Mister Exam

Graphing y = arctanx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = atan(x)
f(x)=atan(x)f{\left(x \right)} = \operatorname{atan}{\left(x \right)}
f = atan(x)
The graph of the function
0.001.000.100.200.300.400.500.600.700.800.9002
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(x)=0\operatorname{atan}{\left(x \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).
atan(0)\operatorname{atan}{\left(0 \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
1x2+1=0\frac{1}{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
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
2x(x2+1)2=0- \frac{2 x}{\left(x^{2} + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{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
(,0]\left(-\infty, 0\right]
Convex at the intervals
[0,)\left[0, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limxatan(x)=π2\lim_{x \to -\infty} \operatorname{atan}{\left(x \right)} = - \frac{\pi}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=π2y = - \frac{\pi}{2}
limxatan(x)=π2\lim_{x \to \infty} \operatorname{atan}{\left(x \right)} = \frac{\pi}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=π2y = \frac{\pi}{2}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(x), divided by x at x->+oo and x ->-oo
limx(atan(x)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
limx(atan(x)x)=0\lim_{x \to \infty}\left(\frac{\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:
atan(x)=atan(x)\operatorname{atan}{\left(x \right)} = - \operatorname{atan}{\left(x \right)}
- No
atan(x)=atan(x)\operatorname{atan}{\left(x \right)} = \operatorname{atan}{\left(x \right)}
- Yes
so, the function
is
odd