Mister Exam
Graphing y = sqrt(cot(x/2)^2)
The solution
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/ 2/x\
f(x) = / cot |-|
\/ \2/
$$f{\left(x \right)} = \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}$$
f = sqrt(cot(x/2)^2)
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:
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 97.3893722612836$$
$$x_{4} = 78.5398163397448$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -65.9734457253857$$
$$x_{7} = 21.9911485751286$$
$$x_{8} = -21.9911485751286$$
$$x_{9} = -15.707963267949$$
$$x_{10} = -34.5575191894877$$
$$x_{11} = -40.8407044966673$$
$$x_{12} = 9.42477796076938$$
$$x_{13} = 34.5575191894877$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -28.2743338823081$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -9.42477796076938$$
$$x_{18} = 40.8407044966673$$
$$x_{19} = -91.106186954104$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = 47.1238898038469$$
$$x_{22} = 91.106186954104$$
$$x_{23} = 28.2743338823081$$
$$x_{24} = -47.1238898038469$$
$$x_{25} = -3.14159265358979$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -84.8230016469244$$
$$x_{28} = 84.8230016469244$$
$$x_{29} = 72.2566310325652$$
$$x_{30} = -78.5398163397448$$
$$x_{31} = 15.707963267949$$
$$x_{32} = 3.14159265358979$$
so we need to solve the equation:
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 97.3893722612836$$
$$x_{4} = 78.5398163397448$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -65.9734457253857$$
$$x_{7} = 21.9911485751286$$
$$x_{8} = -21.9911485751286$$
$$x_{9} = -15.707963267949$$
$$x_{10} = -34.5575191894877$$
$$x_{11} = -40.8407044966673$$
$$x_{12} = 9.42477796076938$$
$$x_{13} = 34.5575191894877$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -28.2743338823081$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -9.42477796076938$$
$$x_{18} = 40.8407044966673$$
$$x_{19} = -91.106186954104$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = 47.1238898038469$$
$$x_{22} = 91.106186954104$$
$$x_{23} = 28.2743338823081$$
$$x_{24} = -47.1238898038469$$
$$x_{25} = -3.14159265358979$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -84.8230016469244$$
$$x_{28} = 84.8230016469244$$
$$x_{29} = 72.2566310325652$$
$$x_{30} = -78.5398163397448$$
$$x_{31} = 15.707963267949$$
$$x_{32} = 3.14159265358979$$
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
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{2 \cot{\left(\frac{x}{2} \right)}} = 0$$
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
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{2 \cot{\left(\frac{x}{2} \right)}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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:
$$y = \lim_{x \to -\infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(cot(x/2)^2), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}}}{x}\right)$$
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:
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = \left|{\cot{\left(\frac{x}{2} \right)}}\right|$$
- No
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = - \left|{\cot{\left(\frac{x}{2} \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = \left|{\cot{\left(\frac{x}{2} \right)}}\right|$$
- No
$$\sqrt{\cot^{2}{\left(\frac{x}{2} \right)}} = - \left|{\cot{\left(\frac{x}{2} \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd