Mister Exam
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-
How to use it?
- Graphing y =:
- (x+3)^3/(x+1)^2
- (x+2)/(x-1)
- x^2+6x+5
- x^2+5x-4
-
Identical expressions
- four *(cos(x)+ three *x)/ ten
- 4 multiply by ( co sinus of e of (x) plus 3 multiply by x) divide by 10
- four multiply by ( co sinus of e of (x) plus three multiply by x) divide by ten
- 4(cos(x)+3x)/10
- 4cosx+3x/10
- 4*(cos(x)+3*x) divide by 10
-
Similar expressions
- 4*(cos(x)-3*x)/10
- 4*(cosx+3*x)/10
Graphing y = 4*(cos(x)+3*x)/10
The solution
You have entered
[src]
4*(cos(x) + 3*x)
f(x) = ----------------
10
$$f{\left(x \right)} = \frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}$$
f = 4*(3*x + cos(x))/10
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:
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -0.316750828771221$$
so we need to solve the equation:
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -0.316750828771221$$
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
$$\frac{6}{5} - \frac{2 \sin{\left(x \right)}}{5} = 0$$
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
$$\frac{6}{5} - \frac{2 \sin{\left(x \right)}}{5} = 0$$
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
$$- \frac{2 \cos{\left(x \right)}}{5} = 0$$
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
$$- \frac{2 \cos{\left(x \right)}}{5} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\right) = \infty$$
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*(cos(x) + 3*x)/10, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{6 x}{5}$$
$$\lim_{x \to \infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{6 x}{5}$$
$$\lim_{x \to -\infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{6 x}{5}$$
$$\lim_{x \to \infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{6 x}{5}$$
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:
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = - \frac{6 x}{5} + \frac{2 \cos{\left(x \right)}}{5}$$
- No
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = \frac{6 x}{5} - \frac{2 \cos{\left(x \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = - \frac{6 x}{5} + \frac{2 \cos{\left(x \right)}}{5}$$
- No
$$\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = \frac{6 x}{5} - \frac{2 \cos{\left(x \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd
The graph