7543261568
x36=−6578.99190624731x_{36} = -6578.99190624731
x37=−6797.06996765569x_{37} = -6797.06996765569
x38=1269.88829606854x_{38} = 1269.88829606854
x39=−1780.819723217x_{39} = -1780.819723217
x40=8903.55836800822x_{40} = 8903.55836800822
x41=−1562.63658826404x_{41} = -1562.63658826404
x42=7377.02730722189x_{42} = 7377.02730722189
x43=9775.85624596024x_{43} = 9775.85624596024
x44=3887.69431260379x_{44} = 3887.69431260379
x45=−8541.68024495205x_{45} = -8541.68024495205
x46=3451.49714588277x_{46} = 3451.49714588277
x47=1706.40397611746x_{47} = 1706.40397611746
x48=−6142.83414468299x_{48} = -6142.83414468299
x49=−7887.45386781925x_{49} = -7887.45386781925
x50=6286.63607187431x_{50} = 6286.63607187431
x51=−7451.30144134243x_{51} = -7451.30144134243
x52=−8759.75518964189x_{52} = -8759.75518964189
x53=1488.1749959572x_{53} = 1488.1749959572
x54=4978.1429751161x_{54} = 4978.1429751161
x55=−10940.4943998579x_{55} = -10940.4943998579
x56=2142.7615009407x_{56} = 2142.7615009407
x57=5196.22771073776x_{57} = 5196.22771073776
x58=9339.70770697353x_{58} = 9339.70770697353
x59=−3743.89650383627x_{59} = -3743.89650383627
x60=10648.1513206778x_{60} = 10648.1513206778
x61=8031.25676272261x_{61} = 8031.25676272261
x62=−9195.90441597085x_{62} = -9195.90441597085
x63=5414.31125798944x_{63} = 5414.31125798944
x64=8685.48336046549x_{64} = 8685.48336046549
x65=−7669.37782233139x_{65} = -7669.37782233139
x66=3015.28309325583x_{66} = 3015.28309325583
x67=−4616.25684045402x_{67} = -4616.25684045402
x68=−2435.25431861402x_{68} = -2435.25431861402
x69=8249.33257887773x_{69} = 8249.33257887773
x70=−7015.1475496631x_{70} = -7015.1475496631
x71=−8323.60505638478x_{71} = -8323.60505638478
x72=−1998.97984699414x_{72} = -1998.97984699414
x73=4105.78827992645x_{73} = 4105.78827992645
x74=4323.87980301897x_{74} = 4323.87980301897
x75=10212.0040884122x_{75} = 10212.0040884122
x76=7813.18063095634x_{76} = 7813.18063095634
x77=−9413.97872797573x_{77} = -9413.97872797573
x78=−10722.4211477545x_{78} = -10722.4211477545
x79=10866.2247311514x_{79} = 10866.2247311514
x80=6722.79414814944x_{80} = 6722.79414814944
x81=−3307.70116458968x_{81} = -3307.70116458968
x82=−8105.52960451977x_{82} = -8105.52960451977
x83=2360.90916447443x_{83} = 2360.90916447443
x84=−5270.51020816399x_{84} = -5270.51020816399
x85=−9632.052857186x_{85} = -9632.052857186
x86=2797.1671928271x_{86} = 2797.1671928271
x87=−1344.42047948115x_{87} = -1344.42047948115
x88=−3089.59790092034x_{88} = -3089.59790092034
x89=−10068.2006142254x_{89} = -10068.2006142254
x90=−4398.16963415573x_{90} = -4398.16963415573
x91=−6360.9133169898x_{91} = -6360.9133169898
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=1x_{1} = 1

lim⁡x→1−(2(8x(4x2+1)2+2(x−1)(4x2+1)+acot⁡(2x)(x−1)2)x−1)=−∞\lim_{x \to 1^-}\left(\frac{2 \left(\frac{8 x}{\left(4 x^{2} + 1\right)^{2}} + \frac{2}{\left(x - 1\right) \left(4 x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(2 x \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = -\infty
lim⁡x→1+(2(8x(4x2+1)2+2(x−1)(4x2+1)+acot⁡(2x)(x−1)2)x−1)=∞\lim_{x \to 1^+}\left(\frac{2 \left(\frac{8 x}{\left(4 x^{2} + 1\right)^{2}} + \frac{2}{\left(x - 1\right) \left(4 x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(2 x \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = \infty
- los límites no son iguales, signo
x1=1x_{1} = 1
- 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:
Cóncava en los intervalos
(−∞,0.0584278431012344]\left(-\infty, 0.0584278431012344\right]
Convexa en los intervalos
[0.0584278431012344,∞)\left[0.0584278431012344, \infty\right)
Asíntotas verticales
Hay:
x1=1x_{1} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(acot⁡(2x)x−1)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(2 x \right)}}{x - 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(acot⁡(2x)x−1)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(2 x \right)}}{x - 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 acot(2*x)/(x - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(acot⁡(2x)x(x−1))=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(2 x \right)}}{x \left(x - 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(acot⁡(2x)x(x−1))=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(2 x \right)}}{x \left(x - 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:
acot⁡(2x)x−1=−acot⁡(2x)−x−1\frac{\operatorname{acot}{\left(2 x \right)}}{x - 1} = - \frac{\operatorname{acot}{\left(2 x \right)}}{- x - 1}
- No
acot⁡(2x)x−1=acot⁡(2x)−x−1\frac{\operatorname{acot}{\left(2 x \right)}}{x - 1} = \frac{\operatorname{acot}{\left(2 x \right)}}{- x - 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор