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{SECT 0 {PARA 218 "" 0 "" {TEXT 216 62 "MAPLE WORKSHEET:  Guessing th
e Fourier Expansion of a function" }{TEXT 216 0 "" }}{EXCHG {PARA 211 
"> " 0 "" {MPLTEXT 1 215 8 "restart:" }{MPLTEXT 1 215 0 "" }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 218 "" 0 "" 
{TEXT 216 40 "Look at the function f(t) plotted below." }{TEXT 216 0 "
" }}{PARA 218 "" 0 "" {TEXT 216 69 "It is a linear combination of sin(
nx) for different values of n, i.e." }{TEXT 216 0 "" }}{PARA 218 "" 0 
"" {TEXT 216 0 "" }}{PARA 218 "" 0 "" {TEXT 216 51 "     f(t)=b1 sin(t
) + b2 sin(2t) + b3 sin(3t) + ..." }{TEXT 216 0 "" }{TEXT 216 1 "\n" }
}{PARA 218 "" 0 "" {TEXT 216 225 "Your job is to guess the values of t
he coefficients.  To make your job a little easier, I have put in only
 3 nonzero terms, all of the coefficients are positive or negative int
egers, and I have not included any terms for n>5." }{TEXT 216 0 "" }}
{PARA 218 "" 0 "" {TEXT 216 0 "" }}{PARA 218 "" 0 "" {TEXT 216 123 "Th
e function is defined below, but I've disguised it so you won't be abl
e to tell what the coefficients are by inspection! " }{TEXT 216 0 "" }
}{PARA 211 "> " 0 "" {MPLTEXT 1 215 37 "f(t):=sin(t)*4*(1-cos(t))-4*si
n(t)^3;" }{MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 215 33 "plot(f(t),t=
0..4*Pi,color=black);" }{MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 218 "" 0 "
" {TEXT 216 120 "You might want to start by plotting sin(nx) for vario
us values of n if you don't remember what this function looks like:" }
{TEXT 216 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 215 35 "plot(sin(t),t=
0..4*Pi,color=black);" }{MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 218 "" 0 "
" {TEXT 216 32 "Guess the most important term.  " }}{PARA 218 "" 0 "" 
{TEXT 216 56 "Plot your guess and the function f(x) on the same graph.
" }{TEXT 216 0 "" }{TEXT 216 1 "\n" }{TEXT 216 110 "Keep guessing more
 and more terms until the graph for your guess and the graph for f(x) \+
overlap.  For example:" }{TEXT 216 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT
 1 215 69 "myguess:=sin(t)-0*sin(2*t)+0*sin(3*t)+0*cos(t)+0*cos(2*t)+0
*cos(3*t);" }{MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 
216 94 "The unknown function is BLACK and your guess is RED, and the d
ifference between the 3 is BLUE." }{TEXT 216 0 "" }}{PARA 211 "> " 0 "
" {MPLTEXT 1 215 69 "plot([f(t), myguess,f(t)-myguess], t=0..2*Pi,colo
r=[black,red,blue]);" }{MPLTEXT 1 215 0 "" }}}{EXCHG {PARA 220 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 0 45 "in
tegrate((2/(2*Pi))*sin(t)*f(t),t=0..2*Pi);\n" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 82 "myguess:=1/2-2/(Pi*1)*sin(t)-(2/3)*sin(3*t)/Pi-(2
/5)*sin(5*t)/Pi-(2/7)*sin(21)/Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,#
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m)), 1), `*`(t))))), `*`(`+`(`*`(2, `*`(m)), 1), `*`(Pi)))), m = 0 .. \+
8)))], t = 0 .. `+`(`*`(4, `*`(Pi))), color = [black, red, blue]);" "-
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11 "" 1 "" {XPPMATH 20 "-I%listG%*protectedG6$-I$sinG6$F$I(_syslibG6\"
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{XPPEDIT 19 1 "Sum(`+`(`-`(`/`(`*`(2, `*`(sin(`*`(`+`(`*`(2, `*`(m)), \+
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