Mister Exam
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-
How to use it?
- Graphing y =:
- x^5+4x
- (x+5)^2(x-1)+7
- x/(4-x^2)
- -x^4+x^2+5
- Limit of the function:
-
x*(pi/2-atan(1/x))
-
Identical expressions
- x*(pi/ two -atan(one /x))
- x multiply by ( Pi divide by 2 minus arc tangent of gent of (1 divide by x))
- x multiply by ( Pi divide by two minus arc tangent of gent of (one divide by x))
- x(pi/2-atan(1/x))
- xpi/2-atan1/x
- x*(pi divide by 2-atan(1 divide by x))
-
Similar expressions
- x*(pi/2+atan(1/x))
- x*(pi/2-arctan(1/x))
Graphing y = x*(pi/2-atan(1/x))
The solution
You have entered
[src]
/pi /1\\
f(x) = x*|-- - atan|-||
\2 \x//
$$f{\left(x \right)} = x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)$$
f = x*(-atan(1/x) + pi/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:
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = 0$$
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:
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = 0$$
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
$$- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2} + \frac{1}{x \left(1 + \frac{1}{x^{2}}\right)} = 0$$
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
$$- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2} + \frac{1}{x \left(1 + \frac{1}{x^{2}}\right)} = 0$$
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
$$\frac{2}{x^{4} \left(1 + \frac{1}{x^{2}}\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2}{x^{4} \left(1 + \frac{1}{x^{2}}\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
$$\lim_{x \to -\infty}\left(x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)\right) = \infty$$
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 x*(pi/2 - atan(1/x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{\pi x}{2}$$
$$\lim_{x \to \infty}\left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{\pi x}{2}$$
$$\lim_{x \to -\infty}\left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{\pi x}{2}$$
$$\lim_{x \to \infty}\left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = \frac{\pi}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{\pi x}{2}$$
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:
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = - x \left(\operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)$$
- No
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = x \left(\operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = - x \left(\operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)$$
- No
$$x \left(- \operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right) = x \left(\operatorname{atan}{\left(\frac{1}{x} \right)} + \frac{\pi}{2}\right)$$
- No
so, the function
not is
neither even, nor odd