Mister Exam
Graphing y = -2ctgx/3
The solution
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-2*cot(x)
f(x) = ---------
3
$$f{\left(x \right)} = \frac{\left(-1\right) 2 \cot{\left(x \right)}}{3}$$
f = (-2*cot(x))/3
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:
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 4.71238898038469$$
$$x_{2} = 17.2787595947439$$
$$x_{3} = -89.5353906273091$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = 70.6858347057703$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = -98.9601685880785$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = -58.1194640914112$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = 39.2699081698724$$
$$x_{12} = -95.8185759344887$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = -92.6769832808989$$
$$x_{15} = -23.5619449019235$$
$$x_{16} = 23.5619449019235$$
$$x_{17} = 61.261056745001$$
$$x_{18} = 29.845130209103$$
$$x_{19} = -32.9867228626928$$
$$x_{20} = -51.8362787842316$$
$$x_{21} = -80.1106126665397$$
$$x_{22} = -83.2522053201295$$
$$x_{23} = 67.5442420521806$$
$$x_{24} = 98.9601685880785$$
$$x_{25} = 92.6769832808989$$
$$x_{26} = -39.2699081698724$$
$$x_{27} = 86.3937979737193$$
$$x_{28} = 45.553093477052$$
$$x_{29} = -67.5442420521806$$
$$x_{30} = 51.8362787842316$$
$$x_{31} = 76.9690200129499$$
$$x_{32} = -26.7035375555132$$
$$x_{33} = -4.71238898038469$$
$$x_{34} = 95.8185759344887$$
$$x_{35} = -86.3937979737193$$
$$x_{36} = -10.9955742875643$$
$$x_{37} = 83.2522053201295$$
$$x_{38} = -7.85398163397448$$
$$x_{39} = -36.1283155162826$$
$$x_{40} = -17.2787595947439$$
$$x_{41} = -14.1371669411541$$
$$x_{42} = 20.4203522483337$$
$$x_{43} = 54.9778714378214$$
$$x_{44} = -70.6858347057703$$
$$x_{45} = -48.6946861306418$$
$$x_{46} = -54.9778714378214$$
$$x_{47} = -45.553093477052$$
$$x_{48} = 14.1371669411541$$
$$x_{49} = -73.8274273593601$$
$$x_{50} = 26.7035375555132$$
$$x_{51} = 89.5353906273091$$
$$x_{52} = 10.9955742875643$$
$$x_{53} = 80.1106126665397$$
$$x_{54} = 73.8274273593601$$
$$x_{55} = 58.1194640914112$$
$$x_{56} = -61.261056745001$$
$$x_{57} = 1.5707963267949$$
$$x_{58} = -20.4203522483337$$
$$x_{59} = -42.4115008234622$$
$$x_{60} = 32.9867228626928$$
$$x_{61} = 42.4115008234622$$
$$x_{62} = -76.9690200129499$$
$$x_{63} = -64.4026493985908$$
$$x_{64} = -29.845130209103$$
so we need to solve the equation:
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 4.71238898038469$$
$$x_{2} = 17.2787595947439$$
$$x_{3} = -89.5353906273091$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = 70.6858347057703$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = -98.9601685880785$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = -58.1194640914112$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = 39.2699081698724$$
$$x_{12} = -95.8185759344887$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = -92.6769832808989$$
$$x_{15} = -23.5619449019235$$
$$x_{16} = 23.5619449019235$$
$$x_{17} = 61.261056745001$$
$$x_{18} = 29.845130209103$$
$$x_{19} = -32.9867228626928$$
$$x_{20} = -51.8362787842316$$
$$x_{21} = -80.1106126665397$$
$$x_{22} = -83.2522053201295$$
$$x_{23} = 67.5442420521806$$
$$x_{24} = 98.9601685880785$$
$$x_{25} = 92.6769832808989$$
$$x_{26} = -39.2699081698724$$
$$x_{27} = 86.3937979737193$$
$$x_{28} = 45.553093477052$$
$$x_{29} = -67.5442420521806$$
$$x_{30} = 51.8362787842316$$
$$x_{31} = 76.9690200129499$$
$$x_{32} = -26.7035375555132$$
$$x_{33} = -4.71238898038469$$
$$x_{34} = 95.8185759344887$$
$$x_{35} = -86.3937979737193$$
$$x_{36} = -10.9955742875643$$
$$x_{37} = 83.2522053201295$$
$$x_{38} = -7.85398163397448$$
$$x_{39} = -36.1283155162826$$
$$x_{40} = -17.2787595947439$$
$$x_{41} = -14.1371669411541$$
$$x_{42} = 20.4203522483337$$
$$x_{43} = 54.9778714378214$$
$$x_{44} = -70.6858347057703$$
$$x_{45} = -48.6946861306418$$
$$x_{46} = -54.9778714378214$$
$$x_{47} = -45.553093477052$$
$$x_{48} = 14.1371669411541$$
$$x_{49} = -73.8274273593601$$
$$x_{50} = 26.7035375555132$$
$$x_{51} = 89.5353906273091$$
$$x_{52} = 10.9955742875643$$
$$x_{53} = 80.1106126665397$$
$$x_{54} = 73.8274273593601$$
$$x_{55} = 58.1194640914112$$
$$x_{56} = -61.261056745001$$
$$x_{57} = 1.5707963267949$$
$$x_{58} = -20.4203522483337$$
$$x_{59} = -42.4115008234622$$
$$x_{60} = 32.9867228626928$$
$$x_{61} = 42.4115008234622$$
$$x_{62} = -76.9690200129499$$
$$x_{63} = -64.4026493985908$$
$$x_{64} = -29.845130209103$$
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
$$\frac{2 \cot^{2}{\left(x \right)}}{3} + \frac{2}{3} = 0$$
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
$$\frac{2 \cot^{2}{\left(x \right)}}{3} + \frac{2}{3} = 0$$
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
$$- \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{3} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{2}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{3} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{2}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-2*cot(x))/3, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{2 \cot{\left(x \right)}}{3 x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{2 \cot{\left(x \right)}}{3 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{2 \cot{\left(x \right)}}{3 x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{2 \cot{\left(x \right)}}{3 x}\right)$$
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:
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = \frac{2 \cot{\left(x \right)}}{3}$$
- No
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = - \frac{2 \cot{\left(x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = \frac{2 \cot{\left(x \right)}}{3}$$
- No
$$\frac{\left(-1\right) 2 \cot{\left(x \right)}}{3} = - \frac{2 \cot{\left(x \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd