{VERSION 6 1 "Linux" "6.1" } {USTYLETAB {PSTYLE "_pstyle10" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Da sh Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 3 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle9" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 2 }{PSTYLE "_pstyle7" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal256" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyl e4" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle3" -1 206 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pst yle1" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Headin g 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Au thor" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "L eft Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed \+ Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 5 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "D iagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle12" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle11" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "" 0 1 147 0 15 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Ti mes" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Define" -1 200 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "" 0 1 0 128 128 1 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle2" -1 201 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle1" -1 202 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 0 1 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "" 1 8 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 0 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 203 "Time s" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Tim es" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Co urier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "T imes" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle9" -1 204 "Time s" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 205 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 206 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "" 1 10 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 207 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 208 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 209 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 210 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 211 "Times" 1 12 167 17 223 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Highlight" -1 212 "" 0 1 0 255 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 0 1 255 0 0 1 0 0 2 2 1 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 213 "Courier" 1 12 200 0 200 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Help Norm al" -1 30 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Output " -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus " -1 36 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 0 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle18" -1 214 "Times" 1 12 0 255 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle17" -1 215 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle16" -1 216 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle15" -1 217 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 218 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle14" -1 219 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle13" -1 220 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle12" -1 221 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle11" -1 222 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle10" -1 223 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "Cou rier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle259" -1 224 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle258" -1 225 "" 1 14 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 226 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle257" -1 227 "" 1 14 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "" 0 1 104 64 92 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier " 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 167 17 223 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle5" -1 228 "Times" 1 12 167 17 223 1 1 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle8" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle19" -1 229 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle13" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 18 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 }{CSTYLE "_cstyle2 0" -1 230 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle14 " -1 214 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle21" -1 231 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle15" -1 215 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle22" -1 232 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle16" -1 216 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle23" -1 233 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle17" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle24" -1 234 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle25" -1 235 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle26" -1 236 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle18" -1 218 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle27" -1 237 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle19" -1 219 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 }{CSTYLE "_cstyle28" -1 238 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle20" -1 220 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2 1" -1 221 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle22" -1 222 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 }{CSTYLE "_cstyle29" -1 239 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle30" -1 240 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle23" -1 223 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cst yle31" -1 241 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_csty le32" -1 242 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyl e33" -1 243 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle 24" -1 224 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle34" -1 244 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle35" -1 245 "Times" 1 12 0 255 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle25" -1 225 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 }{CSTYLE "_cstyle36" -1 246 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle26" -1 226 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle37" -1 247 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle27" -1 227 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 211 "" 0 "" {TEXT 228 36 " Worksheet 4, Math 152 Fal l 2005" }{TEXT 228 0 "" }}{PARA 212 "" 0 "" {TEXT 229 0 "" }}{PARA 213 "" 0 "" {TEXT 230 22 "Integration with Maple" }{TEXT 230 0 "" }} {PARA 214 "" 0 "" {TEXT 231 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 232 8 "restart;" }{MPLTEXT 1 232 0 "" }}}{PARA 214 "" 0 "" {TEXT 231 0 "" }}{SECT 1 {PARA 216 "" 0 "" {TEXT 233 49 "Introduction \+ to Improper Integrals & Applications" }{TEXT 233 0 "" }}{SECT 1 {PARA 216 "" 0 "" {TEXT 233 12 "Introduction" }{TEXT 233 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 234 65 "One of the most famous curves in mathe matics is the graph of the " }{TEXT 235 23 "normal density function" } {TEXT 234 1 " " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 12 " defined \+ by" }{TEXT 234 0 "" }{TEXT 234 18 "\n " }{XPPEDIT 18 0 "f(x) = 1/(sigma*sqrt(2*Pi))" "6#/-%\"fG6#%\"xG*&\"\"\"F)*&%&sigmaGF )-%%sqrtG6#*&\"\"#F)%#PiGF)F)!\"\"" }{TEXT 234 2 " " }{XPPEDIT 18 0 " exp(-(x-mu)^2/(2*sigma^2))" "6#-%$expG6#,$*&,&%\"xG\"\"\"%#muG!\"\"\" \"#*&F-F**$%&sigmaGF-F*F,F," }{TEXT 234 2 " ," }{TEXT 234 0 "" }{TEXT 234 7 "\nwhere " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT 234 14 " (calle d the " }{TEXT 235 4 "mean" }{TEXT 234 6 ") and " }{XPPEDIT 18 0 "sigm a" "6#%&sigmaG" }{TEXT 234 14 " (called the " }{TEXT 235 18 "standard deviation" }{TEXT 234 67 ") are constants that together determine the shape of the graph of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 2 " \+ ." }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 55 "f:=x->1/(s igma*sqrt(2*Pi))*exp(-(x-mu)^2/(2*sigma^2)); " }{MPLTEXT 1 232 0 "" }} }{EXCHG {PARA 217 "" 0 "" {TEXT 234 25 "The bell-shaped graph of " } {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 130 " is frequently seen in co nnection with data that are said to be \"normally distributed\", such \+ as test scores or heights of people." }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 17 "mu:=0: sigma:=1: " }{MPLTEXT 1 232 0 "" } {MPLTEXT 1 232 22 "\nplot(f(x), x=-5..5); " }{MPLTEXT 1 232 0 "" } {MPLTEXT 1 232 26 "\nmu:='mu': sigma:='sigma':" }{MPLTEXT 1 232 0 "" } }}{EXCHG {PARA 217 "" 0 "" {TEXT 234 9 "In fact, " }{TEXT 234 0 "" } {TEXT 234 19 "\n the graph of " }{XPPEDIT 18 0 "f" "6#%\"fG" } {TEXT 234 40 " is symmetric with respect to the line " }{XPPEDIT 18 0 "x=mu" "6#/%\"xG%#muG" }{TEXT 234 6 " , and" }{TEXT 234 0 "" }{TEXT 234 27 "\n the maximum value of " }{XPPEDIT 18 0 "f" "6#%\"fG" } {TEXT 234 5 " is " }{XPPEDIT 18 0 "f(mu) = 1 /(sigma*sqrt(2*Pi))" "6# /-%\"fG6#%#muG*&\"\"\"F)*&%&sigmaGF)-%%sqrtG6#*&\"\"#F)%#PiGF)F)!\"\"" }{TEXT 234 2 " ." }{TEXT 234 0 "" }{TEXT 234 59 "\nTo verify the latt er claim we find the critical points of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 34 " and evaluate the function there:" }{TEXT 234 0 "" }} {PARA 215 "> " 0 "" {MPLTEXT 1 232 13 "fprime:=D(f);" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 38 "\ncriticalpoints:=solve(fprime(x)=0,x);" } {MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 19 "\nf(criticalpoints);" } {MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 234 53 "Furtherm ore, the graph of f has inflection points at " }{XPPEDIT 18 0 "mu" "6# %#muG" }{TEXT 234 4 " +/-" }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT 234 2 " :" }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 17 "f2 prime:=D(D(f));" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 23 "\nsolve(f2pr ime(x)=0,x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 234 13 "The function " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT 234 12 " d efined by" }{TEXT 234 0 "" }{TEXT 234 16 "\n " } {XPPEDIT 18 0 "F(b) = int(f(x), x=-infinity..b)" "6#/-%\"FG6#%\"bG-%$i ntG6$-%\"fG6#%\"xG/F.;,$%)infinityG!\"\"F'" }{TEXT 234 1 " " }{TEXT 234 0 "" }{TEXT 234 13 "\nis called a " }{TEXT 235 31 "normal probabil ity distribution" }{TEXT 234 1 "." }{TEXT 234 0 "" }{TEXT 234 73 "\nAn integral of this form (with one of its bounds at infinity) is called \+ " }{TEXT 236 18 "improper integral." }{TEXT 234 1 "." }{TEXT 234 0 "" }{TEXT 234 55 "\nIt turns out that the special way in which we defined " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 16 " above leads to" } {TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 67 "assume(sigma>0 ): # we use this command to ensure sigma is positive" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 35 "\nInt(f(x),x=-infinity..infinity) = " } {MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 33 "\nint(f(x),x=-infinity..infini ty);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 234 65 "T his means that the entire area of the region under the graph of " } {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 16 " is equal to 1 " }{TEXT 234 0 "" }{TEXT 234 25 "\n(for any choice of mean " }{XPPEDIT 18 0 "mu " "6#%#muG" }{TEXT 234 25 " and standard deviation " }{XPPEDIT 18 0 " sigma" "6#%&sigmaG" }{TEXT 234 3 " )." }{TEXT 234 0 "" }{TEXT 234 53 " \nThe fraction of quantities whose values are between " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 234 6 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" } {TEXT 234 18 " is then given by" }{TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 4 " " }{XPPEDIT 18 0 "R[ab] = int(f(x), x=a..b)" "6#/&%\" RG6#%#abG-%$intG6$-%\"fG6#%\"xG/F.;%\"aG%\"bG" }{TEXT 234 4 " = " } {XPPEDIT 18 0 "F(b) - F(a)" "6#,&-%\"FG6#%\"bG\"\"\"-F%6#%\"aG!\"\"" } {TEXT 234 2 " ." }{TEXT 234 0 "" }{TEXT 234 19 "\nIn particular, if " }{XPPEDIT 18 0 "mu=0" "6#/%#muG\"\"!" }{TEXT 234 3 " , " }{XPPEDIT 18 0 "sigma=1" "6#/%&sigmaG\"\"\"" }{TEXT 234 3 " , " }{XPPEDIT 18 0 "a=- 1" "6#/%\"aG,$\"\"\"!\"\"" }{TEXT 234 6 " and " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT 234 7 " , then" }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 29 "mu:=0: sigma:=1: a:=-1: b:=1:" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 29 "\nR[ab] := int(f(x), x=a..b); " } {MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 234 116 "We see \+ that an answer of this form is so important that Maple even has its ow n function to denote it (the so-called " }{TEXT 235 14 "error function " }{TEXT 234 16 "). The value of " }{XPPEDIT 18 0 "R[ab]" "6#&%\"RG6#% #abG" }{TEXT 234 4 " is" }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 13 "evalf(R[ab]);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 234 41 "We can interpret this answer as follow s: " }{TEXT 234 0 "" }{TEXT 234 38 "\n68% of the region under the grap h of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 25 " lies between the \+ lines " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT 234 6 " \+ and " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT 234 3 " . " }{TEXT 234 0 "" }{TEXT 234 80 "\nIn other words, the probability for a quanti ty to have value between -1 and +1 " }{TEXT 234 0 "" }{TEXT 234 42 "\n (for normally distributed data with mean " }{XPPEDIT 18 0 "mu=0" "6#/% #muG\"\"!" }{TEXT 234 25 " and standard deviation " }{XPPEDIT 18 0 "s igma=1" "6#/%&sigmaG\"\"\"" }{TEXT 234 10 " ) is 68%." }{TEXT 234 0 "" }{TEXT 234 53 "\nThe previous result holds in a very general setting: " }{TEXT 234 0 "" }{TEXT 234 5 "\nFor " }{TEXT 236 3 "any" }{TEXT 234 6 " mean " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT 234 30 " , and any sta ndard deviation " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT 234 48 " \+ , the region under the curve between the lines " }{XPPEDIT 18 0 "x=mu- sigma" "6#/%\"xG,&%#muG\"\"\"%&sigmaG!\"\"" }{TEXT 234 0 "" }{TEXT 234 7 "\n and " }{XPPEDIT 18 0 "x=mu+sigma" "6#/%\"xG,&%#muG\"\"\"%&s igmaGF'" }{TEXT 234 30 " is 68% of the entire region " }{TEXT 234 0 " " }{TEXT 234 72 "\n(recall that these lines were associated with the i nflection points of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 234 3 " ):" }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 51 "mu:='mu': sig ma:='sigma': a:=mu-sigma: b:=mu+sigma:" }{MPLTEXT 1 232 0 "" } {MPLTEXT 1 232 36 "\nR[ab] := evalf(int(f(x), x=a..b)); " }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 237 115 "Thus, for normally distributed data, one can expect 68% of the data to lie within 1 stan dard deviation of the mean." }{TEXT 237 0 "" }{TEXT 237 203 "\nTo have a concrete example, let's assume that the mean of the heights of all \+ IIT students is 68 inches, the standard deviation is 2 inches, and tha t the heights are (approximately) normally distributed." }{TEXT 237 0 "" }{TEXT 237 60 "\nQ: How many students have heights between 66 and 7 0 inches?" }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 9 "A: Since " }{XPPEDIT 18 0 "mu = 68" "6#/%#muG\"#o" }{TEXT 234 6 " and " } {XPPEDIT 18 0 "sigma=2" "6#/%&sigmaG\"\"#" }{TEXT 234 95 " , by the ob servations made above, 68% of the student should be between 66 and 70 \+ inches tall. " }{TEXT 234 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 31 "mu:=68: sigma:=2: a:=66: b:=70:" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 35 "\nR[ab] := evalf(int(f(x), x=a..b));" }{MPLTEXT 1 232 0 "" }}} {EXCHG {PARA 218 "" 0 "" {TEXT 237 55 "Q: How many students are betwee n 67 and 72 inches tall?" }{TEXT 237 0 "" }}{PARA 215 "> " 0 "" {MPLTEXT 1 232 27 "evalf(int(f(x), x=67..72));" }{MPLTEXT 1 232 0 "" } }}{EXCHG {PARA 214 "" 0 "" {TEXT 231 0 "" }}}}{SECT 1 {PARA 216 "" 0 " " {TEXT 233 37 "Exercise (Please complete #2 and #3.)" }{TEXT 233 0 "" }}{SECT 1 {PARA 219 "" 0 "" {TEXT 238 203 "1. (Sample) Suppose the we ights of newly born babies are normally distributed, and the mean weig ht of 800 babies born in a certain hospital in the US is 7.6 pounds, w ith a standard deviation of 1 pound." }{TEXT 238 0 "" }{TEXT 238 1 " \n" }}{PARA 220 "" 0 "" {TEXT 234 22 "a) Approximately what " }{TEXT 236 11 "percentage " }{TEXT 234 61 "of babies would have weighed betwe en 7 and 8 pounds at birth?" }{TEXT 234 0 "" }}{PARA 221 "" 0 "" {TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 7 "b) What" }{TEXT 236 12 " percentage " }{TEXT 234 45 "of babies would have weighed under 6 \+ pounds? " }{TEXT 234 0 "" }}{PARA 214 "" 0 "" {TEXT 231 0 "" }}{PARA 217 "" 0 "" {TEXT 234 7 "c) What" }{TEXT 236 12 " percentage " }{TEXT 234 44 "of babies would have weighed over 9 pounds? " }{TEXT 234 0 "" }}{PARA 214 "" 0 "" {TEXT 231 0 "" }}{PARA 217 "" 0 "" {TEXT 234 65 "d ) Suppose we gather data from a hospital in another part of the " } {TEXT 236 6 "world?" }{TEXT 234 42 " How would you expect the data to \+ change? " }{TEXT 234 0 "" }}}{SECT 1 {PARA 222 "" 0 "" {TEXT 239 28 "2 . A nonnegative function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 239 15 " is called a " }{TEXT 240 28 "probability density function" } {TEXT 239 3 " if" }{TEXT 239 0 "" }}{PARA 223 "" 0 "" {TEXT 241 0 "" } }{PARA 217 "" 0 "" {TEXT 234 41 " \+ " }{XPPEDIT 18 0 "int(f(t),t=-infinity..infinity)=1" "6#/-%$intG6$ -%\"fG6#%\"tG/F*;,$%)infinityG!\"\"F.\"\"\"" }{TEXT 234 2 " ." }{TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 10 " The " }{TEXT 236 11 " probability" }{TEXT 234 6 " that " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 234 15 " lies between " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 234 6 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 234 13 " is given \+ by" }{TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 26 " \+ P(" }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 234 2 " " }{TEXT 242 1 "<" }{TEXT 234 3 " t " }{TEXT 242 1 "<" }{TEXT 234 1 " " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 234 5 " ) = " }{XPPEDIT 18 0 "int(f(t),t=a.. b)" "6#-%$intG6$-%\"fG6#%\"tG/F);%\"aG%\"bG" }{TEXT 234 3 " ." } {TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 10 " The " }{TEXT 236 14 "expected value" }{TEXT 234 19 " of t is given by " }{TEXT 234 0 "" }}{PARA 217 "" 0 "" {TEXT 234 24 " " } {XPPEDIT 200 0 "E = int(t*f(t),t = -infinity .. infinity);" "6#/%\"EG- %$intG6$*&%\"tG\"\"\"-%\"fG6#F)F*/F);,$%)infinityG!\"\"F1" }{TEXT 243 1 " " }{TEXT 234 1 "." }{TEXT 234 0 "" }}{PARA 218 "" 0 "" {TEXT 237 5 " " }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 7 " " } {TEXT 236 165 "A manufacturer markets a light bulb that has an average life expectancy of 3000 hours. Experience has shown that lifetimes of electrical equipment often follow an " }{TEXT 243 29 "exponential p robability model" }{TEXT 235 1 " " }{TEXT 234 29 " \+ " }{TEXT 234 0 "" }}{PARA 224 "" 0 "" {XPPEDIT 227 0 "f(t) = \+ lambda*exp(-lambda*t);" "6#/-%\"fG6#%\"tG*&%'lambdaG\"\"\"-%$expG6#,$* &F)F*F'F*!\"\"F*" }{TEXT 244 1 " " }{TEXT 236 5 " for " }{XPPEDIT 225 0 "0 <= t;" "6#1\"\"!%\"tG" }{TEXT 244 1 " " }{TEXT 236 5 " and " } {XPPEDIT 224 0 "f(t) = 0;" "6#/-%\"fG6#%\"tG\"\"!" }{TEXT 236 6 " for " }{XPPEDIT 212 0 "t < 0;" "6#2%\"tG\"\"!" }{TEXT 245 1 " " }{TEXT 236 12 ", (1)" }{TEXT 234 0 "" }}{PARA 214 "" 0 "" {TEXT 231 0 "" }}{PARA 222 "" 0 "" {TEXT 239 9 "where 1/ " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 239 66 " is the equipment's average life expe ctancy. In this case, 1/ " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT 239 19 " is equal to 3000." }{TEXT 239 0 "" }}{PARA 218 "" 0 "" {TEXT 237 6 " " }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 24 "(a) Is the integral " }{XPPEDIT 18 0 "int(f(t),t = -infinity . . infinity);" "6#-%$intG6$-%\"fG6#%\"tG/F);,$%)infinityG!\"\"F-" } {TEXT 234 30 " proper or improper, where " }{XPPEDIT 18 0 "f(t)" "6 #-%\"fG6#%\"tG" }{TEXT 234 116 " is given in eq. (1)? Explain your a nswer by citing the definition of a proper/ improper integral in the t extbook." }{TEXT 234 0 "" }}{PARA 214 "" 0 "" {TEXT 231 0 "" }}{PARA 217 "" 0 "" {TEXT 234 17 "(b) Show that " }{XPPEDIT 18 0 "f" "6#%\" fG" }{TEXT 234 63 " , given in eq. (1) above, is a probability densi ty function." }{TEXT 234 0 "" }}{PARA 218 "" 0 "" {TEXT 237 6 " " }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 17 "(c) Find P(500 " }{TEXT 242 1 "<" }{TEXT 234 3 " t " }{TEXT 242 1 "<" }{TEXT 234 76 " 1 000). What does the answer mean, in terms of the lifetime of a lightb ulb?" }{TEXT 234 0 "" }}{PARA 218 "" 0 "" {TEXT 237 6 " " }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 11 "(d) Find " }{TEXT 236 2 " E." }{TEXT 234 12 " What does " }{TEXT 236 2 "E " }{TEXT 234 57 "turn out to be, in terms of the lifetime of a lightbulb?" }{TEXT 234 0 "" }}{PARA 218 "" 0 "" {TEXT 237 6 " " }{TEXT 237 0 "" }}{PARA 218 "" 0 "" {TEXT 237 63 "(e) What is the relationship between integral an d probability? " }{TEXT 237 0 "" }}{PARA 217 "" 0 "" {TEXT 234 70 " \+ Hint: See how probability was defined above, and recall the " } {TEXT 236 9 "geometric" }{TEXT 234 24 " meaning of an integral." } {TEXT 234 0 "" }}{PARA 223 "" 0 "" {TEXT 241 0 "" }}}{SECT 1 {PARA 225 "" 0 "" {TEXT 246 166 " 3. Suppose the annual incomes of the famil ies in Chicago in 2004 were normally distributed with the mean value 5 0K dollars and the standard deviation of 30K dollars." }{TEXT 246 0 "" }}{PARA 223 "" 0 "" {TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 88 " a) What percentage of families would have had annual income between 45 K and 55K dollars?" }{TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 79 "b) What percentage of families woul d actually have had negative income in 2004?" }{TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 129 "c) If th e standard deviation was not 30K but 20K, how would the distribution c urve changed? What could you tell from this change?" }{TEXT 241 0 "" } }{PARA 223 "" 0 "" {TEXT 241 0 "" }}{PARA 223 "" 0 "" {TEXT 241 70 "d) What changes may be expected if predicting about the year of 2014?" }{TEXT 241 0 "" }}}}}{PARA 226 "" 0 "" {TEXT 247 0 "" }}{PARA 227 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }