Mister Exam
Graphing y = 2*sin(x)+10*x
The solution
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f(x) = 2*sin(x) + 10*x
$$f{\left(x \right)} = 10 x + 2 \sin{\left(x \right)}$$
f = 10*x + 2*sin(x)
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:
$$10 x + 2 \sin{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$10 x + 2 \sin{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$2 \cos{\left(x \right)} + 10 = 0$$
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
$$2 \cos{\left(x \right)} + 10 = 0$$
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
$$- 2 \sin{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
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
$$\left(-\infty, 0\right] \cup \left[\pi, \infty\right)$$
Convex at the intervals
$$\left[0, \pi\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \sin{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
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
$$\left(-\infty, 0\right] \cup \left[\pi, \infty\right)$$
Convex at the intervals
$$\left[0, \pi\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(10 x + 2 \sin{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(10 x + 2 \sin{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(10 x + 2 \sin{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(10 x + 2 \sin{\left(x \right)}\right) = \infty$$
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 2*sin(x) + 10*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{10 x + 2 \sin{\left(x \right)}}{x}\right) = 10$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 10 x$$
$$\lim_{x \to \infty}\left(\frac{10 x + 2 \sin{\left(x \right)}}{x}\right) = 10$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 10 x$$
$$\lim_{x \to -\infty}\left(\frac{10 x + 2 \sin{\left(x \right)}}{x}\right) = 10$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 10 x$$
$$\lim_{x \to \infty}\left(\frac{10 x + 2 \sin{\left(x \right)}}{x}\right) = 10$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 10 x$$
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:
$$10 x + 2 \sin{\left(x \right)} = - 10 x - 2 \sin{\left(x \right)}$$
- No
$$10 x + 2 \sin{\left(x \right)} = 10 x + 2 \sin{\left(x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$10 x + 2 \sin{\left(x \right)} = - 10 x - 2 \sin{\left(x \right)}$$
- No
$$10 x + 2 \sin{\left(x \right)} = 10 x + 2 \sin{\left(x \right)}$$
- Yes
so, the function
is
odd