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1 } {CSTYLE "" -1 208 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 209 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 210 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 211 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 201 "" 0 "" {TEXT 206 57 "Math 152 Writing Assig nment - Sections 1,2 Spring 2008 " }{TEXT 207 1 " " }}{PARA 210 "" 0 "" {TEXT 213 0 "" }{TEXT 200 21 "Numerical Integration" }{TEXT 213 0 "" }}}{PARA 207 "" 0 "" {TEXT 214 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT 216 245 "There are many approximate techniques for finding the area under a curve (or numeric al integration). The purpose of this assignment is to study four techn iques of numerical integration. This is a required assignment and has \+ two grade components:" }}{PARA 208 "" 0 "" {TEXT 217 226 "A) The writi ng portion will be graded as to composition style, clarity, and comple teness. The grade is used as a bonus project with value 5 points. The range of scores is [-5, +5]. The -5 is achieved by ignoring the proj ect." }}{PARA 0 "" 0 "" {TEXT 216 67 "B) The computational portion wil l be treated as one lab assignment." }}{PARA 208 "" 0 "" {TEXT 217 48 "The paper should have at least these components:" }}{PARA 208 "" 0 "" {TEXT 217 77 "1) An introduction which indicates the purpose and orga nization of the paper." }}{PARA 208 "" 0 "" {TEXT 217 101 "2) A deriva tion/discussion of the four techniques that are utilized and the expec ted errors for each." }}{PARA 208 "" 0 "" {TEXT 217 44 "3) Computation s illustrating the techniques." }}{PARA 208 "" 0 "" {TEXT 217 2 "4)" } {TEXT 208 1 " " }{TEXT 217 12 "A conclusion" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT 215 16 "The four methods" }}{PARA 0 "" 0 "" {TEXT 216 35 "The \+ problem is to approximate I = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typ esetting:-mi(\"\", italic = \"true\", executable = \"true\", font_styl e_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Ty pesetting:-msubsup(Typesetting:-mo(\"\21053\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"a\", italic = \"true\", mathvariant = \"italic\"), Typesetting :-mi(\"b\", italic = \"true\", mathvariant = \"italic\"), superscripts hift = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\", italic = \+ \"true\", executable = \"true\", font_style_name = \"2D Input\", mathv ariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"f\", italic \+ = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFunctio n;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:- mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"true\", executable = \+ \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), T ypesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \"0. 0ex\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0 .0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"ita lic\")), 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)-I(msubsupGF$6'-I#moGF$6-Q&∫F'/F8Q'normalF'/%& fenceGQ&falseF'/%*separatorGFG/%)stretchyGFG/%*symmetricGFG/%(largeopG FG/%.movablelimitsGFG/%'accentGFG/%'lspaceGQ&0.0emF'/%'rspaceGFV-F,6%Q \"aF'F/F7-F,6%Q\"bF'F/F7/%1superscriptshiftGQ\"2F'/%/subscriptshiftGQ \"0F'F+-F#6%-F,6%Q\"fF'F/F7-F@6-Q0⁡F'FCFEFHFJFLFNFPFRFTF W-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F/F7FCF+-I'mspaceGF$6&/%'heightGQ&0.0e xF'/%&widthGQ&0.3emF'/%&depthGFdp/%*linebreakGQ%autoF'-F@6-Q0&Differen tialD;F'FCFEFHFJFLFNFPFRFTFWF\\pF+" }{TEXT 216 26 " for some function f(x). " }}{PARA 0 "" 0 "" {TEXT 216 87 " You can choose any function \+ (of reasonable complexity!). I suggest you work with I = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(\"\21053 \", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesett ing:-msemantics = \"inert\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\"), Types etting:-mn(\"1\", mathvariant = \"normal\"), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetting:-msup(Typesetting:-mo(\"&Exponen tialE;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1111111em\"), Typesetting:-mi(\"x\", italic = \"true\", \+ mathvariant = \"italic\"), superscriptshift = \"0\"), Typesetting:-mi( \"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mspace(height = \"0. 0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Ty pesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mat hvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", i talic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", it alic = \"true\", executable = \"true\", font_style_name = \"2D Input\" , mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0fo nt_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6(-I(msubsupGF$6 '-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F'/F8Q'normalF'/I+mse manticsGF$Q&inertF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*s ymmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0 emF'/%'rspaceGFfn-I#mnGF$6$Q\"0F'FF-Fjn6$Q\"1F'FF/%1superscriptshiftGQ \"2F'/%/subscriptshiftGQ\"0F'-I%msupGF$6%-F@6-Q/ⅇF'FFFKFN FPFRFTFVFXFZ/FhnQ,0.1111111emF'-F,6%Q\"xF'F/F7/FaoFeoF+-I'mspaceGF$6&/ %'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFgp/%*linebreakGQ%autoF'- F@6/Q0ⅆF'FCFFFHFKFNFPFRFTFVFXFZFgnF^pF+" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 125 "In \+ the first three methods, the interval [a,b] is split into n equal piec es of length h where h=(b-a)/n. x[0] = a, x[n] = b," }}{PARA 0 "" 0 " " {TEXT 216 17 "x[i] = x[0] + ih." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }} {PARA 0 "" 0 "" {TEXT 216 59 "Method 1: The left-hand rule (text, page 554, equation 1). " }}{PARA 0 "" 0 "" {TEXT 216 25 " \+ " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", ital ic = \"true\", executable = \"true\", font_style_name = \"2D Input\", \+ mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mfenced(Type setting:-mrow(Typesetting:-munderover(Typesetting:-mo(\"∑\", mathv ariant = \"normal\", fence = \"false\", separator = \"false\", stretch y = \"true\", symmetric = \"false\", largeop = \"true\", movablelimits = \"true\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1666 667em\"), Typesetting:-mrow(Typesetting:-mi(\"i\", italic = \"true\", \+ mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), Typesetting:-m i(\"n\", italic = \"true\", mathvariant = \"italic\"), accent = \"fals e\", accentunder = \"false\"), Typesetting:-mi(\"\", italic = \"true\" , executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mspace(height = \"0.0ex\", width = \"5.0\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mrow(Typesetti ng:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesettin g:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesettin g:-mrow(Typesetting:-msub(Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"\", itali c = \"true\", executable = \"true\", font_style_name = \"2D Input\", m athvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"i\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"−\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", r space = \"0.2222222em\"), Typesetting:-mn(\"1\", mathvariant = \"norma l\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), subscript shift = \"0\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", ita lic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), mathvariant = \"normal\"), Typesetting:-m o(\"⁢\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \" true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"t rue\", executable = \"true\", font_style_name = \"2D Input\", mathvari ant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0font_style_n ameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6%-I(mfencedGF$6$-F#6'-I+m underoverGF$6'-I#moGF$6-Q&∑F'/F8Q'normalF'/%&fenceGQ&falseF'/%*sep aratorGFL/%)stretchyGF1/%*symmetricGFL/%(largeopGF1/%.movablelimitsGF1 /%'accentGFL/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.1666667emF'-F#6%-F,6%Q\"i F'F/F7-FE6-Q\"=F'FHFJFM/FPFLFQ/FTFL/FVFLFW/FZQ,0.2777778emF'/FgnFeo-I# mnGF$6$Q\"1F'FH-F,6%Q\"nF'F/F7/%'accentGFL/%,accentunderGFLF+-I'mspace GF$6&/%'heightGQ&0.0exF'/%&widthGQ$5.0F'/%&depthGFgp/%*linebreakGQ%aut oF'-F#6%-F,6%Q\"fF'F/F7-FE6-Q0⁡F'FHFJFMFaoFQFboFcoFWFY/F gnFen-F=6$-F#6#-I%msubGF$6%-F,6%Q\"xF'F/F7-F#6%F+-F#6%F[o-FE6-Q(&minus ;F'FHFJFMFaoFQFboFcoFW/FZQ,0.2222222emF'/FgnF[sFgoF+/%/subscriptshiftG Q\"0F'FHF+FH-FE6-Q1⁢F'FHFJFMFaoFQFboFcoFWFYFhq-F,6%Q\"h F'F/F7F+" }{TEXT 216 1 "." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 47 "Method 2: The trapezoidal rule (text, page 555)" }}{PARA 0 "" 0 "" {TEXT 216 24 " " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesett ing:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \"true\", mathva riant = \"italic\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\") , Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\", italic \+ = \"true\", executable = \"true\", font_style_name = \"2D Input\", mat hvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"f\", itali c = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFunct ion;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting :-msub(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"itali c\"), Typesetting:-mrow(Typesetting:-mi(\"a\", italic = \"true\", math variant = \"italic\")), subscriptshift = \"0\")), mathvariant = \"norm al\")), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mr ow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\") , Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenc ed(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-m i(\"b\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", math variant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mrow( Typesetting:-mn(\"2\", mathvariant = \"normal\")), linethickness = \"1 \", denomalign = \"center\", numalign = \"center\", bevelled = \"false \"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow (Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-munderover(Typese tting:-mo(\"∑\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"true\", symmetric = \"false\", largeop = \"true\", movablelimits = \"true\", accent = \"false\", lspace = \" 0.0em\", rspace = \"0.1666667em\"), Typesetting:-mrow(Typesetting:-mi( \"i\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em \", rspace = \"0.2777778em\"), Typesetting:-mn(\"1\", mathvariant = \" normal\")), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mo(\"−\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.22 22222em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), accent \+ = \"false\", accentunder = \"false\"), Typesetting:-mi(\"\", italic = \+ \"true\", executable = \"true\", font_style_name = \"2D Input\", mathv ariant = \"italic\"), Typesetting:-mspace(height = \"0.0ex\", width = \+ \"5.0\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mrow(T ypesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Ty pesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(T ypesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"x\", italic = \"t rue\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\" i\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \+ \"0\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \" true\", executable = \"true\", font_style_name = \"2D Input\", mathvar iant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mo(\"&Inv isibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", \+ executable = \"true\", font_style_name = \"2D Input\", 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'/%,mat hvariantGQ'italicF'-F#6&-I&mfracGF$6(-F#6'-I#mnGF$6$Q\"1F'/F8Q'normalF '-I#moGF$6-Q1⁢F'FE/%&fenceGQ&falseF'/%*separatorGFM/%)s tretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/ %'lspaceGQ&0.0emF'/%'rspaceGFfn-F,6%Q\"hF'F/F7FG-I(mfencedGF$6$-F#6'F+ -F#6%-F,6%Q\"fF'F/F7-FH6-Q0⁡F'FEFKFNFPFRFTFVFXFZFgn-F]o6 $-F#6#-I%msubGF$6%-F,6%Q\"xF'F/F7-F#6#-F,6%Q\"aF'F/F7/%/subscriptshift GQ\"0F'FE-FH6-Q\"+F'FEFKFNFPFRFTFVFX/FenQ,0.2222222emF'/FhnF_q-F#6%Fco Ffo-F]o6$-F#6#-F^p6%F`p-F#6#-F,6%Q\"bF'F/F7FhpFEF+FE-F#6#-FB6$Q\"2F'FE /%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFhr/%)bevel ledGFMF[q-F#6%-F]o6$-F#6'-I+munderoverGF$6'-FH6-Q&∑F'FEFKFN/FQF1FR /FUF1/FWF1FXFZ/FhnQ,0.1666667emF'-F#6%-F,6%Q\"iF'F/F7-FH6-Q\"=F'FEFKFN FPFRFTFVFX/FenQ,0.2777778emF'/FhnFgtFA-F#6%-F,6%Q\"nF'F/F7-FH6-Q(&minu s;F'FEFKFNFPFRFTFVFXF^qF`qFA/%'accentGFM/%,accentunderGFMF+-I'mspaceGF $6&/%'heightGQ&0.0exF'/%&widthGQ$5.0F'/%&depthGFju/%*linebreakGQ%autoF '-F#6%FcoFfo-F]o6$-F#6#-F^p6%F`p-F#6#F`tFhpFEF+FEFGFinF+F+" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 40 "Method 3: Simpson's rule (text page 560)" }}{PARA 0 "" 0 "" {TEXT 216 22 " " }}{PARA 0 "" 0 "" {TEXT 216 0 "" } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true \", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow (Typesetting:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\"& InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\"⁢\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\" \", italic = \"true\", executable = \"true\", font_style_name = \"2D I nput\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi( \"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"⁡\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mro w(Typesetting:-msub(Typesetting:-mi(\"x\", italic = \"true\", mathvari ant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"a\", italic = \+ \"true\", mathvariant = \"italic\")), subscriptshift = \"0\")), mathva riant = \"normal\")), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), T ypesetting:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"⁡\", mathvariant = \"n ormal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mfenced(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( \"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow (Typesetting:-mi(\"b\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\")), mathvariant = \"normal\")), Typesetting:-mi (\"\", italic = \"true\", executable = \"true\", font_style_name = \"2 D Input\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typ esetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"normal\")), line thickness = \"1\", denomalign = \"center\", numalign = \"center\", bev elled = \"false\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typ esetting:-mfrac(Typesetting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mfenced(Typesetting:-mrow(Typesetting:-munderover(Typesetting: -mo(\"∑\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"true\", symmetric = \"false\", largeop = \"t rue\", movablelimits = \"true\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.1666667em\"), Typesetting:-mrow(Typesetting:-mi(\"i\" , italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rs pace = \"0.2777778em\"), Typesetting:-mn(\"1\", mathvariant = \"normal \")), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true\", math variant = \"italic\"), Typesetting:-mo(\"−\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222 em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), accent = \"f alse\", accentunder = \"false\"), Typesetting:-mi(\"\", italic = \"tru e\", executable = \"true\", font_style_name = \"2D Input\", mathvarian t = \"italic\"), Typesetting:-mspace(height = \"0.0ex\", width = \"5.0 \", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mrow(Typese tting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typese tting:-mrow(Typesetting:-msub(Typesetting:-mi(\"x\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"i\", \+ italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\") ), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"true\" , executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvar iant = \"normal\")), linethickness = \"1\", denomalign = \"center\", n umalign = \"center\", bevelled = \"false\"), Typesetting:-mo(\"− \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mrow(Types etting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invisi bleTimes;\", 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:-munderover(Typesetting:-mo(\"∑\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"true\", symmet ric = \"false\", largeop = \"true\", movablelimits = \"true\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.1666667em\"), Typesettin g:-mrow(Typesetting:-mi(\"i\", italic = \"true\", mathvariant = \"ital ic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn (\"1\", mathvariant = \"normal\")), Typesetting:-mrow(Typesetting:-mfr ac(Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\" n\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mrow( Typesetting:-mn(\"2\", mathvariant = \"normal\")), linethickness = \"1 \", denomalign = \"center\", numalign = \"center\", bevelled = \"false \"), Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mn(\"1\", mathvariant = \"normal\")), accent = \"false\", accentunder = \"false\"), Typesetting:-mi(\"\", italic = \"true\", executable = \+ \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), T ypesetting:-mspace(height = \"0.0ex\", width = \"5.0\", depth = \"0.0e x\", linebreak = \"auto\"), Typesetting:-mrow(Typesetting:-mi(\"f\", i talic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyF unction;\", 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:-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:-mn(\"2\", mathvariant = \" normal\"), Typesetting:-mo(\"⁢\", mathvariant = \"norma l\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetti ng:-mi(\"i\", italic = \"true\", mathvariant = \"italic\")), Typesetti ng:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), subscriptshift = \"0\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"true\", \+ executable = \"true\", font_style_name = \"2D Input\", mathvariant = \+ \"italic\")), mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleT imes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvar iant = \"normal\")), linethickness = \"1\", denomalign = \"center\", n umalign = \"center\", bevelled = \"false\")), Typesetting:-mi(\"\", it alic = \"true\", executable = \"true\", font_style_name = \"2D Input\" , mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0fo nt_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6'-I&mfracGF$6(- F#6'-I#mnGF$6$Q\"1F'/F8Q'normalF'-I#moGF$6-Q1⁢F'FE/%&fe nceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*symmetricGFM/%(largeopGFM /%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0emF'/%'rspaceGFfn-F,6%Q \"hF'F/F7FG-I(mfencedGF$6$-F#6'F+-F#6%-F,6%Q\"fF'F/F7-FH6-Q0&ApplyFunc tion;F'FEFKFNFPFRFTFVFXFZFgn-F]o6$-F#6#-I%msubGF$6%-F,6%Q\"xF'F/F7-F#6 #-F,6%Q\"aF'F/F7/%/subscriptshiftGQ\"0F'FE-FH6-Q\"+F'FEFKFNFPFRFTFVFX/ FenQ,0.2222222emF'/FhnF_q-F#6%FcoFfo-F]o6$-F#6#-F^p6%F`p-F#6#-F,6%Q\"b F'F/F7FhpFEF+FE-F#6#-FB6$Q\"3F'FE/%.linethicknessGQ\"1F'/%+denomalignG Q'centerF'/%)numalignGFhr/%)bevelledGFMF[q-F=6(-F#6'-FB6$Q\"4F'FEFG-F] o6$-F#6'-I+munderoverGF$6'-FH6-Q&∑F'FEFKFN/FQF1FR/FUF1/FWF1FXFZ/Fh nQ,0.1666667emF'-F#6%-F,6%Q\"iF'F/F7-FH6-Q\"=F'FEFKFNFPFRFTFVFX/FenQ,0 .2777778emF'/FhnF\\uFA-F#6%-F,6%Q\"nF'F/F7-FH6-Q(−F'FEFKFNFPFRFT FVFXF^qF`qFA/%'accentGFM/%,accentunderGFMF+-I'mspaceGF$6&/%'heightGQ&0 .0exF'/%&widthGQ$5.0F'/%&depthGF_v/%*linebreakGQ%autoF'-F#6%FcoFfo-F]o 6$-F#6#-F^p6%F`p-F#6#FetFhpFEF+FEFGFinF^rFcrFfrFirF[sFcu-F=6(-F#6'-FB6 $Q\"2F'FEFG-F]o6$-F#6'-Fis6'F[tFct-F#6%-F=6(-F#6%FAFGF`u-F#6#FfwFcrFfr FirF[sFcuFAFfuFhuF+Fju-F#6%FcoFfo-F]o6$-F#6#-F^p6%F`p-F#6%F+-F#6%FfwFG FetF+FhpFEF+FEFGFinF^rFcrFfrFirF[sF+" }{TEXT 216 0 "" }}{PARA 0 "" 0 " " {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 28 "Method 4: Monte Carlo Method" }}{PARA 0 "" 0 "" {TEXT 216 131 "This method is not used ofte n in usual area problems, but has many applications in high energy phy sics problems. The idea is this: " }}{PARA 0 "" 0 "" {TEXT 216 95 " I \+ /(b-a) is the average value of f(x) in the interval [a,b] - see text, \+ Section 6.5, page 375." }}{PARA 0 "" 0 "" {TEXT 216 33 " Therefore, w e do the following:" }}{PARA 0 "" 0 "" {TEXT 216 56 " a. Generat e random numbers in the interval [a,b]." }}{PARA 0 "" 0 "" {TEXT 216 48 " b. Evaluate f(x) at these random numbers." }}{PARA 0 "" 0 " " {TEXT 216 67 " c. Average these values of f(x) and then multi ply by (b-a)." }}{PARA 0 "" 0 "" {TEXT 216 39 "This should get an appr ocximation to I." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 216 1 " " }{TEXT 209 16 "Error Analysis. " }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 " " 0 "" {TEXT 216 84 "We will discuss the derivation of these methods i n class. Briefly, the idea is this:" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 85 "Method 1 (the left-hand rule) approxin mates the integral with a series of rectangles." }}{PARA 0 "" 0 "" {TEXT 216 86 "Method 2 (the trapezoidal rule) approximates the integra l with a series of trapezoids." }}{PARA 0 "" 0 "" {TEXT 216 80 "Method 3 (Simpson's rule) approximates the integral with a series of parabo las." }}{PARA 0 "" 0 "" {TEXT 216 1 " " }}{PARA 0 "" 0 "" {TEXT 216 251 "A \"rigorous\" derivation of the errors associated with these in tegration methods requires some work with series expansions. We are n ot yet prepared for this (Chapter 12). However, an intutitive idea of the error can be obtained in the following way:" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 73 "Method 1 (the left hand rule) evaluates only a constant function exactly." }}{PARA 0 "" 0 "" {TEXT 216 74 "Method 2 (the trapezoidal rule) evaluates only a linear \+ function exactrly." }}{PARA 0 "" 0 "" {TEXT 216 66 "Method 3 (Simpson' s rule) evaluate a quadratic polynomial exactly." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 121 "Error with Method 1: I f you reduce h by a factor of 2 (twice as many points), the error shou ld decrease by a factor of 2." }}{PARA 0 "" 0 "" {TEXT 216 98 "Error w ith method 2: If you reduce h by a factor of 2, the error should decre ase by a factor of 4." }}{PARA 0 "" 0 "" {TEXT 216 101 "Error with met hod 3: If you reduce h by a factor of 2, the eror should be reduced b y a factor of 16." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 185 "You need to demonstrate these errors in your paper. It is suggested that, for a relatively simple function, that you use val ues of n = 6,12,24,48 - n must be even for Simpson's rule.." }}{PARA 0 "" 0 "" {TEXT 216 1 " " }}{PARA 0 "" 0 "" {TEXT 216 111 "The error f or the Monte Carlo method can be enormous. In this case, you need to \+ choose n = 100, 200, 400, etc." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }} {PARA 0 "" 0 "" {TEXT 216 43 "Supose we accept the above error estimat es." }}{PARA 0 "" 0 "" {TEXT 216 61 "Let T(n) be the trapezoidal appro ximation to I with n points." }}{PARA 0 "" 0 "" {TEXT 216 50 "Then, \+ T(n) = I + E(n), where E(n) is the error." }}{PARA 0 "" 0 "" {TEXT 216 39 " T(2n) = I + E(2n) = E(n)/4" }}{PARA 0 "" 0 "" {TEXT 216 51 " T(4n) = I + E(4n) = E(2n)/4 = E(n)/16" }} {PARA 0 "" 0 "" {TEXT 216 35 "Therefore, T(n) - T(2n) = (3/4)E(n)" }} {PARA 0 "" 0 "" {TEXT 216 42 "and T(2n) - T(4n) = (3/16)E(n )." }}{PARA 0 "" 0 "" {TEXT 216 11 "This gives," }{TEXT 210 42 " [ T(4 n) - T(2n) ]/ [ T(2n) - T(n) ] = 1/4" }}{PARA 0 "" 0 "" {TEXT 216 42 " Your calculations should verify this. " }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 79 "Also, we can extrapolate any \+ two calculations to perhaps get a better estimate." }}{PARA 0 "" 0 "" {TEXT 216 21 " I* = T(n) - E(n)" }}{PARA 0 "" 0 "" {TEXT 216 22 " \+ 4I* = 4 T(2n) - En" }}{PARA 0 "" 0 "" {TEXT 216 35 "Subtrtacting 3I = 4T(2n) - T(n) or " }{TEXT 211 37 "I* = (4/3)T(2n) - (1/3)T(n) = S( 2n) " }}{PARA 0 "" 0 "" {TEXT 216 99 " where S(2n) = Simpson's. You \+ need to verify the last statement and to calculate the error of I*." } }{PARA 0 "" 0 "" {TEXT 216 62 "The corresponding equations for Simpson .s ruke are as follows:" }}{PARA 0 "" 0 "" {TEXT 216 4 " " }{TEXT 212 40 "[S(4n) - S(2n) / [ S(2n) - S(n) ] = 1/16" }}{PARA 211 "" 0 "" {TEXT 200 37 " I* = (16/15)S(2n) - (1/16) S(n). " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 215 23 "Some Maple instructions" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 212 "" 0 "" {TEXT 200 19 "A. Random numbers." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 140 "The \+ maple instruction rand(m) produces a random number in the range [0, m- 1]. Dividing by( m-1) produces a random number in the range [0,1]." }} {PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 93 "The cal l to rand produces a proc and, for example, exp(i()) evaluates e at t he random point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "m:= 101; i := rand(m)/(m -1.);" }}{PARA 200 "" 1 "" {MPLTEXT 1 0 9 "exp(i());" }}{PARA 11 "" 1 "" {TEXT 218 0 "" }}{PARA 0 "" 0 "" {TEXT 216 111 "You should also con sider the function randomize - this changes the seed number for the ra ndom number generator." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}}{PARA 0 "" 0 "" {TEXT 216 48 "One way to generate the sum might be as follows." }}{PARA 0 "" 0 "" {MPLTEXT 1 0 15 "a:= 0.;n:=1000;" }}{PARA 0 "" 0 "" {MPLTEXT 1 0 47 "for j from 1 to n do a:= a + exp(i()); end do:" }} {PARA 0 "" 0 "" {MPLTEXT 1 0 11 "area:= a/n:" }}{PARA 0 "" 0 "" {MPLTEXT 1 0 12 "evalf(area);" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }} {PARA 0 "" 0 "" {TEXT 216 446 "In terms of accuracy there are two numb ers you need to investigate: n and m. I suggest you use a sequence su ch as 10, 100, 1000. If, for example, you keep m at 10, yout will eva luate the function at only 10 posssible points (0, .1, .2, ..3, .4, .5 , .6, .7, .8, .9). Consequently, regardless of how large a value you c hoose for n, the accuracy will be limited. Using randomize will chang e the ten points, but there will still be only 10 points." }}{PARA 0 " " 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 8 "B. Loops" }} {PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 40 "A simpl e loop can be written as follows:" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }} {PARA 0 "" 0 "" {TEXT 216 0 "" }{MPLTEXT 1 0 8 "restart;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A:= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:= x- > sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for k from 1 to 10 do A := A + f(k/ 10); end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "b:= A/10:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\")4)Q,&!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }