Mister Exam
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-
How to use it?
- Graphing y =:
- x^4-2x^2-3
- x^3-2x^2+x
- (x^2-4)/x
-
x^2-4x
-
Identical expressions
- five *cos(x)-(twenty-four /pi)*x+ nine
- 5 multiply by co sinus of e of (x) minus (24 divide by Pi ) multiply by x plus 9
- five multiply by co sinus of e of (x) minus (twenty minus four divide by Pi ) multiply by x plus nine
- 5cos(x)-(24/pi)x+9
- 5cosx-24/pix+9
- 5*cos(x)-(24 divide by pi)*x+9
-
Similar expressions
- 5*cos(x)+(24/pi)*x+9
- 5*cos(x)-(24/pi)*x-9
- 5*cosx-(24/pi)*x+9
Graphing y = 5*cos(x)-(24/pi)*x+9
The solution
You have entered
[src]
24
f(x) = 5*cos(x) - --*x + 9
pi
$$f{\left(x \right)} = \left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9$$
f = -24/pi*x + 5*cos(x) + 9
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:
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.33255497111503$$
so we need to solve the equation:
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.33255497111503$$
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
$$- 5 \sin{\left(x \right)} - \frac{24}{\pi} = 0$$
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
$$- 5 \sin{\left(x \right)} - \frac{24}{\pi} = 0$$
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
$$- 5 \cos{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
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
$$\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 5 \cos{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
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
$$\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9\right) = -\infty$$
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 5*cos(x) - 24/pi*x + 9, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9}{x}\right) = - \frac{24}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{24 x}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9}{x}\right) = - \frac{24}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{24 x}{\pi}$$
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9}{x}\right) = - \frac{24}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{24 x}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9}{x}\right) = - \frac{24}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{24 x}{\pi}$$
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:
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = \frac{24 x}{\pi} + 5 \cos{\left(x \right)} + 9$$
- No
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = - \frac{24 x}{\pi} - 5 \cos{\left(x \right)} - 9$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = \frac{24 x}{\pi} + 5 \cos{\left(x \right)} + 9$$
- No
$$\left(- \frac{24}{\pi} x + 5 \cos{\left(x \right)}\right) + 9 = - \frac{24 x}{\pi} - 5 \cos{\left(x \right)} - 9$$
- No
so, the function
not is
neither even, nor odd