Mister Exam
Graphing y = sqrt(1-sin^2x/sin^2x)
The solution
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/ sin (x)
f(x) = / 1 - -------
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\/ sin (x)
$$f{\left(x \right)} = \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}$$
f = sqrt(-sin(x)^2/sin(x)^2 + 1)
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:
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
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:
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
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
$$\frac{- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}} = 0$$
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
$$\frac{- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}} = 0$$
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
$$\text{NaN} = 0$$
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
$$\text{NaN} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
$$\lim_{x \to -\infty} \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty} \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(1 - sin(x)^2/sin(x)^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}$$
- Yes
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = - \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}$$
- No
so, the function
is
even
So, check:
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}$$
- Yes
$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1} = - \sqrt{- \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1}$$
- No
so, the function
is
even