Mister Exam
Graphing y = 4x/sqrt(x^2-1)-x/2
The solution
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4*x x
f(x) = ----------- - -
________ 2
/ 2
\/ x - 1
$$f{\left(x \right)} = - \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}$$
f = -x/2 + (4*x)/sqrt(x^2 - 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \sqrt{65}$$
$$x_{3} = \sqrt{65}$$
Numerical solution
$$x_{1} = 8.06225774829855$$
$$x_{2} = -8.06225774829855$$
$$x_{3} = 0$$
so we need to solve the equation:
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \sqrt{65}$$
$$x_{3} = \sqrt{65}$$
Numerical solution
$$x_{1} = 8.06225774829855$$
$$x_{2} = -8.06225774829855$$
$$x_{3} = 0$$
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
$$- \frac{4 x^{2}}{\left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{1}{2} + \frac{4}{\sqrt{x^{2} - 1}} = 0$$
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
$$- \frac{4 x^{2}}{\left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{1}{2} + \frac{4}{\sqrt{x^{2} - 1}} = 0$$
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
$$\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = - \infty i$$
$$\lim_{x \to 1^+}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection 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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = - \infty i$$
$$\lim_{x \to 1^+}\left(\frac{12 x \left(\frac{x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (4*x)/sqrt(x^2 - 1) - x/2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
$$\lim_{x \to -\infty}\left(\frac{- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
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:
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = \frac{x}{2} - \frac{4 x}{\sqrt{x^{2} - 1}}$$
- No
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = - \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = \frac{x}{2} - \frac{4 x}{\sqrt{x^{2} - 1}}$$
- No
$$- \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}} = - \frac{x}{2} + \frac{4 x}{\sqrt{x^{2} - 1}}$$
- No
so, the function
not is
neither even, nor odd