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ctg(2x)
Plotting a function graph/ ctg(2x)

Graphing y = ctg(2x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cot(2*x)
f(x)=cot⁡(2x)f{\left(x \right)} = \cot{\left(2 x \right)} 
f = cot(2*x)
The graph of the function
05-40-35-30-25-20-15-10-510-500500
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:
cot⁡(2x)=0\cot{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
Numerical solution
x1=−43.1968989868597x_{1} = -43.1968989868597
x2=−27.4889357189107x_{2} = -27.4889357189107
x3=−30.6305283725005x_{3} = -30.6305283725005
x4=99.7455667514759x_{4} = 99.7455667514759
x5=33.7721210260903x_{5} = 33.7721210260903
x6=18.0641577581413x_{6} = 18.0641577581413
x7=88.7499924639117x_{7} = 88.7499924639117
x8=27.4889357189107x_{8} = 27.4889357189107
x9=22.776546738526x_{9} = 22.776546738526
x10=−49.4800842940392x_{10} = -49.4800842940392
x11=−13.3517687777566x_{11} = -13.3517687777566
x12=−3.92699081698724x_{12} = -3.92699081698724
x13=68.329640215578x_{13} = 68.329640215578
x14=−79.3252145031423x_{14} = -79.3252145031423
x15=−38.484510006475x_{15} = -38.484510006475
x16=30.6305283725005x_{16} = 30.6305283725005
x17=82.4668071567321x_{17} = 82.4668071567321
x18=36.9137136796801x_{18} = 36.9137136796801
x19=10.2101761241668x_{19} = 10.2101761241668
x20=−19.6349540849362x_{20} = -19.6349540849362
x21=−96.6039740978861x_{21} = -96.6039740978861
x22=19.6349540849362x_{22} = 19.6349540849362
x23=25.9181393921158x_{23} = 25.9181393921158
x24=−7.06858347057703x_{24} = -7.06858347057703
x25=80.8960108299372x_{25} = 80.8960108299372
x26=66.7588438887831x_{26} = 66.7588438887831
x27=−77.7544181763474x_{27} = -77.7544181763474
x28=−57.3340659280137x_{28} = -57.3340659280137
x29=98.174770424681x_{29} = 98.174770424681
x30=−65.1880475619882x_{30} = -65.1880475619882
x31=77.7544181763474x_{31} = 77.7544181763474
x32=−35.3429173528852x_{32} = -35.3429173528852
x33=2.35619449019234x_{33} = 2.35619449019234
x34=−82.4668071567321x_{34} = -82.4668071567321
x35=−32.2013246992954x_{35} = -32.2013246992954
x36=−93.4623814442964x_{36} = -93.4623814442964
x37=−18.0641577581413x_{37} = -18.0641577581413
x38=−76.1836218495525x_{38} = -76.1836218495525
x39=54.1924732744239x_{39} = 54.1924732744239
x40=74.6128255227576x_{40} = 74.6128255227576
x41=60.4756585816035x_{41} = 60.4756585816035
x42=24.3473430653209x_{42} = 24.3473430653209
x43=85.6083998103219x_{43} = 85.6083998103219
x44=−25.9181393921158x_{44} = -25.9181393921158
x45=−41.6261026600648x_{45} = -41.6261026600648
x46=−98.174770424681x_{46} = -98.174770424681
x47=32.2013246992954x_{47} = 32.2013246992954
x48=16.4933614313464x_{48} = 16.4933614313464
x49=−47.9092879672443x_{49} = -47.9092879672443
x50=38.484510006475x_{50} = 38.484510006475
x51=−11.7809724509617x_{51} = -11.7809724509617
x52=5.49778714378214x_{52} = 5.49778714378214
x53=76.1836218495525x_{53} = 76.1836218495525
x54=8.63937979737193x_{54} = 8.63937979737193
x55=−16.4933614313464x_{55} = -16.4933614313464
x56=−69.9004365423729x_{56} = -69.9004365423729
x57=11.7809724509617x_{57} = 11.7809724509617
x58=46.3384916404494x_{58} = 46.3384916404494
x59=−52.621676947629x_{59} = -52.621676947629
x60=91.8915851175014x_{60} = 91.8915851175014
x61=−21.2057504117311x_{61} = -21.2057504117311
x62=−5.49778714378214x_{62} = -5.49778714378214
x63=41.6261026600648x_{63} = 41.6261026600648
x64=71.4712328691678x_{64} = 71.4712328691678
x65=96.6039740978861x_{65} = 96.6039740978861
x66=14.9225651045515x_{66} = 14.9225651045515
x67=−40.0553063332699x_{67} = -40.0553063332699
x68=−63.6172512351933x_{68} = -63.6172512351933
x69=44.7676953136546x_{69} = 44.7676953136546
x70=−55.7632696012188x_{70} = -55.7632696012188
x71=55.7632696012188x_{71} = 55.7632696012188
x72=−60.4756585816035x_{72} = -60.4756585816035
x73=69.9004365423729x_{73} = 69.9004365423729
x74=−91.8915851175014x_{74} = -91.8915851175014
x75=−10.2101761241668x_{75} = -10.2101761241668
x76=−87.1791961371168x_{76} = -87.1791961371168
x77=90.3207887907066x_{77} = 90.3207887907066
x78=−71.4712328691678x_{78} = -71.4712328691678
x79=58.9048622548086x_{79} = 58.9048622548086
x80=40.0553063332699x_{80} = 40.0553063332699
x81=−68.329640215578x_{81} = -68.329640215578
x82=−33.7721210260903x_{82} = -33.7721210260903
x83=−2.35619449019234x_{83} = -2.35619449019234
x84=−85.6083998103219x_{84} = -85.6083998103219
x85=49.4800842940392x_{85} = 49.4800842940392
x86=−99.7455667514759x_{86} = -99.7455667514759
x87=3.92699081698724x_{87} = 3.92699081698724
x88=−54.1924732744239x_{88} = -54.1924732744239
x89=62.0464549083984x_{89} = 62.0464549083984
x90=−46.3384916404494x_{90} = -46.3384916404494
x91=−24.3473430653209x_{91} = -24.3473430653209
x92=47.9092879672443x_{92} = 47.9092879672443
x93=−90.3207887907066x_{93} = -90.3207887907066
x94=52.621676947629x_{94} = 52.621676947629
x95=−84.037603483527x_{95} = -84.037603483527
x96=63.6172512351933x_{96} = 63.6172512351933
x97=93.4623814442964x_{97} = 93.4623814442964
x98=−74.6128255227576x_{98} = -74.6128255227576
x99=84.037603483527x_{99} = 84.037603483527
x100=−62.0464549083984x_{100} = -62.0464549083984
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(2*x).
∞~\tilde{\infty}
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
−2cot⁡2(2x)−2=0- 2 \cot^{2}{\left(2 x \right)} - 2 = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
8(cot⁡2(2x)+1)cot⁡(2x)=08 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}

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
(−∞,π4]\left(-\infty, \frac{\pi}{4}\right]
Convex at the intervals
[π4,∞)\left[\frac{\pi}{4}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cot⁡(2x)=⟨−∞,∞⟩\lim_{x \to -\infty} \cot{\left(2 x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞cot⁡(2x)=⟨−∞,∞⟩\lim_{x \to \infty} \cot{\left(2 x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cot⁡(2x)x)=lim⁡x→−∞(cot⁡(2x)x)\lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(2x)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right)
lim⁡x→∞(cot⁡(2x)x)=lim⁡x→∞(cot⁡(2x)x)\lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(2x)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{x}\right)
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:
cot⁡(2x)=−cot⁡(2x)\cot{\left(2 x \right)} = - \cot{\left(2 x \right)}
- No
cot⁡(2x)=cot⁡(2x)\cot{\left(2 x \right)} = \cot{\left(2 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = ctg(2x)
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