Mister Exam
Graphing y = 2^(-3ctgx)
The solution
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-3*cot(x) f(x) = 2
$$f{\left(x \right)} = 2^{- 3 \cot{\left(x \right)}}$$
f = 2^(-3*cot(x))
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:
$$2^{- 3 \cot{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 87.9993974142847$$
$$x_{2} = 66$$
$$x_{3} = 22$$
$$x_{4} = -6.21017480814802$$
$$x_{5} = 44$$
$$x_{6} = 37.7730044296482$$
$$x_{7} = 15.7814286003869$$
$$x_{8} = -28.2018082843184$$
$$x_{9} = 94.25$$
$$x_{10} = 59.7645549561229$$
$$x_{11} = -72.1851765301635$$
$$x_{12} = 81.7560822582677$$
$$x_{13} = -50.1934745414952$$
$$x_{14} = -94.1769174488304$$
so we need to solve the equation:
$$2^{- 3 \cot{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 87.9993974142847$$
$$x_{2} = 66$$
$$x_{3} = 22$$
$$x_{4} = -6.21017480814802$$
$$x_{5} = 44$$
$$x_{6} = 37.7730044296482$$
$$x_{7} = 15.7814286003869$$
$$x_{8} = -28.2018082843184$$
$$x_{9} = 94.25$$
$$x_{10} = 59.7645549561229$$
$$x_{11} = -72.1851765301635$$
$$x_{12} = 81.7560822582677$$
$$x_{13} = -50.1934745414952$$
$$x_{14} = -94.1769174488304$$
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
$$2^{- 3 \cot{\left(x \right)}} \left(3 \cot^{2}{\left(x \right)} + 3\right) \log{\left(2 \right)} = 0$$
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
$$2^{- 3 \cot{\left(x \right)}} \left(3 \cot^{2}{\left(x \right)} + 3\right) \log{\left(2 \right)} = 0$$
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
$$3 \cdot 2^{- 3 \cot{\left(x \right)}} \left(3 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} - 2 \cot{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} = 0$$
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
$$3 \cdot 2^{- 3 \cot{\left(x \right)}} \left(3 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} - 2 \cot{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} = 0$$
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
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} 2^{- 3 \cot{\left(x \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} 2^{- 3 \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} 2^{- 3 \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} 2^{- 3 \cot{\left(x \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2^(-3*cot(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2^{- 3 \cot{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2^{- 3 \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2^{- 3 \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2^{- 3 \cot{\left(x \right)}}}{x}\right)$$
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:
$$2^{- 3 \cot{\left(x \right)}} = 2^{3 \cot{\left(x \right)}}$$
- No
$$2^{- 3 \cot{\left(x \right)}} = - 2^{3 \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2^{- 3 \cot{\left(x \right)}} = 2^{3 \cot{\left(x \right)}}$$
- No
$$2^{- 3 \cot{\left(x \right)}} = - 2^{3 \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd