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Plotting a function graph/ acotx/(x-3)

Graphing y = acotx/(x-3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       acot(x)
f(x) = -------
        x - 3
f(x)=acot⁡(x)x−3f{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{x - 3} 
f = acot(x)/(x - 3)
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=3x_{1} = 3
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−3=0\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = 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 - 3).
acot⁡(0)−3\frac{\operatorname{acot}{\left(0 \right)}}{-3}
The result:
f(0)=−π6f{\left(0 \right)} = - \frac{\pi}{6}
The point:
(0, -pi/6)
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
−1(x−3)(x2+1)−acot⁡(x)(x−3)2=0- \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=20608.7247596784x_{1} = 20608.7247596784
x2=−24290.9504985193x_{2} = -24290.9504985193
x3=−36160.8404213184x_{3} = -36160.8404213184
x4=−26834.7503365653x_{4} = -26834.7503365653
x5=26544.8446743911x_{5} = 26544.8446743911
x6=−38704.1170790913x_{6} = -38704.1170790913
x7=−37856.3650546609x_{7} = -37856.3650546609
x8=21456.8409729567x_{8} = 21456.8409729567
x9=1.26241821842168x_{9} = 1.26241821842168
x10=32479.8996955549x_{10} = 32479.8996955549
x11=−19202.5816602195x_{11} = -19202.5816602195
x12=37566.6273450369x_{12} = 37566.6273450369
x13=−20898.8503044486x_{13} = -20898.8503044486
x14=27392.7585806149x_{14} = 27392.7585806149
x15=35023.3038583412x_{15} = 35023.3038583412
x16=17215.7481743741x_{16} = 17215.7481743741
x17=12123.544841056x_{17} = 12123.544841056
x18=−18354.3754214744x_{18} = -18354.3754214744
x19=41805.3607928303x_{19} = 41805.3607928303
x20=14670.2260130038x_{20} = 14670.2260130038
x21=36718.860771076x_{21} = 36718.860771076
x22=38414.3867016121x_{22} = 38414.3867016121
x23=−16657.783408065x_{23} = -16657.783408065
x24=−37008.6063243624x_{24} = -37008.6063243624
x25=−15809.3788845165x_{25} = -15809.3788845165
x26=24000.9702280612x_{26} = 24000.9702280612
x27=−14112.2895735892x_{27} = -14112.2895735892
x28=−27682.6438443875x_{28} = -27682.6438443875
x29=−29378.3813978389x_{29} = -29378.3813978389
x30=24848.9527856772x_{30} = 24848.9527856772
x31=−30226.22818334x_{31} = -30226.22818334
x32=19760.5640406041x_{32} = 19760.5640406041
x33=−32769.6940726206x_{33} = -32769.6940726206
x34=12972.6101323084x_{34} = 12972.6101323084
x35=−35313.0668342955x_{35} = -35313.0668342955
x36=−42095.0661510489x_{36} = -42095.0661510489
x37=18912.3527533172x_{37} = 18912.3527533172
x38=−21746.9237165814x_{38} = -21746.9237165814
x39=−31921.8835434698x_{39} = -31921.8835434698
x40=30784.242139886x_{40} = 30784.242139886
x41=−25986.8382213314x_{41} = -25986.8382213314
x42=−33617.4943084304x_{42} = -33617.4943084304
x43=31632.0769879872x_{43} = 31632.0769879872
x44=33327.7111964611x_{44} = 33327.7111964611
x45=18064.0836828544x_{45} = 18064.0836828544
x46=28240.6537163389x_{46} = 28240.6537163389
x47=34175.5123312146x_{47} = 34175.5123312146
x48=40957.6259784778x_{48} = 40957.6259784778
x49=13821.4931681849x_{49} = 13821.4931681849
x50=−20050.7379682616x_{50} = -20050.7379682616
x51=−31074.0618885764x_{51} = -31074.0618885764
x52=−23442.9703742917x_{52} = -23442.9703742917
x53=29088.5317374502x_{53} = 29088.5317374502
x54=−41247.3370154355x_{54} = -41247.3370154355
x55=−40399.6026837339x_{55} = -40399.6026837339
x56=−42942.7903957194x_{56} = -42942.7903957194
x57=−12414.7034718108x_{57} = -12414.7034718108
x58=−14960.8860813988x_{58} = -14960.8860813988
x59=15518.8336328721x_{59} = 15518.8336328721
x60=29936.3941106863x_{60} = 29936.3941106863
x61=25696.9101211369x_{61} = 25696.9101211369
x62=35871.0864641394x_{62} = 35871.0864641394
x63=40109.8856055808x_{63} = 40109.8856055808
x64=22304.9178115051x_{64} = 22304.9178115051
x65=−13263.5701308315x_{65} = -13263.5701308315
x66=−17506.1121672217x_{66} = -17506.1121672217
x67=−34465.2850023106x_{67} = -34465.2850023106
x68=−22594.9625089587x_{68} = -22594.9625089587
x69=−39551.8628248821x_{69} = -39551.8628248821
x70=16367.3357554312x_{70} = 16367.3357554312
x71=−28530.5203814017x_{71} = -28530.5203814017
x72=−25138.9056447972x_{72} = -25138.9056447972
x73=23152.9596471974x_{73} = 23152.9596471974
x74=39262.1393115611x_{74} = 39262.1393115611
The values of the extrema at the points:
(20608.7247596784, 2.35483773297575e-9)

(-24290.95049851926, 1.69456155240711e-9)

(-36160.84042131838, 7.64692692897442e-10)

(-26834.750336565336, 1.38853340464679e-9)

(26544.844674391126, 1.41934740490877e-9)

(-38704.11707909125, 6.67501135843943e-10)

(-37856.365054660906, 6.97730588555291e-10)

(21456.840972956685, 2.17234680811585e-9)

(1.262418218421678, -0.385549636898932)

(32479.89969555486, 9.48005280641194e-10)

(-19202.581660219523, 2.7115206374669e-9)

(37566.62734503691, 7.0864751696245e-10)

(-20898.85030444859, 2.2892481474175e-9)

(27392.758580614878, 1.33283388544353e-9)

(35023.303858341176, 8.15310391916053e-10)

(17215.748174374123, 3.37461228493546e-9)

(12123.544841055986, 6.80531501472155e-9)

(-18354.375421474422, 2.96790366013197e-9)

(41805.36079283031, 5.72225509807911e-10)

(14670.226013003838, 4.64745518220953e-9)

(36718.86077107599, 7.41749200934459e-10)

(38414.38670161214, 6.77713458004886e-10)

(-16657.78340806499, 3.60319171329807e-9)

(-37008.606324362416, 7.30061312882698e-10)

(-15809.37888451648, 4.00025778966488e-9)

(24000.970228061182, 1.73618776467237e-9)

(-14112.289573589214, 5.02010415642315e-9)

(-27682.64384438751, 1.30478154940659e-9)

(-29378.38139783892, 1.15851036688784e-9)

(24848.95278567718, 1.61970623227391e-9)

(-30226.22818334005, 1.09443250786901e-9)

(19760.564040604117, 2.56134019256726e-9)

(-32769.694072620616, 9.31141040812685e-10)

(12972.610132308384, 5.94354713795122e-9)

(-35313.06683429549, 8.01848338181294e-10)

(-42095.066151048944, 5.64295591459207e-10)

(18912.352753317246, 2.79626145482265e-9)

(-21746.923716581445, 2.11419202623933e-9)

(-31921.883543469776, 9.81255647453413e-10)

(30784.242139885995, 1.05532297275285e-9)

(-25986.8382213314, 1.48061785005386e-9)

(-33617.49430843041, 8.84770360411383e-10)

(31632.076987987162, 9.99506844645914e-10)

(33327.71119646112, 9.00384720241422e-10)

(18064.083682854387, 3.06506901790584e-9)

(28240.65371633886, 1.25399742921036e-9)

(34175.512331214624, 8.56264733025152e-10)

(40957.625978477816, 5.96159213298478e-10)

(13821.493168184918, 5.23581565927066e-9)

(-20050.737968261594, 2.48699150728489e-9)

(-31074.061888576398, 1.03552841581763e-9)

(-23442.970374291708, 1.81936212368839e-9)

(29088.531737450157, 1.18195567830066e-9)

(-41247.33701543549, 5.87728273655331e-10)

(-40399.602683733865, 6.12651587531458e-10)

(-42942.790395719414, 5.42236987377855e-10)

(-12414.703471810783, 6.48667834896136e-9)

(-14960.886081398789, 4.46681833383236e-9)

(15518.833632872089, 4.15303707114861e-9)

(29936.394110686302, 1.11594951668977e-9)

(25696.91012113692, 1.51456840774484e-9)

(35871.08646413939, 7.77225893994018e-10)

(40109.885605580756, 6.21626667528365e-10)

(22304.91781150513, 2.01028282114719e-9)

(-13263.570130831458, 5.68304250099865e-9)

(-17506.11216722175, 3.26246730310394e-9)

(-34465.285002310586, 8.41779704251999e-10)

(-22594.962508958713, 1.95847977440683e-9)

(-39551.86282488213, 6.39194713316016e-10)

(16367.335755431184, 3.73356422411508e-9)

(-28530.520381401748, 1.22838625950304e-9)

(-25138.905644797193, 1.58217835799154e-9)

(23152.959647197393, 1.86570616473751e-9)

(39262.1393115611, 6.48761793868724e-10)


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:
x74=1.26241821842168x_{74} = 1.26241821842168
Decreasing at intervals
(−∞,1.26241821842168]\left(-\infty, 1.26241821842168\right]
Increasing at intervals
[1.26241821842168,∞)\left[1.26241821842168, \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(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3=0\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3} = 0
Solve this equation
The roots of this equation
x1=5739.03095087926x_{1} = 5739.03095087926
x2=1805.43955191568x_{2} = 1805.43955191568
x3=−4502.04100865049x_{3} = -4502.04100865049
x4=8138.9045667575x_{4} = 8138.9045667575
x5=−9956.2508846458x_{5} = -9956.2508846458
x6=0.17935688002721x_{6} = 0.17935688002721
x7=−6466.02475936076x_{7} = -6466.02475936076
x8=9665.80990054675x_{8} = 9665.80990054675
x9=−5156.81392122052x_{9} = -5156.81392122052
x10=6393.62489744838x_{10} = 6393.62489744838
x11=−3847.08306824419x_{11} = -3847.08306824419
x12=6175.4360524004x_{12} = 6175.4360524004
x13=2024.84149664708x_{13} = 2024.84149664708
x14=−2754.73556955859x_{14} = -2754.73556955859
x15=9883.92871280749x_{15} = 9883.92871280749
x16=−8647.51899910909x_{16} = -8647.51899910909
x17=−9738.13463310465x_{17} = -9738.13463310465
x18=9447.68882539942x_{18} = 9447.68882539942
x19=−6902.36298248242x_{19} = -6902.36298248242
x20=−1660.00043820182x_{20} = -1660.00043820182
x21=5520.81250035928x_{21} = 5520.81250035928
x22=−2317.3026441142x_{22} = -2317.3026441142
x23=4647.7973577865x_{23} = 4647.7973577865
x24=2681.72337804868x_{24} = 2681.72337804868
x25=5957.23840880638x_{25} = 5957.23840880638
x26=−3628.70599204772x_{26} = -3628.70599204772
x27=−2536.07142944323x_{27} = -2536.07142944323
x28=6829.97959979156x_{28} = 6829.97959979156
x29=−2973.31679944955x_{29} = -2973.31679944955
x30=7266.30829889645x_{30} = 7266.30829889645
x31=10320.1601270208x_{31} = 10320.1601270208
x32=−10828.6975404165x_{32} = -10828.6975404165
x33=−2098.39928129367x_{33} = -2098.39928129367
x34=−3191.8312612551x_{34} = -3191.8312612551
x35=−7556.82816022102x_{35} = -7556.82816022102
x36=7484.46433232432x_{36} = 7484.46433232432
x37=−4720.31551331578x_{37} = -4720.31551331578
x38=3774.43639658305x_{38} = 3774.43639658305
x39=5084.33688450853x_{39} = 5084.33688450853
x40=2462.92383062021x_{40} = 2462.92383062021
x41=8575.17841051419x_{41} = 8575.17841051419
x42=−8865.64726589311x_{42} = -8865.64726589311
x43=−7993.11546158443x_{43} = -7993.11546158443
x44=8357.04324622462x_{44} = 8357.04324622462
x45=−6247.8454847632x_{45} = -6247.8454847632
x46=6611.80582431964x_{46} = 6611.80582431964
x47=3556.01176544918x_{47} = 3556.01176544918
x48=7048.14689478809x_{48} = 7048.14689478809
x49=−6029.65847446908x_{49} = -6029.65847446908
x50=−8429.38776883049x_{50} = -8429.38776883049
x51=9229.56532555109x_{51} = 9229.56532555109
x52=1585.66222122544x_{52} = 1585.66222122544
x53=−7774.97384050646x_{53} = -7774.97384050646
x54=2243.97915038475x_{54} = 2243.97915038475
x55=3992.82165401903x_{55} = 3992.82165401903
x56=−3410.29118322484x_{56} = -3410.29118322484
x57=2900.4110627802x_{57} = 2900.4110627802
x58=−10174.3651844545x_{58} = -10174.3651844545
x59=−1879.31904952349x_{59} = -1879.31904952349
x60=−9301.8957324112x_{60} = -9301.8957324112
x61=4866.07619926592x_{61} = 4866.07619926592
x62=3337.53990547011x_{62} = 3337.53990547011
x63=9011.43922338841x_{63} = 9011.43922338841
x64=3119.01072872497x_{64} = 3119.01072872497
x65=8793.31032352348x_{65} = 8793.31032352348
x66=−8211.2533422616x_{66} = -8211.2533422616
x67=−9083.7727799182x_{67} = -9083.7727799182
x68=−4938.5724258037x_{68} = -4938.5724258037
x69=−4065.42827072866x_{69} = -4065.42827072866
x70=−7338.6780646605x_{70} = -7338.6780646605
x71=10756.3841066263x_{71} = 10756.3841066263
x72=4211.17374472313x_{72} = 4211.17374472313
x73=−9520.01629729564x_{73} = -9520.01629729564
x74=7702.6154558089x_{74} = 7702.6154558089
x75=−6684.19704156285x_{75} = -6684.19704156285
x76=7920.76207887546x_{76} = 7920.76207887546
x77=−7120.52315489378x_{77} = -7120.52315489378
x78=−5375.04183086147x_{78} = -5375.04183086147
x79=−10392.4776540664x_{79} = -10392.4776540664
x80=−10610.5884051352x_{80} = -10610.5884051352
x81=5302.58168270876x_{81} = 5302.58168270876
x82=10538.2729880046x_{82} = 10538.2729880046
x83=−4283.7463084719x_{83} = -4283.7463084719
x84=4429.49763773049x_{84} = 4429.49763773049
x85=−5593.25770745173x_{85} = -5593.25770745173
x86=10102.0454099151x_{86} = 10102.0454099151
x87=−11046.8051546367x_{87} = -11046.8051546367
x88=−5811.46287609826x_{88} = -5811.46287609826
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=3x_{1} = 3

lim⁡x→3−(2(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3)=−∞\lim_{x \to 3^-}\left(\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = -\infty
lim⁡x→3+(2(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3)=∞\lim_{x \to 3^+}\left(\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = \infty
- the limits are not equal, so
x1=3x_{1} = 3
- 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
(−∞,0.17935688002721]\left(-\infty, 0.17935688002721\right]
Convex at the intervals
[0.17935688002721,∞)\left[0.17935688002721, \infty\right)
Vertical asymptotes
Have:
x1=3x_{1} = 3
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−3)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x - 3}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(acot⁡(x)x−3)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x - 3}\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 - 3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(acot⁡(x)x(x−3))=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x - 3\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(acot⁡(x)x(x−3))=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x - 3\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⁡(x)x−3=−acot⁡(x)−x−3\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = - \frac{\operatorname{acot}{\left(x \right)}}{- x - 3}
- No
acot⁡(x)x−3=acot⁡(x)−x−3\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = \frac{\operatorname{acot}{\left(x \right)}}{- x - 3}
- No
so, the function
not is
neither even, nor odd
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