Mister Exam
Graphing y = x+3*tg(4x)-3
The solution
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f(x) = x + 3*tan(4*x) - 3
$$f{\left(x \right)} = \left(x + 3 \tan{\left(4 x \right)}\right) - 3$$
f = x + 3*tan(4*x) - 3
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:
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -5.19284053363664$$
$$x_{2} = 2.40513202522971$$
$$x_{3} = 5.33256618900014$$
$$x_{4} = 19.2877914006771$$
$$x_{5} = 4.59051578193643$$
$$x_{6} = -0.56749957611204$$
$$x_{7} = 14.5931706342245$$
$$x_{8} = 1.67478678235624$$
$$x_{9} = 13.8121322249454$$
$$x_{10} = 7.60561844029796$$
$$x_{11} = 0.188254358692314$$
$$x_{12} = -1.32958468375376$$
$$x_{13} = 1.67478678235626$$
$$x_{14} = 6.8416335551704$$
$$x_{15} = 10.6958008094204$$
$$x_{16} = 3.857396370948$$
$$x_{17} = -10.6569337598293$$
$$x_{18} = 0.936051923882515$$
$$x_{19} = 9.14561023500766$$
$$x_{20} = 12.2520603577693$$
$$x_{21} = 9.91974787536206$$
$$x_{22} = -2.09650141185502$$
$$x_{23} = 15.3747244427629$$
$$x_{24} = 6.08340828556129$$
$$x_{25} = -4.41579383455421$$
$$x_{26} = 3.13070726574642$$
$$x_{27} = -7.53066512399045$$
so we need to solve the equation:
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -5.19284053363664$$
$$x_{2} = 2.40513202522971$$
$$x_{3} = 5.33256618900014$$
$$x_{4} = 19.2877914006771$$
$$x_{5} = 4.59051578193643$$
$$x_{6} = -0.56749957611204$$
$$x_{7} = 14.5931706342245$$
$$x_{8} = 1.67478678235624$$
$$x_{9} = 13.8121322249454$$
$$x_{10} = 7.60561844029796$$
$$x_{11} = 0.188254358692314$$
$$x_{12} = -1.32958468375376$$
$$x_{13} = 1.67478678235626$$
$$x_{14} = 6.8416335551704$$
$$x_{15} = 10.6958008094204$$
$$x_{16} = 3.857396370948$$
$$x_{17} = -10.6569337598293$$
$$x_{18} = 0.936051923882515$$
$$x_{19} = 9.14561023500766$$
$$x_{20} = 12.2520603577693$$
$$x_{21} = 9.91974787536206$$
$$x_{22} = -2.09650141185502$$
$$x_{23} = 15.3747244427629$$
$$x_{24} = 6.08340828556129$$
$$x_{25} = -4.41579383455421$$
$$x_{26} = 3.13070726574642$$
$$x_{27} = -7.53066512399045$$
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
$$12 \tan^{2}{\left(4 x \right)} + 13 = 0$$
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
$$12 \tan^{2}{\left(4 x \right)} + 13 = 0$$
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
$$96 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$96 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
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:
$$y = \lim_{x \to -\infty}\left(\left(x + 3 \tan{\left(4 x \right)}\right) - 3\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(x + 3 \tan{\left(4 x \right)}\right) - 3\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(x + 3 \tan{\left(4 x \right)}\right) - 3\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(x + 3 \tan{\left(4 x \right)}\right) - 3\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x + 3*tan(4*x) - 3, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(x + 3 \tan{\left(4 x \right)}\right) - 3}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x + 3 \tan{\left(4 x \right)}\right) - 3}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(x + 3 \tan{\left(4 x \right)}\right) - 3}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(x + 3 \tan{\left(4 x \right)}\right) - 3}{x}\right)$$
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:
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = - x - 3 \tan{\left(4 x \right)} - 3$$
- No
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = x + 3 \tan{\left(4 x \right)} + 3$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = - x - 3 \tan{\left(4 x \right)} - 3$$
- No
$$\left(x + 3 \tan{\left(4 x \right)}\right) - 3 = x + 3 \tan{\left(4 x \right)} + 3$$
- No
so, the function
not is
neither even, nor odd