Mister Exam
Graphing y = atan(x-1/x)
The solution
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/ 1\
f(x) = atan|x - -|
\ x/
$$f{\left(x \right)} = \operatorname{atan}{\left(x - \frac{1}{x} \right)}$$
f = atan(x - 1/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
so we need to solve the equation:
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\left(x - \frac{1}{x}\right)^{2} + 1} = 0$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\left(x - \frac{1}{x}\right)^{2} + 1} = 0$$
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
$$- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{-1 + \sqrt{3}}$$
$$x_{2} = \sqrt{-1 + \sqrt{3}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1}\right) = 0$$
$$\lim_{x \to 0^+}\left(- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1}\right) = 0$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, - \sqrt{-1 + \sqrt{3}}\right]$$
Convex at the intervals
$$\left[\sqrt{-1 + \sqrt{3}}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt{-1 + \sqrt{3}}$$
$$x_{2} = \sqrt{-1 + \sqrt{3}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1}\right) = 0$$
$$\lim_{x \to 0^+}\left(- \frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{2} \left(x - \frac{1}{x}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1} + \frac{1}{x^{3}}\right)}{\left(x - \frac{1}{x}\right)^{2} + 1}\right) = 0$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, - \sqrt{-1 + \sqrt{3}}\right]$$
Convex at the intervals
$$\left[\sqrt{-1 + \sqrt{3}}, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}{\left(x - \frac{1}{x} \right)} = - \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{\pi}{2}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(x - \frac{1}{x} \right)} = \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi}{2}$$
$$\lim_{x \to -\infty} \operatorname{atan}{\left(x - \frac{1}{x} \right)} = - \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{\pi}{2}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(x - \frac{1}{x} \right)} = \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(x - 1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x - \frac{1}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(x - \frac{1}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x - \frac{1}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(x - \frac{1}{x} \right)}}{x}\right) = 0$$
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:
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = - \operatorname{atan}{\left(x - \frac{1}{x} \right)}$$
- No
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = \operatorname{atan}{\left(x - \frac{1}{x} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = - \operatorname{atan}{\left(x - \frac{1}{x} \right)}$$
- No
$$\operatorname{atan}{\left(x - \frac{1}{x} \right)} = \operatorname{atan}{\left(x - \frac{1}{x} \right)}$$
- Yes
so, the function
is
odd