{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 32 "Some Scratch Paper for Lecture 6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Filen ame: Lecture6.mws" }}{PARA 0 "" 0 "" {TEXT -1 32 "Author: Arthur \+ C. Heinricher" }}{PARA 0 "" 0 "" {TEXT -1 24 "Updated: March 29, 2000 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "This worksheet contains some examples doing numerical in tegration with the Trapezoidal Rule and Simpson's Rule" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 10 "Example #1 " }}{PARA 0 "" 0 "" {TEXT -1 76 "An integral that you cannot do by han d: integrating under the bell curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(exp(-x^2), x=0..1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(exp(-x^2),x=-3..3, view=[-4..4,0..1.5],\n title=`The Bell Curve`);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(exp(-x^2 ), x=-infinity..infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 40 "You can fo rce a number if you need it: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "exact1 :=evalf(int(exp(-x^2), x=0..1),20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Example # 1... Again" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "There is a command in the " }{HYPERLNK 17 "student" 2 "student " "" }{TEXT -1 124 " library that does the trapezoidal rule (command = trapezoid) for you. First open the library and then do a few trapezo ids:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(student):\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "trapezoid(exp(-x^2), x=0..1 ,3);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "Nice formula, but you really wan t the number. So ask. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e valf(trapezoid(exp(-x^2), x=0..1,3));" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(trapezoid(exp(-x^2 ), x=0..1,6));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(tra pezoid(exp(-x^2), x=0..1,12));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(trapezoid(exp(-x^2), x=0..1,24));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "evalf(trapezoid(exp(-x^2), x=0..1,100));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "E rror Estimates" }}{PARA 0 "" 0 "" {TEXT -1 55 "There is a nice error f ormula for the trapezoidal rule:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 " Error = - " }{XPPEDIT 18 0 "(b-a)^3/(12*n^2)" "*&,&%\"bG\"\"\"%\"aG!\"\"\"\"$*&\"#7F%*$%\"nG \"\"#F%F'" }{TEXT -1 3 " . " }{XPPEDIT 18 0 "D^2f(eta)" "*&%\"DG\"\"#- %\"fG6#%$etaG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "Let's test it a little:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ddf :=diff (exp(-x^2),x,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(abs(ddf),x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 83 "The maximum value is 2 and it occurs at x=0. Use it to o btain your error estimate:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "estimate := proc(n) evalf(1^3/(12*n ^2)*2) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 70 "No w compare the \"actual error\" with the error estimate just obtained. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "T3 := evalf(trapezoid(ex p(-x^2), x=0..1,3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E3 \+ := abs(exact1-T3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "estim ate(3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T20 := evalf(tr apezoid(exp(-x^2), x=0..1,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "E20 := abs(exact1-T20), estimate(20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T50 := evalf(trapezoid(exp(-x^2), x=0..1,50));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "E50 := abs(exact1-T50), e stimate(50);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Example #1 : And Again!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Running Simpson's rule is just as easy as running the tra pezoidal rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(student):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simpson(exp(-x^2), x=0..1,4);" }}}{PARA 0 "" 0 "" {TEXT -1 111 "Notice that I had to use an even number of intervals! A s before, you still have to ask if you want a number.. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(simpson(exp(-x^2), x=0..1,4)) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(simpson(exp(-x^2), x=0..1,8));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(simpson(exp(-x^2), x=0..1,100 ));" }}}{PARA 0 "" 0 "" {TEXT -1 111 "You should see that the approxim ation gets better (more digits match the exact solution) quicker with \+ Simpson. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 "You can be more precise and do the same kind of error analysis fo r Simpson as you did for Trapezoids. Basic question: If you double t he number of subintervals, what happens to the error? With trapezoids , it was divided by 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{MARK "11 0 0" 17 }{VIEWOPTS 1 1 0 1 1 1803 }