Mister Exam
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-
How to use it?
- Graphing y =:
- -x^4+6x^2-8
- x^2+6x-1
- 5x^2-3x+4
- 2x^3+3x^2
- Derivative of:
-
1/atan(x)
- Integral of d{x}:
- 1/atan(x)
- Limit of the function:
-
1/atan(x)
-
Identical expressions
- one /atan(x)
- 1 divide by arc tangent of gent of (x)
- one divide by arc tangent of gent of (x)
- 1/atanx
- 1 divide by atan(x)
-
Similar expressions
- 2*sin(pi/4+1)^2*cos(pi*x/3+1)/atan(x/3)
- 1/arctan(x)
- 1/arctanx
Graphing y = 1/atan(x)
The solution
You have entered
[src]
1
f(x) = -------
atan(x)
$$f{\left(x \right)} = \frac{1}{\operatorname{atan}{\left(x \right)}}$$
f = 1/atan(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:
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}^{2}{\left(x \right)}} = 0$$
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
$$- \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}^{2}{\left(x \right)}} = 0$$
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
$$\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -18319.8645144627$$
$$x_{2} = -11543.1715980259$$
$$x_{3} = -29336.0044218641$$
$$x_{4} = 32857.1649540246$$
$$x_{5} = 15062.2208891964$$
$$x_{6} = 13368.0996429442$$
$$x_{7} = 9134.4709178039$$
$$x_{8} = -42048.6824303037$$
$$x_{9} = 39637.2981730849$$
$$x_{10} = -32725.9482143302$$
$$x_{11} = 41332.366969216$$
$$x_{12} = -14083.9802104543$$
$$x_{13} = 34552.1728446784$$
$$x_{14} = -34420.9545622591$$
$$x_{15} = -27641.0752781853$$
$$x_{16} = -22556.5342926467$$
$$x_{17} = -41201.1442796829$$
$$x_{18} = -10696.4376916029$$
$$x_{19} = -17472.6070379068$$
$$x_{20} = -20861.8020386426$$
$$x_{21} = 19298.3384810428$$
$$x_{22} = -21709.1590786967$$
$$x_{23} = -19167.1518781024$$
$$x_{24} = -35268.464920553$$
$$x_{25} = -31878.4529769315$$
$$x_{26} = 28619.7473992733$$
$$x_{27} = 22687.7335199133$$
$$x_{28} = 12521.1451821705$$
$$x_{29} = -13236.9615739558$$
$$x_{30} = 30314.6944972281$$
$$x_{31} = 23535.1273249627$$
$$x_{32} = -25098.7509275295$$
$$x_{33} = 35399.6838935398$$
$$x_{34} = 40484.8311268636$$
$$x_{35} = 38789.7682948189$$
$$x_{36} = 16756.5557280193$$
$$x_{37} = -28488.5357664978$$
$$x_{38} = 20992.9957012548$$
$$x_{39} = -15778.2004179215$$
$$x_{40} = 11674.2812023086$$
$$x_{41} = 25229.9565261797$$
$$x_{42} = -9849.84634056459$$
$$x_{43} = -37811.020918257$$
$$x_{44} = 31162.1785415128$$
$$x_{45} = -9003.43567417919$$
$$x_{46} = 37942.2416954908$$
$$x_{47} = -36115.9796249468$$
$$x_{48} = 10827.5280095206$$
$$x_{49} = -26793.6237177463$$
$$x_{50} = 17603.7844984911$$
$$x_{51} = -39506.0763790627$$
$$x_{52} = 29467.2172519256$$
$$x_{53} = -25946.1819435246$$
$$x_{54} = 42179.905528052$$
$$x_{55} = -30183.4805675208$$
$$x_{56} = 15909.3657459457$$
$$x_{57} = 37094.7185968698$$
$$x_{58} = 33704.6664141607$$
$$x_{59} = 21840.3556823708$$
$$x_{60} = -23403.9257474756$$
$$x_{61} = -24251.331774895$$
$$x_{62} = 18451.0468537562$$
$$x_{63} = -36963.4983804708$$
$$x_{64} = -16625.3838854398$$
$$x_{65} = 24382.5354659295$$
$$x_{66} = 26924.8326135147$$
$$x_{67} = -31030.9635994648$$
$$x_{68} = -40353.6088711755$$
$$x_{69} = 26077.3892697666$$
$$x_{70} = -20014.4654249028$$
$$x_{71} = 14215.1289542549$$
$$x_{72} = -33573.4488743823$$
$$x_{73} = 9980.912474663$$
$$x_{74} = 32009.668853018$$
$$x_{75} = -14931.0631736726$$
$$x_{76} = 27772.2856041243$$
$$x_{77} = 20145.6557756236$$
$$x_{78} = 36247.1992411849$$
$$x_{79} = -12390.0199509551$$
$$x_{80} = -38658.5469927142$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection 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
$$\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -18319.8645144627$$
$$x_{2} = -11543.1715980259$$
$$x_{3} = -29336.0044218641$$
$$x_{4} = 32857.1649540246$$
$$x_{5} = 15062.2208891964$$
$$x_{6} = 13368.0996429442$$
$$x_{7} = 9134.4709178039$$
$$x_{8} = -42048.6824303037$$
$$x_{9} = 39637.2981730849$$
$$x_{10} = -32725.9482143302$$
$$x_{11} = 41332.366969216$$
$$x_{12} = -14083.9802104543$$
$$x_{13} = 34552.1728446784$$
$$x_{14} = -34420.9545622591$$
$$x_{15} = -27641.0752781853$$
$$x_{16} = -22556.5342926467$$
$$x_{17} = -41201.1442796829$$
$$x_{18} = -10696.4376916029$$
$$x_{19} = -17472.6070379068$$
$$x_{20} = -20861.8020386426$$
$$x_{21} = 19298.3384810428$$
$$x_{22} = -21709.1590786967$$
$$x_{23} = -19167.1518781024$$
$$x_{24} = -35268.464920553$$
$$x_{25} = -31878.4529769315$$
$$x_{26} = 28619.7473992733$$
$$x_{27} = 22687.7335199133$$
$$x_{28} = 12521.1451821705$$
$$x_{29} = -13236.9615739558$$
$$x_{30} = 30314.6944972281$$
$$x_{31} = 23535.1273249627$$
$$x_{32} = -25098.7509275295$$
$$x_{33} = 35399.6838935398$$
$$x_{34} = 40484.8311268636$$
$$x_{35} = 38789.7682948189$$
$$x_{36} = 16756.5557280193$$
$$x_{37} = -28488.5357664978$$
$$x_{38} = 20992.9957012548$$
$$x_{39} = -15778.2004179215$$
$$x_{40} = 11674.2812023086$$
$$x_{41} = 25229.9565261797$$
$$x_{42} = -9849.84634056459$$
$$x_{43} = -37811.020918257$$
$$x_{44} = 31162.1785415128$$
$$x_{45} = -9003.43567417919$$
$$x_{46} = 37942.2416954908$$
$$x_{47} = -36115.9796249468$$
$$x_{48} = 10827.5280095206$$
$$x_{49} = -26793.6237177463$$
$$x_{50} = 17603.7844984911$$
$$x_{51} = -39506.0763790627$$
$$x_{52} = 29467.2172519256$$
$$x_{53} = -25946.1819435246$$
$$x_{54} = 42179.905528052$$
$$x_{55} = -30183.4805675208$$
$$x_{56} = 15909.3657459457$$
$$x_{57} = 37094.7185968698$$
$$x_{58} = 33704.6664141607$$
$$x_{59} = 21840.3556823708$$
$$x_{60} = -23403.9257474756$$
$$x_{61} = -24251.331774895$$
$$x_{62} = 18451.0468537562$$
$$x_{63} = -36963.4983804708$$
$$x_{64} = -16625.3838854398$$
$$x_{65} = 24382.5354659295$$
$$x_{66} = 26924.8326135147$$
$$x_{67} = -31030.9635994648$$
$$x_{68} = -40353.6088711755$$
$$x_{69} = 26077.3892697666$$
$$x_{70} = -20014.4654249028$$
$$x_{71} = 14215.1289542549$$
$$x_{72} = -33573.4488743823$$
$$x_{73} = 9980.912474663$$
$$x_{74} = 32009.668853018$$
$$x_{75} = -14931.0631736726$$
$$x_{76} = 27772.2856041243$$
$$x_{77} = 20145.6557756236$$
$$x_{78} = 36247.1992411849$$
$$x_{79} = -12390.0199509551$$
$$x_{80} = -38658.5469927142$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(x + \frac{1}{\operatorname{atan}{\left(x \right)}}\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection 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
$$\lim_{x \to -\infty} \frac{1}{\operatorname{atan}{\left(x \right)}} = - \frac{2}{\pi}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{2}{\pi}$$
$$\lim_{x \to \infty} \frac{1}{\operatorname{atan}{\left(x \right)}} = \frac{2}{\pi}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{2}{\pi}$$
$$\lim_{x \to -\infty} \frac{1}{\operatorname{atan}{\left(x \right)}} = - \frac{2}{\pi}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{2}{\pi}$$
$$\lim_{x \to \infty} \frac{1}{\operatorname{atan}{\left(x \right)}} = \frac{2}{\pi}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{2}{\pi}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/atan(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{1}{x \operatorname{atan}{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1}{x \operatorname{atan}{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{1}{x \operatorname{atan}{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1}{x \operatorname{atan}{\left(x \right)}}\right) = 0$$
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:
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = - \frac{1}{\operatorname{atan}{\left(x \right)}}$$
- No
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = \frac{1}{\operatorname{atan}{\left(x \right)}}$$
- Yes
so, the function
is
odd
So, check:
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = - \frac{1}{\operatorname{atan}{\left(x \right)}}$$
- No
$$\frac{1}{\operatorname{atan}{\left(x \right)}} = \frac{1}{\operatorname{atan}{\left(x \right)}}$$
- Yes
so, the function
is
odd