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Plotting a function graph/ acot(x)/x

Graphing y = acot(x)/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       acot(x)
f(x) = -------
          x
f(x)=acot⁡(x)xf{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{x} 
f = acot(x)/x
The graph of the function
02468-8-6-4-2-1010050
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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⁡(x)x=0\frac{\operatorname{acot}{\left(x \right)}}{x} = 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(x)/x.
acot⁡(0)0\frac{\operatorname{acot}{\left(0 \right)}}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−1x(x2+1)−acot⁡(x)x2=0- \frac{1}{x \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=31428.4779581971x_{1} = 31428.4779581971
x2=29733.3552619912x_{2} = 29733.3552619912
x3=−28754.5720832173x_{3} = -28754.5720832173
x4=32276.0426843694x_{4} = 32276.0426843694
x5=−12652.1837404312x_{5} = -12652.1837404312
x6=22105.4658726159x_{6} = 22105.4658726159
x7=−35535.0921922843x_{7} = -35535.0921922843
x8=39904.2007372773x_{8} = 39904.2007372773
x9=40751.780325776x_{9} = 40751.780325776
x10=22952.9911333361x_{10} = 22952.9911333361
x11=−27907.0177880916x_{11} = -27907.0177880916
x12=−18584.2210520308x_{12} = -18584.2210520308
x13=35666.3203642305x_{13} = 35666.3203642305
x14=−19431.7143467142x_{14} = -19431.7143467142
x15=17867.9518916845x_{15} = 17867.9518916845
x16=23800.5220167199x_{16} = 23800.5220167199
x17=16173.0206711654x_{17} = 16173.0206711654
x18=−34687.5204711935x_{18} = -34687.5204711935
x19=−20279.2169096663x_{19} = -20279.2169096663
x20=−29602.129237463x_{20} = -29602.129237463
x21=25495.5984095976x_{21} = 25495.5984095976
x22=39056.6222184191x_{22} = 39056.6222184191
x23=−24516.8351266811x_{23} = -24516.8351266811
x24=15325.5780406013x_{24} = 15325.5780406013
x25=28885.7976883074x_{25} = 28885.7976883074
x26=−42315.7128464392x_{26} = -42315.7128464392
x27=−15194.3715408x_{27} = -15194.3715408
x28=41599.3609185449x_{28} = 41599.3609185449
x29=33123.6094321483x_{29} = 33123.6094321483
x30=−37230.2400751649x_{30} = -37230.2400751649
x31=34818.7484021429x_{31} = 34818.7484021429
x32=−30449.6890121582x_{32} = -30449.6890121582
x33=36513.8938244144x_{33} = 36513.8938244144
x34=28038.2429350559x_{34} = 28038.2429350559
x35=21257.9469066706x_{35} = 21257.9469066706
x36=−27059.4666206439x_{36} = -27059.4666206439
x37=−11804.8475421597x_{37} = -11804.8475421597
x38=−36382.6654280841x_{38} = -36382.6654280841
x39=−39772.9715820244x_{39} = -39772.9715820244
x40=−16889.2678433082x_{40} = -16889.2678433082
x41=−23669.2999443148x_{41} = -23669.2999443148
x42=13630.7530068279x_{42} = 13630.7530068279
x43=33971.1780502568x_{43} = 33971.1780502568
x44=11936.0367344929x_{44} = 11936.0367344929
x45=17020.4793835477x_{45} = 17020.4793835477
x46=42446.9424554335x_{46} = 42446.9424554335
x47=12783.3785763737x_{47} = 12783.3785763737
x48=38209.044840371x_{48} = 38209.044840371
x49=−26211.9188841492x_{49} = -26211.9188841492
x50=27190.691265887x_{50} = 27190.691265887
x51=−41468.131451574x_{51} = -41468.131451574
x52=−32144.815594025x_{52} = -32144.815594025
x53=−38077.8160393051x_{53} = -38077.8160393051
x54=−22821.7698890071x_{54} = -22821.7698890071
x55=−21974.2455536639x_{55} = -21974.2455536639
x56=24648.0579430374x_{56} = 24648.0579430374
x57=14478.1543122982x_{57} = 14478.1543122982
x58=−14346.9510241183x_{58} = -14346.9510241183
x59=−16041.8114526469x_{59} = -16041.8114526469
x60=37361.4686807589x_{60} = 37361.4686807589
x61=18715.4363230897x_{61} = 18715.4363230897
x62=30580.9154216729x_{62} = 30580.9154216729
x63=−38925.3932344872x_{63} = -38925.3932344872
x64=−32992.3820399739x_{64} = -32992.3820399739
x65=26343.1429783643x_{65} = 26343.1429783643
x66=−33839.9503785982x_{66} = -33839.9503785982
x67=−13499.5535498777x_{67} = -13499.5535498777
x68=−25364.3749223954x_{68} = -25364.3749223954
x69=−21126.7276262566x_{69} = -21126.7276262566
x70=20410.4350191541x_{70} = 20410.4350191541
x71=19562.9311293969x_{71} = 19562.9311293969
x72=−31297.2511944755x_{72} = -31297.2511944755
x73=−40620.5510097953x_{73} = -40620.5510097953
x74=−17736.7383530475x_{74} = -17736.7383530475
The values of the extrema at the points:
(31428.477958197083, 1.01240271575178e-9)

