Mister Exam
Graphing y = abs(ctg(y)*0.0015+0.5)
The solution
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f(y) = |cot(y)*0.0015 + 1/2|
$$f{\left(y \right)} = \left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|$$
f = Abs(0.0015*cot(y) + 1/2)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -0.0029999910000486$$
Numerical solution
$$y_{1} = -0.0029999910000486$$
so we need to solve the equation:
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -0.0029999910000486$$
Numerical solution
$$y_{1} = -0.0029999910000486$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$\left(- 0.0015 \cot^{2}{\left(y \right)} - 0.0015\right) \operatorname{sign}{\left(0.0015 \cot{\left(y \right)} + \frac{1}{2} \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$\left(- 0.0015 \cot^{2}{\left(y \right)} - 0.0015\right) \operatorname{sign}{\left(0.0015 \cot{\left(y \right)} + \frac{1}{2} \right)} = 0$$
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
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\left(4.5 \cdot 10^{-6} \left(\cot^{2}{\left(y \right)} + 1\right) \delta\left(\frac{0.003 \cot{\left(y \right)} + 1}{2}\right) + 0.003 \cot{\left(y \right)} \operatorname{sign}{\left(0.003 \cot{\left(y \right)} + 1 \right)}\right) \left(\cot^{2}{\left(y \right)} + 1\right) = 0$$
Solve this equation
The roots of this equation
$$y_{1} = 1.5707963267949$$
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
$$\left(-\infty, 1.5707963267949\right]$$
Convex at the intervals
$$\left[1.5707963267949, \infty\right)$$
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\left(4.5 \cdot 10^{-6} \left(\cot^{2}{\left(y \right)} + 1\right) \delta\left(\frac{0.003 \cot{\left(y \right)} + 1}{2}\right) + 0.003 \cot{\left(y \right)} \operatorname{sign}{\left(0.003 \cot{\left(y \right)} + 1 \right)}\right) \left(\cot^{2}{\left(y \right)} + 1\right) = 0$$
Solve this equation
The roots of this equation
$$y_{1} = 1.5707963267949$$
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
$$\left(-\infty, 1.5707963267949\right]$$
Convex at the intervals
$$\left[1.5707963267949, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{y \to -\infty} \left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{y \to \infty} \left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{y \to -\infty} \left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{y \to \infty} \left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(cot(y)*0.0015 + 1/2), divided by y at y->+oo and y ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = y \lim_{y \to -\infty}\left(\frac{\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|}{y}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = y \lim_{y \to \infty}\left(\frac{\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|}{y}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = y \lim_{y \to -\infty}\left(\frac{\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|}{y}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = y \lim_{y \to \infty}\left(\frac{\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right|}{y}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-y) и f = -f(-y).
So, check:
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = \left|{0.0015 \cot{\left(y \right)} - \frac{1}{2}}\right|$$
- No
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = - \left|{0.0015 \cot{\left(y \right)} - \frac{1}{2}}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = \left|{0.0015 \cot{\left(y \right)} - \frac{1}{2}}\right|$$
- No
$$\left|{0.0015 \cot{\left(y \right)} + \frac{1}{2}}\right| = - \left|{0.0015 \cot{\left(y \right)} - \frac{1}{2}}\right|$$
- No
so, the function
not is
neither even, nor odd