Mister Exam
Graphing y = 7*log(x)*atan(x)
The solution
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f(x) = 7*log(x)*atan(x)
$$f{\left(x \right)} = 7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
f = (7*log(x))*atan(x)
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:
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$\frac{7 \log{\left(x \right)}}{x^{2} + 1} + \frac{7 \operatorname{atan}{\left(x \right)}}{x} = 0$$
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
$$\frac{7 \log{\left(x \right)}}{x^{2} + 1} + \frac{7 \operatorname{atan}{\left(x \right)}}{x} = 0$$
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
$$7 \left(- \frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$7 \left(- \frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \infty$$
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 (7*log(x))*atan(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
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:
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = - 7 \log{\left(- x \right)} \operatorname{atan}{\left(x \right)}$$
- No
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = 7 \log{\left(- x \right)} \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = - 7 \log{\left(- x \right)} \operatorname{atan}{\left(x \right)}$$
- No
$$7 \log{\left(x \right)} \operatorname{atan}{\left(x \right)} = 7 \log{\left(- x \right)} \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd