Mister Exam
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-
How to use it?
- Graphing y =:
-
x^3-3x+1
-
x/cos(x)
-
x^4+8x^2-9
- y=x^3+1.5x^2-6x
-
Identical expressions
- pi/ two +arcsin(x/ two)
- Pi divide by 2 plus arc sinus of (x divide by 2)
- Pi divide by two plus arc sinus of (x divide by two)
- pi/2+arcsinx/2
- pi divide by 2+arcsin(x divide by 2)
-
Similar expressions
- pi/2-arcsin(x/2)
Graphing y = pi/2+arcsin(x/2)
The solution
You have entered
[src]
pi /x\
f(x) = -- + asin|-|
2 \2/
$$f{\left(x \right)} = \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}$$
f = asin(x/2) + pi/2
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:
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
so we need to solve the equation:
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
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
$$\frac{1}{2 \sqrt{1 - \frac{x^{2}}{4}}} = 0$$
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
$$\frac{1}{2 \sqrt{1 - \frac{x^{2}}{4}}} = 0$$
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
$$\frac{x}{8 \left(1 - \frac{x^{2}}{4}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x}{8 \left(1 - \frac{x^{2}}{4}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of pi/2 + asin(x/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}}{x}\right) = 0$$
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:
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = - \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}$$
- No
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = \operatorname{asin}{\left(\frac{x}{2} \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = - \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2}$$
- No
$$\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\pi}{2} = \operatorname{asin}{\left(\frac{x}{2} \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd
The graph