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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 55 "High School Modul
es > Trigonometry by Gregory A. Moore " }}{PARA 3 "" 0 "" {TEXT -1 4 "
    " }{TEXT 256 16 "The Cosine Graph" }}{PARA 0 "" 0 "" {TEXT -1 60 "
\nAn exploration of the many variations of the cosine graph.\n" }}
{PARA 0 "" 0 "" {TEXT 258 327 "[Directions : Execute the Code Resource
 section first. Although there will be no output immediately, these de
finitions are used later in this worksheet. Note that this worksheet i
s graphic intensive at times. If you experience problems it may be nec
essary to re-execute the \"0. Code\" section before executing each sub
section. ]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rest
art; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Soli
dPlot := proc(f,a,b)\nlocal c;\nc := COLOR(RGB, .4, .4, .8);\nplot( f(
x), x = a..b, filled = true, color=c, style=patchnogrid);\011\nend:\n
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "SolidPlotRCol := pro
c(f,a,b)\nlocal c;\nc := COLOR(RGB, 5*evalf(rand()/10^13,2), \n       \+
         5*evalf(rand()/10^13,2), \n                5*evalf(rand()/10^
13,2)  );\nplot( f(x), x = a..b, filled = true, color=c, style=patchno
grid);\011\011\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
174 "SolidPlotCol := proc(f,a,b, COL)\nlocal  box,i,n,x1,x2,xmid,delta
,y1,y2,A,B, m,M,slope, concav, col: \nplot( f(x), x = a..b, filled = t
rue, color=COL, style=patchnogrid);\011\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 965 "CosPlot := proc(f)\nlocal  A, B,G,start, endpt,
 speed, period, lift, amp, M,m : \nM := maximize(f(x), x = 0..10); m :
= minimize(f(x), x = 0..10);\nlift := (M + m)/2 ;\nG := f(x) - lift;\n
\n\nif (nops(G) = 2) then\n   amp := op(1,G);\n   if (amp > 0) then st
art := solve(G = M - lift, x);\n                else start := solve(G \+
= m - lift, x); fi;\n   speed := subs( x = 1, op(1, op(1, op(2,G))));
\n   period := 2*Pi/speed;\n   amp := evalf(subs(x = start, G)); \nels
e \n   amp := 1; \n   start := evalf(solve(G = 1, x));\n   speed := su
bs( x= 1, op(1,op(1,G))); \n   period := evalf(2*Pi/speed);\nfi;\nendp
t := start + period;\n\nplots[display](\n    SolidPlot( x->cos(x), 0, \+
2*Pi),\n    plot( f(x), x = start..endpt, color = blue, thickness = 2)
,\n    plot( \{ [[start,lift],[endpt,lift]],[[start,m],[endpt,m]],\n  \+
          [[start,M],[endpt,M]] ,[[start,m],[start,M]] ,[[endpt,m],[en
dpt,M]]\}  \n           ,color = red, thickness = 1, linestyle = 2), \+
\n    plot( 0, x = 0..2*Pi)  );\011\nend:\n\n\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 
"SinCosTable := proc(n)\nlocal i, A;\nA := array( [seq( [ 2*i*Pi/n,   \+
        evalf(sin(2*i*Pi/n)),` `, \n                    2*i*Pi/n - Pi/
2,    evalf(cos(2*i*Pi/n - Pi/2)) ],   i = 0..n ) ]):\nA[1,1] := theta
:  A[1,2] := sin(theta): A[1,4] := theta- Pi/2: A[1,5] := cos(theta):
\n\nprint(A);\nend: " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 " 1. The
 Basic Cosine Graph" }}{PARA 0 "" 0 "" {TEXT -1 138 "\nMuch like the s
in(x), y = cos(x) is also a function of x and can be plotted. The valu
es are different than those of the sine, of course.\n" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "cos( 3*Pi/7): % = evalf(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "n := 12;\narray( [seq( [ 2*i*Pi/n, \+
evalf(cos(2*i*Pi/n)) ],   i = 1..n ) ]);" }}}{PARA 0 "" 0 "" {TEXT -1 
44 "\nThis is the basic shape of the cosine graph" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "plot( cos(x), x = 0..2*Pi);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "SolidPlot( x-> cos(x) , 0, 2*Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "display( [seq(  \n  SolidPl
otCol( x-> (k/16)*cos(x),0,2*Pi,\n  COLOR(RGB, .35 - k/32, .35 - k/32,
 1 - k/28 )),k = 1..16) ]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 " 2. Connection with Sine Graph" }
}{PARA 0 "" 0 "" {TEXT -1 50 "\nThe sine and cosine graphs are closely
 related. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot( \{sin
(x), cos(x)\},x = 0..2*Pi, color = [red, blue], thickness = 2);\n`Red \+
is sine, Blue is cosine`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "plot( \{sin(x), cos(x)\},x = 0..4*Pi, color = [red, blue], thickne
ss = 2);" }}}{PARA 0 "" 0 "" {TEXT -1 197 " \nThey actually have the s
ame shape - with a horizontal shift. You can think of the cosine graph
 following the sine graph in a sense. Notice that sine and cosine take
 the same values - at x values " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"
\"\"\"#!\"\"" }{TEXT -1 7 " apart." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "sin(Pi/2); cos(0);" }}}{PARA 0 "" 0 "" {TEXT -1 75 "
\nThis is not only true for these two values, but for any two values o
f x. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "SinCosTable(12);
" }}}{PARA 0 "" 0 "" {TEXT -1 101 "\nThis explains the similarity in t
he graphs. Note that both the sine and cosine have the same period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 257 " and same
 amplitude (1). In a sense they are the same graph, starting from diff
erent values : sin starts at 0 while cosine starts at 1. If we think o
f it this way, then all of the factors which influence the sine graph \+
will work the same way on the cosine.\n" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 13 " 3. Amplitude" }}{PARA 0 "" 0 "" {TEXT -1 123 "\nThe ampl
itude of the cosine function is the height of its hill (and depth of i
ts valley). For cos(x), the amplitude is 1.\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 239 "display( plot(cos(x), x = 0..2*Pi, thickness = \+
2),\n         plot( \{[[0,0],[0,1]],[[2*Pi,0],[2*Pi,1]]\} , \n        \+
 color = blue, linestyle = 2, thickness = 3),\n         plot( [[Pi,0],
[Pi,-1]], color = yellow, linestyle = 2, thickness = 3));" }}}{PARA 0 
"" 0 "" {TEXT -1 94 "\nA constant in front of the cos(x) function alte
rs the amplitude just as the sine is altered.\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "f := x -> 1.5*cos(x);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "CosPlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 85 "
\n\nIf the constant is positive but less than 1, the graph contracts t
oward the x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "f := x -> (3/4)*cos(x);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 13 "CosPlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 
105 "\n\nIf the constant is negative, the stretching factor is the sam
e, but the cosine is flipped upside down.\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "f := x -> -3*cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "CosPlot( f );" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
22 " 4. Period & Frequency" }}{PARA 0 "" 0 "" {TEXT -1 144 "\n\nThe co
sine graph is periodic - it repeats. Each section that repeats is call
ed a \"period.\" The fundamental \"piece\" that repeats has length of \+
" }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 56 " - this
 is the period of the cosine. \n\nThe frequency is " }{XPPEDIT 18 0 "1
/(2*Pi)" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 75 " - that is,
 the cos(x) function completes 1 cycle every \"time\" interval of " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 65 " - or, ano
ther way to think of it is that the function completes " }{XPPEDIT 18 
0 "1/(2*Pi)" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 37 " of a p
eriod every one unit of time.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 250 "display( plot( cos(x), x=0..6*Pi, color = blue),\n         pl
ot( cos(x), x=0..2*Pi, color = red, thickness = 5),\n         plot( co
s(x), x=2*Pi..4*Pi, color = green, thickness = 5),\n         plot( cos
(x), x=4*Pi..6*Pi, color = violet, thickness = 5) );" }}}{PARA 0 "" 0 
"" {TEXT -1 223 "\n\nIf there is a constant times x inside of the cos(
x) function, this affects the period and frequency. For example, cos(2
x) will complete its cycles twice as fast - each one taking half as lo
ng. So the period of cos(2x) is " }{XPPEDIT 18 0 "2*Pi/2 = Pi" "6#/*(
\"\"#\"\"\"%#PiGF&F%!\"\"F'" }{TEXT -1 39 ", and the frequency is 1 ov
er that, or " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT 
-1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> cos(2
*x);\nCosPlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 167 "\nNote that the \+
horizonatal shifting is affected, but the amplitude remains 1. If the \+
constant is less than one, the graph has a smaller frequency and a lon
ger period.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := x -> c
os((1/2)*x);\nCosPlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 104 "\nIf the
 constant is negative, the stretching factor is the same, but the cosi
ne is flipped upside down.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "f := x -> -.3*cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "CosPlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 30 " 5. Amplitude & Period Changes" }}{PARA 
0 "" 0 "" {TEXT -1 229 "\nThe changes to period and amplitude can both
 operate indepently on a single cosine function. The period is a featu
re of the x coordiate and the amplitude is a feature of the y coordina
te, so neither has any impact on the other.\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 80 "f := x -> 5*cos(2*x);\n`Half the period, but 5 t
imes the amplitude`; \nCosPlot(f);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "f := x -> (1/4)*cos(.5*x); \n`Twice the period, and 1
/4 of the amplitude`; \nCosPlot(f);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "f := x -> 3*cos((1/2)*x); \n`Twice the period, and 3 \+
times the amplitude`; \nCosPlot(f);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n
 " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 " 6.  Shifting & Lifting" }}
{PARA 0 "" 0 "" {TEXT -1 235 "\nIn algebra we learn that y = f(x) and \+
(y-b) = f(x-a), or y = f(x-a) + b, have the same graphs except the lat
ter is shifted a units right, and b units up. The same concept applies
 to all other functions, including the cosine function.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( \{x^2, (x-3)^2 + 4\} , x = -3
..6, y = 0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f := x -
> cos(x - Pi/6) + 1 ;\nCosPlot(f);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "f := x -> cos(x -  Pi/2) - 2 ;\nCosPlot(f);" }}}
{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
38 " 7.  Shifting/Lifting/Amplitude/Period" }}{PARA 0 "" 0 "" {TEXT 
-1 87 "\nHere are some examples where all four transformation take pla
ce in the same function.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
150 "f := x -> 3*cos( 2*(x - Pi/12)) + 1;\n`Amplitude = 3, speed = 2, \+
period = Pi, vertical lift = 1, horizontal shift (phase shift) = Pi/12
`;\nCosPlot( f );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( \+
f(x), x = 0.. 8*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 16 "\nAnother exmapl
e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f := x -> 5*cos( (1/2)*
x + Pi) + 5;\nCosPlot( f );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "plot( f(x), x = 0..20*Pi );" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
33 " 8. Composite Functions of Cosine" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 137 "f := x -> abs(cos(x));\ndisplay(SolidPlot( x->cos(x)
, 0, 2*Pi),\n        plot(f(x), x = 0..2*Pi, thickness = 2, color = re
d, thickness=3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "f := \+
x -> cos(x)^2;\ndisplay(SolidPlot( x->cos(x), 0, 2*Pi),plot(f(x), x = \+
0..2*Pi, \n        color = red, thickness=3));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 119 "f := x -> cos(x)^3;\ndisplay(SolidPlot( x->co
s(x), 0, 2*Pi),plot(f(x), x = 0..2*Pi, \n        color = red, thicknes
s=3));" }}}{PARA 0 "" 0 "" {TEXT -1 62 "\nWe can look at families of s
imilar cosine functions.  Here's " }{XPPEDIT 18 0 "(k/10)*cos(x)" "6#*
(%\"kG\"\"\"\"#5!\"\"-%$cosG6#%\"xGF%" }{TEXT -1 27 " where k goes fro
m 1 to 10." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "k := 'k':\nplo
t( \{ (k/10)*cos(x) $ k = 1..10\}, x = 0..2*Pi);" }}}{EXCHG }{EXCHG }
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "And " }{XPPEDIT 18 0 "cos(x)^k" "6#
)-%$cosG6#%\"xG%\"kG" }{TEXT -1 12 " for k=1..10" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 44 "plot( \{ cos(x)^k $ k = 1..10\}, x = 0..2*Pi)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "display( [ seq(  \n  \+
SolidPlotCol(  x-> sqrt(k)*cos(x)^(2*k),0,2*Pi,\n  COLOR(RGB, .1 + k/2
5, .2 +  k/25, .4 +  k/25 )),k = 1..12),\n  plot( [sqrt(k)*cos(x)^(2*k
) $ k = 1..12],  x = 0..2*Pi, \n         color = gray, axes = none)]);
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "k + (k/10)*cos(k*x)" "6#,&%
\"kG\"\"\"*(F$F%\"#5!\"\"-%$cosG6#*&F$F%%\"xGF%F%F%" }{TEXT -1 14 " fo
r k = 1..10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot( \{ k +
 (k/10)*cos(k*x) $ k = 1..10\}, x = 0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 4 "And " }{XPPEDIT 18 0 "Sum( cos((2*j+1)*x)/(2*j+1), j= 1
..k)" "6#-%$SumG6$*&-%$cosG6#*&,&*&\"\"#\"\"\"%\"jGF.F.F.F.F.%\"xGF.F.
,&*&F-F.F/F.F.F.F.!\"\"/F/;F.%\"kG" }{TEXT -1 14 " for k = 1..10" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "display( [ seq(  \n  SolidP
lotCol(  x-> sum( cos((2*j+1)*x)/(2*j+1), j= 1..k),0,2*Pi,\n  COLOR(RG
B, .2 + k/25, .4 +  k/25, .1 +  k/25 )),k = 1..10),\n  plot( \{sum( co
s((2*j+1)*x)/(2*j+1), j= 1..k) $ k = 1..10\}, \n                    x \+
= 0..2*Pi, color = wheat, axes = none)]);" }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 39 " 9. Graphs of Sums of Sines and Cosines" }}{PARA 0 "" 
0 "" {TEXT -1 338 "\nThe last example above was not merely a nice draw
ing.  If we add up sines and cosines with differing amplitudes and per
iods, we get some extremely useful periodic functions.  Physicists and
 engineers often use sums of sines and cosines to model periodic elect
rical impulses, such as those seen on a heart monitor.  These sums are
 called " }{TEXT 261 14 "Fourier series" }{TEXT -1 68 ", which you wil
l learn about if you study mathematics in university." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 45 "Here's perhaps the simplest kind of example: " }{XPPEDIT 18 0 "
Sum(cos(k*x),k = 1 .. 15);" "6#-%$SumG6$-%$cosG6#*&%\"kG\"\"\"%\"xGF+/
F*;F+\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( sum( cos
(k*x) , k = 1..15 ), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Sum( cos(k*x)/k, k = 1..20)" "6#-%$SumG6$*&-%$cosG6#*&%
\"kG\"\"\"%\"xGF,F,F+!\"\"/F+;F,\"#?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "plot( sum( cos(k*x)/k, k = 1..20), x = 0..4*Pi);" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum( sin(x*k)/k, k = 1..20)" "6#
-%$SumG6$*&-%$sinG6#*&%\"xG\"\"\"%\"kGF,F,F-!\"\"/F-;F,\"#?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( sum( sin(x*k)/k, k = 1
..20), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum( si
n(x*k)/k - cos(x*k)/k, k = 1..20)" "6#-%$SumG6$,&*&-%$sinG6#*&%\"xG\"
\"\"%\"kGF-F-F.!\"\"F-*&-%$cosG6#*&F,F-F.F-F-F.F/F//F.;F-\"#?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot( sum( sin(x*k)/k - cos(
x*k)/k, k = 1..20), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
18 0 "Sum( sin(x*k)/k + ((-1)^k)*cos(x*k)/k, k = 1..20)" "6#-%$SumG6$,
&*&-%$sinG6#*&%\"xG\"\"\"%\"kGF-F-F.!\"\"F-*(),$F-F/F.F--%$cosG6#*&F,F
-F.F-F-F.F/F-/F.;F-\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "
plot( sum( sin(x*k)/k + ((-1)^k)*cos(x*k)/k, k = 1..20), x = 0..4*Pi);
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(sin(k*x)/(k^2)+cos(k*x)
/(k^2),k = 1 .. 10);" "6#-%$SumG6$,&*&-%$sinG6#*&%\"kG\"\"\"%\"xGF-F-*
$)F,\"\"#F-!\"\"F-*&-%$cosG6#*&F,F-F.F-F-*$F0F-F2F-/F,;F-\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot( sum( sin(x*k)/k^2 + co
s(x*k)/k^2, k = 1..10), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Sum( sin(k^2*x)/k^2 + ((-1)^k)*cos(k*x)/k^4, k = 1..10)
" "6#-%$SumG6$,&*&-%$sinG6#*&%\"kG\"\"#%\"xG\"\"\"F/*$F,F-!\"\"F/*(),$
F/F1F,F/-%$cosG6#*&F,F/F.F/F/*$F,\"\"%F1F//F,;F/\"#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 76 "plot( sum( sin(x*k^2)/k^2 + ((-1)^k)*cos(
x*k)/k^4, k = 1..10), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Sum( sin(2*k*x)/k - cos(x*(2*k+1))/k, k = 1..50)" "6#-%
$SumG6$,&*&-%$sinG6#*(\"\"#\"\"\"%\"kGF-%\"xGF-F-F.!\"\"F-*&-%$cosG6#*
&F/F-,&*&F,F-F.F-F-F-F-F-F-F.F0F0/F.;F-\"#]" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 71 "plot( sum( sin(x*(2*k))/k - cos(x*(2*k+1))/k, k = \+
1..50), x = 0..4*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum( s
in((3*k+1)*x)/(2*k+1)^2 - cos(3*k*x)/(2*k)^2, k = 1..40)" "6#-%$SumG6$
,&*&-%$sinG6#*&,&*&\"\"$\"\"\"%\"kGF/F/F/F/F/%\"xGF/F/*$,&*&\"\"#F/F0F
/F/F/F/F5!\"\"F/*&-%$cosG6#*(F.F/F0F/F1F/F/*$*&F5F/F0F/F5F6F6/F0;F/\"#
S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot( sum( sin(x*(3*k+
1))/(2*k+1)^2 - cos(x*(3*k))/(2*k)^2, k = 1..40), x = 0..4*Pi);" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum( sin(3*k*x)/k + cos(3*k*x)/k
, k = 1..40)" "6#-%$SumG6$,&*&-%$sinG6#*(\"\"$\"\"\"%\"kGF-%\"xGF-F-F.
!\"\"F-*&-%$cosG6#*(F,F-F.F-F/F-F-F.F0F-/F.;F-\"#S" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "plot( sum( sin(x*(3*k))/k + cos(x*(3*k))/k,
 k = 1..40), x = 0..4*Pi);" }}}{EXCHG }{EXCHG }{EXCHG }{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 259 34 " \n       \251 2002 W
aterloo Maple Inc" }}}{MARK "0 1" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }