{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 1 " " }{TEXT 263 63 "Simpso n's Rule: Using the Error Estimate to Determine Accuracy " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }}{PARA 0 "" 0 "" {TEXT 258 19 "Calculus II Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 259 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "Approximate areas to a given accuracy using Simpso n's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Apply the error estimate for Simpson's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Using Maple allows stude nts to quickly and accurately graph the fourth derivatives needed to a pply the error estimate for Simpson's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Using Simpson's rule, find " }{XPPEDIT 18 0 "int(sin(x^2),x = 0 \+ .. 1);" "6#-%$intG6$-%$sinG6#*$%\"xG\"\"#/F*;\"\"!\"\"\"" }{TEXT -1 29 " to within an accuracy of " }{XPPEDIT 18 0 "10^(-4);" "6#)\"#5, $\"\"%!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " " }{TEXT 2 0 "" }{TEXT -1 47 " \+ " }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 265 103 "U se the error bound for Simpson's rule to compute the number of subdivi sions needed for Simpson's rule." }}{PARA 0 "" 0 "" {TEXT -1 37 " 1. \+ Error bound for Simpson's rule: " }}{PARA 0 "" 0 "" {TEXT -1 20 " \+ If " }{XPPEDIT 271 0 "f^` (4)`;" "6#)%\"fG%%~(4)G" }{TEXT -1 19 " is continuous and " }{TEXT 274 1 "M" }{TEXT -1 41 " is an upp er bound for the values of | " }{XPPEDIT 18 0 "f^` (4)`;" "6#)%\"fG%% ~(4)G" }{TEXT -1 17 "| on [a, b], then" }}{PARA 0 "" 0 "" {TEXT -1 77 " | Error| " }{TEXT 264 2 "< " }{XPPEDIT 18 0 "(b-a)^5*M/(180*n^4);" "6#* (,&%\"bG\"\"\"%\"aG!\"\"\"\"&%\"MGF&*&\"$!=F&*$%\"nG\"\"%F&F(" }} {PARA 0 "" 0 "" {TEXT -1 15 "2. Determine " }{TEXT 273 1 "M" }{TEXT -1 24 " f rom the graph of | " }{XPPEDIT 18 0 "f^` (4)`;" "6#)%\"fG% %~(4)G" }{TEXT -1 3 "| ." }}{PARA 0 "" 0 "" {TEXT -1 13 "3. Compute \+ " }{TEXT 272 1 "n" }{TEXT -1 31 " from the error bound formula." }} {PARA 0 "" 0 "" {TEXT -1 53 "4. Evaluate the integral using Simpson's rule with " }{TEXT 275 1 "n" }{TEXT -1 31 " subdivisions. (Remember that " }{TEXT 266 1 "n" }{TEXT -1 15 " must be even.)" }}{PARA 0 "" 0 "" {TEXT -1 92 "5. Check your accuracy by letting Maple compute the integral and comparing the two answers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 14 "Maple commands" }{TEXT -1 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 39 "To use Simpson's rule: with(student); " }}{PARA 0 "" 0 "" {TEXT -1 76 " e valf(simpson(expression, x = a..b, n));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "To find the fourth derivative: diff (expression, x$4); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f d:= diff(sin(x^2),x$4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG,(*&- %$sinG6#*$)%\"xG\"\"#\"\"\"F.)F,\"\"%F.\"#;*&-%$cosGF)F.F+F.!#[F'!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(abs(fd),x = 0..1, y \+ = 0..50);" }}{PARA 13 "" 1 "" {GLPLOT2D 344 123 123 {PLOTDATA 2 "6%-%' CURVESG6$7U7$\"\"!F(7$$\"1nmm;arz@!#<$\"1'3T\"G^p]GF,7$$\"1LL$e9ui2%F, $\"1%)>^zsep**F,7$$\"1nmm\"z_\"4iF,$\"1A'p_j!>8B!