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atan(x)^2

  • How to use it?

  • Graphing y =:
  • x√2-x
  • -x^2+5*x+4
  • x^2+4x+2
  • x^2-6x-7
  • Limit of the function:
  • atan(x)^2 atan(x)^2

  • Identical expressions

  • atan(x)^ two
  • arc tangent of gent of (x) squared
  • arc tangent of gent of (x) to the power of two
  • atan(x)2
  • atanx2
  • atan(x)²
  • atan(x) to the power of 2
  • atanx^2

  • Similar expressions

  • arctan(x)^2
  • arctanx^2
Plotting a function graph/ atan(x)^2

Graphing y = atan(x)^2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           2   
f(x) = atan (x)
f(x)=atan⁡2(x)f{\left(x \right)} = \operatorname{atan}^{2}{\left(x \right)} 
f = atan(x)^2
The graph of the function
02468-8-6-4-2-10100.02.5
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:
atan⁡2(x)=0\operatorname{atan}^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(x)^2.
atan⁡2(0)\operatorname{atan}^{2}{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
2atan⁡(x)x2+1=0\frac{2 \operatorname{atan}{\left(x \right)}}{x^{2} + 1} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(0, 0)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=0x_{1} = 0
The function has no maxima
Decreasing at intervals
[0,∞)\left[0, \infty\right)
Increasing at intervals
(−∞,0]\left(-\infty, 0\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(−2xatan⁡(x)+1)(x2+1)2=0\frac{2 \left(- 2 x \operatorname{atan}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−22962.2226401045x_{1} = -22962.2226401045
x2=−31436.1015983531x_{2} = -31436.1015983531
x3=−41605.9287947859x_{3} = -41605.9287947859
x4=−33978.4777406911x_{4} = -33978.4777406911
x5=26482.8037290628x_{5} = 26482.8037290628
x6=−24656.8778783254x_{6} = -24656.8778783254
x7=−32283.5525952437x_{7} = -32283.5525952437
x8=−39063.4019253875x_{8} = -39063.4019253875
x9=29872.4321264636x_{9} = 29872.4321264636
x10=−25504.2334265512x_{10} = -25504.2334265512
x11=−28893.8031518313x_{11} = -28893.8031518313
x12=18857.1946299244x_{12} = 18857.1946299244
x13=−35673.4300548609x_{13} = -35673.4300548609
x14=17162.962929885x_{14} = 17162.962929885
x15=−39910.9068025809x_{15} = -39910.9068025809
x16=33262.2228288295x_{16} = 33262.2228288295
x17=30719.8667071727x_{17} = 30719.8667071727
x18=−36520.9152075726x_{18} = -36520.9152075726
x19=25635.4296319425x_{19} = 25635.4296319425
x20=−28046.391296672x_{20} = -28046.391296672
x21=−29741.2260909927x_{21} = -29741.2260909927
x22=−42453.4454174628x_{22} = -42453.4454174628
x23=24788.0714796005x_{23} = 24788.0714796005
x24=21398.8367920726x_{24} = 21398.8367920726
x25=−18726.030446616x_{25} = -18726.030446616
x26=20551.5900405008x_{26} = 20551.5900405008
x27=0.765378926665789x_{27} = 0.765378926665789
x28=−34825.9507519323x_{28} = -34825.9507519323
x29=38347.1182450815x_{29} = 38347.1182450815
x30=−22114.9272950956x_{30} = -22114.9272950956
x31=−30588.6591830513x_{31} = -30588.6591830513
x32=23093.4101312038x_{32} = 23093.4101312038
x33=31567.3104913969x_{33} = 31567.3104913969
x34=40042.1249189752x_{34} = 40042.1249189752
x35=36652.130400909x_{35} = 36652.130400909
x36=34109.6901394521x_{36} = 34109.6901394521
x37=−33131.0115087019x_{37} = -33131.0115087019
x38=−23809.5402776119x_{38} = -23809.5402776119
x39=34957.164150976x_{39} = 34957.164150976
x40=−20420.4147645461x_{40} = -20420.4147645461
x41=27330.1922698542x_{41} = 27330.1922698542
x42=41737.1481106533x_{42} = 41737.1481106533
x43=22246.1111845123x_{43} = 22246.1111845123
x44=−37368.4058086007x_{44} = -37368.4058086007
x45=−26351.6051691354x_{45} = -26351.6051691354
x46=42584.6652795986x_{46} = 42584.6652795986
x47=−40758.4158445424x_{47} = -40758.4158445424
x48=35804.6443832452x_{48} = 35804.6443832452
x49=−17878.8979748856x_{49} = -17878.8979748856
x50=28177.5939359089x_{50} = 28177.5939359089
x51=19704.3745564126x_{51} = 19704.3745564126
x52=−21267.6569489348x_{52} = -21267.6569489348
x53=−17031.8133699325x_{53} = -17031.8133699325
x54=−27198.9915740665x_{54} = -27198.9915740665
x55=−38215.9014923937x_{55} = -38215.9014923937
x56=29025.007564543x_{56} = 29025.007564543
x57=39194.6193822603x_{57} = 39194.6193822603
x58=18010.055373674x_{58} = 18010.055373674
x59=32414.7627499852x_{59} = 32414.7627499852
x60=−19573.2044619574x_{60} = -19573.2044619574
x61=40889.6345794937x_{61} = 40889.6345794937
x62=23940.7309886273x_{62} = 23940.7309886273
x63=37499.6218083141x_{63} = 37499.6218083141

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
(−∞,0.765378926665789]\left(-\infty, 0.765378926665789\right]
Convex at the intervals
[0.765378926665789,∞)\left[0.765378926665789, \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→−∞atan⁡2(x)=π24\lim_{x \to -\infty} \operatorname{atan}^{2}{\left(x \right)} = \frac{\pi^{2}}{4}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=π24y = \frac{\pi^{2}}{4}
lim⁡x→∞atan⁡2(x)=π24\lim_{x \to \infty} \operatorname{atan}^{2}{\left(x \right)} = \frac{\pi^{2}}{4}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=π24y = \frac{\pi^{2}}{4}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{2}{\left(x \right)}}{x}\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:
atan⁡2(x)=atan⁡2(x)\operatorname{atan}^{2}{\left(x \right)} = \operatorname{atan}^{2}{\left(x \right)}
- Yes
atan⁡2(x)=−atan⁡2(x)\operatorname{atan}^{2}{\left(x \right)} = - \operatorname{atan}^{2}{\left(x \right)}
- No
so, the function
is
even
The graph
Graphing y = atan(x)^2
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