Mister Exam
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-
How to use it?
- Graphing y =:
-
x^3-3x+1
- y=2x^3+3x^2-4
- cos(x-pi/3)
- x^4/4
-
Identical expressions
- ctg(two *x)-x
- ctg(2 multiply by x) minus x
- ctg(two multiply by x) minus x
- ctg(2x)-x
- ctg2x-x
-
Similar expressions
- arctg(2x)-((x-1)^4)/5+(sin(5x))^2
- ctg(2*x)+x
Graphing y = ctg(2*x)-x
The solution
You have entered
[src]
f(x) = cot(2*x) - x
$$f{\left(x \right)} = - x + \cot{\left(2 x \right)}$$
f = -x + cot(2*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:
$$- x + \cot{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -9.47734088326452$$
$$x_{2} = 1.8217985837127$$
$$x_{3} = -1.8217985837127$$
$$x_{4} = -0.538436993155902$$
$$x_{5} = 0.538436993155902$$
$$x_{6} = -3.28916686636117$$
$$x_{7} = 6.36114938588332$$
so we need to solve the equation:
$$- x + \cot{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -9.47734088326452$$
$$x_{2} = 1.8217985837127$$
$$x_{3} = -1.8217985837127$$
$$x_{4} = -0.538436993155902$$
$$x_{5} = 0.538436993155902$$
$$x_{6} = -3.28916686636117$$
$$x_{7} = 6.36114938588332$$
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
$$- 2 \cot^{2}{\left(2 x \right)} - 3 = 0$$
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
$$- 2 \cot^{2}{\left(2 x \right)} - 3 = 0$$
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
$$8 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{4}$$
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
$$\left(-\infty, \frac{\pi}{4}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{4}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$8 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{4}$$
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
$$\left(-\infty, \frac{\pi}{4}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{4}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- x + \cot{\left(2 x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(- x + \cot{\left(2 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- x + \cot{\left(2 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(- x + \cot{\left(2 x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(2*x) - x, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- x + \cot{\left(2 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- x + \cot{\left(2 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{- x + \cot{\left(2 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{- x + \cot{\left(2 x \right)}}{x}\right)$$
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:
$$- x + \cot{\left(2 x \right)} = x - \cot{\left(2 x \right)}$$
- No
$$- x + \cot{\left(2 x \right)} = - x + \cot{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- x + \cot{\left(2 x \right)} = x - \cot{\left(2 x \right)}$$
- No
$$- x + \cot{\left(2 x \right)} = - x + \cot{\left(2 x \right)}$$
- No
so, the function
not is
neither even, nor odd