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The grade is used as a bonus project with value 5 points . The range of scores is [-5, +5]. The -5 is assigned only if the as signment is essentially ignored." }}{PARA 213 "" 0 "" {TEXT 214 67 "B) The computational portion will be treated as one lab assignment." }} {PARA 213 "" 0 "" {TEXT 244 0 "" }}{PARA 213 "" 0 "" {TEXT 214 48 "The paper should have at least these components:" }}{PARA 213 "" 0 "" {TEXT 214 77 "1) An introduction which indicates the purpose and organ ization of the paper." }}{PARA 213 "" 0 "" {TEXT 244 0 "" }}{PARA 213 "" 0 "" {TEXT 214 68 "2) A derivation/discussion of the four technique s that are utilized." }}{PARA 213 "" 0 "" {TEXT 244 0 "" }}{PARA 213 " " 0 "" {TEXT 214 195 "3) Computations illustrating the techniques. Th e problem to be solved is given below. It is suggested that you includ e the computations as appendices so as not to impede the \"flow\" of t he paper." }}{PARA 213 "" 0 "" {TEXT 214 2 "4)" }{TEXT 231 1 " " } {TEXT 214 12 "A conclusion" }}{PARA 213 "" 0 "" {TEXT 214 1 " " }} {PARA 214 "" 0 "" {TEXT 243 25 "Method 1: Newton-Raphson" }}{PARA 213 "" 0 "" {TEXT 214 25 "Problem: Solve F(x) = 0." }}{PARA 213 "" 0 "" {TEXT 214 49 "Algorithm: x(n+1) = x(n) - F[(x(n)]) / F'[x(n)]." }} {PARA 213 "" 0 "" {TEXT 231 3 " " }}{PARA 213 "" 0 "" {TEXT 230 19 " Method 2: Iteration" }}{PARA 213 "" 0 "" {TEXT 214 24 "Problem Solve \+ x = F(x)." }}{PARA 207 "" 0 "" {TEXT 212 28 "Algorithm: x(n+1) = F[x(n )]." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 229 30 "Method 3: Halving the interval" }}{PARA 207 "" 0 "" {TEXT 212 23 " Problem: Solve F(x) =0." }}{PARA 207 "" 0 "" {TEXT 212 85 "Algorithm: \+ decrease the region containing a solution by halving it at each iterat ion." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 229 19 "Method 4: Graphical" }}{PARA 207 "" 0 "" {TEXT 212 23 "Problem : Solve F(x) =0." }}{PARA 207 "" 0 "" {TEXT 212 51 "Algorithm: Graph y = F(x) and zoom in on the root.." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 227 45 "Example: The square root us ing Newton-Raphson" }}{PARA 213 "" 0 "" {TEXT 214 73 "To find the squa re root of a number b we can use the following procedure." }}{EXCHG {PARA 213 "" 0 "" {TEXT 214 15 "Let f(x) = y = " }{XPPEDIT 18 0 "Types etting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"t rue\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Type setting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \" true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typ esetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \+ \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"−\", \+ mathvariant = \"normal\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rsp ace = \"0.2222222em\"), Typesetting:-mi(\"b\", italic = \"true\", math variant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~Inp utF'/%,mathvariantGQ'italicF'-F#6&F+-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-I #mnGF$6$Q\"2F'/F8Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(&minu s;F'FH/%&fenceGQ&falseF'/%*separatorGFS/%)stretchyGFS/%*symmetricGFS/% (largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceGQ,0.2222222emF'/%' rspaceGF\\o-F,6%Q\"bF'F/F7F+" }{TEXT 244 26 ". 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font_style_name = \"2D Input\", mathvariant = \"italic\"), Typeset ting:-mrow(Typesetting:-msup(Typesetting:-mrow(Typesetting:-mi(\"x\", \+ italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&Apply Function;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typese tting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), mathva riant = \"normal\")), Typesetting:-mn(\"2\", mathvariant = \"normal\") , superscriptshift = \"0\")), Typesetting:-mo(\"+\", mathvariant = \"n ormal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.222222 2em\"), Typesetting:-mi(\"b\", italic = \"true\", mathvariant = \"ital ic\")), 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\"true\", mathvariant = \"italic \")), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"tru e\", executable = \"true\", font_style_name = \"2D Input\", mathvarian t = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", executable \+ = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")) , linethickness = \"1\", denomalign = \"center\", numalign = \"center \", bevelled = \"false\")), Typesetting:-mi(\"\", italic = \"true\", e xecutable = \"true\", font_style_name = \"2D Input\", mathvariant = \" italic\")), mathvariant = \"normal\", open = \"[\", close = \"]\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I(mfencedGF$6 &-F#6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF6/%0font_style_na meGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6&F0-F#6%-F16%Q\"xF'F4F<-I# moGF$6-Q0⁡F'/F=Q'normalF'/%&fenceGQ&falseF'/%*separatorG FN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN/%.movablelimitsGFN/%'acce ntGFN/%'lspaceGQ&0.0emF'/%'rspaceGFgn-F,6$-F#6%F0-F#6%-F16%Q\"nF'F4F<- FG6-Q\"+F'FJFLFOFQFSFUFWFY/FfnQ,0.2222222emF'/FinFgo-I#mnGF$6$Q\"1F'FJ F0FJ-FG6-Q\"=F'FJFLFOFQFSFUFWFY/FfnQ,0.2777778emF'/FinFap-I&mfracGF$6( -F#6%F0-F#6&F0-F#6#-I%msupGF$6%-F#6%FCFF-F,6$-F#6#F`oFJ-Fjo6$Q\"2F'FJ/ %1superscriptshiftGQ\"0F'Fco-F16%Q\"bF'F4F " 0 "" {MPLTEXT 1 224 8 "restart;" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 6 "b:=.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\" \"&!\"\"" }}}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 16 "x[i], i = 1,100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%&%\"xG6#%\"iG/F&\"\"\"\"$+\"" }}}{PARA 200 "" 0 "" {TEXT 247 0 " " }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 11 "x[1] := 1.;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$\"\"\"\"\"!" }}}{PARA 200 "" 0 "" {TEXT 247 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 61 "for k from 1 to 10 do x[k+1] := (x[k]^2.+b)/(2.*x[k]);end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$\"+++++v!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$\"+NLL$3(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$\"+I%y52(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"+:y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#6$\"+5y1rq!#5" }}}{PARA 210 "" 0 "" {TEXT 248 0 "" } }}{SECT 1 {PARA 210 "" 0 "" {TEXT 227 53 "Example: The square root usi ng \"halving the interval\"" }}{EXCHG {PARA 213 "" 0 "" {TEXT 214 255 "Another procedure for finding the square root of a number x0 would be to halve the interval. Suppose we want to find the square root of an y number between 0 and 100, say 5.. We know the answer is in [0,10]. \+ A maple iteration could be set up as follows." }}{PARA 213 "" 0 "" {TEXT 214 1 " " }{TEXT 244 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:= 5.;" }}{PARA 11 "" 1 "" {TEXT 249 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 13 "right := 10.;" }}{PARA 11 "" 1 "" {TEXT 249 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "left:=0.;" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 112 "for k from 1 to 20 do mid := (left+right)*(1/2); if mid^2 < b then left := mid else right := mid end if; end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 205 "" 0 "" {TEXT 201 54 "Examp le: The square root using the method of iteration" }}{PARA 213 "" 0 "" {TEXT 214 21 "The problem is still " }{XPPEDIT 18 0 "Typesetting:-mro w(Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font _style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mr ow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", fon t_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-m row(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"=\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778 em\"), Typesetting:-mi(\"b\", italic = \"true\", mathvariant = \"itali c\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrow G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'itali cGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvarian tGQ'italicF'-F#6&F+-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-I#mnGF$6$Q\"2F'/F8 Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q\"=F'FH/%&fenceGQ&false F'/%*separatorGFS/%)stretchyGFS/%*symmetricGFS/%(largeopGFS/%.movablel imitsGFS/%'accentGFS/%'lspaceGQ,0.2777778emF'/%'rspaceGF\\o-F,6%Q\"bF' F/F7F+" }{TEXT 244 35 ". We change the form to x= f(x) = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", exec utable = \"true\", font_style_name = \"2D Input\", mathvariant = \"ita lic\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", math variant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"& minus;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222 2em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"b\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"it alic\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true \", font_style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'it alicGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvar iantGQ'italicF'-F#6(F+-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-I#mnGF$6$Q\"2F' /F8Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(−F'FH/%&fence GQ&falseF'/%*separatorGFS/%)stretchyGFS/%*symmetricGFS/%(largeopGFS/%. movablelimitsGFS/%'accentGFS/%'lspaceGQ,0.2222222emF'/%'rspaceGF\\o-F, 6%Q\"bF'F/F7-FN6-Q\"+F'FHFQFTFVFXFZFfnFhnFjnF]oFAF+" }{TEXT 244 1 "." }}{PARA 213 "" 0 "" {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mfra c(Typesetting:-mrow(Typesetting:-mi(\"df\", italic = \"true\", mathvar iant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\"dx\", italic \+ = \"true\", mathvariant = \"italic\")), linethickness = \"1\", denomal ign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I% mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I&mfracGF$6(-F#6# -I#miGF$6%Q#dfF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F16%Q #dxF'F4F7/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFD /%)bevelledGQ&falseF'" }{TEXT 244 46 "= 2x+1. |2x+1| is less than 1 f or -1 < x < 0." }}{PARA 213 "" 0 "" {TEXT 244 0 "" }{TEXT 214 1 "." }} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 213 26 "restart; x[i], i = 1..10 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"xG6#%\"iG/F&;\"\"\"\"$+\"" } }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 213 10 "b := 1/9.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+66666!#5" }}}{EXCHG {PARA 207 "" 0 " " {TEXT 245 0 "" }}{PARA 11 "" 1 "" {TEXT 249 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "x[1] := -.3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$!\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executa ble = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic \"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execut able = \"true\", font_style_name = \"2D Input\", mathvariant = \"itali c\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", exec utable = \"true\", font_style_name = \"2D Input\", mathvariant = \"ita lic\"), Typesetting:-mrow(Typesetting:-mo(\"for\", bold = \"true\", ma thvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mspace(height = \"0.0ex\" , width = \"0.5em\", depth = \"0.0ex\", linebreak = \"auto\"), Typeset ting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mspace(height = \"0.0ex\", width = \"0.5em\", depth = \"0.0ex\", \+ linebreak = \"auto\"), Typesetting:-mo(\"from\", bold = \"true\", math variant = \"bold\", fontweight = \"bold\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mspace(height = \"0.0ex\", w idth = \"0.5em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesettin g:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mspace(height = \+ \"0.0ex\", width = \"0.5em\", depth = \"0.0ex\", linebreak = \"auto\") , Typesetting:-mo(\"to\", bold = \"true\", mathvariant = \"bold\", fon tweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.5em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mn(\"10\", mathvaria nt = \"normal\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0. 5em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"do \", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.0em\")), Typesetting:-mi (\"\", italic = \"true\", executable = \"true\", font_style_name = \"2 D Input\", mathvariant = \"italic\"), Typesetting:-mspace(height = \"0 .0ex\", width = \"0.5em\", depth = \"0.0ex\", linebreak = \"increasein dentnewline\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\" x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(T ypesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\")), s ubscriptshift = \"0\"), Typesetting:-mo(\":=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executabl e = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\" ), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi(\"x\", itali c = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting :-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesettin g:-mi(\"k\", italic = \"true\", mathvariant = \"italic\"), Typesetting :-mo(\"−\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"1\", mathv ariant = \"normal\")), Typesetting:-mi(\"\", italic = \"true\", execut able = \"true\", font_style_name = \"2D Input\", mathvariant = \"itali c\")), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptsh ift = \"0\", subscriptshift = \"0\")), Typesetting:-mo(\"−\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspac e = \"0.2222222em\"), Typesetting:-mi(\"b\", italic = \"true\", mathva riant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Ty pesetting:-msub(Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true \", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"k\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mo(\"−\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \" 0.2222222em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), Typ esetting:-mi(\"\", italic = \"true\", executable = \"true\", font_styl e_name = \"2D Input\", mathvariant = \"italic\")), subscriptshift = \" 0\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Typesetti ng:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.5em\", depth = \"0.0ex\", linebreak = \"decr easeindentnewline\"), Typesetting:-mo(\"end do\", bold = \"true\", mat hvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.0em\", rspace = \"0.0em\")), Typesetting:-mo(\";\", mathvariant = \" normal\", fence = \"false\", separator = \"true\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.2777778em\") , Typesetting:-mspace(height = \"0.0ex\", width = \"0.5em\", depth = \+ \"0.0ex\", linebreak = \"firstprocnewline\"), Typesetting:-mi(\"\", it alic = \"true\", executable = \"true\", font_style_name = \"2D Input\" , mathvariant = \"italic\")), Typesetting:-mi(\"?\", italic = \"true\" , mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \+ \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0font_style_nameGQ) 2D~InputF'/%,mathvariantGQ'italicF'-F#6%F+-F#6'F+-F#6*F+-F#6/-I#moGF$6 /Q$forF'/%%boldGF1/F8Q%boldF'/%+fontweightGFI/%&fenceGQ&falseF'/%*sepa ratorGFN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN/%.movablelimitsGFN/ %'accentGFN/%'lspaceGQ&0.0emF'/%'rspaceGFgn-I'mspaceGF$6&/%'heightGQ&0 .0exF'/%&widthGQ&0.5emF'/%&depthGF_o/%*linebreakGQ%autoF'-F,6%Q\"kF'F/ F7Fjn-FC6/Q%fromF'FFFHFJFLFOFQFSFUFWFYFenFhnFjn-I#mnGF$6$Q\"2F'/F8Q'no rmalF'Fjn-FC6/Q#toF'FFFHFJFLFOFQFSFUFWFYFenFhnFjn-F_p6$Q#10F'FbpFjn-FC 6/Q#doF'FFFHFJFLFOFQFSFUFWFYFenFhnF+-F[o6&F]oF`oFco/FfoQ6increaseinden tnewlineF'-F#6&-I%msubGF$6%-F,6%Q\"xF'F/F7-F#6#Fho/%/subscriptshiftGQ \"0F'-FC6-Q#:=F'FbpFLFOFQFSFUFWFY/FfnQ,0.2777778emF'/FinFbr-F#6(F+-F#6 #-I(msubsupGF$6'Ffq-F#6%F+-F#6%Fho-FC6-Q(−F'FbpFLFOFQFSFUFWFY/Ff nQ,0.2222222emF'/FinFcs-F_p6$Q\"1F'FbpF+F^p/%1superscriptshiftGF]rF[rF _s-F,6%Q\"bF'F/F7-FC6-Q\"+F'FbpFLFOFQFSFUFWFYFbsFds-Fdq6%FfqF[sF[rF+F+ -F[o6&F]oF`oFco/FfoQ6decreaseindentnewlineF'-FC6/Q'end~doF'FFFHFJFLFOF QFSFUFWFYFenFhn-FC6-Q\";F'FbpFL/FPF1FQFSFUFWFYFenFcr-F[o6&F]oF`oFco/Ff oQ1firstprocnewlineF'F+-F,6%Q\"?F'F/F7F+" }}{PARA 0 "" 0 "" {TEXT 246 0 "" }{XPPEDIT 20 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \+ \"true\", executable = \"true\", font_style_name = \"2D Input\", mathv ariant = \"italic\"), Typesetting:-mrow(Typesetting:-msub(Typesetting: -mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:- mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\")), subscriptshift = \"0\"), Typesetting:-mo(\":=\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesettin g:-mn(\"&uminus0;.3211111111\", mathvariant = \"normal\")), Typesettin g:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name \+ = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+exec utableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6% -I%msubGF$6%-F,6%Q\"xF'F/F7-F#6#-I#mnGF$6$Q\"2F'/F8Q'normalF'/%/subscr iptshiftGQ\"0F'-I#moGF$6-Q#:=F'FH/%&fenceGQ&falseF'/%*separatorGFS/%)s tretchyGFS/%*symmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/ %'lspaceGQ,0.2777778emF'/%'rspaceGF\\o-FE6$Q5&uminus0;.3211111111F'FHF +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$!+6666K!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$!+l()4\"H$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$!+on2>L!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"xG6#\"\"&$!+?3cGL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$!+=-uJL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($!+T?!GL$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"xG6#\"\")$!+3i:LL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\" \"*$!+)GuKL$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$!+^OJ LL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 246 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restar t;b:=1/9;x[i], i = 1..100; x[1] := .2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"xG6#%\"i G/F&;\"\"\"\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$ \"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for k from 2 \+ to 10 do x[k] := -x[k-1]^2+b+x[k-1] end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$\"+ 6666F!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$$\"+w)4s3$!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$\"+0XBXK!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"+<4>.L!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"+0]>BL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"($\"+7O%*HL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")$\"+xA?KL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*$\"+(=cHL$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5$\"+/w?LL!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 240 0 "" }{TEXT 245 0 "" }}}{SECT 0 {PARA 210 "" 0 "" {TEXT 227 21 "Problem to be solved." }} {PARA 210 "" 0 "" {TEXT 242 50 "Find the roots of f(x) = Nx + 1/x -(N+ 1) = 0 where" }}{PARA 207 "" 0 "" {TEXT 212 74 " N = 1 +k/9 and \+ k corresponds to the first letter of your last name." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 212 97 "Suppose your name is Sally Crain. Then k= 3, N = 1+1/3 and you are to solve (4/3)x + l/ x - 7 = 0." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 210 20 "Graphical technique." }}{PARA 207 "" 0 "" {TEXT 212 110 "Graph the equation. You should find two roots. By zooming in, deter mine each root to several decimal places." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 210 16 "Newton-Raphson. " }}{PARA 207 "" 0 "" {TEXT 212 93 " df/dx = N - 1/x^2 . Therefore, x[k+1] = x [k] - \{ x[k] N +1/x[k] -(N+1 \}/( N -1/x[k]^2)) \}." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 210 "" 0 "" {TEXT 242 20 "Halving the inter val" }}{PARA 207 "" 0 "" {TEXT 212 90 "From the graph, you should obta in two values - for each root- which straddle the solution." }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 210 9 "Iteration" }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 212 178 "Write f(x) = 0 as x = (N + 1 - 1/x)/N = g(x). Using a \"good\" firs t guess for x[1]), try the iteration x[i+1] = g(x(i)). You will not b e able to use this method for both roots." }}{PARA 204 "" 0 "" {TEXT 250 0 "" }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}{PARA 207 "" 0 "" {TEXT 212 2 " " }}{PARA 207 "" 0 "" {TEXT 245 0 "" }}}{PARA 216 "" 0 "" {TEXT 251 0 "" }}{PARA 217 "" 0 "" {TEXT 252 0 "" }}{PARA 215 "" 0 "" {TEXT 253 0 "" }}{PARA 208 "" 0 "" {TEXT 254 0 "" }}{PARA 211 "" 0 "" {TEXT 255 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }