Mister Exam
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-
How to use it?
- Graphing y =:
-
x^3-3x^2-9x
- x^4-2*x^2-3
- (x^2-1)/(x-1)
- x^2-2x-3
-
Identical expressions
- ctg(one /2x)
- ctg(1 divide by 2x)
- ctg(one divide by 2x)
- ctg1/2x
-
Similar expressions
- 2ctg1/2(x/2-pi/6)
Graphing y = ctg(1/2x)
The solution
You have entered
[src]
/x\
f(x) = cot|-|
\2/
$$f{\left(x \right)} = \cot{\left(\frac{x}{2} \right)}$$
f = cot(x/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:
$$\cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = -40.8407044966673$$
$$x_{2} = 47.1238898038469$$
$$x_{3} = -53.4070751110265$$
$$x_{4} = -47.1238898038469$$
$$x_{5} = -65.9734457253857$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = -15.707963267949$$
$$x_{8} = 78.5398163397448$$
$$x_{9} = 72.2566310325652$$
$$x_{10} = 53.4070751110265$$
$$x_{11} = 15.707963267949$$
$$x_{12} = 59.6902604182061$$
$$x_{13} = -3.14159265358979$$
$$x_{14} = -91.106186954104$$
$$x_{15} = -72.2566310325652$$
$$x_{16} = -34.5575191894877$$
$$x_{17} = -21.9911485751286$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = 40.8407044966673$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = 21.9911485751286$$
$$x_{23} = 97.3893722612836$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = -84.8230016469244$$
$$x_{26} = 91.106186954104$$
$$x_{27} = 3.14159265358979$$
$$x_{28} = -78.5398163397448$$
$$x_{29} = 84.8230016469244$$
$$x_{30} = 34.5575191894877$$
$$x_{31} = 65.9734457253857$$
$$x_{32} = -28.2743338823081$$
so we need to solve the equation:
$$\cot{\left(\frac{x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = -40.8407044966673$$
$$x_{2} = 47.1238898038469$$
$$x_{3} = -53.4070751110265$$
$$x_{4} = -47.1238898038469$$
$$x_{5} = -65.9734457253857$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = -15.707963267949$$
$$x_{8} = 78.5398163397448$$
$$x_{9} = 72.2566310325652$$
$$x_{10} = 53.4070751110265$$
$$x_{11} = 15.707963267949$$
$$x_{12} = 59.6902604182061$$
$$x_{13} = -3.14159265358979$$
$$x_{14} = -91.106186954104$$
$$x_{15} = -72.2566310325652$$
$$x_{16} = -34.5575191894877$$
$$x_{17} = -21.9911485751286$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = 40.8407044966673$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = 21.9911485751286$$
$$x_{23} = 97.3893722612836$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = -84.8230016469244$$
$$x_{26} = 91.106186954104$$
$$x_{27} = 3.14159265358979$$
$$x_{28} = -78.5398163397448$$
$$x_{29} = 84.8230016469244$$
$$x_{30} = 34.5575191894877$$
$$x_{31} = 65.9734457253857$$
$$x_{32} = -28.2743338823081$$
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
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2} = 0$$
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
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2} = 0$$
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
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \pi$$
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
$$\left(-\infty, \pi\right]$$
Convex at the intervals
$$\left[\pi, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \pi$$
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
$$\left(-\infty, \pi\right]$$
Convex at the intervals
$$\left[\pi, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \cot{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \cot{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \cot{\left(\frac{x}{2} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{2} \right)}}{x}\right)$$
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:
$$\cot{\left(\frac{x}{2} \right)} = - \cot{\left(\frac{x}{2} \right)}$$
- No
$$\cot{\left(\frac{x}{2} \right)} = \cot{\left(\frac{x}{2} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\cot{\left(\frac{x}{2} \right)} = - \cot{\left(\frac{x}{2} \right)}$$
- No
$$\cot{\left(\frac{x}{2} \right)} = \cot{\left(\frac{x}{2} \right)}$$
- Yes
so, the function
is
odd
The graph