Mister Exam
Graphing y = arctg((x-1)/x)
The solution
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/x - 1\
f(x) = atan|-----|
\ x /
$$f{\left(x \right)} = \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
f = atan((x - 1*1)/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = 0$$
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:
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = 0$$
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
$$\frac{\frac{1}{x} - \frac{x - 1}{x^{2}}}{1 + \frac{\left(x - 1\right)^{2}}{x^{2}}} = 0$$
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
$$\frac{\frac{1}{x} - \frac{x - 1}{x^{2}}}{1 + \frac{\left(x - 1\right)^{2}}{x^{2}}} = 0$$
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
$$- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right) = 2$$
Let's take the limit
$$\lim_{x \to 0^+}\left(- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right) = 2$$
Let's take the limit
- limits are equal, then skip the corresponding 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
$$\left(-\infty, \frac{1}{2}\right]$$
Convex at the intervals
$$\left[\frac{1}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right) = 2$$
Let's take the limit
$$\lim_{x \to 0^+}\left(- \frac{2 \cdot \left(1 - \frac{x - 1}{x}\right) \left(1 + \frac{\left(1 - \frac{x - 1}{x}\right) \left(x - 1\right)}{x \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right)}{x^{2} \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x^{2}}\right)}\right) = 2$$
Let's take the limit
- limits are equal, then skip the corresponding 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
$$\left(-\infty, \frac{1}{2}\right]$$
Convex at the intervals
$$\left[\frac{1}{2}, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x - 1}{x} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan((x - 1*1)/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x - 1}{x} \right)}}{x}\right)$$
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:
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = - \operatorname{atan}{\left(\frac{- x - 1}{x} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \operatorname{atan}{\left(\frac{- x - 1}{x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = - \operatorname{atan}{\left(\frac{- x - 1}{x} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{x - 1}{x} \right)} = \operatorname{atan}{\left(\frac{- x - 1}{x} \right)}$$
- No
so, the function
not is
neither even, nor odd