(29733.355261991168, 1.13112905968158e-9)

(-28754.57208321733, 1.20944516293517e-9)

(32276.042684369393, 9.59929719294004e-10)

(-12652.183740431175, 6.24696421999817e-9)

(22105.46587261586, 2.04644772560039e-9)

(-35535.09219228433, 7.91926921190877e-10)

(39904.200737277286, 6.28004516156724e-10)

(40751.780325776024, 6.02152961766238e-10)

(22952.99113333609, 1.89811019523146e-9)

(-27907.017788091554, 1.28402400029464e-9)

(-18584.221052030778, 2.89541802966794e-9)

(35666.320364230516, 7.86110120503169e-10)

(-19431.714346714212, 2.64836454119158e-9)

(17867.951891684497, 3.13220696217645e-9)

(23800.52201671992, 1.76533460543662e-9)

(16173.020671165448, 3.82311811900536e-9)

(-34687.5204711935, 8.3110039044608e-10)

(-20279.216909666342, 2.43163081627016e-9)

(-29602.12923746302, 1.14117986183505e-9)

(25495.59840959757, 1.53840109445342e-9)

(39056.62221841914, 6.55557270003213e-10)

(-24516.835126681057, 1.66368532879824e-9)

(15325.57804060126, 4.25761386490084e-9)

(28885.797688307437, 1.19848132029617e-9)

(-42315.71284643916, 5.58465921890688e-10)

(-15194.371540799999, 4.33146207083914e-9)

(41599.36091854488, 5.77865387842876e-10)

(33123.60943214828, 9.11432875514495e-10)

(-37230.24007516492, 7.21453469740323e-10)

(34818.74840214285, 8.2484754877614e-10)

(-30449.689012158226, 1.07853508385086e-9)

(36513.89382441435, 7.50038753065707e-10)

(28038.24293505593, 1.27203309639058e-9)

(21257.946906670644, 2.21287745209561e-9)

(-27059.466620643874, 1.36571958114469e-9)

(-11804.84754215968, 7.17594718693526e-9)

(-36382.66542808411, 7.55459130506513e-10)

(-39772.971582024365, 6.32155498949648e-10)

(-16889.26784330823, 3.50572909699655e-9)

(-23669.2999443148, 1.78496281577192e-9)

(13630.753006827903, 5.38220583289574e-9)

(33971.17805025678, 8.66520386476298e-10)

(11936.036734492856, 7.01907213279033e-9)

(17020.47938354767, 3.45188583500149e-9)

(42446.94245543354, 5.55018136381889e-10)

(12783.378576373669, 6.11939796664849e-9)

(38209.044840371025, 6.84963827783871e-10)

(-26211.918884149185, 1.45546702043703e-9)

(27190.691265887024, 1.35256922689075e-9)

(-41468.13145157402, 5.81528584332591e-10)

(-32144.815594025018, 9.67783297380155e-10)

(-38077.816039305086, 6.89693188941407e-10)

(-22821.769889007064, 1.92000056082403e-9)

(-21974.245553663874, 2.07096163295244e-9)

(24648.057943037446, 1.64601802701379e-9)

(14478.154312298198, 4.77060651575018e-9)

(-14346.951024118276, 4.85826016207709e-9)

(-16041.811452646887, 3.88591399481171e-9)

(37361.468680758895, 7.16394296902737e-10)

(18715.4363230897, 2.85496038596343e-9)

(30580.9154216729, 1.06929869452076e-9)

(-38925.39323448724, 6.59984874728492e-10)

(-32992.38203997393, 9.18697754050756e-10)

(26343.14297836428, 1.44100279019725e-9)

(-33839.95037859824, 8.73253962135754e-10)

(-13499.553549877683, 5.48733137015487e-9)

(-25364.374922395407, 1.554360214983e-9)

(-21126.727626256612, 2.24045142860024e-9)

(20410.435019154116, 2.40046554818317e-9)

(19562.931129396926, 2.61295630664365e-9)

(-31297.25119447549, 1.02091035462809e-9)

(-40620.55100979533, 6.06049893789816e-10)

(-17736.73835304749, 3.17872150865334e-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
The function has no maxima
Decreasing at the entire real axis
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(1(x2+1)2+1x2(x2+1)+acot⁡(x)x3)=02 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{1}{x^{2} \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{x^{3}}\right) = 0
Solve this equation
The roots of this equation
x1=5251.30137550908x_{1} = 5251.30137550908
x2=−8488.3067437525x_{2} = -8488.3067437525
x3=6341.52684619151x_{3} = 6341.52684619151
x4=4815.22711267691x_{4} = 4815.22711267691
x5=4597.19468744241x_{5} = 4597.19468744241
x6=−10668.9165856714x_{6} = -10668.9165856714
x7=−4999.49869173587x_{7} = -4999.49869173587
x8=−1947.71724494957x_{8} = -1947.71724494957
x9=7431.78917927949x_{9} = 7431.78917927949
x10=8740.13322183241x_{10} = 8740.13322183241
x11=−8270.24844071797x_{11} = -8270.24844071797
x12=7867.90100292932x_{12} = 7867.90100292932
x13=7213.73456552556x_{13} = 7213.73456552556
x14=9394.31331980633x_{14} = 9394.31331980633
x15=10920.747799808x_{15} = 10920.747799808
x16=−3037.36245670878x_{16} = -3037.36245670878
x17=9176.25280257723x_{17} = 9176.25280257723
x18=8304.01583378425x_{18} = 8304.01583378425
x19=−5871.66555552497x_{19} = -5871.66555552497
x20=−5653.62054203212x_{20} = -5653.62054203212
x21=2635.18392023345x_{21} = 2635.18392023345
x22=−2383.49808265034x_{22} = -2383.49808265034
x23=−10886.9796067325x_{23} = -10886.9796067325
x24=2853.14299321083x_{24} = 2853.14299321083
x25=10048.4974004532x_{25} = 10048.4974004532
x26=−9796.66771034085x_{26} = -9796.66771034085
x27=−6307.76081059899x_{27} = -6307.76081059899
x28=−7179.96778971602x_{28} = -7179.96778971602
x29=10702.6847336344x_{29} = 10702.6847336344
x30=4379.16602380878x_{30} = 4379.16602380878
x31=−3691.34936334623x_{31} = -3691.34936334623
x32=8522.0742326882x_{32} = 8522.0742326882
x33=−3255.34811569675x_{33} = -3255.34811569675
x34=−4127.37999482338x_{34} = -4127.37999482338
x35=3943.12247393109x_{35} = 3943.12247393109
x36=1545.85652440415x_{36} = 1545.85652440415
x37=−10232.7914652025x_{37} = -10232.7914652025
x38=4161.14171232771x_{38} = 4161.14171232771
x39=5033.26281115033x_{39} = 5033.26281115033
x40=10266.5595141513x_{40} = 10266.5595141513
x41=6777.62835158876x_{41} = 6777.62835158876
x42=1981.45294630499x_{42} = 1981.45294630499
x43=−7616.07762853286x_{43} = -7616.07762853286
x44=−4781.46347328923x_{44} = -4781.46347328923
x45=−4345.40356910782x_{45} = -4345.40356910782
x46=3071.11781687027x_{46} = 3071.11781687027
x47=−2601.43357295479x_{47} = -2601.43357295479
x48=−9360.54551191929x_{48} = -9360.54551191929
x49=8958.19275812722x_{49} = 8958.19275812722
x50=−6743.86191014379x_{50} = -6743.86191014379
x51=8085.95807285625x_{51} = 8085.95807285625
x52=5469.34246328408x_{52} = 5469.34246328408
x53=−8052.19078353361x_{53} = -8052.19078353361
x54=−9142.48506596214x_{54} = -9142.48506596214
x55=10484.6219654214x_{55} = 10484.6219654214
x56=−1512.14254711129x_{56} = -1512.14254711129
x57=9612.37427764737x_{57} = 9612.37427764737
x58=−10014.7294059226x_{58} = -10014.7294059226
x59=−5217.53683474655x_{59} = -5217.53683474655
x60=−7834.13382611099x_{60} = -7834.13382611099
x61=−3473.34437617912x_{61} = -3473.34437617912
x62=6995.68092499917x_{62} = 6995.68092499917
x63=−2819.38985192754x_{63} = -2819.38985192754
x64=−9578.60640329083x_{64} = -9578.60640329083
x65=−6089.71237407404x_{65} = -6089.71237407404
x66=2417.24484370879x_{66} = 2417.24484370879
x67=−6961.91430854258x_{67} = -6961.91430854258
x68=−4563.4315981799x_{68} = -4563.4315981799
x69=5905.43109185005x_{69} = 5905.43109185005
x70=3507.10299454414x_{70} = 3507.10299454414
x71=−8706.36564412441x_{71} = -8706.36564412441
x72=7649.84468307146x_{72} = 7649.84468307146
x73=2199.33168337407x_{73} = 2199.33168337407
x74=6123.47817338971x_{74} = 6123.47817338971
x75=−5435.57755066611x_{75} = -5435.57755066611
x76=−10450.8538654191x_{76} = -10450.8538654191
x77=−1729.8944644263x_{77} = -1729.8944644263
x78=−7398.02225796209x_{78} = -7398.02225796209
x79=3289.10526717558x_{79} = 3289.10526717558
x80=−2165.58963064479x_{80} = -2165.58963064479
x81=−6525.81070300431x_{81} = -6525.81070300431
x82=3725.10919799312x_{82} = 3725.10919799312
x83=5687.38578446966x_{83} = 5687.38578446966
x84=9830.4356467864x_{84} = 9830.4356467864
x85=−8924.4250980605x_{85} = -8924.4250980605
x86=6559.57695165855x_{86} = 6559.57695165855
x87=1763.62130986744x_{87} = 1763.62130986744
x88=−3909.36161963539x_{88} = -3909.36161963539
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=0x_{1} = 0

lim⁡x→0−(2(1(x2+1)2+1x2(x2+1)+acot⁡(x)x3))=∞\lim_{x \to 0^-}\left(2 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{1}{x^{2} \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{x^{3}}\right)\right) = \infty
lim⁡x→0+(2(1(x2+1)2+1x2(x2+1)+acot⁡(x)x3))=∞\lim_{x \to 0^+}\left(2 \left(\frac{1}{\left(x^{2} + 1\right)^{2}} + \frac{1}{x^{2} \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{x^{3}}\right)\right) = \infty
- limits are equal, then skip the corresponding 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(acot⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(acot⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x}\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(x)/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(acot⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(acot⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2}}\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⁡(x)x=acot⁡(x)x\frac{\operatorname{acot}{\left(x \right)}}{x} = \frac{\operatorname{acot}{\left(x \right)}}{x}
- No
acot⁡(x)x=−acot⁡(x)x\frac{\operatorname{acot}{\left(x \right)}}{x} = - \frac{\operatorname{acot}{\left(x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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