Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-6x^2+9x-3
- x^3+3x
-
x^2-4x
- (x^3-5x^2-6x)/x
-
Identical expressions
- two ^arctan(x)
- 2 to the power of arc tangent of (x)
- two to the power of arc tangent of (x)
- 2arctan(x)
- 2arctanx
- 2^arctanx
Graphing y = 2^arctan(x)
The solution
You have entered
[src]
atan(x) f(x) = 2
$$f{\left(x \right)} = 2^{\operatorname{atan}{\left(x \right)}}$$
f = 2^atan(x)
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:
$$2^{\operatorname{atan}{\left(x \right)}} = 0$$
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:
$$2^{\operatorname{atan}{\left(x \right)}} = 0$$
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
$$\frac{2^{\operatorname{atan}{\left(x \right)}} \log{\left(2 \right)}}{x^{2} + 1} = 0$$
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
$$\frac{2^{\operatorname{atan}{\left(x \right)}} \log{\left(2 \right)}}{x^{2} + 1} = 0$$
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
$$\frac{2^{\operatorname{atan}{\left(x \right)}} \left(- 2 x + \log{\left(2 \right)}\right) \log{\left(2 \right)}}{\left(x^{2} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\log{\left(2 \right)}}{2}$$
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
$$\left(-\infty, \frac{\log{\left(2 \right)}}{2}\right]$$
Convex at the intervals
$$\left[\frac{\log{\left(2 \right)}}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2^{\operatorname{atan}{\left(x \right)}} \left(- 2 x + \log{\left(2 \right)}\right) \log{\left(2 \right)}}{\left(x^{2} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\log{\left(2 \right)}}{2}$$
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
$$\left(-\infty, \frac{\log{\left(2 \right)}}{2}\right]$$
Convex at the intervals
$$\left[\frac{\log{\left(2 \right)}}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} 2^{\operatorname{atan}{\left(x \right)}} = 2^{- \frac{\pi}{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2^{- \frac{\pi}{2}}$$
$$\lim_{x \to \infty} 2^{\operatorname{atan}{\left(x \right)}} = 2^{\frac{\pi}{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2^{\frac{\pi}{2}}$$
$$\lim_{x \to -\infty} 2^{\operatorname{atan}{\left(x \right)}} = 2^{- \frac{\pi}{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2^{- \frac{\pi}{2}}$$
$$\lim_{x \to \infty} 2^{\operatorname{atan}{\left(x \right)}} = 2^{\frac{\pi}{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2^{\frac{\pi}{2}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2^atan(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2^{\operatorname{atan}{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2^{\operatorname{atan}{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2^{\operatorname{atan}{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2^{\operatorname{atan}{\left(x \right)}}}{x}\right) = 0$$
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:
$$2^{\operatorname{atan}{\left(x \right)}} = 2^{- \operatorname{atan}{\left(x \right)}}$$
- No
$$2^{\operatorname{atan}{\left(x \right)}} = - 2^{- \operatorname{atan}{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2^{\operatorname{atan}{\left(x \right)}} = 2^{- \operatorname{atan}{\left(x \right)}}$$
- No
$$2^{\operatorname{atan}{\left(x \right)}} = - 2^{- \operatorname{atan}{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd