Mister Exam
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How to use it?
- Graphing y =:
- x*|x|
-
y=|x-2|-|x+1|+x-2
- x|x|-|x|-6x
- x^2-2x+3
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Identical expressions
- (one / two)^(3cos^2x-4tgxctgx)
- (1 divide by 2) to the power of (3 co sinus of e of squared x minus 4tgxctgx)
- (one divide by two) to the power of (3 co sinus of e of squared x minus 4tgxctgx)
- (1/2)(3cos2x-4tgxctgx)
- 1/23cos2x-4tgxctgx
- (1/2)^(3cos²x-4tgxctgx)
- (1/2) to the power of (3cos to the power of 2x-4tgxctgx)
- 1/2^3cos^2x-4tgxctgx
- (1 divide by 2)^(3cos^2x-4tgxctgx)
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Similar expressions
- (1/2)^(3cos^2x+4tgxctgx)
Graphing y = (1/2)^(3cos^2x-4tgxctgx)
The solution
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2
- 3*cos (x) + 4*tan(x)*cot(x)
f(x) = 2
$$f{\left(x \right)} = \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
f = (1/2)^(3*cos(x)^2 - 4*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:
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- 2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(- 4 \tan^{2}{\left(x \right)} - 4\right) \cot{\left(x \right)} - 4 \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} - 6 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(2 \right)} = 0$$
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
$$- 2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(- 4 \tan^{2}{\left(x \right)} - 4\right) \cot{\left(x \right)} - 4 \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} - 6 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(2 \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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:
$$y = \lim_{x \to -\infty} \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1/2)^(3*cos(x)^2 - 4*tan(x)*cot(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2^{- 3 \cos^{2}{\left(x \right)} + 4 \tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
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:
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = - \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
$$\left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}} = - \left(\frac{1}{2}\right)^{3 \cos^{2}{\left(x \right)} - 4 \tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd