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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 49 "APPLICATIONS OF INTEGRATION - F
OURIER POLYNOMIALS" }}{PARA 4 "" 0 "" {TEXT -1 19 "Calculus II Project
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 256 
11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 80 "To graph and integrate
  sin mx  and  cos nx  and their products and powers on [ " }{XPPEDIT 
18 0 "-Pi,Pi;" "6$,$%#PiG!\"\"F$" }{TEXT -1 25 "  ] for  m, n = 1, 2, \+
..." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}{TEXT 257 79 "To apply integration to approximate a given function us
ing Fourier polynomials." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT 258 110 "To use Maple to evaluate the Fourier coefficient
s of  f(x) and to plot  f(x) and a set of Fourier polynomials." }
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 
0 "" {TEXT -1 1 " " }{TEXT 261 34 "Information on Fourier Polynomials
" }}{PARA 5 "" 0 "" {TEXT 296 65 "Just as functions under some restric
tions can be approximated by " }}{PARA 0 "" 0 "" {TEXT -1 19 "Taylor p
olynomials " }{TEXT 297 7 "locally" }{TEXT -1 41 ", meaning in an inte
rval around a point, " }}{PARA 0 "" 0 "" {TEXT -1 62 "periodic functio
ns can be approximated by Fourier polynomials " }{TEXT 298 8 "globally
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "   f(x) ~ " }
{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 
18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 9 " cos x + " }{XPPEDIT 18 
0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 9 "cos 2x + " }{XPPEDIT 18 0 "a[
3];" "6#&%\"aG6#\"\"$" }{TEXT -1 10 " cos 3x + " }{XPPEDIT 18 0 "a[4];
" "6#&%\"aG6#\"\"%" }{TEXT -1 15 " cos 4x + \205+   " }{XPPEDIT 18 0 "
a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 7 " cos nx" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "            + " }{XPPEDIT 18 0 "b[1];" "6#&%\"bG6#\"\"\"
" }{TEXT -1 9 " sin x + " }{XPPEDIT 18 0 "b[2];" "6#&%\"bG6#\"\"#" }
{TEXT -1 9 "sin 2x + " }{XPPEDIT 18 0 "b[3];" "6#&%\"bG6#\"\"$" }
{TEXT -1 10 " sin 3x + " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }
{TEXT -1 15 " sin 4x + \205+   " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"
nG" }{TEXT -1 8 " sin nx," }}{PARA 0 "" 0 "" {TEXT -1 55 "where we ass
ume the period of the function  f   to be 2" }{XPPEDIT 18 0 "Pi;" "6#%
#PiG" }{TEXT -1 24 ".  We consider  f  over " }}{PARA 0 "" 0 "" {TEXT 
-1 16 "the interval  [-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ",
 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 "].   " }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 
11 "=    (1/(2*" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 "))" }
{XPPEDIT 18 0 "int(f(x),x = -Pi .. Pi);" "6#-%$intG6$-%\"fG6#%\"xG/F);
,$%#PiG!\"\"F-" }{TEXT -1 4 ";   " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[k];" "6#&%\"aG6#%\"kG" }{TEXT -1 9 " =    (1/" }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "int(f(x)
*cos(k*x),x = -Pi .. Pi);" "6#-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#*
&%\"kGF+F*F+F+/F*;,$%#PiG!\"\"F4" }{TEXT -1 4 "   (" }{TEXT 279 1 "k" 
}{TEXT -1 5 " > 0)" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "
b[k];" "6#&%\"bG6#%\"kG" }{TEXT -1 9 " =    (1/" }{XPPEDIT 18 0 "Pi;" 
"6#%#PiG" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "int(f(x)*sin(k*x),x = -Pi ..
 Pi);" "6#-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&%\"kGF+F*F+F+/F*;,$
%#PiG!\"\"F4" }{TEXT -1 3 "  (" }{TEXT 280 1 "k" }{TEXT -1 5 " > 0)" }
}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#
\"\"!" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "a[k];" "6#&%\"aG6#%\"kG" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6#%\"kG" }{TEXT -1 
56 "  are called Fourier coefficients for the function f(x)." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "These results a
rise from the following facts." }}{PARA 0 "" 0 "" {TEXT -1 9 "   1.   \+
 " }{XPPEDIT 18 0 "int(cos(k*x),x = -Pi .. Pi) = 0;" "6#/-%$intG6$-%$c
osG6#*&%\"kG\"\"\"%\"xGF,/F-;,$%#PiG!\"\"F1\"\"!" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "   2.     " }{XPPEDIT 18 0 "int(sin(k*x),x = -Pi .. Pi) =
 0;" "6#/-%$intG6$-%$sinG6#*&%\"kG\"\"\"%\"xGF,/F-;,$%#PiG!\"\"F1\"\"!
" }{TEXT -1 11 "           " }}{PARA 0 "" 0 "" {TEXT -1 9 "    3.   " 
}{XPPEDIT 18 0 "int(cos(k*x)*sin(m*x),x = -Pi .. Pi) = 0;" "6#/-%$intG
6$*&-%$cosG6#*&%\"kG\"\"\"%\"xGF-F--%$sinG6#*&%\"mGF-F.F-F-/F.;,$%#PiG
!\"\"F7\"\"!" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 12 "    4.
      " }{XPPEDIT 18 0 "int(cos(k*x)*cos(m*x),x = -Pi .. Pi) = 0;" "6#
/-%$intG6$*&-%$cosG6#*&%\"kG\"\"\"%\"xGF-F--F)6#*&%\"mGF-F.F-F-/F.;,$%
#PiG!\"\"F6\"\"!" }{TEXT -1 8 "    if  " }{TEXT 299 1 "k" }{TEXT -1 5 
" and " }{TEXT 300 1 "m" }{TEXT -1 17 " are not equal.  " }}{PARA 0 "
" 0 "" {TEXT -1 10 "     5.   " }{XPPEDIT 18 0 "int(\{cos(k*x)\}^2,x =
 -Pi .. Pi) = Pi;" "6#/-%$intG6$*$<#-%$cosG6#*&%\"kG\"\"\"%\"xGF.\"\"#
/F/;,$%#PiG!\"\"F4F4" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 9 "
     6.  " }{XPPEDIT 18 0 "int(sin(k*x)*sin(m*x),x = -Pi .. Pi) = 0;" 
"6#/-%$intG6$*&-%$sinG6#*&%\"kG\"\"\"%\"xGF-F--F)6#*&%\"mGF-F.F-F-/F.;
,$%#PiG!\"\"F6\"\"!" }{TEXT -1 8 "    if  " }{TEXT 301 1 "k" }{TEXT 
-1 5 " and " }{TEXT 302 2 " m" }{TEXT -1 17 " are not equal.  " }}
{PARA 0 "" 0 "" {TEXT -1 9 "     7.  " }{XPPEDIT 18 0 "int(\{sin(k*x)
\}^2,x = -Pi .. Pi) = Pi;" "6#/-%$intG6$*$<#-%$sinG6#*&%\"kG\"\"\"%\"x
GF.\"\"#/F/;,$%#PiG!\"\"F4F4" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" 
{TEXT -1 42 "If we multiply both sides of the equation " }}{PARA 0 "" 
0 "" {TEXT -1 9 "  f(x) ~ " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 
9 " cos x + " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 9 "co
s 2x + " }{XPPEDIT 18 0 "a[3];" "6#&%\"aG6#\"\"$" }{TEXT -1 10 " cos 3
x + " }{XPPEDIT 18 0 "a[4];" "6#&%\"aG6#\"\"%" }{TEXT -1 15 " cos 4x +
 \205+   " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 7 " cos \+
nx" }}{PARA 0 "" 0 "" {TEXT -1 20 "                  + " }{XPPEDIT 18 
0 "b[1];" "6#&%\"bG6#\"\"\"" }{TEXT -1 9 " sin x + " }{XPPEDIT 18 0 "b
[2];" "6#&%\"bG6#\"\"#" }{TEXT -1 9 "sin 2x + " }{XPPEDIT 18 0 "b[3];
" "6#&%\"bG6#\"\"$" }{TEXT -1 10 " sin 3x + " }{XPPEDIT 18 0 "b[4];" "
6#&%\"bG6#\"\"%" }{TEXT -1 15 " sin 4x + \205+   " }{XPPEDIT 18 0 "b[n
];" "6#&%\"bG6#%\"nG" }{TEXT -1 7 " sin nx" }}{PARA 0 "" 0 "" {TEXT 
-1 8 "by  cos(" }{TEXT 303 5 "kx)  " }{TEXT -1 11 "or by  sin(" }
{TEXT 304 4 "kx) " }{TEXT -1 32 "and use the results 1 through 7," }}
{PARA 0 "" 0 "" {TEXT -1 66 "we get the value of the various Fourier c
oefficients given above. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 282 17 "Solved Example 1:" }}
{PARA 0 "" 0 "" {TEXT -1 8 "(a) For " }{TEXT 307 1 "k" }{TEXT -1 19 " \+
= 1, 2  graph cos " }{TEXT 305 2 "kx" }{TEXT -1 16 " and show that  " 
}{XPPEDIT 18 0 "int(cos(k*x),x = -Pi .. Pi) = 0;" "6#/-%$intG6$-%$cosG
6#*&%\"kG\"\"\"%\"xGF,/F-;,$%#PiG!\"\"F1\"\"!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 283 0 "" }{TEXT -1 8 "(b) For " }{TEXT 308 1 "k" 
}{TEXT -1 19 " = 1, 2  graph sin " }{TEXT 306 2 "kx" }{TEXT -1 16 " an
d show that  " }{XPPEDIT 18 0 "int(sin(k*x),x = -Pi .. Pi) = 0;" "6#/-
%$intG6$-%$sinG6#*&%\"kG\"\"\"%\"xGF,/F-;,$%#PiG!\"\"F1\"\"!" }}{PARA 
0 "" 0 "" {TEXT -1 8 "(c) For " }{TEXT 309 1 "k" }{TEXT -1 12 " = 1, 2
 and " }{TEXT 310 1 "m" }{TEXT -1 20 " = 1, 2  graph cos (" }{TEXT 
311 2 "kx" }{TEXT -1 7 ") sin (" }{TEXT 312 2 "mx" }{TEXT -1 10 ") ,  \+
sin (" }{TEXT 313 2 "kx" }{TEXT -1 7 ") sin (" }{TEXT 314 2 "mx" }
{TEXT -1 7 "), cos(" }{TEXT 315 2 "kx" }{TEXT -1 5 ")cos(" }{TEXT 316 
2 "mx" }{TEXT -1 12 ") show that " }}{PARA 0 "" 0 "" {TEXT -1 8 "   wh
en " }{TEXT 317 1 "k" }{TEXT -1 2 ", " }{TEXT 318 1 "m" }{TEXT -1 17 "
 are not equal,  " }{XPPEDIT 18 0 "int(cos(k*x)*sin(m*x),x = -Pi .. Pi
) = 0;" "6#/-%$intG6$*&-%$cosG6#*&%\"kG\"\"\"%\"xGF-F--%$sinG6#*&%\"mG
F-F.F-F-/F.;,$%#PiG!\"\"F7\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 41 "    an
d                                  " }{XPPEDIT 18 0 "int(cos(k*x)^2,x \+
= -Pi .. Pi) = Pi;" "6#/-%$intG6$*$-%$cosG6#*&%\"kG\"\"\"%\"xGF-\"\"#/
F.;,$%#PiG!\"\"F3F3" }}{PARA 0 "" 0 "" {TEXT -1 44 "                  \+
                          " }{XPPEDIT 18 0 "int(sin(k*x)^2,x = -Pi .. \+
Pi) = Pi;" "6#/-%$intG6$*$-%$sinG6#*&%\"kG\"\"\"%\"xGF-\"\"#/F.;,$%#Pi
G!\"\"F3F3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 284 0 "" }}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 285 0 "" }{TEXT -1 10 " Solution:
" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 
1 " " }{TEXT 286 14 "Graph of cos (" }{TEXT -1 2 "kx" }{TEXT 319 26 ")
 and its integral over [-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 287 
2 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 288 2 "] " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{
cos(x), cos(2*x)\}, x = -Pi..Pi);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 309 
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ve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "int((cos
(x)^2),x=-Pi..Pi);\nint((cos(2*x)^2),x=-Pi..Pi);\n" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 259 17 "Solved Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 18 "Let
 f(x) = x,   - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 8 " < x <= \+
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 64 "(a) Construct the nth Fourier polynomial for the given fu
nction," }}{PARA 0 "" 0 "" {TEXT -1 56 "      assuming the function to
 be periodic with period 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 
3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 64 "(b) Draw the graphs of  f  and \+
of its Fourier polynomials up to " }{TEXT 321 1 "n" }{TEXT -1 10 " = 4
 on [-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 2 "]." }}{PARA 257 "" 0 "" {TEXT 260 0 "" 
}{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 
0 "" {TEXT -1 9 "Solution:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }
{TEXT 262 16 " Define function" }}{PARA 0 "" 0 "" {TEXT -1 80 "We cons
truct Fourier polynomials by using the definition.                    \+
   " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f := x -> x;\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(9$F(F(F(" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 263 44 " Evaluate first
 three Fourier coefficients: " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!
" }{TEXT 264 3 ",  " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT 
266 4 ",   " }{XPPEDIT 18 0 "b[1];" "6#&%\"bG6#\"\"\"" }{TEXT 265 1 ".
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a0 :=  (1/(2*Pi))*int (f
(x), x = -Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G\"\"!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a1 := 1/Pi* int(f(x)*cos(x
), x = -Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b1 := 1/Pi*int(f(x)*sin(x), \+
x = -Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G\"\"#" }}}}
{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 267 33 "Define the gener
al coefficients  " }{XPPEDIT 18 0 "a[k];" "6#&%\"aG6#%\"kG" }{TEXT 
268 3 " , " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6#%\"kG" }{TEXT 269 22 "  \+
 as functions of  f(" }{TEXT -1 1 "x" }{TEXT 329 8 "), sin (" }{TEXT 
-1 2 "kx" }{TEXT 330 11 ") and cos (" }{TEXT -1 2 "kx" }{TEXT 331 2 ")
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a :=  k ->(1/(Pi))*int \+
(f (x)*cos (k*x), x = -Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"aGf*6#%\"kG6\"6$%)operatorG%&arrowGF(,$-%$intG6$*&-%\"fG6#%\"xG\"\"
\"-%$cosG6#*&9$F5F4F5F5/F4;,$%#PiG!\"\"F>*$F>F?F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "b :=  k -> (1/(Pi))*int (f (x)*sin \+
(k*x), x = -Pi..Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%
\"kG6\"6$%)operatorG%&arrowGF(,$-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6
#*&9$F5F4F5F5/F4;,$%#PiG!\"\"F>*$F>F?F(F(F(" }}}{PARA 0 "" 0 "" {TEXT 
-1 36 "Note that we have not evaluated the " }{XPPEDIT 18 0 "a[k];" "6
#&%\"aG6#%\"kG" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "b[k];" "6#&%\"bG6
#%\"kG" }{TEXT -1 13 "  functions. " }}{PARA 0 "" 0 "" {TEXT 270 35 " \+
                                  " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
276 39 " Define the general Fourier polynomial." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 79 "F:= (f,n) -> a0 + sum(a(k)*cos(k*x), k = 1..n)
 + sum(b(k)*sin(k*x), k = 1..n);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"FGf*6$%\"fG%\"nG6\"6$%)operatorG%&arrowGF),(%#a0G\"\"\"-%$sumG6$*&
-%\"aG6#%\"kGF/-%$cosG6#*&F7F/%\"xGF/F//F7;F/9%F/-F16$*&-%\"bGF6F/-%$s
inGF:F/F=F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
5 "" 0 "" {TEXT 271 10 " Evaluate " }{TEXT 275 0 "" }{XPPEDIT 18 0 "a[
k];" "6#&%\"aG6#%\"kG" }{TEXT 274 1 " " }{TEXT 272 2 ", " }{XPPEDIT 
18 0 "b[k];" "6#&%\"bG6#%\"kG" }{TEXT 273 78 "  for the given function
 and plot the function together with a set of Fourier " }}{PARA 5 "" 
0 "" {TEXT 281 13 " polynomials." }}{PARA 0 "" 0 "" {TEXT -1 6 "For f(
" }{TEXT 328 1 "x" }{TEXT -1 4 ") = " }{TEXT 332 1 "x" }{TEXT -1 7 "  \+
and  " }{TEXT 333 1 "n" }{TEXT -1 60 " = 1, 2, 3, 4 we evaluate the Fo
urier polynomials and graph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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*F6F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
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}{PARA 0 "" 0 "" {TEXT -1 67 " _______________________________________
___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT 295 12 "ASSIGNMENT  " }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 1 " " }{TEXT 277 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT -1 14 "
Construct the " }{TEXT 334 1 "n" }{TEXT -1 29 "th Fourier polynomial f
or  f(" }{TEXT 278 1 "x" }{TEXT -1 12 ") =  1 on [-" }{XPPEDIT 18 0 "P
i;" "6#%#PiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 
-1 16 "] and  0 on [0, " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 "]
.   " }}{PARA 0 "" 0 "" {TEXT -1 56 "Draw the graphs of  f and its Fou
rier polynomials up to " }{TEXT 335 1 "n" }{TEXT -1 8 " = 10.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 67 "________
___________________________________________________________" }}{PARA 
0 "" 0 "" {TEXT -1 96 "MSIP Grant #P120A80089-98:  \"Three Urban Calcu
lus Reform Programs: Adopting the Best\" 1998-2001 " }}}{MARK "15 0" 
67 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }