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  • How to use it?

  • Graphing y =:
  • -x^2+4x+5
  • x^2+3x+3
  • x^2-3x-4
  • x^2-2x-2
  • Limit of the function:
  • cos(2*x)^3 cos(2*x)^3
  • Derivative of:
  • cos(2*x)^3

  • Identical expressions

  • cos(two *x)^ three
  • co sinus of e of (2 multiply by x) cubed
  • co sinus of e of (two multiply by x) to the power of three
  • cos(2*x)3
  • cos2*x3
  • cos(2*x)³
  • cos(2*x) to the power of 3
  • cos(2x)^3
  • cos(2x)3
  • cos2x3
  • cos2x^3
Plotting a function graph/ cos(2*x)^3

Graphing y = cos(2*x)^3

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          3     
f(x) = cos (2*x)
f(x)=cos⁡3(2x)f{\left(x \right)} = \cos^{3}{\left(2 x \right)} 
f = cos(2*x)^3
The graph of the function
02468-8-6-4-2-10102-2
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:
cos⁡3(2x)=0\cos^{3}{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Numerical solution
x1=22.7765299742415x_{1} = 22.7765299742415
x2=−38.4844614652797x_{2} = -38.4844614652797
x3=40.0552901985337x_{3} = 40.0552901985337
x4=−60.4756065596791x_{4} = -60.4756065596791
x5=−55.7632661328603x_{5} = -55.7632661328603
x6=71.4712468772624x_{6} = 71.4712468772624
x7=49.4801084694813x_{7} = 49.4801084694813
x8=−62.0464394517635x_{8} = -62.0464394517635
x9=18.0641405401649x_{9} = 18.0641405401649
x10=82.4667599080185x_{10} = 82.4667599080185
x11=−40.0552881765552x_{11} = -40.0552881765552
x12=−99.7455679128678x_{12} = -99.7455679128678
x13=−65.1880051060871x_{13} = -65.1880051060871
x14=93.4623829732543x_{14} = 93.4623829732543
x15=0.785393273933224x_{15} = 0.785393273933224
x16=47.9093160339062x_{16} = 47.9093160339062
x17=38.4844600184581x_{17} = 38.4844600184581
x18=46.3384695953885x_{18} = 46.3384695953885
x19=−84.0375908161324x_{19} = -84.0375908161324
x20=−16.4933171765021x_{20} = -16.4933171765021
x21=−3.9270084103161x_{21} = -3.9270084103161
x22=25.9181651117762x_{22} = 25.9181651117762
x23=76.1836370585162x_{23} = 76.1836370585162
x24=44.7676689135041x_{24} = 44.7676689135041
x25=66.758809738519x_{25} = 66.758809738519
x26=−33.7721152742823x_{26} = -33.7721152742823
x27=−54.19242365084x_{27} = -54.19242365084
x28=−43.1968731975691x_{28} = -43.1968731975691
x29=−47.9093039196703x_{29} = -47.9093039196703
x30=−18.0641369915069x_{30} = -18.0641369915069
x31=91.891617631336x_{31} = 91.891617631336
x32=3.92701410404678x_{32} = 3.92701410404678
x33=10.2101862358417x_{33} = 10.2101862358417
x34=−11.7809644712633x_{34} = -11.7809644712633
x35=−5.4978396683363x_{35} = -5.4978396683363
x36=68.3296202261614x_{36} = 68.3296202261614
x37=24.3473190031204x_{37} = 24.3473190031204
x38=−69.9004501993631x_{38} = -69.9004501993631
x39=−27.4889894320297x_{39} = -27.4889894320297
x40=−32.2012736277405x_{40} = -32.2012736277405
x41=27.4889680868262x_{41} = 27.4889680868262
x42=11.7810262246701x_{42} = 11.7810262246701
x43=90.320770894749x_{43} = 90.320770894749
x44=2.35616844993271x_{44} = 2.35616844993271
x45=−76.183573738864x_{45} = -76.183573738864
x46=−98.1747238922947x_{46} = -98.1747238922947
x47=−96.603943131047x_{47} = -96.603943131047
x48=62.046440478226x_{48} = 62.046440478226
x49=−41.626132537197x_{49} = -41.626132537197
x50=−10.2101236692732x_{50} = -10.2101236692732
x51=88.7499520940251x_{51} = 88.7499520940251
x52=−77.7544170189269x_{52} = -77.7544170189269
x53=−25.9181565927284x_{53} = -25.9181565927284
x54=−21.2057409948195x_{54} = -21.2057409948195
x55=−91.8915951676306x_{55} = -91.8915951676306
x56=−19.6349820636656x_{56} = -19.6349820636656
x57=−33.7721254392715x_{57} = -33.7721254392715
x58=84.0375912615573x_{58} = 84.0375912615573
x59=−11.7809638202413x_{59} = -11.7809638202413
x60=54.1924871034185x_{60} = 54.1924871034185
x61=5.49782609634756x_{61} = 5.49782609634756
x62=60.4756099385565x_{62} = 60.4756099385565
x63=32.2013367887941x_{63} = 32.2013367887941
x64=−63.6172829701901x_{64} = -63.6172829701901
x65=16.4933101490183x_{65} = 16.4933101490183
x66=69.900466872915x_{66} = 69.900466872915
x67=8.63933453425596x_{67} = 8.63933453425596
x68=98.1747864821732x_{68} = 98.1747864821732
x69=−85.6084333621604x_{69} = -85.6084333621604
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(2*x)^3.
cos⁡3(0⋅2)\cos^{3}{\left(0 \cdot 2 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−6sin⁡(2x)cos⁡2(2x)=0- 6 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π4x_{2} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
The values of the extrema at the points:
(0, 1)

 -pi     
(----, 0)
  4      

 pi    
(--, 0)
 4


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:
The function has no minima
Maxima of the function at points:
x3=0x_{3} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \infty\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
12(2sin⁡2(2x)−cos⁡2(2x))cos⁡(2x)=012 \left(2 \sin^{2}{\left(2 x \right)} - \cos^{2}{\left(2 x \right)}\right) \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
x3=−atan⁡(5−26)x_{3} = - \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x4=atan⁡(5−26)x_{4} = \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x5=−atan⁡(26+5)x_{5} = - \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
x6=atan⁡(26+5)x_{6} = \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}

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
[atan⁡(26+5),∞)\left[\operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}, \infty\right)
Convex at the intervals
(−∞,−π4]\left(-\infty, - \frac{\pi}{4}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cos⁡3(2x)=⟨−1,1⟩\lim_{x \to -\infty} \cos^{3}{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡3(2x)=⟨−1,1⟩\lim_{x \to \infty} \cos^{3}{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(2*x)^3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡3(2x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{3}{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡3(2x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{3}{\left(2 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:
cos⁡3(2x)=cos⁡3(2x)\cos^{3}{\left(2 x \right)} = \cos^{3}{\left(2 x \right)}
- Yes
cos⁡3(2x)=−cos⁡3(2x)\cos^{3}{\left(2 x \right)} = - \cos^{3}{\left(2 x \right)}
- No
so, the function
is
even
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