Mister Exam
Graphing y = asin(sqrt(sqrt(2))-2/sqrt(5*x))
The solution
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/ _______ \
| / ___ 2 |
f(x) = asin|\/ \/ 2 - -------|
| _____|
\ \/ 5*x /
$$f{\left(x \right)} = \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)}$$
f = asin(sqrt(sqrt(2)) - 2*sqrt(5)/(5*sqrt(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:
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2 \sqrt{2}}{5}$$
Numerical solution
$$x_{1} = 0.565685424949238$$
so we need to solve the equation:
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2 \sqrt{2}}{5}$$
Numerical solution
$$x_{1} = 0.565685424949238$$
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
$$\frac{\sqrt{5}}{5 x^{\frac{3}{2}} \sqrt{1 - \left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}}\right)^{2}}} = 0$$
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
$$\frac{\sqrt{5}}{5 x^{\frac{3}{2}} \sqrt{1 - \left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}}\right)^{2}}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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
$$\lim_{x \to -\infty} \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
$$\lim_{x \to \infty} \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
$$\lim_{x \to -\infty} \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
$$\lim_{x \to \infty} \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \operatorname{asin}{\left(\sqrt[4]{2} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(sqrt(sqrt(2)) - 2*sqrt(5)/(5*sqrt(x))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)}}{x}\right) = 0$$
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:
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2 \sqrt{5}}{5 \sqrt{- x}} \right)}$$
- No
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = - \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2 \sqrt{5}}{5 \sqrt{- x}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2 \sqrt{5}}{5 \sqrt{- x}} \right)}$$
- No
$$\operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2}{\sqrt{5 x}} \right)} = - \operatorname{asin}{\left(\sqrt{\sqrt{2}} - \frac{2 \sqrt{5}}{5 \sqrt{- x}} \right)}$$
- No
so, the function
not is
neither even, nor odd