Mister Exam
Graphing y = -exp(asin(x/9))
The solution
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/x\
asin|-|
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f(x) = -e
$$f{\left(x \right)} = - e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}$$
f = -exp(asin(x/9))
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:
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = 0$$
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:
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = 0$$
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
$$- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{9 \sqrt{1 - \frac{x^{2}}{81}}} = 0$$
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
$$- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{9 \sqrt{1 - \frac{x^{2}}{81}}} = 0$$
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
$$- \frac{\left(\frac{x}{\left(1 - \frac{x^{2}}{81}\right)^{\frac{3}{2}}} - \frac{729}{x^{2} - 81}\right) e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{729} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{9 \sqrt{2}}{2}$$
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
$$\left(-\infty, - \frac{9 \sqrt{2}}{2}\right]$$
Convex at the intervals
$$\left[- \frac{9 \sqrt{2}}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\left(\frac{x}{\left(1 - \frac{x^{2}}{81}\right)^{\frac{3}{2}}} - \frac{729}{x^{2} - 81}\right) e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{729} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{9 \sqrt{2}}{2}$$
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
$$\left(-\infty, - \frac{9 \sqrt{2}}{2}\right]$$
Convex at the intervals
$$\left[- \frac{9 \sqrt{2}}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}\right)$$
$$\lim_{x \to \infty}\left(- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}\right) = - e^{- \infty i}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = - e^{- \infty i}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}\right)$$
$$\lim_{x \to \infty}\left(- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}\right) = - e^{- \infty i}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = - e^{- \infty i}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -exp(asin(x/9)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{e^{\operatorname{asin}{\left(\frac{x}{9} \right)}}}{x}\right) = 0$$
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:
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = - e^{- \operatorname{asin}{\left(\frac{x}{9} \right)}}$$
- No
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = e^{- \operatorname{asin}{\left(\frac{x}{9} \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = - e^{- \operatorname{asin}{\left(\frac{x}{9} \right)}}$$
- No
$$- e^{\operatorname{asin}{\left(\frac{x}{9} \right)}} = e^{- \operatorname{asin}{\left(\frac{x}{9} \right)}}$$
- No
so, the function
not is
neither even, nor odd