Mister Exam
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-
How to use it?
- Graphing y =:
- (x+3)^3/(x+1)^2
-
|x^2-x-2|
- x^2+4x-5
- x^4+2x^2+5
-
Identical expressions
- -ctg(pix)
- minus ctg( Pi x)
- -ctgpix
-
Similar expressions
- ctg(pix)
Graphing y = -ctg(pix)
The solution
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f(x) = -cot(pi*x)
$$f{\left(x \right)} = - \cot{\left(\pi x \right)}$$
f = -cot(pi*x)
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:
$$- \cot{\left(\pi x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.5$$
so we need to solve the equation:
$$- \cot{\left(\pi x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.5$$
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
$$- \pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right) = 0$$
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
$$- \pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right) = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \pi^{2} \left(\cot^{2}{\left(\pi x \right)} + 1\right) \cot{\left(\pi x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
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
$$\left[\frac{1}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{1}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \pi^{2} \left(\cot^{2}{\left(\pi x \right)} + 1\right) \cot{\left(\pi x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
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
$$\left[\frac{1}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{1}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \cot{\left(\pi x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- \cot{\left(\pi x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(- \cot{\left(\pi x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- \cot{\left(\pi x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
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(pi*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right) = \lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right) = \lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right) = \lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right) = \lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{\cot{\left(\pi x \right)}}{x}\right)$$
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:
$$- \cot{\left(\pi x \right)} = \cot{\left(\pi x \right)}$$
- No
$$- \cot{\left(\pi x \right)} = - \cot{\left(\pi x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$- \cot{\left(\pi x \right)} = \cot{\left(\pi x \right)}$$
- No
$$- \cot{\left(\pi x \right)} = - \cot{\left(\pi x \right)}$$
- Yes
so, the function
is
odd
The graph