Mister Exam
Graphing y = (cosx)/(sinx)-8
The solution
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cos(x)
f(x) = ------ - 8
sin(x)
$$f{\left(x \right)} = -8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$
f = -8 + cos(x)/sin(x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
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:
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(\frac{1}{8} \right)}$$
Numerical solution
$$x_{1} = -21.8667935805818$$
$$x_{2} = -43.8579421557103$$
$$x_{3} = -116.114573188276$$
$$x_{4} = -65.8490907308389$$
$$x_{5} = -87.8402393059674$$
so we need to solve the equation:
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(\frac{1}{8} \right)}$$
Numerical solution
$$x_{1} = -21.8667935805818$$
$$x_{2} = -43.8579421557103$$
$$x_{3} = -116.114573188276$$
$$x_{4} = -65.8490907308389$$
$$x_{5} = -87.8402393059674$$
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
$$-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0$$
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
$$-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0$$
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
$$\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 3.14159265358979^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}$$
$$\lim_{x \to 3.14159265358979^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}$$
- 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
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 3.14159265358979^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}$$
$$\lim_{x \to 3.14159265358979^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}$$
- 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
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left[\frac{\pi}{2}, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
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:
$$y = \lim_{x \to -\infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/sin(x) - 8, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)$$
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:
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = -8 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$
- No
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = -8 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$
- No
$$-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd