Mister Exam
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How to use it?
- Graphing y =:
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y=log0.7x
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tgx×ctgx
-
sqrt(3)sinx+cosx
- cos(ln(x))
-
Identical expressions
- tgx×ctgx
-
Similar expressions
- y=-x^2/2tg(x)×ctg(x)
Graphing y = tgx×ctgx
The solution
You have entered
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f(x) = tan(x)*cot(x)
$$f{\left(x \right)} = \tan{\left(x \right)} \cot{\left(x \right)}$$
f = tan(x)*cot(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:
$$\tan{\left(x \right)} \cot{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
so we need to solve the equation:
$$\tan{\left(x \right)} \cot{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 0$$
$$x_{2} = 1.5707963267949$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} = 0$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} = 0$$
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
$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) = 0$$
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
$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) = 0$$
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
$$\lim_{x \to -\infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\tan{\left(x \right)} \cot{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x)*cot(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)$$
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:
$$\tan{\left(x \right)} \cot{\left(x \right)} = \tan{\left(x \right)} \cot{\left(x \right)}$$
- No
$$\tan{\left(x \right)} \cot{\left(x \right)} = - \tan{\left(x \right)} \cot{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} \cot{\left(x \right)} = \tan{\left(x \right)} \cot{\left(x \right)}$$
- No
$$\tan{\left(x \right)} \cot{\left(x \right)} = - \tan{\left(x \right)} \cot{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph