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cos(4*x)^3

  • How to use it?

  • Graphing y =:
  • x^3-((x^4)/4)
  • -x^3+3x+2
  • x^3-3x^2+6
  • (x^2-1)/(x-1)

  • Identical expressions

  • cos(four *x)^ three
  • co sinus of e of (4 multiply by x) cubed
  • co sinus of e of (four multiply by x) to the power of three
  • cos(4*x)3
  • cos4*x3
  • cos(4*x)³
  • cos(4*x) to the power of 3
  • cos(4x)^3
  • cos(4x)3
  • cos4x3
  • cos4x^3
Plotting a function graph/ cos(4*x)^3

Graphing y = cos(4*x)^3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          3     
f(x) = cos (4*x)
f(x)=cos⁡3(4x)f{\left(x \right)} = \cos^{3}{\left(4 x \right)} 
f = cos(4*x)^3
The graph of the function
0-50-40-30-20-1010203040502-2
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(4x)=0\cos^{3}{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π8x_{1} = \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}
Numerical solution
x1=53.7997853211765x_{1} = 53.7997853211765
x2=26.3108234823209x_{2} = 26.3108234823209
x3=−52.2289964711527x_{3} = -52.2289964711527
x4=5.89051323025993x_{4} = 5.89051323025993
x5=62.4391347445363x_{5} = 62.4391347445363
x6=82.07409161375x_{6} = 82.07409161375
x7=100.138273383671x_{7} = 100.138273383671
x8=−16.1006363484365x_{8} = -16.1006363484365
x9=−79.7179218360511x_{9} = -79.7179218360511
x10=40.4479922392116x_{10} = 40.4479922392116
x11=−12.1736465614658x_{11} = -12.1736465614658
x12=30.2378324690299x_{12} = 30.2378324690299
x13=−27.8816387571644x_{13} = -27.8816387571644
x14=−9.81746232653608x_{14} = -9.81746232653608
x15=12.1736868103765x_{15} = 12.1736868103765
x16=22.3838213254975x_{16} = 22.3838213254975
x17=34.1648351716409x_{17} = 34.1648351716409
x18=−93.8550699930765x_{18} = -93.8550699930765
x19=142596.497846589x_{19} = 142596.497846589
x20=52.2289833146706x_{20} = 52.2289833146706
x21=−20.027643859159x_{21} = -20.027643859159
x22=−65.5807721275388x_{22} = -65.5807721275388
x23=−90.7134911844256x_{23} = -90.7134911844256
x24=88.3572698819983x_{24} = 88.3572698819983
x25=61.6537436502142x_{25} = 61.6537436502142
x26=64.0099468805818x_{26} = 64.0099468805818
x27=−75.7909125593499x_{27} = -75.7909125593499
x28=4.31967302423842x_{28} = 4.31967302423842
x29=−1.96350706117559x_{29} = -1.96350706117559
x30=−38.0917861662877x_{30} = -38.0917861662877
x31=−21.5984763497616x_{31} = -21.5984763497616
x32=−35.735620188184x_{32} = -35.735620188184
x33=86.0010985105937x_{33} = 86.0010985105937
x34=−87.5719188066833x_{34} = -87.5719188066833
x35=23.9546585885569x_{35} = 23.9546585885569
x36=−23.9546576988554x_{36} = -23.9546576988554
x37=−67.9369573345193x_{37} = -67.9369573345193
x38=−34.1647934351719x_{38} = -34.1647934351719
x39=−96.2112783976881x_{39} = -96.2112783976881
x40=−69.5077450247164x_{40} = -69.5077450247164
x41=18.4568534820342x_{41} = 18.4568534820342
x42=1.96350744933497x_{42} = 1.96350744933497
x43=−31.8086120334863x_{43} = -31.8086120334863
x44=−25.5254155517343x_{44} = -25.5254155517343
x45=16.1006501839457x_{45} = 16.1006501839457
x46=97.7820559471952x_{46} = 97.7820559471952
x47=42.0187953414244x_{47} = 42.0187953414244
x48=67.9369602944665x_{48} = 67.9369602944665
x49=78.1471286142162x_{49} = 78.1471286142162
x50=−61.6537776443278x_{50} = -61.6537776443278
x51=−74.2201457084879x_{51} = -74.2201457084879
x52=−82.0740864891433x_{52} = -82.0740864891433
x53=70.293124683485x_{53} = 70.293124683485
x54=31.8086436305876x_{54} = 31.8086436305876
x55=60.0829436123968x_{55} = 60.0829436123968
x56=0.392672311431238x_{56} = 0.392672311431238
x57=−39.6626275514554x_{57} = -39.6626275514554
x58=−5.89049485527301x_{58} = -5.89049485527301
x59=9.81749932569655x_{59} = 9.81749932569655
x60=31.0232402626803x_{60} = 31.0232402626803
x61=96.2112847352841x_{61} = 96.2112847352841
x62=45.9458095411353x_{62} = 45.9458095411353
x63=84.4302796341733x_{63} = 84.4302796341733
x64=−53.7997621509517x_{64} = -53.7997621509517
x65=−45.9458078168629x_{65} = -45.9458078168629
x66=−86.0010985105934x_{66} = -86.0010985105934
x67=89.9281108391142x_{67} = 89.9281108391142
x68=−71.8639255328187x_{68} = -71.8639255328187
x69=−64.0099468783114x_{69} = -64.0099468783114
x70=38.0917964061703x_{70} = 38.0917964061703
x71=82.0741026637191x_{71} = 82.0741026637191
x72=−89.9281062023017x_{72} = -89.9281062023017
x73=−17.6714773242508x_{73} = -17.6714773242508
x74=−97.782063169724x_{74} = -97.782063169724
x75=66.3661201522979x_{75} = 66.3661201522979
x76=−60.0829362154557x_{76} = -60.0829362154557
x77=−47.5165805457209x_{77} = -47.5165805457209
x78=−13.7444693081471x_{78} = -13.7444693081471
x79=48.3019740390089x_{79} = 48.3019740390089
x80=20.027643990615x_{80} = 20.027643990615
x81=56.1559825151659x_{81} = 56.1559825151659
x82=−42.0187953127979x_{82} = -42.0187953127979
x83=−3.53428046316638x_{83} = -3.53428046316638
x84=8.2466815852306x_{84} = 8.2466815852306
x85=−83.6449275875978x_{85} = -83.6449275875978
x86=75.7909228991636x_{86} = 75.7909228991636
x87=44.3749706219384x_{87} = 44.3749706219384
x88=−57.7267710360857x_{88} = -57.7267710360857
x89=74.2201340857909x_{89} = 74.2201340857909
x90=−49.8727820349293x_{90} = -49.8727820349293
x91=92.2842754046216x_{91} = 92.2842754046216
x92=−43.5896245516526x_{92} = -43.5896245516526
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(4*x)^3.
cos⁡3(4⋅0)\cos^{3}{\left(4 \cdot 0 \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
−12sin⁡(4x)cos⁡2(4x)=0- 12 \sin{\left(4 x \right)} \cos^{2}{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π8x_{2} = \frac{\pi}{8}
x3=π4x_{3} = \frac{\pi}{4}
x4=3π8x_{4} = \frac{3 \pi}{8}
The values of the extrema at the points:
(0, 1)

 pi    
(--, 0)
 8     

 pi     
(--, -1)
 4      

 3*pi    
(----, 0)
  8


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=π4x_{1} = \frac{\pi}{4}
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π4,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{4}, \infty\right)
Increasing at intervals
[0,π4]\left[0, \frac{\pi}{4}\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
48⋅(2sin⁡2(4x)−cos⁡2(4x))cos⁡(4x)=048 \cdot \left(2 \sin^{2}{\left(4 x \right)} - \cos^{2}{\left(4 x \right)}\right) \cos{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=π8x_{1} = \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}
x3=−atan⁡(22)4x_{3} = - \frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{4}
x4=atan⁡(22)4x_{4} = \frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{4}

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
[3π8,∞)\left[\frac{3 \pi}{8}, \infty\right)
Convex at the intervals
(−∞,atan⁡(22)4]∪[π8,3π8]\left(-\infty, \frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{4}\right] \cup \left[\frac{\pi}{8}, \frac{3 \pi}{8}\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(4x)=⟨−1,1⟩\lim_{x \to -\infty} \cos^{3}{\left(4 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(4x)=⟨−1,1⟩\lim_{x \to \infty} \cos^{3}{\left(4 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(4*x)^3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡3(4x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{3}{\left(4 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡3(4x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{3}{\left(4 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(4x)=cos⁡3(4x)\cos^{3}{\left(4 x \right)} = \cos^{3}{\left(4 x \right)}
- Yes
cos⁡3(4x)=−cos⁡3(4x)\cos^{3}{\left(4 x \right)} = - \cos^{3}{\left(4 x \right)}
- No
so, the function
is
even
The graph
Graphing y = cos(4*x)^3
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