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-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 21 "Introduction to Maple" }}{PARA 258 "" 0 "" {TEXT 258 9 "Exercises" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expand a section. Click on a \+ [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That can be done by clicking on the toolbar \+ icon that looks like \"[>\". It can also be done via the Insert menu. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "As you will see, the software package " }{TEXT 318 5 "MAPLE" }{TEXT -1 158 " couples som e rudimentary word processing capabilities with a very powerful mathem atical engine. You can use as a very powerful calculator. You can also use " }{TEXT 319 5 "MAPLE" }{TEXT -1 67 " as a very high level progr amming language. Most often, however, " }{TEXT 320 5 "MAPLE" }{TEXT -1 58 " is used interactively to analyze mathematical questions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "In intera ctive form, commands are typed at a " }{TEXT 321 5 "MAPLE" }{TEXT -1 11 " prompt. " }{TEXT 322 5 "MAPLE" }{TEXT -1 142 " commands are \+ organized like mathematical functions: the command name followed by a \+ comma-separated list of arguments enclosed in parentheses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 314 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet " "" }{TEXT -1 28 ", you will be asked to use " }{TEXT 313 5 "MAPLE" }{TEXT -1 40 " to study some limits and derivatives.." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Re ports that you prepare with " }{TEXT 306 5 "MAPLE" }{TEXT -1 156 " sho uld be prepared with the same care that you would devote to laboratory reports in biology and chemistry. A report should not be a diary or \+ history of a" }{TEXT 259 1 " " }{TEXT 307 5 "MAPLE" }{TEXT -1 77 " ses sion. Delete what is not needed for the report. All lines of the for m " }{TEXT 260 6 "?topic" }{TEXT -1 74 " should be erased. All erro rs should be erased. When you are printing a" }{TEXT 261 2 " " } {TEXT 308 5 "MAPLE" }{TEXT -1 113 "report, think about the toner and p aper resources that you are using. When you assign a variable, for exa mple \"" }{TEXT 262 8 " x := 5;" }{TEXT -1 29 " \", there is no nee d to have" }{TEXT 263 1 " " }{TEXT 309 5 "MAPLE" }{TEXT 310 1 " " } {TEXT -1 11 "echo back " }{TEXT 264 6 "x := 5" }{TEXT -1 89 ". When \+ this is printed, it simply wastes paper and ink. Choose the silent te rminator \"" }{TEXT 265 9 " x := 5: " }{TEXT -1 38 " \" instead. Wh en you load a package," }{TEXT 266 2 " " }{TEXT 311 5 "MAPLE" }{TEXT -1 95 " may list the commands that become available with the package. Delete these before printing. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "Some of the text in this worksheet should be deleted. Delete the sections the " }{TEXT 267 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 268 8 "Keywords" }{TEXT -1 34 " sections. D elete this section on " }{TEXT 269 7 "Reports" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your worksheet should execute in the order that it has been writ ten. In particular, remember that the ditto refers to the result of t he last executed command - not the result of the command that physical ly precedes it. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT -1 186 "You may not make a cop y of all or part of your work for the purpose of transferring the copy to another student. You may consult with other class members but you \+ may not copy their work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Key Words " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT -1 34 "On-line information about using a " }{TEXT 305 5 "MAPLE" }{TEXT -1 199 " function can be obtained at a prompt by preceding the functio n by a question mark and executing that line. For example, execute the next line to see the information on the differentiation function " } {MPLTEXT 1 0 4 "diff" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?diff" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 27 "For he lp on \"help\", enter " }{MPLTEXT 1 0 2 "??" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "??" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 152 "The words and symbols that are used in this workshee t have been collected here as hypertext links. Click on any word to go to the correspond help page." