Mister Exam
Graphing y = sin(3x/2)+tg(7x)
The solution
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/3*x\
f(x) = sin|---| + tan(7*x)
\ 2 /
$$f{\left(x \right)} = \sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}$$
f = sin((3*x)/2) + tan(7*x)
The graph of the function
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
$$\frac{3 \cos{\left(\frac{3 x}{2} \right)}}{2} + 7 \tan^{2}{\left(7 x \right)} + 7 = 0$$
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
$$\frac{3 \cos{\left(\frac{3 x}{2} \right)}}{2} + 7 \tan^{2}{\left(7 x \right)} + 7 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin((3*x)/2) + tan(7*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}}{x}\right)$$
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:
$$\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)} = - \sin{\left(\frac{3 x}{2} \right)} - \tan{\left(7 x \right)}$$
- No
$$\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)} = \sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)} = - \sin{\left(\frac{3 x}{2} \right)} - \tan{\left(7 x \right)}$$
- No
$$\sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)} = \sin{\left(\frac{3 x}{2} \right)} + \tan{\left(7 x \right)}$$
- No
so, the function
not is
neither even, nor odd