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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Maple Lab 4" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Example with numeric
al approximation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 73 "This example uses differentials to approximate a function
 f at a point p." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 80 "The algorithm stops when successive approximations are wi
thin .01 of each other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:= x -> x^2:" }{TEXT -1 47 "  Just
 as in newton's method you need f and f'." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "f_prime:= D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%(f_primeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"#\"\"\"9$F/F/F(F
(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "p := 2.1:" }{TEXT -1 
32 "  We want to appoximate (2.1)^2." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "IV:= 2:  " }{TEXT -1 68 "An initial value, Similiar to
 the initial guess for Newton's method." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 41 "Now I generate a couple of approxima
tions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "approx1:= f(IV);  \+
" }{TEXT -1 33 "This is just f at a nearby point." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(approx1G\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "approx2:= f(IV) + f_prime(p-IV);" }{TEXT -1 38 "This is the di
fferential approximation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(approx2
G$\"#U!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Here is a loop wh
ich uses points closer and closer to 2.1 to get improved differential \+
approximations." }}{PARA 0 "" 0 "" {TEXT -1 74 "The loop stops when su
ccessive approximations are within .01 of each other" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "while abs(approx
2 - approx1) > .01 do  IV := IV + (p - IV)/2:\n  approx1 := approx2:\n
  approx2 := f(IV) + f_prime(p-IV):\nend do:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "approx2;  " }{TEXT -1 24 "This is our final resu
lt" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+mZ8.W!\"*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "actual:= f(p);" }{TEXT -1 37 "  This is t
he actual value of (2.1)^2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'actua
lG$\"$T%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }