Mister Exam
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-
How to use it?
- Graphing y =:
- x^4-2x^2+1
- x^2+2
- x^4
-
y=-tg(x/3+pi/6)
-
Identical expressions
- y=-tg(x/ three +pi/ six)
- y equally minus tg(x divide by 3 plus Pi divide by 6)
- y equally minus tg(x divide by three plus Pi divide by six)
- y=-tgx/3+pi/6
- y=-tg(x divide by 3+pi divide by 6)
-
Similar expressions
- y=+tg(x/3+pi/6)
- y=-tg(x/3-pi/6)
Graphing y = y=-tg(x/3+pi/6)
The solution
You have entered
[src]
/x pi\
f(x) = -tan|- + --|
\3 6 /
$$f{\left(x \right)} = - \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$$
f = -tan(x/3 + pi/6)
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:
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 64.4026493985908$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = 73.8274273593601$$
$$x_{4} = -20.4203522483337$$
$$x_{5} = 102.101761241668$$
$$x_{6} = -10.9955742875643$$
$$x_{7} = 7.85398163397448$$
$$x_{8} = -67.5442420521806$$
$$x_{9} = -48.6946861306418$$
$$x_{10} = -76.9690200129499$$
$$x_{11} = -29.845130209103$$
$$x_{12} = 17.2787595947439$$
$$x_{13} = 45.553093477052$$
$$x_{14} = 83.2522053201295$$
$$x_{15} = -58.1194640914112$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = -86.3937979737193$$
$$x_{18} = 92.6769832808989$$
$$x_{19} = -95.8185759344887$$
$$x_{20} = 36.1283155162826$$
$$x_{21} = 26.7035375555132$$
$$x_{22} = -1.5707963267949$$
so we need to solve the equation:
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 64.4026493985908$$
$$x_{2} = 54.9778714378214$$
$$x_{3} = 73.8274273593601$$
$$x_{4} = -20.4203522483337$$
$$x_{5} = 102.101761241668$$
$$x_{6} = -10.9955742875643$$
$$x_{7} = 7.85398163397448$$
$$x_{8} = -67.5442420521806$$
$$x_{9} = -48.6946861306418$$
$$x_{10} = -76.9690200129499$$
$$x_{11} = -29.845130209103$$
$$x_{12} = 17.2787595947439$$
$$x_{13} = 45.553093477052$$
$$x_{14} = 83.2522053201295$$
$$x_{15} = -58.1194640914112$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = -86.3937979737193$$
$$x_{18} = 92.6769832808989$$
$$x_{19} = -95.8185759344887$$
$$x_{20} = 36.1283155162826$$
$$x_{21} = 26.7035375555132$$
$$x_{22} = -1.5707963267949$$
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
$$- \frac{\tan^{2}{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{3} - \frac{1}{3} = 0$$
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
$$- \frac{\tan^{2}{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{3} - \frac{1}{3} = 0$$
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
$$- \frac{2 \left(\tan^{2}{\left(\frac{2 x + \pi}{6} \right)} + 1\right) \tan{\left(\frac{2 x + \pi}{6} \right)}}{9} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
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
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(\tan^{2}{\left(\frac{2 x + \pi}{6} \right)} + 1\right) \tan{\left(\frac{2 x + \pi}{6} \right)}}{9} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
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
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -tan(x/3 + pi/6), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)$$
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:
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = - \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}$$
- No
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = - \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}$$
- No
$$- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph