Mister Exam
Graphing y = tg(4x)+4
The solution
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f(x) = tan(4*x) + 4
$$f{\left(x \right)} = \tan{\left(4 x \right)} + 4$$
f = tan(4*x) + 4
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:
$$\tan{\left(4 x \right)} + 4 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\operatorname{atan}{\left(4 \right)}}{4}$$
Numerical solution
$$x_{1} = 67.998185799661$$
$$x_{2} = -50.5969368733537$$
$$x_{3} = 52.290222531712$$
$$x_{4} = -53.7385295269435$$
$$x_{5} = -60.8071129975205$$
$$x_{6} = 46.0070372245324$$
$$x_{7} = 34.2260647735707$$
$$x_{8} = -31.7473809518149$$
$$x_{9} = 16.1619070154294$$
$$x_{10} = 38.153055590558$$
$$x_{11} = 23.2304904860064$$
$$x_{12} = 40.5092500807503$$
$$x_{13} = -9.75623237668639$$
$$x_{14} = 96.2725196819691$$
$$x_{15} = 44.4362408977375$$
$$x_{16} = 12.2349161984422$$
$$x_{17} = -18.3956121740583$$
$$x_{18} = -75.729678102072$$
so we need to solve the equation:
$$\tan{\left(4 x \right)} + 4 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\operatorname{atan}{\left(4 \right)}}{4}$$
Numerical solution
$$x_{1} = 67.998185799661$$
$$x_{2} = -50.5969368733537$$
$$x_{3} = 52.290222531712$$
$$x_{4} = -53.7385295269435$$
$$x_{5} = -60.8071129975205$$
$$x_{6} = 46.0070372245324$$
$$x_{7} = 34.2260647735707$$
$$x_{8} = -31.7473809518149$$
$$x_{9} = 16.1619070154294$$
$$x_{10} = 38.153055590558$$
$$x_{11} = 23.2304904860064$$
$$x_{12} = 40.5092500807503$$
$$x_{13} = -9.75623237668639$$
$$x_{14} = 96.2725196819691$$
$$x_{15} = 44.4362408977375$$
$$x_{16} = 12.2349161984422$$
$$x_{17} = -18.3956121740583$$
$$x_{18} = -75.729678102072$$
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
$$4 \tan^{2}{\left(4 x \right)} + 4 = 0$$
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
$$4 \tan^{2}{\left(4 x \right)} + 4 = 0$$
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
$$32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
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
$$32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
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:
$$y = \lim_{x \to -\infty}\left(\tan{\left(4 x \right)} + 4\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(4 x \right)} + 4\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(4 x \right)} + 4\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(4 x \right)} + 4\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(4*x) + 4, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)$$
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:
$$\tan{\left(4 x \right)} + 4 = 4 - \tan{\left(4 x \right)}$$
- No
$$\tan{\left(4 x \right)} + 4 = \tan{\left(4 x \right)} - 4$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(4 x \right)} + 4 = 4 - \tan{\left(4 x \right)}$$
- No
$$\tan{\left(4 x \right)} + 4 = \tan{\left(4 x \right)} - 4$$
- No
so, the function
not is
neither even, nor odd