Mister Exam
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-
How to use it?
- Graphing y =:
- x/(1-x^2)
- x^2-4x+4
- x/(1+x)
- x^3+1
-
Identical expressions
- pi/ two +acot(x)
- Pi divide by 2 plus arcco tangent of gent of (x)
- Pi divide by two plus arcco tangent of gent of (x)
- pi/2+acotx
- pi divide by 2+acot(x)
-
Similar expressions
- pi/2-acot(x)
- pi/2+arccot(x)
- pi/2+arccotx
Graphing y = pi/2+acot(x)
The solution
You have entered
[src]
pi
f(x) = -- + acot(x)
2
$$f{\left(x \right)} = \operatorname{acot}{\left(x \right)} + \frac{\pi}{2}$$
f = acot(x) + 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:
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = 0$$
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:
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = 0$$
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
$$- \frac{1}{x^{2} + 1} = 0$$
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
$$- \frac{1}{x^{2} + 1} = 0$$
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
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} = 0$$
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
$$\frac{2 x}{\left(x^{2} + 1\right)^{2}} = 0$$
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
$$\lim_{x \to -\infty}\left(\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi}{2}$$
$$\lim_{x \to \infty}\left(\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi}{2}$$
$$\lim_{x \to -\infty}\left(\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi}{2}$$
$$\lim_{x \to \infty}\left(\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
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 pi/2 + acot(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)} + \frac{\pi}{2}}{x}\right) = 0$$
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:
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = - \operatorname{acot}{\left(x \right)} + \frac{\pi}{2}$$
- No
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = \operatorname{acot}{\left(x \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = - \operatorname{acot}{\left(x \right)} + \frac{\pi}{2}$$
- No
$$\operatorname{acot}{\left(x \right)} + \frac{\pi}{2} = \operatorname{acot}{\left(x \right)} - \frac{\pi}{2}$$
- No
so, the function
not is
neither even, nor odd
The graph