Mister Exam
Graphing y = ctg((x+1)/2)
The solution
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/x + 1\
f(x) = cot|-----|
\ 2 /
$$f{\left(x \right)} = \cot{\left(\frac{x + 1}{2} \right)}$$
f = cot((x + 1)/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:
$$\cot{\left(\frac{x + 1}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1 + \pi$$
Numerical solution
$$x_{1} = 77.5398163397448$$
$$x_{2} = 96.3893722612836$$
$$x_{3} = -54.4070751110265$$
$$x_{4} = 71.2566310325652$$
$$x_{5} = -60.6902604182061$$
$$x_{6} = 39.8407044966673$$
$$x_{7} = -85.8230016469244$$
$$x_{8} = -79.5398163397448$$
$$x_{9} = 102.672557568463$$
$$x_{10} = -10.4247779607694$$
$$x_{11} = -35.5575191894877$$
$$x_{12} = 90.106186954104$$
$$x_{13} = -48.1238898038469$$
$$x_{14} = 64.9734457253857$$
$$x_{15} = -73.2566310325652$$
$$x_{16} = -29.2743338823081$$
$$x_{17} = 58.6902604182061$$
$$x_{18} = 8.42477796076938$$
$$x_{19} = -98.3893722612836$$
$$x_{20} = 52.4070751110265$$
$$x_{21} = -22.9911485751286$$
$$x_{22} = 46.1238898038469$$
$$x_{23} = 2.14159265358979$$
$$x_{24} = -66.9734457253857$$
$$x_{25} = -16.707963267949$$
$$x_{26} = -92.106186954104$$
$$x_{27} = -41.8407044966673$$
$$x_{28} = 83.8230016469244$$
$$x_{29} = 27.2743338823081$$
$$x_{30} = 14.707963267949$$
$$x_{31} = 20.9911485751286$$
$$x_{32} = -4.14159265358979$$
$$x_{33} = 33.5575191894877$$
so we need to solve the equation:
$$\cot{\left(\frac{x + 1}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1 + \pi$$
Numerical solution
$$x_{1} = 77.5398163397448$$
$$x_{2} = 96.3893722612836$$
$$x_{3} = -54.4070751110265$$
$$x_{4} = 71.2566310325652$$
$$x_{5} = -60.6902604182061$$
$$x_{6} = 39.8407044966673$$
$$x_{7} = -85.8230016469244$$
$$x_{8} = -79.5398163397448$$
$$x_{9} = 102.672557568463$$
$$x_{10} = -10.4247779607694$$
$$x_{11} = -35.5575191894877$$
$$x_{12} = 90.106186954104$$
$$x_{13} = -48.1238898038469$$
$$x_{14} = 64.9734457253857$$
$$x_{15} = -73.2566310325652$$
$$x_{16} = -29.2743338823081$$
$$x_{17} = 58.6902604182061$$
$$x_{18} = 8.42477796076938$$
$$x_{19} = -98.3893722612836$$
$$x_{20} = 52.4070751110265$$
$$x_{21} = -22.9911485751286$$
$$x_{22} = 46.1238898038469$$
$$x_{23} = 2.14159265358979$$
$$x_{24} = -66.9734457253857$$
$$x_{25} = -16.707963267949$$
$$x_{26} = -92.106186954104$$
$$x_{27} = -41.8407044966673$$
$$x_{28} = 83.8230016469244$$
$$x_{29} = 27.2743338823081$$
$$x_{30} = 14.707963267949$$
$$x_{31} = 20.9911485751286$$
$$x_{32} = -4.14159265358979$$
$$x_{33} = 33.5575191894877$$
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
$$- \frac{\cot^{2}{\left(\frac{x + 1}{2} \right)}}{2} - \frac{1}{2} = 0$$
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
$$- \frac{\cot^{2}{\left(\frac{x + 1}{2} \right)}}{2} - \frac{1}{2} = 0$$
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
$$\frac{\left(\cot^{2}{\left(\frac{x + 1}{2} \right)} + 1\right) \cot{\left(\frac{x + 1}{2} \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1 + \pi$$
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
$$\left(-\infty, -1 + \pi\right]$$
Convex at the intervals
$$\left[-1 + \pi, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\cot^{2}{\left(\frac{x + 1}{2} \right)} + 1\right) \cot{\left(\frac{x + 1}{2} \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1 + \pi$$
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
$$\left(-\infty, -1 + \pi\right]$$
Convex at the intervals
$$\left[-1 + \pi, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \cot{\left(\infty \right)}$$
$$\lim_{x \to -\infty} \cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\infty \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \cot{\left(\infty \right)}$$
$$\lim_{x \to \infty} \cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\infty \right)}$$
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((x + 1)/2), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\right)$$
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:
$$\cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}$$
- No
$$\cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}$$
- No
$$\cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd