{VERSION 7 1 "Windows XP" "7.1" } {USTYLETAB {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 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 }{PSTYLE "Ord ered List 5" -1 200 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 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 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 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 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 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Lef t Justified Maple Output" -1 12 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 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 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 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Headi ng 2" -1 4 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 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 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 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Tim es" 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 } {PSTYLE "Normal" -1 0 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 }{PSTYLE "Annotation Title" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 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 3 3 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 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 204 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 3 3 2 108 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 1 0 0 1 }3 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 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 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 1 0 0 1 }1 1 0 0 3 3 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 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 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 1 0 0 1 }1 1 0 0 8 4 2 0 2 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 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 205 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 3 3 2 36 2 0 2 2 -1 1 } {CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small" -1 201 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 202 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 203 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Bold \+ Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math " -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Annotation T ext" -1 204 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help No tes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Und erlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 } {CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 205 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Tim es" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 206 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 207 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 208 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 209 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 210 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 211 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Tim es" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 214 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 216 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 } {CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "Times" 1 12 255 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 222 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 223 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 224 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 225 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 226 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 227 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 228 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 229 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 230 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 231 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 232 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 233 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 234 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 235 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 236 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 237 "Times" 1 12 255 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 238 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 239 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 240 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 241 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 242 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 243 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 244 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 245 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 246 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 247 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 248 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 249 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 250 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 251 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 252 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 253 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 254 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 255 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 256 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 257 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 258 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 259 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 260 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 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 "" -1 261 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 262 "Ti mes" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 263 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 264 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 265 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 266 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 267 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 268 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 269 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 270 "Times" 1 12 0 0 128 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 271 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 272 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 273 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 274 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 275 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 276 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 277 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 278 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 279 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 280 "Tim es" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 281 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 282 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 283 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 284 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 285 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 286 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 287 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 288 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 289 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 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 "" -1 290 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 291 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 292 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 293 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 294 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 295 "Tim es" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 296 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 297 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 298 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 299 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 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 "" -1 300 "Ti mes" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 301 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 302 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 303 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 304 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 305 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 210 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 "" -1 306 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 307 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 308 "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 2 2 2 2 2 1 0 0 1 }3 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 }{CSTYLE "" -1 309 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 213 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 }{PSTYLE "" -1 214 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 }{PSTYLE "" -1 215 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 "" -1 310 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 311 " Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 312 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -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 2 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 313 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 217 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 "" -1 314 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 218 1 {CSTYLE "" -1 -1 "Tim es" 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 } {PSTYLE "" -1 219 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 }{PSTYLE "" -1 220 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 "" -1 315 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 316 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 200 36 "Introduction to Maple in Calculus II" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 4 "" 0 "" {TEXT 318 16 "Maple Worksheets" }}{PARA 0 " " 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 117 "In MTH 142 we s hall continue our work with Maple. Similarly as last semester, our wor k is going to be organized into " }{TEXT 206 10 "worksheets" }{TEXT 317 216 " like this one. A worksheet consists of text and Maple comman d lines. Portions of a worksheet containing text and portions containi ng Maple command lines marked by vertical lines on the left of the scr een are called " }{TEXT 207 16 "execution groups" }{TEXT 317 246 ". It is very important for you to learn how to toggle between the text mod e and the command line mode. Your homework problems will require of yo u opening a new Maple worksheet, using Maple to plot functions, perfor m calculations etc., as well as " }{TEXT 208 57 "entering your answers and comments in complete sentences " }{TEXT 317 162 "using the text m ode. As most of you already know, you toggle between the text mode and the Maple command mode by pressing the two buttons on the toolbar mar ked \"" }{TEXT 209 1 "T" }{TEXT 317 19 "\" for text, and \"" }{TEXT 210 2 "[>" }{TEXT 317 165 "\" for Maple command prompt. Right now we a re, of course, in the text mode. As an excercise, place the cursor at \+ the end of this sentence, press the button marked \"" }{TEXT 211 2 "[> " }{TEXT 317 44 "\" and see a new Maple prompt \">\" appear. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 29 "Type a simple command at the " }{TEXT 212 5 "first" } {TEXT 317 13 " prompt, say " }{TEXT 213 4 "2+2;" }{TEXT 317 221 " (Don 't forget the semicolon!) Press \"Enter\". Maple output, hopefully 4, \+ appears and the cursor jumps to the next prompt. (If there was none be low, Maple would create one). With the cursor at the new prompt press \+ the \"" }{TEXT 214 1 "T" }{TEXT 317 155 "\" button. The prompt disappe ars and the mode changes to the text mode. Type something, say your na me. Then switch back to the command mode by pressing \"" }{TEXT 215 2 "[>" }{TEXT 317 484 "\" again. And so it goes. Should you ever want \+ to insert a command line before already typed text go to the \"Insert \" menu, choose \"Execution Group\" and click on \"Before Cursor\". Be aware of one thing though, you can't insert a command prompt in the m iddle of an execution group which consists of already typed text. If y ou are within already typed worksheet, after you click on any command, the cursor jumps to the next command line. If there is none below, on e will be created." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 145 "If you want Maple to do something, you type your comma nd at a command prompt and press \"Enter\". Your success with Maple de pends on your use of " }{TEXT 216 26 "correct and precise syntax" } {TEXT 317 386 ". Syntax is a way of entering your commands. Last semes ter we learned quite a bit of Maple syntax and we shall continue using it this semster, as well as introduce new commands. For those of you \+ who want to refresh your memory or who didn't take MTH 141 last semest er, we review below Maple syntax used in Calculus I. Remember, also, t hat whenever in doubt, you can refer to the manual " }{TEXT 217 26 "Ge tting Started with Maple" }{TEXT 317 315 " that came with your textboo k or to this introductory worksheet. Those of you who have never worke d with Maple should not get discouraged by the amount of syntax review ed in the next section. You don't have to memorize it all at once. You shall learn it as you work on examples in this and in subsequent work sheets." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 218 6 "Note: " }{TEXT 317 198 "As you probably learned the hard way last s emester, whenever you open a worksheet and want to do more with it tha n only viewing, if you want to change commands, enter new commands etc ., you have to " }{TEXT 219 10 "re-execute" }{TEXT 220 1 " " }{TEXT 317 294 "all the commands in the worksheet; that is you have to click \+ \"Enter\" on all command lines, even if they have been saved and their outputs appear already on screen. Otherwise, Maple will not recognize functions and expressions defined in the worksheet and you will get s trange error messages. " }}{PARA 0 "" 0 "" {TEXT 317 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 4 " " 0 "" {TEXT 318 33 "Review of Calculus I Maple Syntax" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 366 "Each of the sectio ns below describes syntax related to a specific topic. If you want to \+ view any of the sections just click on the \"+\" button (right-pointin g arrow in Maple 10). The \"+\" will be replaced with \"-\" (down-poin ting arrow in Maple 10) and the section will open. If a section contai ns material familiar to you, click on \"-\". The section will close." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 35 "In al l sections below the symbol \"" }{TEXT 221 1 "%" }{TEXT 317 37 "\" is \+ used. In Release 5 of Maple, \"" }{TEXT 222 1 "%" }{TEXT 317 20 "\" de notes the last " }{TEXT 223 8 "executed" }{TEXT 317 84 " output. That \+ is, the output of the last command on which you have clicked. (It was \+ " }{TEXT 224 2 "\"" }{TEXT 317 36 " in earlier releases.) The symbol \+ \"" }{TEXT 225 1 "%" }{TEXT 317 121 "\" easily leads to confusion and \+ it should preferably be used on the same command line as the command t o which it refers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 12 "Basic syntax" } }{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 544 "To ma ke Maple do something you type your command at the Maple prompt \">\". Your commands will usually appear in red on screen. You have to end e ach command by a semicolon and then press \"Enter\". Maple will return its output that usually appears in blue. (Ocassionally, when you do n ot want Maple to print out the output, you end your command with a col on.) The best way to learn the correct Maple syntax is by examples. W e shall switch to the command mode now and ask Maple to perform some s imple calculations. Let's ask Maple to calculate " }{XPPEDIT 18 0 "(2) (359+23)-(5+21)/(2+11)+2^9/32456789999;" "6#,(-\"\"#6#,&\"$f$\"\"\"\"# BF)F)*&,&\"\"&F)\"#@F)F),&F%F)\"#6F)!\"\"F1*&)F%\"\"*F)\",****ycC$F1F) " }{TEXT 317 107 " . We type our command at the prompt, end it with a \+ semicolon, press \"Enter\" and Maple returns an answer." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "2*(359 +23)-(5+21)/(2+11)+25^9/32456789999;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6##\"/j[CrnaG\",****ycC$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 142 "As you see, the rules of e ntering the arithmetical operations are similar to those on your calcu lator. Multiplication is always denoted by \"" }{TEXT 226 1 "*" }{TEXT 317 19 "\" , division by \"" }{TEXT 227 1 "/" }{TEXT 317 25 "\" , exp onentiation by \"" }{TEXT 228 1 "^" }{TEXT 317 250 "\", addition and s ubtraction in an obvious way. The proper use of paretheses is, of cour se, crucial to the correct syntax. As in the above example, whenever p ossible Maple will return an exact answer. If you want a decimal form, or in other words, a " }{TEXT 229 28 "floating point approximation" } {TEXT 317 47 " of your answer, you have to use the command \"" }{TEXT 230 5 "evalf" }{TEXT 317 100 "\" .( The \"f\" at the end of the comman d stands for \"floating point\", \"eval\" for \"evaluate\".)" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+KcJ&z)!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 32 "As mentioned above the symbol \"" }{TEXT 231 1 "%" }{TEXT 317 28 "\" that stands for the last " }{TEXT 232 8 "executed" }{TEXT 317 285 " output, that is, the output of the last command that you clicked on, should preferably be used on the same command line as the command to which it refers. You can enter several commands on one line, end e ach with semicolon, and then press \"Enter\" to execute all of them. F or example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "(2*7+23)/(17-2); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#P\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmmC! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 317 88 "Maple gave the exact version of the answer follow ed by its floating point approximation." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 114 "As a little excercise, press \"E nter\" on the command lines below. Do you expect the same output for b oth of them?" }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "3*5+2/7; evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(3*5+2)/7; evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 50 "Functions and Expressions. Evaluating Expressions." }}{EXCHG {PARA 0 "" 0 "" {TEXT 317 14 "Recall that a " }{TEXT 233 8 "function" }{TEXT 317 14 ", for example " }{XPPEDIT 18 0 "f(x) = exp(2*x)+x*sin(x );" "6#/-%\"fG6#%\"xG,&-%$expG6#*&\"\"#\"\"\"F'F.F.*&F'F.-%$sinGF&F.F. " }{TEXT 317 274 " , is a rule which to each input, say x, prescribes the output given by the formula for a function. Maple's syntax for de fining a function reflects this interpretation of a function. If we wa nt to define the function f(x) in Maple, we type the following at the \+ command line" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=x->exp(2*x)+x*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#,$* &\"\"#\"\"\"F'F3F3F3*&F'F3-%$sinGF&F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 44 "Observe that you have to type in an arrow \"" }{TEXT 234 2 "->" }{TEXT 317 81 "\" which is entered using the \"minus\" key and the \"greater\" k ey. The symbol " }{TEXT 235 2 ":=" }{TEXT 317 95 "in Maple means alway s \"define\". Note also that the proper syntax for the natural exponen tial " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 317 7 " is \+ \"" }{TEXT 236 6 "exp(x)" }{TEXT 317 4 "\" ." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 244 "Since a function is a rule w hich to each input prescribes the output given by the formula for the \+ function, you can apply a function to anything in place of x and obtai n the corresponding value. The Maple syntax for doing that is simple. \+ Namely" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(s); f(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$ expG6#,$*&\"\"#\"\"\"%\"sGF*F*F**&F+F*-%$sinG6#F+F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"#?\"\"\"*&\"#5F(-%$sinG6#F*F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 14 " If you define" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m:=exp(2* x)+x*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG,&-%$expG6#,$*& \"\"#\"\"\"%\"xGF,F,F,*&F-F,-%$sinG6#F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 52 "you have defined not a function but a formula \+ or an " }{TEXT 237 10 "expression" }{TEXT 317 160 " m in terms of x. M aple can process expressions, as well as functions, but, as you see be low, the appropriate syntax may look a little different. For example \+ \"" }{TEXT 238 4 "m(s)" }{TEXT 317 104 "\" is meaningless to Maple. If you want to substitute s for x in the expression m you have to use th e \"" }{TEXT 239 4 "subs" }{TEXT 317 54 "\" command that, of course, s tands for \"substitute\"." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=s,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$*&\"\"#\"\"\"%\"sGF*F*F**&F+F*-%$sinG6#F+F *F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 70 "There are basica lly three commands to evalute expressions. They are \"" }{TEXT 240 4 " eval" }{TEXT 317 6 "\", \"" }{TEXT 241 5 "evalf" }{TEXT 317 9 "\" and \+ \"" }{TEXT 242 5 "value" }{TEXT 317 19 "\". The command \"" }{TEXT 243 4 "eval" }{TEXT 317 72 "\" will attempt to give you an exact value for your expression, while \"" }{TEXT 244 5 "evalf" }{TEXT 317 99 "\" will evaluate an expression numerically and return a decimal point ap proximation. The command \"" }{TEXT 245 5 "value" }{TEXT 317 41 "\" is most often used in conjuction with " }{TEXT 246 5 "inert" }{TEXT 317 18 " commands, like \"" }{TEXT 247 3 "Int" }{TEXT 317 6 "\", \"" } {TEXT 248 5 "Limit" }{TEXT 317 100 "\" which do not evaluate the input but only print it out. You will see in subsequent sections how \"" } {TEXT 249 5 "value" }{TEXT 317 170 "\" works. The best way to learn th e differences between the evaluation commands is by practice and examp le. Click on the lines below and see if you can predict an output." }} {PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eval(exp(x),x=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eval(exp(x),x=3));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "eval(sin(h)/h,h=2); evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(m,x=10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 314 "If \+ the last command gave you the answer \"m\", which obviously doesn't ma ke sense, that is because you forgot to re-execute commands in this wo rksheet and you didn't click on the line defining m. Hence, to Maple, \+ m is just some constant. Click on the line on which m is defined and c lick again on the last command." }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }} {PARA 0 "" 0 "" {TEXT 250 5 "Note:" }{TEXT 317 36 " You should always \+ use the syntax \"" }{TEXT 251 6 "exp(x)" }{TEXT 317 183 "\" for the na tural exponential. You should not use exp(1)^x, or define e:=exp(1) an d then use e^x. Although all of these options seem equivalent, Maple w orks better with the syntax \"" }{TEXT 252 6 "exp(x)" }{TEXT 317 4 "\" . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 6 "Limits" }}{EXCHG {PARA 0 "" 0 "" {TEXT 317 91 "Syntax for finding l imits of functions is very simple. Let's define a function, for exampl e" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=t->(cos(t)-1)/t;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&,&-%$cosGF&\"\"\"F0!\"\"F 0F'F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 53 "We find the limit of g(t) as t approaches 0 by typing" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(g(t),t=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 64 "The limit is 0. If you hadn't defined g(t) you could simply t ype" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit((cos(t)-1)/t,t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 76 "You coul d also find limits of expressions. Define an expression a as follows" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:=(cos(t)-1)/t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&,& -%$cosG6#%\"tG\"\"\"F+!\"\"F+F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 47 "The syntax f or finding limits of expressions is" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(a,t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 296 "This is an example of \+ when the syntax involving expressions is somewhat different than the s yntax involving functions. You couldn't use the syntax \"limit(g,t=0); \" You have to type \"g(t)\". In most commands involving functions yo u have to enter \"g(t)\" and not just the name of the function. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 54 " If you want Maple to print out a given limit, use the" }{TEXT 253 1 " " }{TEXT 254 5 "inert" }{TEXT 317 26 " version of the command \"" } {TEXT 255 5 "limit" }{TEXT 317 15 "\", that is, \"" }{TEXT 256 5 "Limi t" }{TEXT 317 26 "\". Inert commands like \"" }{TEXT 257 5 "Limit" } {TEXT 317 6 "\", \"" }{TEXT 258 3 "Int" }{TEXT 317 214 "\" print out a n input without evaluating it. They are useful if you want to verify i f you entered a correct expression. If you want to evaluate your limi t or integral right away, follow an inert command by the \"" }{TEXT 259 5 "value" }{TEXT 317 16 "\" command. The " }{TEXT 260 7 "\"value" }{TEXT 317 105 "\" command simply changes the upper case inert command s to lower case commands and evaluates. For example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(g(t ),t=0); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&-% $cosG6#%\"tG\"\"\"F,!\"\"F,F+F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 62 "Maple c an find limits at infinity and infinite limits as well." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit (ln(x)/x,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(exp(-x)/x^2,x=-infinit y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 126 "You can use L'Hospital rules to see that Maple \+ gave us correct answers. If a given limit does not exist, Maple will t ell us so" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "limit(1/x,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %*undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 69 "You can find one -sided limits using Maple. The appropriate syntax is:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(1 /x,x=0,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(1/x,x=0,right);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 46 "If you want the limits to be printed out, use:" }} {PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Limit(1/x,x=0,left); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%&LimitG6%*&\"\"\"F'%\"xG!\"\"/F(\"\"!%%leftG" }}{PARA 11 "" 1 "" {TEXT 320 0 "" }}}{PARA 4 "" 0 "" {TEXT 318 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " } {TEXT 200 11 "Derivatives" }}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }} {PARA 0 "" 0 "" {TEXT 317 69 "There are several ways of finding deriva tives. Let's have a function" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "h:=x->x*sin(x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&F '\"\"\"-%$sinG6#*$)F'\"\"#F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 37 "We can find its derivative as follows" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(h(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#*$)%\"xG\"\"#\"\"\"F+*(F* F+F(F+-%$cosGF&F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 77 "If we hadn't defined the function h(x) above, we could simply use the syntax" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(x*sin(x^2),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#*$)%\"xG\"\"#\"\"\"F+*(F*F+ F(F+-%$cosGF&F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 150 "The syntax \"diff(h,x);\", however, does not work. You have to type \+ \"h(x)\". For expressions the syntax is a little different. Define an \+ expression b" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b:=x*sin(x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"bG*&%\"xG\"\"\"-%$sinG6#*$)F&\"\"#F'F'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 317 32 "We find its derivative by typing" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(b,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#*$)%\"xG\"\"#\"\"\"F+*( F*F+F(F+-%$cosGF&F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }} {PARA 0 "" 0 "" {TEXT 317 87 "The simplest syntax for finding the seco nd order and other higher oreder derivatives is" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(h(x), x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"'\"\"\"-%$cosG6#*$)% \"xG\"\"#F&F&F,F&F&*(\"\"%F&)F,\"\"$F&-%$sinGF)F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 72 "In an i nstant, Maple can calculate for you the tenth derivative of h(x)" }} {PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(h(x),x$10);" }}{PARA 207 "" 1 "" {XPPMATH 20 "6#,.*(\"'SEL \"\"\"-%$cosG6#*$)%\"xG\"\"#F&F&F,F&F&*(\"(+)36F&)F,\"\"$F&-%$sinGF)F& !\"\"*(\"'Sq))F&)F,\"\"&F&F'F&F4*(\"'SMDF&)F,\"\"(F&F2F&F&*(\"&g\"GF&) F,\"\"*F&F'F&F&*(\"%C5F&)F,\"#6F&F2F&F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 53 "Another way of finding \+ derivatives is by using the \"" }{TEXT 261 1 "D" }{TEXT 317 33 "\" ope rator. It works as follows:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&,&-%$sinG6#*$)F%\"\" #\"\"\"F1*(F0F1F/F1-%$cosGF-F1F1F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 42 "To find the second deri vative using the \"" }{TEXT 262 1 "D" }{TEXT 317 19 "\" syntax, you ty pe" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(D(h));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6 \"6$%)operatorG%&arrowGF&,&*(\"\"'\"\"\"-%$cosG6#*$)F%\"\"#F-F-F%F-F-* (\"\"%F-)F%\"\"$F--%$sinGF0F-!\"\"F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 " " 0 "" {TEXT 263 5 "Note:" }{TEXT 317 63 " There is an important diffe rence between the outputs of the \"" }{TEXT 264 4 "D(h)" }{TEXT 317 13 "\" and the \"" }{TEXT 265 4 "diff" }{TEXT 317 27 "\" commands. The command \"" }{TEXT 266 5 "D(h);" }{TEXT 317 36 "\" returns the deriva tive of h as a " }{TEXT 267 10 " function " }{TEXT 317 20 "of x. The c ommand \"" }{TEXT 268 13 "diff(h(x),x);" }{TEXT 317 30 "\" gives the d erivative as an " }{TEXT 269 10 "expression" }{TEXT 270 1 " " }{TEXT 317 138 "in terms of x. Both forms have advantages and disadvantages. \+ For example, if you want to simplify the derivative, the important com mand \"" }{TEXT 271 8 "simplify" }{TEXT 317 97 "\" which tells Maple t o simplify a given expression, works, in general, better applied the way " }{TEXT 272 23 "simplify(diff(h(x),x));" }{TEXT 317 21 " than in the context " }{TEXT 273 18 "simplify(D(h)(x));" }{TEXT 317 170 ". Fo r other purposes, it may be more convenient to have the derivative D(h ) as a function. There is a way of turning an expression into a functi on, using the so called \"" }{TEXT 274 7 "unapply" }{TEXT 317 48 "\" c ommand. We shall look at this command later." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 9 "Integrals" }}{PARA 3 "" 0 "" {TEXT 319 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 317 79 "Maple can f ind indefinite integrals, as well as definite intergals. For example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(exp(3*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\" \"\"\"$F&-%$expG6#,$*&F'F&%\"xGF&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 50 "Maple has found an antiderivative of the function " } {XPPEDIT 18 0 "exp(3*x);" "6#-%$expG6#*&\"\"$\"\"\"%\"xGF(" }{TEXT 317 243 ". You know that the indefinite integral is equal to an antide rivative plus an arbitrary constant. Maple doesn't bother to add an ar bitrary constant. It does the hard part of the job. Namely, finds anti derivatives. If you define a function, say" }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "p:=x->x^2*exp(( 1/3)*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(*&)F'\"\"#\"\"\"-%$expG6#,$*&#F/\"\"$F/F'F/F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 53 "you can find its indefinite in tegral using the syntax" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(p(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"$\"\"\")%\"xG\"\"#F&-%$expG6#,$*&#F&F%F&F(F&F&F &F&*(\"#=F&F(F&F*F&!\"\"*&\"#aF&F*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 123 "If an antid erivative cannot be found in a closed form, which as you know may happ en, Maple will simply print out the input." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(sin(cos(x^2) ),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$sinG6#-%$cosG6#*$ )%\"xG\"\"#\"\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 84 "If you want a given integral prin ted out, as well as found, use the inert version \"" }{TEXT 275 3 "Int " }{TEXT 317 12 "\" of the \"" }{TEXT 276 3 "int" }{TEXT 317 29 "\" co mmand together with the " }{TEXT 277 7 "\"value" }{TEXT 317 10 "\" com mand" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(x^3,x); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)%\"xG\"\"$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&#\"\"\"\"\"%F&)%\"xGF'F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 45 "Similar syntax applies to definite integrals." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(p(x),x =0..2); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#I\"\"\"-%$ expG6##\"\"#\"\"$F&F&\"#a!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"* B@?V%!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 237 "We have just found the exact and the decimal value of d efinite integral of the function p(x) from 0 to 2. It is usually a go od idea to have a given integral printed out as well as evaluated. Aga in, we accomplish that with the command \"" }{TEXT 278 3 "Int" }{TEXT 317 3 "\"." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(cos(3*x),x=-1..3); value(%); evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#,$*&\"\"$\"\"\"%\"xG F,F,/F-;!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"$F&- %$sinG6#\"\"*F&F&*&F%F&-F)6#F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+6$GT%=!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 175 "If Maple prints out a given definite integral instead of evaluating, \+ that means that an antiderivative cannot be found and the Fundamental \+ Theorem cannot be used. For example:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(sin(cos(x^2)),x=0..2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$sinG6#-%$cosG6#*$)% \"xG\"\"#\"\"\"/F.;\"\"!F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 74 "In such case, we can ask Maple to find the integral numerically by typin g:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(int(sin(cos(x^2)),x=0..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JG\"*RR!#5" }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 8 "Plot ting" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 349 "We made an extensive use of Maple's plotting facility last semest er and we shall continue using it. The basic plotting syntax is very s imple. You type in a formula for a function you wanted plotted and the range for the independent variable. Maple automatically adjusts the r ange for the dependent variable so that you can see your plot. For exa mple" }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(cos(3.5*x),x=0..Pi);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 34 "If you have defined a function say" }} {PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k:=t->3*exp(.5*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGf*6# %\"tG6\"6$%)operatorG%&arrowGF(,$*&\"\"$\"\"\"-%$expG6#,$*&$\"\"&!\"\" F/F'F/F/F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 32 "you can pl ot it using the syntax" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(k(t),t=0..4);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 81 "You can add a lot of options to y our plots with appropriate commands. For example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot(k(t), t=0..4,color=blue,thickness=2,labels=[\"time in seconds\",\"position i n feet\"]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 132 "You can plot several functions i n one coordinate system and assign attributes to each graph to disting uish between them. For example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([sin(x),sin(2*x)],x=0.. 2*Pi,color=[black,blue]);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 146 "If you are assigning attributes to your graphs, like co lors, it is important to enter the functions, as well as colors in bet ween square brackets " }{TEXT 279 5 "[..] " }{TEXT 317 56 ". Objects i nside square brackets are read by Maple as a " }{TEXT 280 4 "list" } {TEXT 317 85 " in which the order of elements matters. If you enter ob jects between curly brackets " }{TEXT 281 6 "\{..\}" }{TEXT 317 131 " \+ Maple considers it a set in which the order does not matter. In order \+ to use more advanced plotting commands you need to load the " }{TEXT 282 7 "package" }{TEXT 317 3 " \"" }{TEXT 283 5 "plots" }{TEXT 317 50 "\" which is discussed in the section \"Packages\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 17 "Solving Equations" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 319 "As you know, solving equations is not an easy matter. In many cases there are no reliable methods of findin g exact values of solutions and numerical methods have to be used. Map le knows all the standard techniques and algorithms, as well as numeri cal methods. There are essentially two commands for solving equations \+ \"" }{TEXT 284 5 "solve" }{TEXT 317 9 "\" and \"" }{TEXT 285 6 "fsolve " }{TEXT 317 18 "\". The command \"" }{TEXT 286 5 "solve" }{TEXT 317 194 "\" attempts to find exact values of as many solutions as possible . It works very well with polynomial equations up to the order four an d equations which can be reduced to such form. For example:" }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "s olve(x^2-4*x-8=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"#\"\"\"* &F$F%-%%sqrtG6#\"\"$F%F%,&F$F%F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 10 "Command \"" }{TEXT 287 5 "solve" }{TEXT 317 92 "\" can be app lied to equations containing letter constants, that is, parameters. Fo r example" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "solve(c*x^2+d*x+3=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*(#\"\"\"\"\"#F&,&%\"dG!\"\"*$-%%sqrtG6#,&*$)F)F'F&F& *&\"#7F&%\"cGF&F*F&F&F&F4F*F&,$*(F%F&,&F)F*F+F*F&F4F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 33 "For more complicated equation s \"" }{TEXT 288 5 "solve" }{TEXT 317 43 "\" will return essentially t he input itself" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^5-x+1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,(*$)%#_ZG\"\"&\"\"\"F+F)!\"\"F+F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 288 "Maple's response indicates that it does not know how to solve this eq uation exactly. It is not surprising. As you may know there are no for mulas for finding roots of polynamials of degree five or higher. In th at case, we can ask Maple to solve a given equation numerically. The c ommand \"" }{TEXT 289 9 "allvalues" }{TEXT 317 88 "\" applied to an ou tput \"RootOf\" will produce approximate solutions, real and complex:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "allvalues(%);" }}{PARA 208 "" 1 "" {XPPMATH 20 "6'$!+yRIn6!\" *,&$\"+XWK7=!#5!\"\"*&$\"+,T&R3\"F%\"\"\"%\"IGF.F*,&F'F*F+F.,&$\"+OV%) [wF)F.*&$\"+garCNF)F.F/F.F*,&F2F.F4F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 65 "The best way to solve equati ons numerically is by applying the \"" }{TEXT 290 6 "fsolve" }{TEXT 317 14 "\" command. \"" }{TEXT 291 6 "fsolve" }{TEXT 317 97 "\" will a ttempt to find a solution numerically, and in general, it will return \+ one real solution." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "w:=t->sin(3*t)+cos(2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&-%$sinG6# ,$*&\"\"$\"\"\"F'F3F3F3-%$cosG6#,$*&\"\"#F3F'F3F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(w(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+Fjzq:!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 18 "A solution that \"" }{TEXT 292 6 "f solve" }{TEXT 317 113 "\" returns may not be located in a range of int erest to you. The situation can be remedied by adding under the \"" } {TEXT 293 6 "fsolve" }{TEXT 317 366 "\" command a range in which you \+ want Maple to find a solution. The latter capability of Maple equation solver combined with Maple's plotting facility provide a very powerfu l tool of finding solutions relevant to your problem. Suppose that you are studying a process modeled by the function w(t) and the range of \+ interest for t is [0,3]. You plot w(t) in that range:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(w( t),t=0..3);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 108 "You se e from the plot approximately where the zeros of w(t) are located. You ask Maple to find them using \"" }{TEXT 294 6 "fsolve" }{TEXT 317 42 "\" with the range for t specified. Namely:" }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(w(t)=0,t ,0.5..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hzxC%*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(w(t)=0,t,2..2.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e[6*>#!\"*" }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 8 "Packages" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 287 "When you start Map le, it loads into the computer's memory only the so called kernel, whi ch contains the basic commands and functions. More advanced commands \+ are contained in packages and libraries, which can be loaded with appr opriate commands. So far we have used two such packages: \"" }{TEXT 295 5 "plots" }{TEXT 317 9 "\" and \"" }{TEXT 296 7 "student" }{TEXT 317 29 "\". To load a package, say \"" }{TEXT 297 5 "plots" }{TEXT 317 26 "\", you use the command \"" }{TEXT 298 12 "with(plots):" } {TEXT 317 139 "\". If you end your command with a colon, the package w ill be loaded but its content will not be printed. If you end it with \+ a semicolon \"" }{TEXT 299 12 "with(plots);" }{TEXT 317 63 "\" Maple \+ will load the package, as well as print its content. " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plo ts);" }}{PARA 209 "" 1 "" {XPPMATH 20 "6#7U%(animateG%*animate3dG%-ani matecurveG%-changecoordsG%,complexplotG%.complexplot3dG%*conformalG%,c ontourplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%, densityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplot G%+gradplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%-listcontplot G%/listcontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogpl otG%(logplotG%+matrixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3d G%*polarplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.pol yhedraplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoption s3dG%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplot G%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }} {PARA 0 "" 0 "" {TEXT 317 55 "Among the commands which the package con tains we see \"" }{TEXT 300 9 "pointplot" }{TEXT 317 9 "\" and \"" } {TEXT 301 7 "display" }{TEXT 317 28 "\", which we used before. \"" } {TEXT 302 9 "pointplot" }{TEXT 317 85 "\" allows you to print numerica l data, that is, a bunch of points on the xy-plane. \"" }{TEXT 303 7 " display" }{TEXT 317 125 "\" allows you to print in one coordinate syst ems graphs of different types or in different ranges. We also see the \+ command \"" }{TEXT 304 12 "implicitplot" }{TEXT 317 67 "\" which allow s plotting curves given by xy-equations. For example:" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "implici tplot(3*x^2+y^2=4,x=-3..3,y=-3..3);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{PARA 0 "" 0 "" {TEXT 317 15 " The package \"" }{TEXT 305 5 "plots " }{TEXT 317 91 "\" contains many important plotting commands and we s hall use it extensively in the future." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 319 1 " " }{TEXT 200 18 "Using Maple's Help" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 135 "Maple has an e xtensive on-line help and it may be a good idea to learn how to use it . For example, you want to factor a polynomial, say" }}{PARA 210 "" 0 "" {XPPEDIT 18 0 "x^3-3*x-2;" "6#,(*$)%\"xG\"\"$\"\"\"F(*&F'F(F&F(!\" \"\"\"#F*" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 317 598 "To find \+ a Maple command to do this, select Topic Search...from the Help menu. \+ Then type \"factor\" because this is want you want to do. Among the ch oices that come up, you see \"Factor\" and \"factor\". Both look promi sing. Say you decide to try \"Factor\" first. Click on it to select it , and then double click to open the corresponding Help menu. A long pa ge comes up. Scroll down to examples. The examples look nothing like w hat you want. Try \"factor\" instead. That's it! The first example tha t you see looks almost exactly like what you want. It is an example of how to factor the polynomial " }{XPPEDIT 18 0 "6*x^2+18*x-24;" "6#,(* &\"\"'\"\"\"*$)%\"xG\"\"#F&F&F&*&\"#=F&F)F&F&\"#C!\"\"" }{TEXT 317 351 " . You can try to remember the syntax and use it in your example, or, if the syntax is complicated, you can copy an example from the He lp menu into your worksheet as follows. Highlight the command line tha t you want copied. Click on Copy under the Edit menu and close the Hel p screen. Open a new execution group in your worksheet by clicking on \+ the \"" }{TEXT 306 2 "[>" }{TEXT 317 120 "\" button. Move the cursor t o the new \">\" prompt and click on Paste under the Edit menu. Below w e have done just that." }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(6*x^2+18*x-24);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 62 "Now we \+ can simply follow the syntax to factor our polynomial. " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "facto r(x^3-3*x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"#! \"\"F&),&F%F&F&F&F'F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 218 "Done. \+ You could have arrived at the right Help menu by typing \"polynomial\" instead of \"factor\" and then navigating your way through a series o f menus that appears on the top of the screen. Try this as an excercis e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 4 "" 0 "" {TEXT 318 8 "Examples" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 " " 0 "" {TEXT 317 350 "The first example, which is very similar to some examples that we looked at last semester, will help you refresh your \+ memory about the syntax. We use it to illustrate how you are expected \+ to do your homework problems in terms of proper comments and explanati ons. Remember, it is usually not sufficient to hand in a bunch of Mapl e inputs and outputs. " }{TEXT 307 113 "In most examples, you have to \+ interpret the results and write your comments and answers in complete \+ sentences. " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 308 11 "Example 1. " }{TEXT 317 234 "A rumor is spreading among \+ a group of 200 people in an isolated region. The number of people, N(t ), who have heard the rumor by time t, measured in hours since the rum or started to spread, can be approximated by the following function" } }{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 211 "" 0 "" {XPPEDIT 18 0 "N(t ) = 200/(1+199*exp(-Float(17, -2)*t));" "6#/-%\"NG6#%\"tG*&\"$+#\"\"\" ,&F*F**&\"$*>F*-%$expG6#,$*&-%&FloatG6$\"# " 0 "" {MPLTEXT 1 0 30 "N:=t->200/ (1+199*exp(-.17*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6#%\"t G6\"6$%)operatorG%&arrowGF(,$*(\"$+#\"\"\"F/F/,&F/F/*&\"$*>F/-%$expG6# ,$*&$\"# " 0 "" {MPLTEXT 1 0 13 "N(20); N(30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ )Gith#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+EVZO!*!\")" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 121 "After 20 hours approximately 26 \+ people have heard the rumor. After 30 hours approximately 90 people ha ve heard the rumor." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "N(40);N(50);N(60);N(65);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I=9P;!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+14!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+R[J&)>!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Uop$*>!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 63 "After about 65 hours practically everyone has heard \+ the rumor. " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(N(t),t=0..70);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 253 "At first the rumor is spreading slowly \+ as the graph climbs slowly, then the rumor is spreading faster and fas ter. After about 40 hours it begins to spread slower again. After abou t 60 hours the graph levels off. Practically everyone has heard the ru mor." }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6 $%)operatorG%&arrowGF&,$*($\"'+mn!\"#\"\"\"-%$expG6#,$*&$\"#F/F0F/F/\"\"#F/F7F/F&F&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "plot(D(N)(t),t=0..70);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 317 180 "The rumor is spreading fastest w hen the derivative is maximal. It seems to be around t=30. To find the point exactly, we shall find the corresponding zero of the second der ivative." }}{PARA 0 "" 0 "" {TEXT 317 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "D(D(N));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"t G6\"6$%)operatorG%&arrowGF&,&*($\"++c(yd%!\"%\"\"\"*$)-%$expG6#,$*&$\" #F/F2F/F/\"\"$F/F:F/*($\")+A] 6F.F/F2F/*$)F>F;F/F:F:F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve(D(D(N))(t)=0,t,20..40);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+?(3P6$!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 58 "The rumor is spreading fastest at approximately t = 31.14." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(N(t) =150,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V7&*fP!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 111 "150 people have heard the rumor at t approximately 37.6, that is, 37.6 hours after the rumor began spreading. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 126 "This was a model homework problem solution. Usually , examples in the worksheets will contain much more elaborate explanat ions." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 121 "Since the first part of the course is devoted to integration, our next example involves indefinte and definite integrals." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 309 12 "Example 2. " } {TEXT 317 21 "Consider the function" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 213 "" 0 "" {XPPEDIT 18 0 "r(t) = sin(t)/(2+ln(t)^2);" "6#/-% \"rG6#%\"tG*&-%$sinGF&\"\"\",&\"\"#F+*$)-%#lnGF&F-F+F+!\"\"" }{TEXT 200 4 " ." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 86 "(a) Try to find the indefinite integral of r(t). What does Ma ple's response indicate?" }}{PARA 0 "" 0 "" {TEXT 317 31 "(b) Find th e definite integral" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 214 "" 0 "" {XPPEDIT 18 0 "int(r(t), t = 1 .. 2);" "6#-%$intG6$-%\"rG6#%\"tG/F );\"\"\"\"\"#" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 317 34 "(c) \+ Define in Maple the function " }}{PARA 215 "" 0 "" {XPPEDIT 18 0 "R(x) = int(r(t), t = 1 .. x);" "6#/-%\"RG6#%\"xG-%$intG6$-%\"rG6#%\"tG/F.; \"\"\"F'" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 317 87 "(d) Find t he derivative of R(x). Are you surprised with the result? Why not or w hy yes?" }}{PARA 0 "" 0 "" {TEXT 317 123 "(e) Plot graphs of r(x) and \+ R(x) in one coordinate system for x between 1 and 8. Comment on how th e two graphs are related." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 50 "We start from defining the function r(t) in Mapl e." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r:=t->sin(t)/(2+(ln(t))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%$sinGF&\" \"\",&\"\"#F/*$)-%#lnGF&F1F/F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 57 "Now we ask Maple to find the indefinite integral of r(t) ." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(r(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$in tG6$*&-%$sinG6#%\"tG\"\"\",&\"\"#F+*$)-%#lnGF)F-F+F+!\"\"F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 490 "Maple simply printed out the input. This means that an \+ antiderivative of r(t) cannot be found in terms of elementary (and eve n not so elementary) functions. Believe it or not, it is hard to come \+ up with a function whose antiderivative cannot be found by Maple! Neve rtheless, the definite integral of r(t) can be found numerically in an y interval in the positive half-line. (Remember, ln(t) is defined only for positive t.). The integral can be found numerically, thus we need the command \"" }{TEXT 310 5 "evalf" }{TEXT 317 17 "\" in front of \" " }{TEXT 311 3 "int" }{TEXT 317 3 "\"." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(int(r(t),t=1..2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C[M&Q%!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 113 "The integral is well-defined and can be found \+ numerically in any interval [1,x], hence we can define the function" } }{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R:=x->int(r(t),t=1..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"RGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%\"rG6#%\"tG/F2;\"\" \"F'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 155 "Of course, we cannot find a more explic it formula for R(x). Nonetheless, there is plenty that we can do with \+ the function. For example, find its derivative" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(R(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#%\"xG\"\"\",&\"\"#F( *$)-%#lnGF&F*F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 102 "Are we surprised? Of course n ot! The 2nd Fundamental Theorem of Calculus says exactly that R'(x)=r( x)." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 312 9 "A mazingly" }{TEXT 317 261 ", Maple can also plot the function R(x) (by \+ evaluating numerically the corresponding definite integral for many va lues of x). We shall plot both functions r and R in one coordinate sys tem. (Observe that we have to enter r(x) not r(t) to plot them both to gether." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([r(x),R(x)],x=1..8,color=[red,blue]);" }}{PARA 13 "" 1 "" {TEXT 321 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 279 "How are the graphs related \+ the red graph r(x) is the derivative of the blue graph R(x). Hence, wh ere r(x) is positive, R(x) increases. Where r(x) in negative, R(x) dec reases. Local minima and maxima of R(x) correspond to zeros of the red graph, which happens around 3.2 and 6.2. " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 216 "" 0 "" {TEXT 200 17 "Homework Problems" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 296 "To solve your homework problems you should open a new worksheet, enter your name and the title of the worksheet to wh ich your homework corresponds in the text mode. You should perform all the necessary Maple operations and supplement them by your explanatio ns and answers as in the example above." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 313 12 "Problem 1. " }{TEXT 317 150 "Sup pose that the total number of people, M, in a small town, who have con tracted a contagious disease by a time t days after its outbreak is gi ven by " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 217 "" 0 "" {XPPEDIT 18 0 "M(t) = 1000/(1+24*exp(-Float(25, -2)*t));" "6#/-%\"MG6# %\"tG*&\"%+5\"\"\",&F*F**&\"#CF*-%$expG6#,$*&-%&FloatG6$\"#D!\"#F*F'F* !\"\"F*F*F8" }{TEXT 200 3 " ." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }} {PARA 0 "" 0 "" {TEXT 317 66 "(a) How many people have become sick af ter 5 days? After 10 days?" }}{PARA 0 "" 0 "" {TEXT 317 56 "(b) Graph the function M(t) and describe its behavior. " }}{PARA 0 "" 0 "" {TEXT 317 62 "(c) How long will it take for 300 people to have become sick?" }}{PARA 0 "" 0 "" {TEXT 317 43 "(d) When is the disease sprea ding fastest?" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 314 11 "Problem 2. " }{TEXT 317 21 "Consider the function" }} {PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 218 "" 0 "" {XPPEDIT 18 0 "p(t) = 2*cos(t)/(1+ln(t));" "6#/-%\"pG6#%\"tG*(\"\"#\"\"\"-%$cosGF&F*,&F*F *-%#lnGF&F*!\"\"" }{TEXT 200 3 " ." }}{PARA 0 "" 0 "" {TEXT 317 85 "( a) Try finding the indefinite integral of p(t). What does Maple's resp onse indicate?" }}{PARA 0 "" 0 "" {TEXT 317 33 "(b) Find numerically t he integral" }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 219 "" 0 "" {XPPEDIT 18 0 "int(p(t), t = 1 .. 4);" "6#-%$intG6$-%\"pG6#%\"tG/F);\" \"\"\"\"%" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 317 24 "(c) Defin e the function " }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 220 "" 0 "" {XPPEDIT 18 0 "P(x) = int(p(t), t = 1 .. x);" "6#/-%\"PG6#%\"xG-%$intG 6$-%\"pG6#%\"tG/F.;\"\"\"F'" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 317 63 "(d) Find the derivative P'(x). Is the answer what you expecte d?" }}{PARA 0 "" 0 "" {TEXT 317 120 "(e) Plot p(x) and P(x) in one coo rdinate system between x=1 and x=9. Comment on the relationship betwee n the two graphs." }}{PARA 0 "" 0 "" {TEXT 317 0 "" }}{PARA 0 "" 0 "" {TEXT 317 2 " " }{TEXT 315 77 "MTH 142 Maple Worksheets written by B. Kaskosz and L. Pakula, Copyright 1998." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 2 " " }{TEXT 316 26 "Last modified August 1999." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }