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arctan(x/sqrt(1-x*x))

Graphing y = arctan(x/sqrt(1-x*x))

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The graph:

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Intersection points:

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Piecewise:

The solution

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           /     x     \
f(x) = atan|-----------|
           |  _________|
           \\/ 1 - x*x /
f(x)=atan(xxx+1)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
02468-8-6-4-2-10102.5-2.5
The domain of the function
The points at which the function is not precisely defined:
x1=1x_{1} = -1
x2=1x_{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:
atan(xxx+1)=0\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(x/(sqrt(1 - x*x))).
atan(000+1)\operatorname{atan}{\left(\frac{0}{\sqrt{- 0 \cdot 0 + 1}} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
x2(xx+1)32+1xx+1x2xx+1+1=0\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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
x(x2x2+1+1)(2x21+3x2+1)x2+1(x2x211)=0- \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
x1=0x_{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:
x1=1x_{1} = -1
x2=1x_{2} = 1

limx1(x(x2x2+1+1)(2x21+3x2+1)x2+1(x2x211))=i\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
limx1+(x(x2x2+1+1)(2x21+3x2+1)x2+1(x2x211))=\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
x1=1x_{1} = -1
- is an inflection point
limx1(x(x2x2+1+1)(2x21+3x2+1)x2+1(x2x211))=\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
limx1+(x(x2x2+1+1)(2x21+3x2+1)x2+1(x2x211))=i\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
x2=1x_{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
[0,)\left[0, \infty\right)
Convex at the intervals
(,0]\left(-\infty, 0\right]
Vertical asymptotes
Have:
x1=1x_{1} = -1
x2=1x_{2} = 1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limxatan(xxx+1)=limxatan(xxx+1)\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=limxatan(xxx+1)y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}
limxatan(xxx+1)=limxatan(xxx+1)\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=limxatan(xxx+1)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
limx(atan(xxx+1)x)=limx(atan(xxx+1)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)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlimx(atan(xxx+1)x)y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)}}{x}\right)
limx(atan(xxx+1)x)=limx(atan(xxx+1)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)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlimx(atan(xxx+1)x)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:
atan(xxx+1)=atan(xx2+1)\operatorname{atan}{\left(\frac{x}{\sqrt{- x x + 1}} \right)} = - \operatorname{atan}{\left(\frac{x}{\sqrt{- x^{2} + 1}} \right)}
- No
atan(xxx+1)=atan(xx2+1)\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
Graphing y = arctan(x/sqrt(1-x*x))