Mister Exam
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How to use it?
- Graphing y =:
-
arctan(x/sqrt(1-x*x))
-
(x^2-7x+12)
-
4x²-4x+1
- f(x)=x^4-5x^2+4
-
Identical expressions
- arctan(x/sqrt(one -x*x))
- arc tangent of (x divide by square root of (1 minus x multiply by x))
- arc tangent of (x divide by square root of (one minus x multiply by x))
- arctan(x/√(1-x*x))
- arctan(x/sqrt(1-xx))
- arctanx/sqrt1-xx
- arctan(x divide by sqrt(1-x*x))
-
Similar expressions
- arctan(x/sqrt(1+x*x))
Graphing y = arctan(x/sqrt(1-x*x))
The solution
You have entered
[src]
/ x \
f(x) = atan|-----------|
| _________|
\\/ 1 - x*x /
$$f{\left(x \right)} = \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
f = atan(x/(sqrt(1 - x*x)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$\frac{\frac{x^{2}}{\left(- x x + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{- x x + 1}}}{\frac{x^{2}}{- x x + 1} + 1} = 0$$
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
$$\frac{\frac{x^{2}}{\left(- x x + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{- x x + 1}}}{\frac{x^{2}}{- x x + 1} + 1} = 0$$
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
$$- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = - \infty i$$
Let's take the limit
$$\lim_{x \to -1^+}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = -\infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = \infty$$
Let's take the limit
$$\lim_{x \to 1^+}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = \infty i$$
Let's take the limit
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection point
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
$$- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = - \infty i$$
Let's take the limit
$$\lim_{x \to -1^+}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = -\infty$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = \infty$$
Let's take the limit
$$\lim_{x \to 1^+}\left(- \frac{x \left(\frac{x^{2}}{- x^{2} + 1} + 1\right) \left(\frac{2}{x^{2} - 1} + \frac{3}{- x^{2} + 1}\right)}{\sqrt{- x^{2} + 1} \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}\right) = \infty i$$
Let's take the limit
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection point
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
$$\left(-\infty, 0\right]$$
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(x/(sqrt(1 - x*x))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)$$
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:
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = - \operatorname{atan}{\left(\frac{x}{\sqrt{- x^{2} + 1}} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \operatorname{atan}{\left(\frac{x}{\sqrt{- x^{2} + 1}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = - \operatorname{atan}{\left(\frac{x}{\sqrt{- x^{2} + 1}} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = \operatorname{atan}{\left(\frac{x}{\sqrt{- x^{2} + 1}} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph