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  • x²-2x+8
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  • Identical expressions

  • arctan((two 0x)/(one hundred -x^2))
  • arc tangent of ((20x) divide by (100 minus x squared ))
  • arc tangent of ((two 0x) divide by (one hundred minus x squared ))
  • arctan((20x)/(100-x2))
  • arctan20x/100-x2
  • arctan((20x)/(100-x²))
  • arctan((20x)/(100-x to the power of 2))
  • arctan20x/100-x^2
  • arctan((20x) divide by (100-x^2))

  • Similar expressions

  • arctan((20x)/(100+x^2))
Plotting a function graph/ arctan((20x)/(100-x^2))

Graphing y = arctan((20x)/(100-x^2))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /  20*x  \
f(x) = atan|--------|
           |       2|
           \100 - x /
f(x)=atan⁡(20x100−x2)f{\left(x \right)} = \operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} 
f = atan((20*x)/(100 - x^2))
The domain of the function
The points at which the function is not precisely defined:
x1=−10x_{1} = -10
x2=10x_{2} = 10
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⁡(20x100−x2)=0\operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} = 0
Solve this equation
The points of intersection with the axis X:

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((20*x)/(100 - x^2)).
atan⁡(0⋅20100−02)\operatorname{atan}{\left(\frac{0 \cdot 20}{100 - 0^{2}} \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
40x2(100−x2)2+20100−x2400x2(100−x2)2+1=0\frac{\frac{40 x^{2}}{\left(100 - x^{2}\right)^{2}} + \frac{20}{100 - x^{2}}}{\frac{400 x^{2}}{\left(100 - x^{2}\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
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
40x(−4x2x2−100+3+400(2x2x2−100−1)2(x2−100)(400x2(x2−100)2+1))(x2−100)2(400x2(x2−100)2+1)=0\frac{40 x \left(- \frac{4 x^{2}}{x^{2} - 100} + 3 + \frac{400 \left(\frac{2 x^{2}}{x^{2} - 100} - 1\right)^{2}}{\left(x^{2} - 100\right) \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right)}{\left(x^{2} - 100\right)^{2} \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 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=−10x_{1} = -10
x2=10x_{2} = 10

lim⁡x→−10−(40x(−4x2x2−100+3+400(2x2x2−100−1)2(x2−100)(400x2(x2−100)2+1))(x2−100)2(400x2(x2−100)2+1))=−∞\lim_{x \to -10^-}\left(\frac{40 x \left(- \frac{4 x^{2}}{x^{2} - 100} + 3 + \frac{400 \left(\frac{2 x^{2}}{x^{2} - 100} - 1\right)^{2}}{\left(x^{2} - 100\right) \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right)}{\left(x^{2} - 100\right)^{2} \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right) = -\infty
lim⁡x→−10+(40x(−4x2x2−100+3+400(2x2x2−100−1)2(x2−100)(400x2(x2−100)2+1))(x2−100)2(400x2(x2−100)2+1))=∞\lim_{x \to -10^+}\left(\frac{40 x \left(- \frac{4 x^{2}}{x^{2} - 100} + 3 + \frac{400 \left(\frac{2 x^{2}}{x^{2} - 100} - 1\right)^{2}}{\left(x^{2} - 100\right) \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right)}{\left(x^{2} - 100\right)^{2} \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right) = \infty
- the limits are not equal, so
x1=−10x_{1} = -10
- is an inflection point
lim⁡x→10−(40x(−4x2x2−100+3+400(2x2x2−100−1)2(x2−100)(400x2(x2−100)2+1))(x2−100)2(400x2(x2−100)2+1))=−∞\lim_{x \to 10^-}\left(\frac{40 x \left(- \frac{4 x^{2}}{x^{2} - 100} + 3 + \frac{400 \left(\frac{2 x^{2}}{x^{2} - 100} - 1\right)^{2}}{\left(x^{2} - 100\right) \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right)}{\left(x^{2} - 100\right)^{2} \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right) = -\infty
lim⁡x→10+(40x(−4x2x2−100+3+400(2x2x2−100−1)2(x2−100)(400x2(x2−100)2+1))(x2−100)2(400x2(x2−100)2+1))=∞\lim_{x \to 10^+}\left(\frac{40 x \left(- \frac{4 x^{2}}{x^{2} - 100} + 3 + \frac{400 \left(\frac{2 x^{2}}{x^{2} - 100} - 1\right)^{2}}{\left(x^{2} - 100\right) \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right)}{\left(x^{2} - 100\right)^{2} \left(\frac{400 x^{2}}{\left(x^{2} - 100\right)^{2}} + 1\right)}\right) = \infty
- the limits are not equal, so
x2=10x_{2} = 10
- 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(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\right)
Vertical asymptotes
Have:
x1=−10x_{1} = -10
x2=10x_{2} = 10
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞atan⁡(20x100−x2)=0\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞atan⁡(20x100−x2)=0\lim_{x \to \infty} \operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan((20*x)/(100 - x^2)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡(20x100−x2)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡(20x100−x2)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \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:
atan⁡(20x100−x2)=−atan⁡(20x100−x2)\operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} = - \operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)}
- No
atan⁡(20x100−x2)=atan⁡(20x100−x2)\operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)} = \operatorname{atan}{\left(\frac{20 x}{100 - x^{2}} \right)}
- No
so, the function
not is
neither even, nor odd
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