Mister Exam

# Graphing y = ctg(2x)

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### The solution

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f(x) = cot(2*x)
$$f{\left(x \right)} = \cot{\left(2 x \right)}$$
f = cot(2*x)
The graph of the function
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:
$$\cot{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = \frac{\pi}{4}$$
Numerical solution
$$x_{1} = -43.1968989868597$$
$$x_{2} = -27.4889357189107$$
$$x_{3} = -30.6305283725005$$
$$x_{4} = 99.7455667514759$$
$$x_{5} = 33.7721210260903$$
$$x_{6} = 18.0641577581413$$
$$x_{7} = 88.7499924639117$$
$$x_{8} = 27.4889357189107$$
$$x_{9} = 22.776546738526$$
$$x_{10} = -49.4800842940392$$
$$x_{11} = -13.3517687777566$$
$$x_{12} = -3.92699081698724$$
$$x_{13} = 68.329640215578$$
$$x_{14} = -79.3252145031423$$
$$x_{15} = -38.484510006475$$
$$x_{16} = 30.6305283725005$$
$$x_{17} = 82.4668071567321$$
$$x_{18} = 36.9137136796801$$
$$x_{19} = 10.2101761241668$$
$$x_{20} = -19.6349540849362$$
$$x_{21} = -96.6039740978861$$
$$x_{22} = 19.6349540849362$$
$$x_{23} = 25.9181393921158$$
$$x_{24} = -7.06858347057703$$
$$x_{25} = 80.8960108299372$$
$$x_{26} = 66.7588438887831$$
$$x_{27} = -77.7544181763474$$
$$x_{28} = -57.3340659280137$$
$$x_{29} = 98.174770424681$$
$$x_{30} = -65.1880475619882$$
$$x_{31} = 77.7544181763474$$
$$x_{32} = -35.3429173528852$$
$$x_{33} = 2.35619449019234$$
$$x_{34} = -82.4668071567321$$
$$x_{35} = -32.2013246992954$$
$$x_{36} = -93.4623814442964$$
$$x_{37} = -18.0641577581413$$
$$x_{38} = -76.1836218495525$$
$$x_{39} = 54.1924732744239$$
$$x_{40} = 74.6128255227576$$
$$x_{41} = 60.4756585816035$$
$$x_{42} = 24.3473430653209$$
$$x_{43} = 85.6083998103219$$
$$x_{44} = -25.9181393921158$$
$$x_{45} = -41.6261026600648$$
$$x_{46} = -98.174770424681$$
$$x_{47} = 32.2013246992954$$
$$x_{48} = 16.4933614313464$$
$$x_{49} = -47.9092879672443$$
$$x_{50} = 38.484510006475$$
$$x_{51} = -11.7809724509617$$
$$x_{52} = 5.49778714378214$$
$$x_{53} = 76.1836218495525$$
$$x_{54} = 8.63937979737193$$
$$x_{55} = -16.4933614313464$$
$$x_{56} = -69.9004365423729$$
$$x_{57} = 11.7809724509617$$
$$x_{58} = 46.3384916404494$$
$$x_{59} = -52.621676947629$$
$$x_{60} = 91.8915851175014$$
$$x_{61} = -21.2057504117311$$
$$x_{62} = -5.49778714378214$$
$$x_{63} = 41.6261026600648$$
$$x_{64} = 71.4712328691678$$
$$x_{65} = 96.6039740978861$$
$$x_{66} = 14.9225651045515$$
$$x_{67} = -40.0553063332699$$
$$x_{68} = -63.6172512351933$$
$$x_{69} = 44.7676953136546$$
$$x_{70} = -55.7632696012188$$
$$x_{71} = 55.7632696012188$$
$$x_{72} = -60.4756585816035$$
$$x_{73} = 69.9004365423729$$
$$x_{74} = -91.8915851175014$$
$$x_{75} = -10.2101761241668$$
$$x_{76} = -87.1791961371168$$
$$x_{77} = 90.3207887907066$$
$$x_{78} = -71.4712328691678$$
$$x_{79} = 58.9048622548086$$
$$x_{80} = 40.0553063332699$$
$$x_{81} = -68.329640215578$$
$$x_{82} = -33.7721210260903$$
$$x_{83} = -2.35619449019234$$
$$x_{84} = -85.6083998103219$$
$$x_{85} = 49.4800842940392$$
$$x_{86} = -99.7455667514759$$
$$x_{87} = 3.92699081698724$$
$$x_{88} = -54.1924732744239$$
$$x_{89} = 62.0464549083984$$
$$x_{90} = -46.3384916404494$$
$$x_{91} = -24.3473430653209$$
$$x_{92} = 47.9092879672443$$
$$x_{93} = -90.3207887907066$$
$$x_{94} = 52.621676947629$$
$$x_{95} = -84.037603483527$$
$$x_{96} = 63.6172512351933$$
$$x_{97} = 93.4623814442964$$
$$x_{98} = -74.6128255227576$$
$$x_{99} = 84.037603483527$$
$$x_{100} = -62.0464549083984$$
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:
$$f{\left(0 \right)} = \tilde{\infty}$$
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} =$$
the first derivative
$$- 2 \cot^{2}{\left(2 x \right)} - 2 = 0$$
Solve this equation
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} =$$
the second derivative
$$8 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{4}$$

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