Mister Exam
Graphing y = atan(tan(x/2)+1)
The solution
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/ /x\ \
f(x) = atan|tan|-| + 1|
\ \2/ /
$$f{\left(x \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}$$
f = atan(tan(x/2) + 1)
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:
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 42.4115008234622$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = 29.845130209103$$
$$x_{4} = -95.8185759344887$$
$$x_{5} = -64.4026493985908$$
$$x_{6} = 48.6946861306418$$
$$x_{7} = -32.9867228626928$$
$$x_{8} = -7.85398163397448$$
$$x_{9} = 105.243353895258$$
$$x_{10} = 10.9955742875643$$
$$x_{11} = -76.9690200129499$$
$$x_{12} = 98.9601685880785$$
$$x_{13} = 306.305283725005$$
$$x_{14} = 36.1283155162826$$
$$x_{15} = 23.5619449019235$$
$$x_{16} = -45.553093477052$$
$$x_{17} = -1.5707963267949$$
$$x_{18} = -102.101761241668$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = 92.6769832808989$$
$$x_{21} = -58.1194640914112$$
$$x_{22} = 73.8274273593601$$
$$x_{23} = -39.2699081698724$$
$$x_{24} = -70.6858347057703$$
$$x_{25} = 80.1106126665397$$
$$x_{26} = -14.1371669411541$$
$$x_{27} = -83.2522053201295$$
$$x_{28} = -89.5353906273091$$
$$x_{29} = 86.3937979737193$$
$$x_{30} = -26.7035375555132$$
$$x_{31} = -51.8362787842316$$
$$x_{32} = 17.2787595947439$$
$$x_{33} = 4.71238898038469$$
$$x_{34} = -20.4203522483337$$
$$x_{35} = 312.588469032184$$
$$x_{36} = 61.261056745001$$
so we need to solve the equation:
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 42.4115008234622$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = 29.845130209103$$
$$x_{4} = -95.8185759344887$$
$$x_{5} = -64.4026493985908$$
$$x_{6} = 48.6946861306418$$
$$x_{7} = -32.9867228626928$$
$$x_{8} = -7.85398163397448$$
$$x_{9} = 105.243353895258$$
$$x_{10} = 10.9955742875643$$
$$x_{11} = -76.9690200129499$$
$$x_{12} = 98.9601685880785$$
$$x_{13} = 306.305283725005$$
$$x_{14} = 36.1283155162826$$
$$x_{15} = 23.5619449019235$$
$$x_{16} = -45.553093477052$$
$$x_{17} = -1.5707963267949$$
$$x_{18} = -102.101761241668$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = 92.6769832808989$$
$$x_{21} = -58.1194640914112$$
$$x_{22} = 73.8274273593601$$
$$x_{23} = -39.2699081698724$$
$$x_{24} = -70.6858347057703$$
$$x_{25} = 80.1106126665397$$
$$x_{26} = -14.1371669411541$$
$$x_{27} = -83.2522053201295$$
$$x_{28} = -89.5353906273091$$
$$x_{29} = 86.3937979737193$$
$$x_{30} = -26.7035375555132$$
$$x_{31} = -51.8362787842316$$
$$x_{32} = 17.2787595947439$$
$$x_{33} = 4.71238898038469$$
$$x_{34} = -20.4203522483337$$
$$x_{35} = 312.588469032184$$
$$x_{36} = 61.261056745001$$
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
$$\frac{\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1} = 0$$
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
$$\frac{\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1} = 0$$
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
$$\frac{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
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
$$\left(-\infty, - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}, \infty\right)$$
Convex at the intervals
$$\left[- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}, - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}$$
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
$$\left(-\infty, - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}, \infty\right)$$
Convex at the intervals
$$\left[- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}, - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}\right]$$
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:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(tan(x/2) + 1), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{x}\right)$$
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:
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = - \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}$$
- No
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = - \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}$$
- No
$$\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd