Mister Exam
Graphing y = ctg(3x/5)
The solution
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/3*x\
f(x) = cot|---|
\ 5 /
$$f{\left(x \right)} = \cot{\left(\frac{3 x}{5} \right)}$$
f = cot((3*x)/5)
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:
$$\cot{\left(\frac{3 x}{5} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{5 \pi}{6}$$
Numerical solution
$$x_{1} = 7.85398163397448$$
$$x_{2} = -54.9778714378214$$
$$x_{3} = 60.2138591938044$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -75.9218224617533$$
$$x_{6} = 102.101761241668$$
$$x_{7} = -44.5058959258554$$
$$x_{8} = -39.2699081698724$$
$$x_{9} = 2.61799387799149$$
$$x_{10} = 28.7979326579064$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = -91.6297857297023$$
$$x_{13} = 54.9778714378214$$
$$x_{14} = 70.6858347057703$$
$$x_{15} = -49.7418836818384$$
$$x_{16} = 23.5619449019235$$
$$x_{17} = 65.4498469497874$$
$$x_{18} = 75.9218224617533$$
$$x_{19} = 81.1578102177363$$
$$x_{20} = 91.6297857297023$$
$$x_{21} = -86.3937979737193$$
$$x_{22} = 39.2699081698724$$
$$x_{23} = -81.1578102177363$$
$$x_{24} = 96.8657734856853$$
$$x_{25} = -7.85398163397448$$
$$x_{26} = 86.3937979737193$$
$$x_{27} = -2.61799387799149$$
$$x_{28} = -13.0899693899575$$
$$x_{29} = 34.0339204138894$$
$$x_{30} = 44.5058959258554$$
$$x_{31} = -28.7979326579064$$
$$x_{32} = 49.7418836818384$$
$$x_{33} = 13.0899693899575$$
$$x_{34} = 18.3259571459405$$
$$x_{35} = -18.3259571459405$$
$$x_{36} = -23.5619449019235$$
$$x_{37} = -102.101761241668$$
$$x_{38} = -60.2138591938044$$
$$x_{39} = -70.6858347057703$$
$$x_{40} = -65.4498469497874$$
so we need to solve the equation:
$$\cot{\left(\frac{3 x}{5} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{5 \pi}{6}$$
Numerical solution
$$x_{1} = 7.85398163397448$$
$$x_{2} = -54.9778714378214$$
$$x_{3} = 60.2138591938044$$
$$x_{4} = -96.8657734856853$$
$$x_{5} = -75.9218224617533$$
$$x_{6} = 102.101761241668$$
$$x_{7} = -44.5058959258554$$
$$x_{8} = -39.2699081698724$$
$$x_{9} = 2.61799387799149$$
$$x_{10} = 28.7979326579064$$
$$x_{11} = -34.0339204138894$$
$$x_{12} = -91.6297857297023$$
$$x_{13} = 54.9778714378214$$
$$x_{14} = 70.6858347057703$$
$$x_{15} = -49.7418836818384$$
$$x_{16} = 23.5619449019235$$
$$x_{17} = 65.4498469497874$$
$$x_{18} = 75.9218224617533$$
$$x_{19} = 81.1578102177363$$
$$x_{20} = 91.6297857297023$$
$$x_{21} = -86.3937979737193$$
$$x_{22} = 39.2699081698724$$
$$x_{23} = -81.1578102177363$$
$$x_{24} = 96.8657734856853$$
$$x_{25} = -7.85398163397448$$
$$x_{26} = 86.3937979737193$$
$$x_{27} = -2.61799387799149$$
$$x_{28} = -13.0899693899575$$
$$x_{29} = 34.0339204138894$$
$$x_{30} = 44.5058959258554$$
$$x_{31} = -28.7979326579064$$
$$x_{32} = 49.7418836818384$$
$$x_{33} = 13.0899693899575$$
$$x_{34} = 18.3259571459405$$
$$x_{35} = -18.3259571459405$$
$$x_{36} = -23.5619449019235$$
$$x_{37} = -102.101761241668$$
$$x_{38} = -60.2138591938044$$
$$x_{39} = -70.6858347057703$$
$$x_{40} = -65.4498469497874$$
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
$$- \frac{3 \cot^{2}{\left(\frac{3 x}{5} \right)}}{5} - \frac{3}{5} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{3 \cot^{2}{\left(\frac{3 x}{5} \right)}}{5} - \frac{3}{5} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{18 \left(\cot^{2}{\left(\frac{3 x}{5} \right)} + 1\right) \cot{\left(\frac{3 x}{5} \right)}}{25} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{5 \pi}{6}$$
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
$$\left(-\infty, \frac{5 \pi}{6}\right]$$
Convex at the intervals
$$\left[\frac{5 \pi}{6}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{18 \left(\cot^{2}{\left(\frac{3 x}{5} \right)} + 1\right) \cot{\left(\frac{3 x}{5} \right)}}{25} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{5 \pi}{6}$$
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
$$\left(-\infty, \frac{5 \pi}{6}\right]$$
Convex at the intervals
$$\left[\frac{5 \pi}{6}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
$$\lim_{x \to -\infty} \cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot((3*x)/5), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{3 x}{5} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{3 x}{5} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{3 x}{5} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{3 x}{5} \right)}}{x}\right)$$
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:
$$\cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\frac{3 x}{5} \right)}$$
- No
$$\cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\frac{3 x}{5} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\frac{3 x}{5} \right)}$$
- No
$$\cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\frac{3 x}{5} \right)}$$
- No
so, the function
not is
neither even, nor odd