#;7$$\"1mmmT&phN)F,$ \"1%)*)ez6R*=%F97$$\"1LLe*=)H\\5F9$\"1od$\\c*f0mF97$$\"1nm\"z/3uC\"F9$ \"1M7&)G(yXL*F97$$\"1++DJ$RDX\"F9$\"1F%fF&y_l7!#:7$$\"1nm\"zR'ok;F9$\" 15oo_Z\"=m\"FN7$$\"1++D1J:w=F9$\"1_())\\#)Q,6#FN7$$\"1LLL3En$4#F9$\"1% )>O@AaEEFN7$$\"1nm;/RE&G#F9$\"12^pG#yu7$FN7$$\"1+++D.&4]#F9$\"1Z#zQwyD u$FN7$$\"1+++vB_5!#97$$\"1++vV&R Y2aF9$\"11`*G))[/l\"Fjq7$$\"1mm;zXu9cF9$\" 1!\\`)zuSh#=()>Fjq7$$\"1***\\7y%3TiF9$\"1&)G6`,6$4#Fjq7$$\"1****\\P![hY'F9 $\"1\"Gu&42s2AFjq7$$\"1LLL$Qx$omF9$\"1xqhb6'pI#Fjq7$$\"1+++v.I%)oF9$\" 1fv(*H8!yS#Fjq7$$\"1mm\"zpe*zqF9$\"1[t)=!**\\$\\#Fjq7$$\"1+++D\\'QH(F9 $\"1D4mJPpzDFjq7$$\"1KLe9S8&\\(F9$\"1&Rt\"y;L_EFjq7$$\"1***\\i?=bq(F9$ \"1Dq^')F9$\"182ERX 6RGFjq7$$\"1LLe9tOc()F9$\"1Qu\\A[XIGFjq7$$\"1+++]Qk\\*)F9$\"1tEa[0_+GF jq7$$\"1LL$3dg6<*F9$\"15[RD'o@u#Fjq7$$\"1mmmmxGp$*F9$\"1f\"\\akxjm#Fjq 7$$\"1++D\"oK0e*F9$\"1q>G&oS!fDFjq7$$\"1++v=5s#y*F9$\"1+(QqO<)GCFjq7$$ \"\"\"F($\"17RCuE'oD#Fjq-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG 6$Q\"x6\"Q\"yF\\\\l-%%VIEWG6$;F(F][l;F($\"#]F(" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "From the graph, it is s afe to choose " }{TEXT 276 1 "M" }{TEXT -1 6 " = 30." }}{PARA 0 "" 0 " " {TEXT -1 57 "Substitute this value in the error formula and solve fo r " }{TEXT 278 1 "n" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "n:=((1-0)^5*30/(1 80*10^(-4)))^(1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG,$*&)\"%+ ]#\"\"\"\"\"%F*)\"\"$#F-F+F*#F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 279 1 "n" }{TEXT -1 6 " = 8. " }{TEXT 280 4 "Why?" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 6 "Check:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "est:=evalf(simpson(sin(x^2), x = 0..1,8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"+C`[-J!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "act:=evalf(int(sin(x^2),x = 0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$actG$\"+8Io-J!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Error:=act-est;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &ErrorG$\"'*o(>!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "___________ _______________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" } {TEXT 268 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 269 10 "Problem \+ 1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Using Simpson's rule, find " }{XPPEDIT 18 0 "int(exp(x^ 2),x = 0 .. 1);" "6#-%$intG6$-%$expG6#*$%\"xG\"\"#/F*;\"\"!\"\"\"" } {TEXT -1 29 " to within an accuracy of " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Using Simpson's rule, find " }{XPPEDIT 18 0 "int(exp(cos(x)),x = 0 . . Pi);" "6#-%$intG6$-%$expG6#-%$cosG6#%\"xG/F,;\"\"!%#PiG" }{TEXT -1 29 " to within an accuracy of " }{XPPEDIT 18 0 "10^(-7);" "6#)\"#5, $\"\"(!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "_____________________________________ __________________________________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Refor m programs: Adopting the Best\" 1998-2001, MSEIP Grant #P120A010031: \+ \"Four Colleges: Calculus + Enhancements\" 2001-04" }}{PARA 11 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "2 0 1" 10 }{VIEWOPTS 1 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }