Mister Exam
Graphing y = (x-arctg(2x))/x
The solution
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x - atan(2*x)
f(x) = -------------
x
$$f{\left(x \right)} = \frac{x - \operatorname{atan}{\left(2 x \right)}}{x}$$
f = (x - atan(2*x))/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.16556118520721$$
$$x_{2} = -1.16556118520721$$
so we need to solve the equation:
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.16556118520721$$
$$x_{2} = -1.16556118520721$$
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
$$\frac{1 - \frac{2}{4 x^{2} + 1}}{x} - \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}} = 0$$
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
$$\frac{1 - \frac{2}{4 x^{2} + 1}}{x} - \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}} = 0$$
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
$$2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29020.249035495$$
$$x_{2} = -30584.1466713136$$
$$x_{3} = 15459.9279005977$$
$$x_{4} = 24782.4884954453$$
$$x_{5} = 23087.4114110973$$
$$x_{6} = -13633.8767787572$$
$$x_{7} = -33974.4198323863$$
$$x_{8} = -40755.0384320226$$
$$x_{9} = 25630.0336293753$$
$$x_{10} = 37495.9482372872$$
$$x_{11} = 31562.9392656591$$
$$x_{12} = -37364.7192890114$$
$$x_{13} = -35669.5667462001$$
$$x_{14} = -21261.135176979$$
$$x_{15} = -23803.7247300519$$
$$x_{16} = 17154.8496093912$$
$$x_{17} = -19566.1081803417$$
$$x_{18} = 18002.3310464122$$
$$x_{19} = -34821.99253386$$
$$x_{20} = 23934.9475985011$$
$$x_{21} = 16307.3811680521$$
$$x_{22} = 40038.6865345901$$
$$x_{23} = -39907.4570770544$$
$$x_{24} = -36517.1423633654$$
$$x_{25} = -29736.5835540329$$
$$x_{26} = 18849.8236987768$$
$$x_{27} = 19697.3260971377$$
$$x_{28} = 27325.1350142595$$
$$x_{29} = -17871.1161770063$$
$$x_{30} = 34953.2208565225$$
$$x_{31} = -18718.6072010202$$
$$x_{32} = 21392.3554347137$$
$$x_{33} = 42581.4337794408$$
$$x_{34} = 34105.6479144478$$
$$x_{35} = -42450.2039018375$$
$$x_{36} = 13765.0783402001$$
$$x_{37} = -41602.6207258237$$
$$x_{38} = -38212.2974354065$$
$$x_{39} = -24651.2649324947$$
$$x_{40} = 26477.5825889113$$
$$x_{41} = -14481.2873067429$$
$$x_{42} = -22956.1893162643$$
$$x_{43} = -20413.6178571567$$
$$x_{44} = 41733.8504719762$$
$$x_{45} = -17023.6366209311$$
$$x_{46} = 29867.8101058239$$
$$x_{47} = 39191.1060213421$$
$$x_{48} = 40886.2680383924$$
$$x_{49} = 22239.8804787742$$
$$x_{50} = -26346.3578342418$$
$$x_{51} = -28041.4648541161$$
$$x_{52} = -31431.7120267314$$
$$x_{53} = 33258.076581398$$
$$x_{54} = -28889.0228738707$$
$$x_{55} = 14612.4925008121$$
$$x_{56} = 30715.3735810553$$
$$x_{57} = -27193.9097461941$$
$$x_{58} = 20544.8370180755$$
$$x_{59} = -32279.2794423922$$
$$x_{60} = 28172.6905893722$$
$$x_{61} = -25498.8094406069$$
$$x_{62} = 38343.5265648352$$
$$x_{63} = -33126.8487587679$$
$$x_{64} = 36648.3711176498$$
$$x_{65} = -15328.7196680046$$
$$x_{66} = 32410.5069847427$$
$$x_{67} = -39059.8767224905$$
$$x_{68} = -16176.170368212$$
$$x_{69} = -22108.6592491612$$
$$x_{70} = 35800.7952924122$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right)\right) = \frac{16}{3}$$
$$\lim_{x \to 0^+}\left(2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right)\right) = \frac{16}{3}$$
- limits are equal, then skip the corresponding point
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29020.249035495$$
$$x_{2} = -30584.1466713136$$
$$x_{3} = 15459.9279005977$$
$$x_{4} = 24782.4884954453$$
$$x_{5} = 23087.4114110973$$
$$x_{6} = -13633.8767787572$$
$$x_{7} = -33974.4198323863$$
$$x_{8} = -40755.0384320226$$
$$x_{9} = 25630.0336293753$$
$$x_{10} = 37495.9482372872$$
$$x_{11} = 31562.9392656591$$
$$x_{12} = -37364.7192890114$$
$$x_{13} = -35669.5667462001$$
$$x_{14} = -21261.135176979$$
$$x_{15} = -23803.7247300519$$
$$x_{16} = 17154.8496093912$$
$$x_{17} = -19566.1081803417$$
$$x_{18} = 18002.3310464122$$
$$x_{19} = -34821.99253386$$
$$x_{20} = 23934.9475985011$$
$$x_{21} = 16307.3811680521$$
$$x_{22} = 40038.6865345901$$
$$x_{23} = -39907.4570770544$$
$$x_{24} = -36517.1423633654$$
$$x_{25} = -29736.5835540329$$
$$x_{26} = 18849.8236987768$$
$$x_{27} = 19697.3260971377$$
$$x_{28} = 27325.1350142595$$
$$x_{29} = -17871.1161770063$$
$$x_{30} = 34953.2208565225$$
$$x_{31} = -18718.6072010202$$
$$x_{32} = 21392.3554347137$$
$$x_{33} = 42581.4337794408$$
$$x_{34} = 34105.6479144478$$
$$x_{35} = -42450.2039018375$$
$$x_{36} = 13765.0783402001$$
$$x_{37} = -41602.6207258237$$
$$x_{38} = -38212.2974354065$$
$$x_{39} = -24651.2649324947$$
$$x_{40} = 26477.5825889113$$
$$x_{41} = -14481.2873067429$$
$$x_{42} = -22956.1893162643$$
$$x_{43} = -20413.6178571567$$
$$x_{44} = 41733.8504719762$$
$$x_{45} = -17023.6366209311$$
$$x_{46} = 29867.8101058239$$
$$x_{47} = 39191.1060213421$$
$$x_{48} = 40886.2680383924$$
$$x_{49} = 22239.8804787742$$
$$x_{50} = -26346.3578342418$$
$$x_{51} = -28041.4648541161$$
$$x_{52} = -31431.7120267314$$
$$x_{53} = 33258.076581398$$
$$x_{54} = -28889.0228738707$$
$$x_{55} = 14612.4925008121$$
$$x_{56} = 30715.3735810553$$
$$x_{57} = -27193.9097461941$$
$$x_{58} = 20544.8370180755$$
$$x_{59} = -32279.2794423922$$
$$x_{60} = 28172.6905893722$$
$$x_{61} = -25498.8094406069$$
$$x_{62} = 38343.5265648352$$
$$x_{63} = -33126.8487587679$$
$$x_{64} = 36648.3711176498$$
$$x_{65} = -15328.7196680046$$
$$x_{66} = 32410.5069847427$$
$$x_{67} = -39059.8767224905$$
$$x_{68} = -16176.170368212$$
$$x_{69} = -22108.6592491612$$
$$x_{70} = 35800.7952924122$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right)\right) = \frac{16}{3}$$
$$\lim_{x \to 0^+}\left(2 \left(\frac{8}{\left(4 x^{2} + 1\right)^{2}} - \frac{1 - \frac{2}{4 x^{2} + 1}}{x^{2}} + \frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{3}}\right)\right) = \frac{16}{3}$$
- limits are equal, then skip the corresponding point
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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x - atan(2*x))/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(2 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = - \frac{- x + \operatorname{atan}{\left(2 x \right)}}{x}$$
- No
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = \frac{- x + \operatorname{atan}{\left(2 x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = - \frac{- x + \operatorname{atan}{\left(2 x \right)}}{x}$$
- No
$$\frac{x - \operatorname{atan}{\left(2 x \right)}}{x} = \frac{- x + \operatorname{atan}{\left(2 x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd