Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • 9^(1/(x-3))
  • 5x^2-3x-1
  • 3x^2+4x-7

  • Identical expressions

  • acot(two *x)/(x- one)
  • arcco tangent of gent of (2 multiply by x) divide by (x minus 1)
  • arcco tangent of gent of (two multiply by x) divide by (x minus one)
  • acot(2x)/(x-1)
  • acot2x/x-1
  • acot(2*x) divide by (x-1)

  • Similar expressions

  • acot(2*x)/(x+1)
  • arccot(2*x)/(x-1)
Plotting a function graph/ acot(2*x)/(x-1)

Graphing y = acot(2*x)/(x-1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [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)
The graph of the function
02468-8-6-4-2-1010-2020
The domain of the function
The points at which the function is not precisely defined:
x1=1x_{1} = 1
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
acot⁡(2x)x−1=0\frac{\operatorname{acot}{\left(2 x \right)}}{x - 1} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to acot(2*x)/(x - 1).
acot⁡(0⋅2)−1\frac{\operatorname{acot}{\left(0 \cdot 2 \right)}}{-1}
The result:
f(0)=−π2f{\left(0 \right)} = - \frac{\pi}{2}
The point:
(0, -pi/2)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−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
Solve this equation
The roots of this equation
x1=38843.4653933966x_{1} = 38843.4653933966
x2=15109.673041667x_{2} = 15109.673041667
x3=36300.6141788641x_{3} = 36300.6141788641
x4=32062.515115973x_{4} = 32062.515115973
x5=21043.3085720763x_{5} = 21043.3085720763
x6=14261.9826775785x_{6} = 14261.9826775785
x7=35452.9958690241x_{7} = 35452.9958690241
x8=−20487.2780980658x_{8} = -20487.2780980658
x9=−34896.9623277814x_{9} = -34896.9623277814
x10=−25573.1013303947x_{10} = -25573.1013303947
x11=−9467.2926099517x_{11} = -9467.2926099517
x12=23586.2319179501x_{12} = 23586.2319179501
x13=−27268.359552755x_{13} = -27268.359552755
x14=−28963.6120206595x_{14} = -28963.6120206595
x15=−17096.6652306606x_{15} = -17096.6652306606
x16=19348.0099308358x_{16} = 19348.0099308358
x17=−21334.9217457585x_{17} = -21334.9217457585
x18=13414.2811900563x_{18} = 13414.2811900563
x19=−31506.4819421501x_{19} = -31506.4819421501
x20=−37439.8151451701x_{20} = -37439.8151451701
x21=17652.6936745703x_{21} = 17652.6936745703
x22=−12010.5446730764x_{22} = -12010.5446730764
x23=39691.0813598296x_{23} = 39691.0813598296
x24=−26420.7312291888x_{24} = -26420.7312291888
x25=−36592.1981530122x_{25} = -36592.1981530122
x26=37995.8488992771x_{26} = 37995.8488992771
x27=30367.2693415002x_{27} = 30367.2693415002
x28=−39135.0474861716x_{28} = -39135.0474861716
x29=−16248.99951668x_{29} = -16248.99951668
x30=−35744.5805616267x_{30} = -35744.5805616267
x31=−41677.892386554x_{31} = -41677.892386554
x32=−12858.2578651732x_{32} = -12858.2578651732
x33=−23030.2004700538x_{33} = -23030.2004700538
x34=−10315.0664908658x_{34} = -10315.0664908658
x35=−32354.1032719495x_{35} = -32354.1032719495
x36=−24725.4696966835x_{36} = -24725.4696966835
x37=10871.0832955839x_{37} = 10871.0832955839
x38=−11162.8154801548x_{38} = -11162.8154801548
x39=10023.3059886798x_{39} = 10023.3059886798
x40=−22182.5624290172x_{40} = -22182.5624290172
x41=16805.0272459016x_{41} = 16805.0272459016
x42=−13705.9579412476x_{42} = -13705.9579412476
x43=−39982.6629040901x_{43} = -39982.6629040901
x44=31214.8927402376x_{44} = 31214.8927402376
x45=−42525.5065046146x_{45} = -42525.5065046146
x46=42233.9264130482x_{46} = 42233.9264130482
x47=−23877.8361460348x_{47} = -23877.8361460348
x48=20195.661172063x_{48} = 20195.661172063
x49=12566.5662777284x_{49} = 12566.5662777284
x50=−34049.343403984x_{50} = -34049.343403984
x51=−38287.431577539x_{51} = -38287.431577539
x52=−40830.2778615979x_{52} = -40830.2778615979
x53=32910.1365484833x_{53} = 32910.1365484833
x54=26129.1334776658x_{54} = 26129.1334776658
x55=27824.3920621207x_{55} = 27824.3920621207
x56=25281.5016345177x_{56} = 25281.5016345177
x57=40538.6968319284x_{57} = 40538.6968319284
x58=−15401.3271925303x_{58} = -15401.3271925303
x59=37148.2318410501x_{59} = 37148.2318410501
x60=−18791.980335574x_{60} = -18791.980335574
x61=−28115.9864417731x_{61} = -28115.9864417731
x62=24433.8678518666x_{62} = 24433.8678518666
x63=34605.3768635164x_{63} = 34605.3768635164
x64=15957.3540863923x_{64} = 15957.3540863923
x65=21890.9525830866x_{65} = 21890.9525830866
x66=−29811.2363998511x_{66} = -29811.2363998511
x67=41386.3118402928x_{67} = 41386.3118402928
x68=0.299468985102263x_{68} = 0.299468985102263
x69=18500.3543120234x_{69} = 18500.3543120234
x70=−17944.3252518746x_{70} = -17944.3252518746
x71=22738.5935890233x_{71} = 22738.5935890233
x72=11718.8349580713x_{72} = 11718.8349580713
x73=29519.6448307279x_{73} = 29519.6448307279
x74=−14553.6471327275x_{74} = -14553.6471327275
x75=26976.7635661688x_{75} = 26976.7635661688
x76=−30658.8596777143x_{76} = -30658.8596777143
x77=33757.7571094617x_{77} = 33757.7571094617
x78=28672.0191082501x_{78} = 28672.0191082501
x79=−33201.7237379436x_{79} = -33201.7237379436
x80=−19639.6311091475x_{80} = -19639.6311091475
The values of the extrema at the points:
(38843.465393396575, 3.3139446354388e-10)

(15109.673041667047, 2.19022440802021e-9)

(36300.61417886415, 3.79449532632734e-10)

(32062.515115973005, 4.86394186318491e-10)

(21043.30857207633, 1.12917848775663e-9)

(14261.98267757852, 2.45833263849057e-9)

(35452.99586902411, 3.97810621184273e-10)

(-20487.278098065755, 1.19118792177403e-9)

(-34896.96232778141, 4.10565365367217e-10)

(-25573.101330394697, 7.64515373854258e-10)

(-9467.292609951697, 5.57792322985131e-9)

(23586.23191795006, 8.98817022075509e-10)

(-27268.35955275504, 6.72412927209326e-10)

(-28963.612020659526, 5.96004539352621e-10)

(-17096.665230660605, 1.71049491297e-9)

(19348.009930835764, 1.33573359189659e-9)

(-21334.92174575853, 1.09841774463209e-9)

(13414.2811900563, 2.77886577091334e-9)

(-31506.481942150105, 5.03681959987282e-10)

(-37439.81514517007, 3.56690065091881e-10)

(17652.693674570288, 1.60462161135569e-9)

(-12010.544673076394, 3.46583944632635e-9)

(39691.0813598296, 3.17391348115915e-10)

(-26420.73122918883, 7.1624878901441e-10)

(-36592.19815301222, 3.7340586744676e-10)

(37995.84889927713, 3.46345166149953e-10)

(30367.269341500167, 5.42216617057071e-10)

(-39135.04748617162, 3.26457895680577e-10)

(-16248.999516679962, 1.89360776601023e-9)

(-35744.580561626666, 3.91324864896379e-10)

(-41677.89238655404, 2.87837972075299e-10)

(-12858.257865173153, 3.02393163075937e-9)

(-23030.200470053816, 9.42661370342373e-10)

(-10315.06649086584, 4.69876623793991e-9)

(-32354.103271949505, 4.77636873128713e-10)

(-24725.46969668346, 8.17830568060985e-10)

(10871.083295583927, 4.23120735206181e-9)

(-11162.815480154843, 4.01220955995748e-9)

(10023.30598867983, 4.97727179998685e-9)

(-22182.56242901715, 1.01607789110859e-9)

(16805.02724590162, 1.77058755052948e-9)

(-13705.957941247598, 2.66145746157632e-9)

(-39982.66290409008, 3.12763245855605e-10)

(31214.892740237614, 5.13168776007676e-10)

(-42525.506504614576, 2.76478141728713e-10)

(42233.92641304821, 2.80322120504772e-10)

(-23877.83614603476, 8.7692384847109e-10)

(20195.66117206302, 1.22595733811094e-9)

(12566.56627772839, 3.16644036216381e-9)

(-34049.343403984, 4.31260583883113e-10)

(-38287.431577539006, 3.41072099113022e-10)

(-40830.27786159789, 2.99912589764952e-10)

(32910.13654848328, 4.6166168504241e-10)

(26129.133477665782, 7.32380227407848e-10)

(27824.39206212073, 6.45853837877993e-10)

(25281.501634517706, 7.82314628168441e-10)

(40538.696831928406, 3.04257400376883e-10)

(-15401.327192530276, 2.10778112918953e-9)

(37148.23184105011, 3.62330923326709e-10)

(-18791.980335574022, 1.41579963358164e-9)

(-28115.986441773104, 6.32481617412191e-10)

(24433.86785186661, 8.37535718469471e-10)

(34605.37686351635, 4.17537405235419e-10)

(15957.354086392263, 1.96370143326721e-9)

(21890.95258308659, 1.04342328253237e-9)

(-29811.236399851103, 5.62594470647764e-10)

(41386.311840292765, 2.91922148402099e-10)

(0.29946898510226344, -1.47196637230352)

(18500.35431202337, 1.46094339039383e-9)

(-17944.325251874623, 1.55271424577534e-9)

(22738.593589023312, 9.67078893909473e-10)

(11718.834958071282, 3.64114680508727e-9)

(29519.644830727913, 5.73802532598462e-10)

(-14553.647132727501, 2.36045911551622e-9)

(26976.763566168815, 6.87078584294831e-10)

(-30658.859677714296, 5.31916967142086e-10)

(33757.7571094617, 4.38768762856835e-10)

(28672.019108250137, 6.08231063639072e-10)

(-33201.723737943634, 4.53560954733933e-10)

(-19639.631109147533, 1.296227524837e-9)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x80=0.299468985102263x_{80} = 0.299468985102263
Decreasing at intervals
(−∞,0.299468985102263]\left(-\infty, 0.299468985102263\right]
Increasing at intervals
[0.299468985102263,∞)\left[0.299468985102263, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
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
Solve this equation
The roots of this equation
x1=−1562.63658826404x_{1} = -1562.63658826404
x2=−5924.7543261568x_{2} = -5924.7543261568
x3=1924.59554434664x_{3} = 1924.59554434664
x4=8903.55836800822x_{4} = 8903.55836800822
x5=−3525.80047619857x_{5} = -3525.80047619857
x6=−5052.42695344654x_{6} = -5052.42695344654
x7=8249.33257887773x_{7} = 8249.33257887773
x8=−8541.68024495205x_{8} = -8541.68024495205
x9=−7887.45386781925x_{9} = -7887.45386781925
x10=−7451.30144134243x_{10} = -7451.30144134243
x11=−5706.6737887799x_{11} = -5706.6737887799
x12=−2653.37580051623x_{12} = -2653.37580051623
x13=2360.90916447443x_{13} = 2360.90916447443
x14=5414.31125798944x_{14} = 5414.31125798944
x15=−3089.59790092034x_{15} = -3089.59790092034
x16=2579.04331201249x_{16} = 2579.04331201249
x17=−8323.60505638478x_{17} = -8323.60505638478
x18=−6797.06996765569x_{18} = -6797.06996765569
x19=−6578.99190624731x_{19} = -6578.99190624731
x20=4760.05688473916x_{20} = 4760.05688473916
x21=10212.0040884122x_{21} = 10212.0040884122
x22=6722.79414814944x_{22} = 6722.79414814944
x23=3669.59745505432x_{23} = 3669.59745505432
x24=−9632.052857186x_{24} = -9632.052857186
x25=−10068.2006142254x_{25} = -10068.2006142254
x26=5196.22771073776x_{26} = 5196.22771073776
x27=7158.95004982661x_{27} = 7158.95004982661
x28=3015.28309325583x_{28} = 3015.28309325583
x29=−5488.59244862096x_{29} = -5488.59244862096
x30=9775.85624596024x_{30} = 9775.85624596024
x31=9121.6331447341x_{31} = 9121.6331447341
x32=6068.55606953298x_{32} = 6068.55606953298
x33=−1344.42047948115x_{33} = -1344.42047948115
x34=−7669.37782233139x_{34} = -7669.37782233139
x35=6504.71540956074x_{35} = 6504.71540956074
x36=9993.93024849973x_{36} = 9993.93024849973
x37=1706.40397611746x_{37} = 1706.40397611746
x38=5632.39375709389x_{38} = 5632.39375709389
x39=7377.02730722189x_{39} = 7377.02730722189
x40=−3743.89650383627x_{40} = -3743.89650383627
x41=−2435.25431861402x_{41} = -2435.25431861402
x42=−10504.3477714382x_{42} = -10504.3477714382
x43=4105.78827992645x_{43} = 4105.78827992645
x44=−5270.51020816399x_{44} = -5270.51020816399
x45=−9195.90441597085x_{45} = -9195.90441597085
x46=−4180.0807038557x_{46} = -4180.0807038557
x47=−8105.52960451977x_{47} = -8105.52960451977
x48=4978.1429751161x_{48} = 4978.1429751161
x49=−7015.1475496631x_{49} = -7015.1475496631
x50=−4834.34255044352x_{50} = -4834.34255044352
x51=2142.7615009407x_{51} = 2142.7615009407
x52=−6142.83414468299x_{52} = -6142.83414468299
x53=3233.39266844105x_{53} = 3233.39266844105
x54=−2871.48982329053x_{54} = -2871.48982329053
x55=1269.88829606854x_{55} = 1269.88829606854
x56=−8759.75518964189x_{56} = -8759.75518964189
x57=7595.1041560603x_{57} = 7595.1041560603
x58=4323.87980301897x_{58} = 4323.87980301897
x59=6940.87234485317x_{59} = 6940.87234485317
x60=−9413.97872797573x_{60} = -9413.97872797573
x61=−4398.16963415573x_{61} = -4398.16963415573
x62=6286.63607187431x_{62} = 6286.63607187431
x63=8685.48336046549x_{63} = 8685.48336046549
x64=−2217.12330500479x_{64} = -2217.12330500479
x65=−1998.97984699414x_{65} = -1998.97984699414
x66=−3961.98977326656x_{66} = -3961.98977326656
x67=9339.70770697353x_{67} = 9339.70770697353
x68=4541.96924066321x_{68} = 4541.96924066321
x69=−3307.70116458968x_{69} = -3307.70116458968
x70=−4616.25684045402x_{70} = -4616.25684045402
x71=−7233.22469496698x_{71} = -7233.22469496698
x72=10866.2247311514x_{72} = 10866.2247311514
x73=−9850.12681560958x_{73} = -9850.12681560958
x74=−8977.82990801102x_{74} = -8977.82990801102
x75=8031.25676272261x_{75} = 8031.25676272261
x76=−10286.2742630911x_{76} = -10286.2742630911
x77=10648.1513206778x_{77} = 10648.1513206778
x78=2797.1671928271x_{78} = 2797.1671928271
x79=−10940.4943998579x_{79} = -10940.4943998579
x80=3887.69431260379x_{80} = 3887.69431260379
x81=3451.49714588277x_{81} = 3451.49714588277
x82=8467.40810407493x_{82} = 8467.40810407493
x83=7813.18063095634x_{83} = 7813.18063095634
x84=9557.78206955215x_{84} = 9557.78206955215
x85=0.0584278431012344x_{85} = 0.0584278431012344
x86=−1780.819723217x_{86} = -1780.819723217
x87=10430.0777759907x_{87} = 10430.0777759907
x88=−6360.9133169898x_{88} = -6360.9133169898
x89=5850.47532705702x_{89} = 5850.47532705702
x90=−10722.4211477545x_{90} = -10722.4211477545
x91=1488.1749959572x_{91} = 1488.1749959572
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
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
- the limits are not equal, so
x1=1x_{1} = 1
- is an inflection point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,0.0584278431012344]\left(-\infty, 0.0584278431012344\right]
Convex at the intervals
[0.0584278431012344,∞)\left[0.0584278431012344, \infty\right)
Vertical asymptotes
Have:
x1=1x_{1} = 1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and 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
Let's take the limit
so,
equation of the horizontal asymptote on the left:
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
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of acot(2*x)/(x - 1), divided by x at x->+oo and 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
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
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
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
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
so, the function
not is
neither even, nor odd
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