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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 15 "Newton's Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 18 "Calc
ulus I Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 11 "
Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 53 "Approximate zeros of funct
ions using Newton's method." }}{PARA 0 "" 0 "" {TEXT -1 53 "Understand
 that Newton's method does not always work." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Maple allows students to quick
ly make tables and graphs of the values calculated using Newton's meth
od." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }{TEXT 260 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 59 
"Find the positive fourth root of 2 using Newton's method.  " }}{PARA 
0 "" 0 "" {TEXT -1 105 "a) Plot the graph to find an estimate for the \+
root (zero of the function) and then apply Newton's method." }}{PARA 
0 "" 0 "" {TEXT -1 70 "b) Use 0 for the estimate of the root and then \+
apply Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 49 "c) Discuss why
 Newton's method fails in part (b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 9 "
Solution:" }}{PARA 0 "" 0 "" {TEXT -1 24 "a) Solve the equation   " }
{XPPEDIT 18 0 "x^4-2 = 0;" "6#/,&*$%\"xG\"\"%\"\"\"\"\"#!\"\"\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "    First graph the fun
ction   " }{TEXT 266 1 "f" }{TEXT -1 1 "(" }{TEXT 267 1 "x" }{TEXT -1 
4 ") = " }{XPPEDIT 18 0 "x^4-2;" "6#,&*$%\"xG\"\"%\"\"\"\"\"#!\"\"" }
{TEXT -1 27 "  and approximate the zero." }}{PARA 0 "" 0 "" {TEXT -1 
62 "    Use this approximation in the formula for Newton's method." }}
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->x^4 - 2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$
\"\"%\"\"\"F1!\"#F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "plot(f(x),x = 0..2, y =-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 
74 74 {PLOTDATA 2 "6%-%'CURVESG6$7S7$\"\"!$!\"#F(7$$\"1LLLL3VfV!#<$!1`
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$\"1#z:Y#fB1uF17$$\"1+++q(G**y\"F1$\"1J'\\ZzAYE)F17$$\"1nm;9@BM=F1$\"1
kkY![P#>$*F17$$\"1LLL`v&Q(=F1$\"1#y?bN_H.\"!#97$$\"1++DOl5;>F1$\"1X!o*
z\\'z9\"Fiy7$$\"1++v.Uac>F1$\"1CtzL$3aE\"Fiy7$$\"\"#F($\"#9F(-%'COLOUR
G6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q\"yFd[l-%%VIEWG6$;F(Fez
;F)Fez" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=diff(f(x),x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,$*$)%\"xG\"\"$\"\"\"\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fprime:=x->4*x^3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'fprimeGf*6#%\"xG6\"6$%)operatorG%&arrowGF
(,$*$)9$\"\"$\"\"\"\"\"%F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "Newt:=x->x-f(x)/fprime(x);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%NewtGf*6#%\"xG6\"6$
%)operatorG%&arrowGF(,&9$\"\"\"*&-%\"fG6#F-F.-%'fprimeGF2!\"\"F5F(F(F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 76 "Pick an initial point (estimate of the zero of the functi
on) from the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "x[0]:=1.2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"xG6#\"\"!$\"#7!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Using a \"do\" loop, have Map
le compute a table of estimations" }}}{EXCHG {PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for nn from 0 to \+
7 do x[nn+1]:= evalf(Newt(x[nn])) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%\"xG6#\"\"\"$\"+_=N*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%\"xG6#\"\"#$\"+Tr?*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"x
G6#\"\"$$\"+:r?*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"
\"%$\"+:r?*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$
\"+:r?*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"+:r
?*=\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"+:r?*=\"
!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$\"+:r?*=\"!\"*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "A positive fourth root of 2 s
eems to converge to 1.189207115." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "b)  Use 0 as the first estimate of
 the positive fourth root of 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x[0]:=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"!F'" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for nn from 0 to 5 d
o x[nn+1]:=evalf(Newt(x[nn]))  od;" }}{PARA 8 "" 1 "" {TEXT -1 33 "Err
or, (in Newt) division by zero" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "
c) At  " }{TEXT 268 1 "x" }{TEXT -1 7 " = 0,  " }{TEXT 269 1 "f" }
{TEXT -1 3 " '(" }{TEXT 270 1 "x" }{TEXT -1 21 ") =0  and therefore  \+
" }{TEXT 271 1 "f" }{TEXT -1 1 "(" }{TEXT 272 1 "x" }{TEXT -1 2 ")/" }
{TEXT 273 1 "f" }{TEXT -1 3 " '(" }{TEXT 274 1 "x" }{TEXT -1 16 ")  is
 undefined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 78 "_______________________________________________________________
_______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 0 "" }{TEXT 263 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Problem 1:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Use N
ewton's method to find all the roots of  " }{XPPEDIT 18 0 "x^3-4*x+1;
" "6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F'F%F'!\"\"F'F'" }{TEXT -1 36 " = 0  c
orrect to six decimal places." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "
" }{TEXT 265 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 50 "a) Use Newton's method to find all the roots of
   " }{XPPEDIT 18 0 "sin(2*x)-x+1 = 0;" "6#/,(-%$sinG6#*&\"\"#\"\"\"%
\"xGF*F*F+!\"\"F*F*\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 
58 "b) Try Newton's method again using the first estimate of  " }
{TEXT 275 1 "x" }{TEXT -1 27 " = 0.  Describe the output." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "_________
______________________________________________________________________
__" }}{PARA 0 "" 0 "" {TEXT -1 172 "MSIP Grant #P120A80089-98:  \"Thre
e Urban Calculus Reform programs: Adopting the Best\" 1998-2001, MSEIP
 Grant #P120A010031: \"Four Colleges: Calculus + Enhancements\"  2001-
04" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "4 2 0" 0 }{VIEWOPTS 1 
0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }