Mister Exam
Graphing y = y=7cosx+sin6x−14x
The solution
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f(x) = 7*cos(x) + sin(6*x) - 14*x
$$f{\left(x \right)} = - 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)$$
f = -14*x + sin(6*x) + 7*cos(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:
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.468996049013888$$
so we need to solve the equation:
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.468996049013888$$
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
$$- 7 \sin{\left(x \right)} + 6 \cos{\left(6 x \right)} - 14 = 0$$
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
$$- 7 \sin{\left(x \right)} + 6 \cos{\left(6 x \right)} - 14 = 0$$
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
$$- (36 \sin{\left(6 x \right)} + 7 \cos{\left(x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- (36 \sin{\left(6 x \right)} + 7 \cos{\left(x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left[\frac{\pi}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)\right) = -\infty$$
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 7*cos(x) + sin(6*x) - 14*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)}{x}\right) = -14$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - 14 x$$
$$\lim_{x \to \infty}\left(\frac{- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)}{x}\right) = -14$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - 14 x$$
$$\lim_{x \to -\infty}\left(\frac{- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)}{x}\right) = -14$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - 14 x$$
$$\lim_{x \to \infty}\left(\frac{- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right)}{x}\right) = -14$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - 14 x$$
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:
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = 14 x - \sin{\left(6 x \right)} + 7 \cos{\left(x \right)}$$
- No
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = - 14 x + \sin{\left(6 x \right)} - 7 \cos{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = 14 x - \sin{\left(6 x \right)} + 7 \cos{\left(x \right)}$$
- No
$$- 14 x + \left(\sin{\left(6 x \right)} + 7 \cos{\left(x \right)}\right) = - 14 x + \sin{\left(6 x \right)} - 7 \cos{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd