Mister Exam
Graphing y = asin(log(x))/x^2
The solution
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asin(log(x))
f(x) = ------------
2
x
$$f{\left(x \right)} = \frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}$$
f = asin(log(x))/x^2
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:
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$\frac{1}{x x^{2} \sqrt{1 - \log{\left(x \right)}^{2}}} - \frac{2 \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{3}} = 0$$
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
$$\frac{1}{x x^{2} \sqrt{1 - \log{\left(x \right)}^{2}}} - \frac{2 \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{3}} = 0$$
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
$$\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2.09653484237811$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i$$
- 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
$$\left[2.09653484237811, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.09653484237811\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2.09653484237811$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{\sqrt{1 - \log{\left(x \right)}^{2}}} + 6 \operatorname{asin}{\left(\log{\left(x \right)} \right)} - \frac{4}{\sqrt{1 - \log{\left(x \right)}^{2}}}}{x^{4}}\right) = \infty i$$
- 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
$$\left[2.09653484237811, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.09653484237811\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(log(x))/x^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x x^{2}}\right) = 0$$
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:
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}$$
- No
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = - \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}$$
- No
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{x^{2}} = - \frac{\operatorname{asin}{\left(\log{\left(- x \right)} \right)}}{x^{2}}$$
- No
so, the function
not is
neither even, nor odd