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{HYPERLNK 17 "$" 2 "$" "" }{TEXT -1 4 ", " }{HYPERLNK 17 ":=" 2 " :=" "" }{TEXT -1 1 " " }{TEXT 272 3 ", " }{HYPERLNK 17 "assignment" 2 ":=" "" }{TEXT 273 1 "," }{TEXT -1 2 " " }{HYPERLNK 17 "assume" 2 " assume" "" }{TEXT -1 5 " , " }{HYPERLNK 17 "D" 2 "D" "" }{TEXT -1 1 "," }{TEXT 23 1 " " }{TEXT -1 4 " " }{HYPERLNK 17 "display" 2 "disp lay" "" }{TEXT 274 1 "," }{TEXT -1 1 " " }{TEXT 275 2 " " }{HYPERLNK 17 "diff" 2 "diff" "" }{TEXT -1 1 "," }{TEXT 276 2 " " }{TEXT -1 1 " \+ " }{HYPERLNK 17 "ditto" 2 "ditto" "" }{TEXT -1 5 " , " }{HYPERLNK 17 "evalf" 2 "evalf" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "function" 2 "operators,functional" "" }{TEXT -1 1 " " }{TEXT 277 1 "," }{TEXT -1 3 " " }{HYPERLNK 17 "help" 2 "?" "" }{TEXT 271 1 "," }{TEXT -1 3 " " }{HYPERLNK 17 "implicitplot" 2 "implicitplot" "" }{TEXT -1 4 ", \+ " }{HYPERLNK 17 "limit" 2 "limit" "" }{TEXT -1 4 ", " }{TEXT 278 1 " " }{HYPERLNK 17 "list" 2 "list" "" }{TEXT 279 1 "," }{TEXT -1 2 " \+ " }{HYPERLNK 17 "name" 2 "name" "" }{TEXT 283 3 ", " }{TEXT -1 2 " \+ " }{HYPERLNK 17 "packages" 2 "packages" "" }{TEXT 285 3 ", " } {HYPERLNK 17 "plot" 2 "plot" "" }{TEXT 281 3 ", " }{HYPERLNK 17 "plot s" 2 "plots" "" }{TEXT 23 3 ", " }{HYPERLNK 17 "plot,structure" 2 "pl ot,structure" "" }{TEXT 282 4 ", " }{HYPERLNK 17 "plotting an implic it equation" 2 "implicitplot" "" }{TEXT 290 4 ", " }{HYPERLNK 17 "ra dsimp" 2 "radsimp" "" }{TEXT 287 2 ", " }{TEXT -1 1 " " }{HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 4 " , " }{TEXT 286 1 " " } {HYPERLNK 17 "rhs" 2 "rhs" "" }{TEXT 284 2 ", " }{TEXT -1 2 " " } {HYPERLNK 17 "selection" 2 "selection" "" }{TEXT -1 2 ", " }{TEXT 288 2 " " }{HYPERLNK 17 "simplify a radical" 2 "radsimp" "" }{TEXT 289 2 ", " }{TEXT -1 2 " " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT 280 3 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 270 2 ", " }{TEXT -1 1 " " }{HYPERLNK 17 "with" 2 "with" "" }{TEXT -1 3 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 10 "Exercise 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 " The Difference between Assignmen t and Equality" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "In " }{TEXT 332 5 "MAPLE" }{TEXT -1 98 " there is an im portant distinction between equality and assignment. For example, when we write " }{TEXT 323 5 "x = 2" }{TEXT -1 25 " we may be stating t hat" }}{PARA 0 "" 0 "" {TEXT 324 1 "x" }{TEXT -1 44 " has a value and \+ that its value is equal to " }{TEXT 325 1 "2" }{TEXT -1 74 ". Such a s tatement may be true or false. With this meaning the value of " } {TEXT 326 1 "x" }{TEXT -1 19 " does not change. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The statement " }{TEXT 330 5 "x = 2" }{TEXT -1 102 " may also have a quite different meanin g: namely, we may be stating that we are assigning the value " }{TEXT 331 1 "2" }{TEXT -1 17 " to the variable " }{TEXT 327 1 "x" }{TEXT -1 16 ". The variable " }{TEXT 328 1 "x" }{TEXT -1 101 " may have alrea dy had a value before the assignment. If so, the assignment may change the value of " }{TEXT 329 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "When we read and write m athematics, we must rely on our understanding of the context to discer n the meaning of mathematical notation. In using " }{TEXT 333 5 "MAPL E" }{TEXT -1 158 " we must avoid ambiguous meanings - the software do es not have human intuition. In particular, the notations for equality and assignment are different. In " }{TEXT 334 5 "MAPLE" }{TEXT -1 336 " we use the \"equals\" sign for the statement of an requality, \+ true or false. We precede the equals sign by a colon (as in the progra mming language PASCAL) to indicate assignment. The following group of \+ commands will illustrate. Execute the first \"restart\" line. Then exe cute each line that follows, trying to predict in advance what " } {TEXT 335 5 "MAPLE" }{TEXT -1 22 "'s response will be." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x = 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "testeq( x^2 = 4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x \+ := 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "testeq( x^2 = 4 );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Maple as a Calculator" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 337 5 "MAPLE" } {TEXT -1 65 " can be used as a calculator, but it is no ordinary calc ulator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "For example, unless you indicate that you want a decimal approxim ation, " }{TEXT 345 5 "MAPLE" }{TEXT -1 148 " will treat algebraic c ombinations of rational numbers (and certain constants such as Pi) in \+ an exact manner. For example, suppose you substitute " }{MPLTEXT 1 0 15 "x = 29 , y = 15" }{TEXT -1 22 " in the expression " }{MPLTEXT 1 0 11 "(x+y)/(x-y)" }{TEXT -1 40 ". A calculator will return the resu lt as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf( (29 + 15 )/(29 - \+ 15) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "As you see, by using the " } {HYPERLNK 17 "evalf" 2 "evalf" "" }{TEXT -1 22 " command we can have \+ " }{TEXT 348 5 "MAPLE" }{TEXT -1 85 " emulate a calculator. Unless p rompted with the floating point evaluation command " }{HYPERLNK 17 "e valf" 2 "evalf" "" }{TEXT -1 3 ", " }{TEXT 349 5 "MAPLE" }{TEXT -1 50 " does not destroy the exactness of an expression:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs( \{ x = 29 , y = 15 \} , (x+y)/(x-y) ) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "Of course a floating point evaluation of \+ an exact expression can always be forced:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf( \" );" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 23 "In Releases 1 to 4 of " }{TEXT 346 5 "MA PLE" }{TEXT -1 7 ", the " }{TEXT 347 1 " " }{TEXT -1 1 " " } {HYPERLNK 17 "ditto" 2 "ditto" "" }{TEXT -1 125 " ( \" ) has stood for the result of the last execution. Starting with Release 5, the pe r cent sign has replaced the ditto." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 50 "In the next execution line, assign to a \+ variable " }{TEXT 339 1 "N" }{TEXT -1 210 " the number of digits tha t your hand-held calculator displays. Remember to terminate the line w ith a semi-colon. Remember also to execute the line by pressing . The typing of a line does not send it to " }{TEXT 338 5 "MAPLE" } {TEXT -1 23 "'s mathematical kernel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N := ?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Execute the following assignmen t:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "alpha := 1728148040 - 140634 693*sqrt(151);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Your calculator evaluates " }{MPLTEXT 1 0 5 "alpha" } {TEXT -1 12 " as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf( alpha , N );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 38 "Is alpha really 0 ? How do you know?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Continue with the follo wing assignments:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "beta := 1728148040 + 140634693*sqrt(151);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x := alpha*beta;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "This is what yo ur calculator thinks " }{MPLTEXT 1 0 1 "x" }{TEXT -1 15 " is equal \+ to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf( x , N );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The second argu ment of " }{HYPERLNK 17 "evalf" 2 "evalf" "" }{TEXT -1 81 " ref lects the number of significant digits that are retained in calculatio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Un like a calculator, you can use " }{TEXT 343 5 "MAPLE" }{TEXT -1 122 " to increase the accuracy of this answer by increasing the number of \+ significant digits that are used in the computation:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf( x , 75 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Or, \+ you can use " }{TEXT 342 5 "MAPLE" }{TEXT -1 19 " to calculate the \+ " }{TEXT 344 5 "exact" }{TEXT -1 12 " value of " }{MPLTEXT 1 0 1 "x " }{TEXT -1 14 " as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(x);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "Not convinced? Work it out yourself by writing " }{MPLTEXT 1 0 1 "x" }{TEXT -1 31 " as the difference of squares:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "is ( x = 1728148040^2 - (140634693^ 2)*151 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "1728148040^2 - 140634693^2*151;" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 " The Difference Betwee n Functions and Expressions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 268 "A function is a rule that produces a definite \+ output for each inputted value. When the output is determined by a spe cific expression involving the input, we can symbolize the function by means of arrow notation. For example, the squaring function can be re presented as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x -> x^2;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "It is us ually convenient to name the functions that we work with. We can assig n a name to a function in the same way we have used assignment above. \+ The syntax is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "function_nam e := ( comma_separated_variable_list ) -> expression_involving_var iables ; " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For example," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We ca n then work with the function named " }{MPLTEXT 1 0 1 "f" }{TEXT -1 19 " in the usual way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(x);\nf(a);\nf(3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The following is most often an error:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g(x) := x -> x^3;" }}}{PARA 0 "" 0 "" {TEXT -1 41 "\nIt attempts to give an illegal name, " }{MPLTEXT 1 0 4 "g(x) " }{TEXT -1 25 ", to the cubing function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 301 1 " " }{TEXT 350 5 "MAPLE" }{TEXT -1 44 " there are important distinctions between " } {TEXT 302 1 "f" }{TEXT -1 4 ", " }{TEXT 303 4 "f(x)" }{TEXT -1 8 ", \+ and " }{TEXT 304 9 "x -> f(x)" }{TEXT -1 140 ". The first object is the name of a function, the second is an expression that defines the \+ function, and the third is the actual function. " }{TEXT 352 5 "MAPLE " }{TEXT -1 129 " always distinguishes between functions and expressio ns. Most commands require expressions as input, but some require func tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 228 "The advantage of using a function ov er an expression is that functions allow for easy evaluation. For exam ple, to calculate the square roots of several numbers, we may name the square-root function and use functional evaluation:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := x -> evalf( sqrt( x ) ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F(2);\nF(3);\nF(5); " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "With the square-r oot expression, we would have to use the more cumbersome substitution \+ technique:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "G := sqrt(x) ;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "evalf( subs( x = 2 , G )); \nevalf( subs( x = 3 , G ));\nevalf( subs( x = 5 , G ));" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 17 "Notice that the " }{HYPERLNK 17 "subs" 2 "subs" "" } {TEXT -1 40 " command does not assign a value to " }{MPLTEXT 1 0 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " Maple knows Calculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "limit( (f(x+h)-f(x))/h \+ , h = 0 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "In general:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "restart;\nlimit( (F(x+h)-F(x))/h , h = 0 );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Notice th e syntax for derivative:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "D(F);\nD(F)(x);\nD( x -> F(x) );\n D( x -> F(x) )(c);\n\nD( x -> x^3 );\nD( x -> x^3 )(x);\nD( x -> x^3 ) (c);\nD( x -> x^3 )(2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 44 "The argument of the differential operator " } {HYPERLNK 17 "D" 2 "D" "" }{TEXT -1 218 " must be a function (or a \+ name of a function) of one variable. The result is a function which ma y then be evaluated at a point. Make sure that you understand exactly what each of the lines above calls for and why " }{TEXT 355 5 "MAPLE " }{TEXT -1 25 " responds as it does.\n\n" }{TEXT 354 43 "The follo wing is most often an error:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D( x^2 )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Why is it an error?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 353 5 "MAPLE" }{TEXT -1 175 " does have a command for differentiating expressions. It requires two arguments. The first is \+ the expression to be differentiated. The second is the differentiation variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "diff( x^2 , x );\ndiff( x^2 , t );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "As remarked above expressions are better for some things , functions for others. To repeat, you evaluate an expression by using the substitute command " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 35 ". The following do the same things:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "D(f)(5);\nsubs( x = 5 , diff(f(x),x) );" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 " A Calculus Li mit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Wh at is the important calculus limit " }{TEXT 358 1 " " }{XPPEDIT 356 0 "Limit((1+1/n)^n,n = infinity)" "-%&LimitG6$),&\"\"\"F'*&F'F'%\"nG! \"\"F'F)/F)%)infinityG" }{TEXT 357 1 " " }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Let us get some nume rical evidence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for n from 100 to 200 by 20 do value( (1+1/n)^n ) \+ od;" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Notice that " }{TEXT 359 5 "MAPLE" } {TEXT -1 120 " does exact arithmetic unless signaled not to. Often t hat is a useful feature but in this case it is not. Try instead:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for n from 100 to 200 by 20 \+ do value( (1.+1/n)^n ) od;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "What was the difference in commands?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Let us restore \+ the variable n:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n;\nn \+ := 'n';\nn;\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Let's try some larger values of n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for k from 10 to 20 do evalf( subs( x = 10^k , (1. +1/x)^x ) , 30 ) od;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "We can't study this limit graphically (because n tends to infinity) but we can study the \+ limit by making the change of variable " }{TEXT 360 8 " x = 1/n" } {TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot( subs( \+ n = 1/x , (1+1/n)^n ) , x = 0 .. 1 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "Finally, let us ask " }{TEXT 317 5 "M APLE" }{TEXT -1 24 " to just do this limit:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "limit( (1+1/n)^n , n = infinity );" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "limit( (1+x) ^(1/x) , x = 0 , right );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 10 "Exercise 2" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 18 " A Tangent Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "In this exercise we will see that the tan gent lines to the curve " }{XPPEDIT 18 0 "x^(1/2)+y^(1/2) = a" "/,&)% \"xG*&\"\"\"F'\"\"#!\"\"F')%\"yG*&F'F'F(F)F'%\"aG" }{TEXT -1 32 " \+ have a remarkable property." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "An equation is \+ a mathematical object and" }{TEXT 291 1 " " }{TEXT -1 2 " " }{TEXT 361 5 "MAPLE" }{TEXT -1 144 " treats it as such. You can assign a na me to an equation - it is frequently convenient to do so. Always kee p in mind that the equal sign \" " }{TEXT 292 1 "=" }{TEXT -1 57 " \+ \" is used in an equation and the assignment sign \" " }{TEXT 293 3 ":= " }{TEXT -1 33 " \" is used for assignment. Thus," }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "resta rt:\neqn := sqrt(x) + sqrt(y) = a ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "gives the name " }{TEXT 294 3 "eqn" } {TEXT -1 20 " to the equation " }{XPPEDIT 18 0 "x^(1/2)+y^(1/2) = a " "/,&)%\"xG*&\"\"\"F'\"\"#!\"\"F')%\"yG*&F'F'F(F)F'%\"aG" }{TEXT -1 98 " . Remember that writing down an equation does not make it true. \+ In the preceding input line, " }{TEXT 364 5 "MAPLE" }{TEXT -1 77 " \+ was asked to ame an equation and it did just that. If we want to enl ist " }{TEXT 362 7 "MAPLE's" }{TEXT -1 67 " aid in testing the vali dity of an equation, the commands and " }{HYPERLNK 17 "testeq" 2 " testeq" "" }{TEXT -1 14 " can be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "testeq( eqn );\ntesteq ( subs( \{ x = a^2 , y = 0 \} , eqn ) );\nassume( a > 0 );\ntesteq( su bs( \{ x = a^2 , y = 0 \} , eqn ) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Let u s solve for y in terms of x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "formula_for_y := solve( eqn \+ , y );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Y := X -> subs( x = X , formula_for_y );\nY(x);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Fill in t he question marks to obtain the tangent line to the curve at " } {TEXT 299 8 "(c,Y(c))" }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "assume( c > 0 );\nT_c := x -> subs( t = c , diff( Y(t ) , t ) )*? + ?;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Now solve for the interce pts of the tangent line:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x_intercept := solve( ? , x );\ny_intercept := ? ;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "S := x _intercept + y_intercept;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "radsimp( S );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Illustration" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "Use the value a = 5. Find the point \+ " }{TEXT 297 1 "P" }{TEXT -1 46 " on the curve such that the tangent \+ line at " }{TEXT 298 1 "P" }{TEXT -1 67 " has x-intercept 16.\nFi rst, let us plot the curve with a = 5 :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "with(plots):\ncurve_plot := implicitplot( subs( a = 5 , eqn ) , x = 0 .. 16, y = 0 .. 16 ):\ndisplay( \{ curve_plot \} \+ );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Next we find " }{TEXT 300 1 "P" }{TEXT -1 53 " . You must su bstitute an appropriate equation for " }{MPLTEXT 1 0 1 "?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve( \{ subs( a = 5 , ? \} , c );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "tangent _plot := plot( subs( \{ c = ? , a = 5 \} , T_c(x) ) , x = 0.. 16 , y = 0 .. 16 , color = blue ):\ndisplay(tangent_plot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "display( \{tangent_plot , curve_plot \} ); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 10 "Exercise 3" }}{PARA 0 "" 0 "" {TEXT -1 66 "As practice follow the \+ previous exercise. Consider the astroid " }{XPPEDIT 19 1 "x^(2/3)+y ^(2/3) = a^(2/3)" "/,&)%\"xG*&\"\"#\"\"\"\"\"$!\"\"F()%\"yG*&F'F(F)F*F ()%\"aG*&F'F(F)F*" }{TEXT -1 177 ". Show that the distance between the intercepts of any tangent does not depend on the selected point of ta ngency. Illustrate with a graph of the astroid with two tangent lines ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Exercise 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the equation " } {XPPEDIT 18 0 "exp(x) = 3*x" "/-%$expG6#%\"xG*&\"\"$\"\"\"F&F)" } {TEXT -1 34 " using the Newton-Raphson Method:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f := x -> ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Phi := x -> evalf( x - f(x)/D(f)(x) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Au thor Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 46 "M132-1R4.mws A MapleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank (14 January \+ 1999)" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, with out" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the au thor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Departmen t of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Uni versity in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, M O 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 199 7 Brian E. Blank, All Rights Reserved." }}}}}{MARK "3 8 0" 120 } {VIEWOPTS 1 1 0 3 4 1802 }