Mister Exam
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How to use it?
- Graphing y =:
- x^3-((x^4)/4)
- -x^3+3x+2
- x^3-3*x-2
- -x^3+3x^2+9x-12
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Identical expressions
- x/ two +arccos(two x/(one +x^2))
- x divide by 2 plus arc co sinus of e of (2x divide by (1 plus x squared ))
- x divide by two plus arc co sinus of e of (two x divide by (one plus x squared ))
- x/2+arccos(2x/(1+x2))
- x/2+arccos2x/1+x2
- x/2+arccos(2x/(1+x²))
- x/2+arccos(2x/(1+x to the power of 2))
- x/2+arccos2x/1+x^2
- x divide by 2+arccos(2x divide by (1+x^2))
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Similar expressions
- x/2+arccos(2x/(1-x^2))
- x/2-arccos(2x/(1+x^2))
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What you mean?
- x/2 + acos(2*x/(1 + x^2))
- x/2 + acos(2*x*2/(1 + x))
- x/2 + acos(2*x/((1 + x)^2))
- x/2 + acos(2*x/1 + x^2)
- x/2 + acos(2*x/(1 + x^2))
- x/2 + acos(2*x*2/(1 + x))
- x/2 + acos(2*x/1 + x^2)
You entered:
x/2+arccos(2x/(1+x^2))
What you mean?
Graphing y = x/2+arccos(2x/(1+x^2))
The solution
You have entered
[src]
x / 2*x \
f(x) = - + acos|------|
2 | 2|
\1 + x /
$$f{\left(x \right)} = \frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}$$
f = x/2 + acos(2*x/(x^2 + 1))
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:
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -4.09879072392429$$
so we need to solve the equation:
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -4.09879072392429$$
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
$$\frac{1}{2} - \frac{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x^{2} + 1}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = 0$$
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
$$\frac{1}{2} - \frac{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x^{2} + 1}}{\sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = 0$$
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
$$- \frac{4 x \left(\frac{4 x^{2}}{x^{2} + 1} - 3 + \frac{2 \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)^{2}}{\left(x^{2} + 1\right) \left(- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)}\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = 0$$
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
$$- \frac{4 x \left(\frac{4 x^{2}}{x^{2} + 1} - 3 + \frac{2 \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)^{2}}{\left(x^{2} + 1\right) \left(- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1\right)}\right)}{\left(x^{2} + 1\right)^{2} \sqrt{- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + 1}} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}\right) = \infty$$
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/2 + acos(2*x/(1 + x^2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{x}{2}$$
$$\lim_{x \to -\infty}\left(\frac{\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{x}{2}$$
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:
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = - \frac{x}{2} + \operatorname{acos}{\left(- \frac{2 x}{x^{2} + 1} \right)}$$
- No
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = \frac{x}{2} - \operatorname{acos}{\left(- \frac{2 x}{x^{2} + 1} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = - \frac{x}{2} + \operatorname{acos}{\left(- \frac{2 x}{x^{2} + 1} \right)}$$
- No
$$\frac{x}{2} + \operatorname{acos}{\left(\frac{2 x}{x^{2} + 1} \right)} = \frac{x}{2} - \operatorname{acos}{\left(- \frac{2 x}{x^{2} + 1} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph