Mister Exam
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-
How to use it?
- Graphing y =:
- x^2-6
-
y=-tg(x/3+pi/6)
-
tgx+2
- x²
-
Identical expressions
- tgx+ two
- tgx plus 2
- tgx plus two
-
Similar expressions
- tgx-2
- tg(x+(2pi)/3)
- arctg(x+2)
- 0.05arctg(x)+2tg(x)/x^2+5
- arctg((x+2)/(x-3))
- f(x)=|x+2|/arctg(x+2)
Graphing y = tgx+2
The solution
You have entered
[src]
f(x) = tan(x) + 2
$$f{\left(x \right)} = \tan{\left(x \right)} + 2$$
f = tan(x) + 2
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:
$$\tan{\left(x \right)} + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(2 \right)}$$
Numerical solution
$$x_{1} = 14.6008145501549$$
$$x_{2} = 83.7158529291303$$
$$x_{3} = -29.3814826001022$$
$$x_{4} = 30.3087778181038$$
$$x_{5} = 2.0344439357957$$
$$x_{6} = 99.4238161970793$$
$$x_{7} = -19.9567046393328$$
$$x_{8} = -101.638113632667$$
$$x_{9} = -13.6735193321533$$
$$x_{10} = -4.24874137138388$$
$$x_{11} = -57.6558164824104$$
$$x_{12} = -117.346076900616$$
$$x_{13} = 39.7335557788732$$
$$x_{14} = -48.231038521641$$
$$x_{15} = -35.6646679072818$$
$$x_{16} = 74.2910749683609$$
$$x_{17} = -16.8151119857431$$
$$x_{18} = 61.7247043540018$$
$$x_{19} = -63.93900178959$$
$$x_{20} = -70.2221870967695$$
$$x_{21} = -26.2398899465124$$
$$x_{22} = -79.6469650575389$$
$$x_{23} = 68.0078896611814$$
$$x_{24} = 52.2999263932324$$
$$x_{25} = -7.39033402497368$$
$$x_{26} = -51.3726311752308$$
$$x_{27} = -126.770854861386$$
$$x_{28} = 36.5919631252834$$
$$x_{29} = -76.5053724039491$$
$$x_{30} = 80.5742602755405$$
$$x_{31} = 17.7424072037447$$
$$x_{32} = 24.0255925109243$$
$$x_{33} = 58.583111700412$$
$$x_{34} = 46.0167410860528$$
$$x_{35} = 96.2822235434895$$
$$x_{36} = -89.0717430183083$$
$$x_{37} = 89.9990382363099$$
$$x_{38} = -92.2133356718981$$
$$x_{39} = 8.31762924297529$$
$$x_{40} = -41.9478532144614$$
$$x_{41} = -85.9301503647185$$
so we need to solve the equation:
$$\tan{\left(x \right)} + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(2 \right)}$$
Numerical solution
$$x_{1} = 14.6008145501549$$
$$x_{2} = 83.7158529291303$$
$$x_{3} = -29.3814826001022$$
$$x_{4} = 30.3087778181038$$
$$x_{5} = 2.0344439357957$$
$$x_{6} = 99.4238161970793$$
$$x_{7} = -19.9567046393328$$
$$x_{8} = -101.638113632667$$
$$x_{9} = -13.6735193321533$$
$$x_{10} = -4.24874137138388$$
$$x_{11} = -57.6558164824104$$
$$x_{12} = -117.346076900616$$
$$x_{13} = 39.7335557788732$$
$$x_{14} = -48.231038521641$$
$$x_{15} = -35.6646679072818$$
$$x_{16} = 74.2910749683609$$
$$x_{17} = -16.8151119857431$$
$$x_{18} = 61.7247043540018$$
$$x_{19} = -63.93900178959$$
$$x_{20} = -70.2221870967695$$
$$x_{21} = -26.2398899465124$$
$$x_{22} = -79.6469650575389$$
$$x_{23} = 68.0078896611814$$
$$x_{24} = 52.2999263932324$$
$$x_{25} = -7.39033402497368$$
$$x_{26} = -51.3726311752308$$
$$x_{27} = -126.770854861386$$
$$x_{28} = 36.5919631252834$$
$$x_{29} = -76.5053724039491$$
$$x_{30} = 80.5742602755405$$
$$x_{31} = 17.7424072037447$$
$$x_{32} = 24.0255925109243$$
$$x_{33} = 58.583111700412$$
$$x_{34} = 46.0167410860528$$
$$x_{35} = 96.2822235434895$$
$$x_{36} = -89.0717430183083$$
$$x_{37} = 89.9990382363099$$
$$x_{38} = -92.2133356718981$$
$$x_{39} = 8.31762924297529$$
$$x_{40} = -41.9478532144614$$
$$x_{41} = -85.9301503647185$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
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
$$\lim_{x \to -\infty}\left(\tan{\left(x \right)} + 2\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\tan{\left(x \right)} + 2\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\tan{\left(x \right)} + 2\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\tan{\left(x \right)} + 2\right) = \left\langle -\infty, \infty\right\rangle$$
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) + 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)$$
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:
$$\tan{\left(x \right)} + 2 = - \tan{\left(x \right)} + 2$$
- No
$$\tan{\left(x \right)} + 2 = \tan{\left(x \right)} - 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} + 2 = - \tan{\left(x \right)} + 2$$
- No
$$\tan{\left(x \right)} + 2 = \tan{\left(x \right)} - 2$$
- No
so, the function
not is
neither even, nor odd
The graph