{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 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 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 "" -1 271 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 }1 0 0 0 8 4 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 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 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 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 256 21 "GRAPHING AN INTEGRAL " }}{PARA 257 "" 0 "" {TEXT 268 1 " " }{TEXT 257 46 "Calculus II Project \+ " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 258 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 116 " To find and grap h an integral of a function whose antiderivative is not found by formu las in a Calculus II course." }}{PARA 0 "" 0 "" {TEXT -1 44 " To use \+ the graphing capabilities of Maple." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 15 "Solved Problem:" }}{PARA 0 "" 0 "" {TEXT -1 28 "Find the antiderivative, g(" }{TEXT 264 1 "x" }{TEXT -1 1 ")" } {TEXT 273 3 ", " }{TEXT -1 7 "of f(" }{TEXT 263 1 "x" }{TEXT -1 8 " ) = sin " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 11 " on [ 0, 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " ]. " }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Verify that g'(" }{TEXT 272 1 "x" }{TEXT -1 6 ") = f(" }{TEXT 274 1 "x" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 26 "(b) Plot the graph of g(" }{TEXT 275 1 "x" }{TEXT -1 28 "), t he antiderivative of f(" }{TEXT 276 1 "x" }{TEXT -1 5 "). " }} {PARA 0 "" 0 "" {TEXT -1 37 "(c) Evaluate g(.1) approximately. " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 " " }{TEXT 260 0 "" }{TEXT 261 10 " Solution:" }}{PARA 0 "" 0 "" {TEXT -1 19 "The integral of f(" }{TEXT 262 1 "x" }{TEXT -1 9 ") = sin " } {XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 113 " cannot be found by any formulas. But by the Fundamental Theorem of Calculus and know ing that the graph of sin " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" } {TEXT -1 99 " over any finite interval lies between -1 and 1, we see that the integral exists and is given by " }{XPPEDIT 18 0 "int(sin*t^ 2,t = 0 .. x);" "6#-%$intG6$*&%$sinG\"\"\"*$%\"tG\"\"#F(/F*;\"\"!%\"xG " }{TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "We define g(" }{TEXT 277 1 "x" }{TEXT -1 38 ") = integral sin( t^2)dt from 0 to " }{TEXT 278 1 "x" } {TEXT -1 1 "." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 265 31 "(a) Verify that g'(x) = f(x)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f : = x -> sin(x^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(-%$sinG6#*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g := x -> int(sin(t^2), t = 0..x); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG% &arrowGF(-%$intG6$-%$sinG6#*$)%\"tG\"\"#\"\"\"/F4;\"\"!9$F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(g(x),x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*$)%\"xG\"\"#\"\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 266 18 "(b) Graph of g(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(g(x), x = 0..2*Pi);\n" }} {PARA 13 "" 1 "" {GLPLOT2D 252 180 180 {PLOTDATA 2 "6%-%'CURVESG6$7\\t 7$$\"\"!F)F(7$$\"+&eb&p8!#5$\"+bHii&)!#87$$\"+!3x`'>F-$\"+h*)GID!#77$$ \"+w&)>hDF-$\"+SHa)f&F67$$\"+j@EJKF-$\"+%[FA7$$\"+`4$ed%F-$\"+oGn$=$FA7$$\"+dhL]_F-$\"+O=C)z%FA7$$\"+`\\$Hf'F- $\"+l`JC%*FA7$$\"+,fpPyF-$\"+vG9i:F-7$$\"+K[8#[)F-$\"+7&=-'>F-7$$\"+jP dE\"*F-$\"+Gm86CF-7$$\"+FM0$z*F-$\"+'3!*3$HF-7$$\"+4L&f/\"!\"*$\"+.'z. ]$F-7$$\"+MvQ76Fgo$\"+90C5TF-7$$\"+g<#)y6Fgo$\"+[8(>v%F-7$$\"+Fv:Z7Fgo $\"+!=+7V&F-7$$\"+%H$\\:8Fgo$\"+qm%>6'F-7$$\"+uMov8Fgo$\"+S:%fp'F-7$$ \"+bO(eV\"Fgo$\"+b1K[sF-7$$\"+UNj.:Fgo$\"+!fC0\"yF-7$$\"+IMRr:Fgo$\"+: k)[G)F-7$$\"+Y:VR;Fgo$\"+XUzZ')F-7$$\"+i'puq\"Fgo$\"+:epu))F-7$$\"+&\\ hQs\"Fgo$\"+!pXp!*)F-7$$\"+GLDSV*)F-7$$\"+g 1#e!=Fgo$\"+v3[G*)F-7$$\"+$\\7A#=Fgo$\"+&)p8/*)F-7$$\"+EVgQ=Fgo$\"+I,; q))F-7$$\"+Wd9)*=Fgo$\"+&\\ymm)F-7$$\"+jrod>Fgo$\"+?5IV$)F-7$$\"+qw[G? 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(This integral can be found by using " }{TEXT 280 3 "int" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the graph of integra l of sin(sqrt(" }{TEXT 281 1 "x" }{TEXT -1 3 "))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 271 11 " Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) Find g(" }{TEXT 282 1 "x" }{TEXT -1 24 ") = integral of exp(-.5*" }{TEXT 283 1 "x" }{TEXT -1 26 "^2) and check your answer." }}{PARA 0 "" 0 "" {TEXT -1 48 "(b) \+ Plot the graph of integral of exp(-.5*x^2)" }}{PARA 0 "" 0 "" {TEXT -1 23 "(c) Evaluate g(2). " }}}{PARA 0 "" 0 "" {TEXT -1 81 "______ ______________________________________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 99 "MSIP Grant # P120A80089-98: \"T hree Urban Calculus Reform Programs: Adopting the Best\" 1998-2001 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 11 " " }}} {MARK "4 6 0 1" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }