Mister Exam
Graphing y = cth(x)/1000
The solution
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coth(x)
f(x) = -------
1000
$$f{\left(x \right)} = \frac{\coth{\left(x \right)}}{1000}$$
f = coth(x)/1000
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:
$$\frac{\coth{\left(x \right)}}{1000} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{\coth{\left(x \right)}}{1000} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{1}{1000 \sinh^{2}{\left(x \right)}} = 0$$
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
$$- \frac{1}{1000 \sinh^{2}{\left(x \right)}} = 0$$
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
$$\frac{\cosh{\left(x \right)}}{500 \sinh^{3}{\left(x \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\cosh{\left(x \right)}}{500 \sinh^{3}{\left(x \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\coth{\left(x \right)}}{1000}\right) = - \frac{1}{1000}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{1}{1000}$$
$$\lim_{x \to \infty}\left(\frac{\coth{\left(x \right)}}{1000}\right) = \frac{1}{1000}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{1000}$$
$$\lim_{x \to -\infty}\left(\frac{\coth{\left(x \right)}}{1000}\right) = - \frac{1}{1000}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{1}{1000}$$
$$\lim_{x \to \infty}\left(\frac{\coth{\left(x \right)}}{1000}\right) = \frac{1}{1000}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{1000}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of coth(x)/1000, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\coth{\left(x \right)}}{1000 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\coth{\left(x \right)}}{1000 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\coth{\left(x \right)}}{1000 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\coth{\left(x \right)}}{1000 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{\coth{\left(x \right)}}{1000} = - \frac{\coth{\left(x \right)}}{1000}$$
- No
$$\frac{\coth{\left(x \right)}}{1000} = \frac{\coth{\left(x \right)}}{1000}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\coth{\left(x \right)}}{1000} = - \frac{\coth{\left(x \right)}}{1000}$$
- No
$$\frac{\coth{\left(x \right)}}{1000} = \frac{\coth{\left(x \right)}}{1000}$$
- No
so, the function
not is
neither even, nor odd