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0 0 0 0 0 -1 0 }{PSTYLE "" 18 258 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 "" 19 259 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 "" 3 270 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 "" 3 273 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 "" 3 274 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 "" 0 277 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 "" 0 278 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 "" 0 279 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 "" 0 280 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 "" 0 281 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 "" 0 282 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 "" 0 283 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 "" 0 284 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 9 "Functions" }}{PARA 258 " " 0 "" {TEXT 259 8 "in Maple" }}{PARA 258 "" 0 "" {TEXT 256 12 "Tutor1 R4.mws" }}{PARA 259 "" 0 "" {TEXT 260 14 "Brian E. Blank" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a se ction." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 " \+ " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 70 " we will l earn to define and differentiate mathematical functions in " }{TEXT 270 6 " MAPLE" }{TEXT -1 3 ". " }}}{SECT 1 {PARA 270 "" 0 "" {TEXT -1 9 "Key Words" }}{PARA 0 "" 0 "" {TEXT -1 152 "The words and symbol s that are used in this worksheet have been collected here as hypertex t links. Click on any word to go to the correspond help page." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {HYPERLNK 17 "@@" 2 "@@" "" }{TEXT 322 2 ", " }{TEXT -1 1 " " } {HYPERLNK 17 "arrow operators" 2 "operators,functional" "" }{TEXT 23 1 "," }{TEXT -1 4 " " }{HYPERLNK 17 "D" 2 "D" "" }{TEXT 318 1 "," } {TEXT -1 1 " " }{TEXT 319 1 " " }{HYPERLNK 17 "diff" 2 "diff" "" } {TEXT -1 1 "," }{TEXT 320 2 " " }{HYPERLNK 17 "procedure" 2 "procedur e" "" }{TEXT 23 3 ", " }{HYPERLNK 17 "unapply" 2 "unapply" "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "On-line i nformation about using a " }{TEXT 325 5 "MAPLE" }{TEXT -1 60 " functio n can also be obtained at a prompt by preceding the " }{TEXT 324 5 "ex act" }{TEXT -1 53 " keyword by a question mark and executing that line . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 23 "Defining Functions 1: " }{TEXT 273 3 " ->" }}{PARA 0 "" 0 "" {TEXT -1 189 "Whenever we do mathematics it is not long before we consider one function or another. It is important to understand what \+ a function is. It is also important to know what a function is not! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "To begi n, " }{TEXT 274 6 " MAPLE" }{TEXT -1 16 " uses the term " }{MPLTEXT 1 0 9 "procedure" }{TEXT -1 102 " for what we think of as a function, reserving the term \"function\" for use in symbolic programming. " } {TEXT 308 1 "A" }{TEXT -1 3 " " }{MPLTEXT 1 0 9 "procedure" }{TEXT -1 2 " " }{TEXT 309 69 "produces one return value as the result of in putting appropriate data" }{TEXT 310 1 "." }{TEXT -1 103 " The data mi ght consist of one item or it might be a comma separated sequence. Eac h entry is called an " }{TEXT 311 8 "argument" }{TEXT -1 124 ". For ex ample, a procedure may\nbe defined by cubing an inputted value and add ing 5 to the result. If we name this function " }{TEXT 275 2 " f" } {TEXT -1 12 ", and if " }{TEXT 277 1 "x" }{TEXT -1 59 " is the inp utted value, then the returned value is denoted" }{TEXT 276 5 " f(x)" }{TEXT -1 49 ". In this case we have\n\n " } {TEXT 278 14 "f(x) = x^3 + 5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "We say that " }{TEXT 280 4 "f (x)" }{TEXT -1 3 " is" }{TEXT 281 2 " f" }{TEXT -1 2 " " }{TEXT 282 7 "applied" }{TEXT -1 9 " to " }{TEXT 279 1 "x" }{TEXT -1 56 ".\n \n\nWhat is the function or procedure here? It is not " }{TEXT 283 4 "f(x)" }{TEXT -1 15 " nor is it " }{TEXT 284 7 "x^3 + 5" }{TEXT -1 40 " - indeed they are the same thing: an " }{TEXT 285 20 "algebr aic expression" }{TEXT -1 3 ".\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "type(x^3 + 5, procedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 71 "The idea of a function is that it transforms data, here symbolized by " }{TEXT 286 1 "x" }{TEXT -1 36 ", into a retur n value, which is " }{TEXT 287 7 "x^3 + 5" }{TEXT -1 58 " in this c ase. The standard mathematical notation of an " }{TEXT 288 5 "arrow" }{TEXT -1 12 " is used in " }{TEXT 289 6 " MAPLE" }{TEXT -1 46 " to d enote this dynamic transformation. Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "type(x -> x^3 + 5, pr ocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "As with any object in " }{TEXT 290 6 " MAPLE" }{TEXT -1 66 " , we can assign a name to it \+ by using the assignment operator " }{MPLTEXT 1 0 4 " := " }{TEXT -1 29 " . Thus, to give the name " }{TEXT 291 1 "f" }{TEXT -1 32 " to the procedure (function) " }{MPLTEXT 1 0 12 "x -> x^3 + 5" }{TEXT -1 26 ", we make the assignment:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> x^3 + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\" xG6\"6$%)operatorG%&arrowGF(,&*$9$\"\"$\"\"\"\"\"&F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "type(f, procedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 12 "To summarize" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(i) " }{MPLTEXT 1 0 1 "f " }{TEXT -1 6 " and " }{MPLTEXT 1 0 12 "x -> x^3 + 5" }{TEXT -1 45 " \+ are the functions that we have created, and" }}{PARA 0 "" 0 "" {TEXT -1 30 "(ii) the algebraic expression " }{MPLTEXT 1 0 8 " x^3 + 5" } {TEXT -1 63 " is not a function: it is the result of applying the fun ction " }{MPLTEXT 1 0 1 "f" }{TEXT -1 6 " to " }{MPLTEXT 1 0 1 "x" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Functions of More than One Variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "It is just as easy to use the arrow notation to make a fu nction of more than one variable:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P := (V,T) -> n*R*T/V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG:6 $%\"VG%\"TG6\"6$%)operatorG%&arrowGF)**%\"nG\"\"\"%\"RGF/9%F/9$!\"\"F) F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 273 "" 0 "" {TEXT -1 23 "Defining Functions 2: " } {TEXT 303 8 " unapply" }}{PARA 0 "" 0 "" {TEXT -1 56 "Suppose that we \+ wish to define a procedure (function) " }{TEXT 293 1 "f" }{TEXT -1 54 " by the formula\n \n " }{TEXT 292 14 "f(x) = x^3 + 5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 " \nSince " }{TEXT 294 8 " x^3 + 5" }{TEXT -1 13 " is to be " } {TEXT 295 1 "f" }{TEXT -1 14 " applied to " }{TEXT 296 1 "x" }{TEXT -1 21 " , we may think of " }{TEXT 313 2 " f" }{TEXT -1 5 " as" } {TEXT 298 8 " x^3 + 5" }{TEXT -1 31 " \"unapplied with respect to \+ " }{TEXT 297 2 " x" }{TEXT -1 12 ".\" That is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := unapply(x^3 + 5 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$9$\"\"$\"\"\"\"\"&F0F(F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "This " } {TEXT 299 5 "MAPLE" }{TEXT -1 69 " idiom is especially useful in the \+ theory of differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Functions of More than One Variable" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "It is jus t as easy to use this idiom to make a function of more than one variab le:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "P := unapply( n*R*T/V, V, T );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG:6$%\"VG%\"TG6\"6$%)operat orG%&arrowGF)**%\"nG\"\"\"%\"RGF/9%F/9$!\"\"F)F)" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 274 "" 0 "" {TEXT -1 23 "Defining Functions 3: " }{TEXT 304 5 " proc" }}{PARA 0 "" 0 " " {TEXT -1 3 "In " }{TEXT 300 5 "MAPLE" }{TEXT -1 121 " we can define procedures that are like functions in programming languages such as C and Java. Here is the basic syntax:" }}{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 42 "f := proc( x)\n RETURN(x^3+5);\n end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fG:6#%\"xG6\"F(F(-%'RETURNG6#,&*$9$\"\"$\"\"\"\"\"&F0F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"$\"\"\"\"\"&F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "type(f, procedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "There is not much reason to use this method of defining mathematical functions. It is general ly used only for creating programming functions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 35 "Functions of More than One Variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{MPLTEXT 1 0 5 " pr oc" }{TEXT -1 69 " technique can also be used for functions of more t han one variable:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P := proc(V, T)\n RETURN(n*R*T/V);\n end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG:6$%\"VG%\"TG6\"F)F)-%'RETURNG6#**%\"nG\"\"\"%\"RGF/9%F/9$!\" \"F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P(V, T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"nG\"\"\"%\"RGF%%\"TGF%%\"VG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 29 "Warning - A very Common Error" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 80 "Only the name of a funct ion should go on the left side of an assignment operator" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "An \+ attempt such as\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f(x) \+ := x^3 + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"xG,&*$F'\" \"$\"\"\"\"\"&F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "is invariably incorrect (for what is intended). It is \+ not syntactically incorrect so no error message is generated. But" }} {PARA 0 "" 0 "" {TEXT -1 93 "this construction will not do what is exp ected. Therefore newcomers cannot be told too often," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 172 "Only the name of a function should go on the left side \+ of an assignment operator. Parentheses and independent variables do no t belong to the left of the assignment operator" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Workin g with Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Functions are convenient because they permit easy evaluat ion. Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> x^3 + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$ 9$\"\"$\"\"\"\"\"&F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " f(-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "f(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$% \"cG\"\"$\"\"\"\"\"&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f( x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&%\"xG\"\"\"!\"\"F'\"\"$ F'\"\"&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f(f(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&*$%\"xG\"\"$\"\"\"\"\"&F)F(F)F*F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "No tice that we do not have to make an assignment in order to evaluate a \+ function. When we evaluate " }{MPLTEXT 1 0 5 "f(-2)" }{TEXT -1 16 ", \+ we do not set " }{MPLTEXT 1 0 9 " x := - 2" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(- 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "testeq( x = - 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Differentiating Functi ons" }}{PARA 0 "" 0 "" {TEXT -1 37 "There are two built-in functions i n " }{TEXT 301 5 "MAPLE" }{TEXT -1 39 " for differentiating. The f unction " }{MPLTEXT 1 0 1 "D" }{TEXT -1 98 " is usd for differentiat ing a function of one variable. That function should be the argument o f " }{MPLTEXT 1 0 1 "D" }{TEXT -1 4 " . " }}{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 18 "f := x -> x^3 + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*$9$\"\"$\"\"\"\"\"&F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D(x -> x^3+5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"xG6\"6$%)operatorG%&arrowGF&,$*$9$\"\"#\"\"$F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#:6#%\"xG6\"6$%)operatorG%&arrowGF&,$*$9$\"\"#\"\"$F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(c);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*$%\"cG\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "D(f)(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Notice t hat if " }{MPLTEXT 1 0 1 "f" }{TEXT -1 30 " is a function then so is " }{MPLTEXT 1 0 4 "D(f)" }{TEXT -1 87 " - the derived function. It can be evaluated in the usual way (as illustrated above)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 302 5 "MAPLE" }{TEXT -1 29 " also provides a function, " }{MPLTEXT 1 0 4 "diff" }{TEXT -1 46 " , for differentiating algebraic expressio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(x^3+5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* $%\"xG\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(f( x),x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%\"xG\"\"#\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),u);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 83 "Finally, the next two lines would almost certainly be errors - but they are co mmon!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%\"DG6#%\"xG\"\"\"F(\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "If you think about them, then you will see that each \+ line above is technically correct, but probably not what the user want ed. In the first example, " }{MPLTEXT 1 0 4 "f(x)" }{TEXT -1 75 " m ust be treated as a function (since it is the argument of a procedure, " }{MPLTEXT 1 0 1 "D" }{TEXT -1 88 ", that takes a function as its o nly argument). The only way to do this is to interpret " }{MPLTEXT 1 0 1 "x" }{TEXT -1 35 " as the name of the function and " }{MPLTEXT 1 0 4 "f(x)" }{TEXT -1 37 " as the composition of the functions " } {MPLTEXT 1 0 1 "f" }{TEXT -1 6 " and " }{MPLTEXT 1 0 2 " x" }{TEXT -1 4 ". " }{TEXT 306 5 "MAPLE" }{TEXT -1 94 " does this and then a pplies the Chain Rule to differentiate the composition. The expression " }{MPLTEXT 1 0 4 "D(x)" }{TEXT -1 106 " arises because of the Chain \+ Rule and is left unevaluated because no functional rule has been assig ned to " }{MPLTEXT 1 0 2 " x" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The second computation is less mysterious. The function " }{MPLTEXT 1 0 4 "diff" }{TEXT -1 91 " expects its first argument to be an algebraic expression. Here we have passed the name " }{MPLTEXT 1 0 1 "f" }{TEXT -1 65 " as the \+ first argument. Syntactically there is no error. Both " }{MPLTEXT 1 0 4 "f(x)" }{TEXT -1 8 " and " }{MPLTEXT 1 0 1 "f" }{TEXT -1 31 " \+ satisfy the same type test: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "type(f(x),algebraic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "type(f,algebraic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Clearly " }{MPLTEXT 1 0 4 "f(x)" } {TEXT -1 47 " is an algebraic expression since it equals " } {MPLTEXT 1 0 7 "x^3 + 5" }{TEXT -1 8 ". But " }{MPLTEXT 1 0 3 " f " }{TEXT -1 143 " is also an algebraic expression since names can be use d for constants and constants are certainly algebraic expressions. In the calculation " }{MPLTEXT 1 0 9 "diff(f,x)" }{TEXT -1 3 ", " } {TEXT 307 5 "MAPLE" }{TEXT -1 58 " treats the first argument as a con stant with respect to " }{MPLTEXT 1 0 1 "x" }{TEXT -1 14 " and returns " }{MPLTEXT 1 0 1 "0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Higher Order Derivatives" }} {PARA 0 "" 0 "" {TEXT -1 28 "Continuing with our example," }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := \+ x -> x^3 + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%) operatorG%&arrowGF(,&*$9$\"\"$\"\"\"\"\"&F0F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "let us use the two functions, " }{MPLTEXT 1 0 4 "diff" } {TEXT -1 7 " and " }{MPLTEXT 1 0 1 "D" }{TEXT -1 110 ", to calculate second and higher order derivatives. First, we obtain the second deri vative using each command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@2)(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(f(x), x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "and hi gher order derivatives:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "(D@@3)(f)(x); (D@@4)(f)(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dif f(f(x), x$3); diff(f(x), x$4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Differentiating Functi ons of Two or More Variables" }}{PARA 0 "" 0 "" {TEXT -1 14 "The comma nds " }{MPLTEXT 1 0 4 "diff" }{TEXT -1 7 " and " }{MPLTEXT 1 0 1 "D " }{TEXT -1 83 " can be used to calculate partial derivatives of func tions of 2 or more variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P := (V,T) -> n*R*T/V;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG:6$%\"VG%\"TG6\"6$%)operatorG%&a rrowGF)**%\"nG\"\"\"%\"RGF/9%F/9$!\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "D[1](P)(V,T); diff(P(V,T), V);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG!\"#!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "D[2](P)(V,T); diff(P(V ,T), T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"nG\"\"\"%\"RGF%%\"VG! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"nG\"\"\"%\"RGF%%\"VG!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 97 "Mix ed partial derivatives are obtained in the obvious way. For example, h ere is how to evaluate " }{TEXT 326 1 " " }{XPPEDIT 327 1 "diff(P(V,T ),V,T)" "-%%diffG6%-%\"PG6$%\"VG%\"TGF(F)" }{TEXT -1 29 " using each type of syntax:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "D[1,2](P)(V,T); \ndiff(P(V,T), V, T);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\"%\"RGF&%\"VG!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\"%\"RGF&%\"VG!\"#!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "High er order partial derivatives are also possible. Consider " }{TEXT 328 2 " " }{XPPEDIT 329 1 "diff(P(V,T),V,V" "-%%diffG6%-%\"PG6$%\"VG% \"TGF(F(" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "D[1,1](P)(V,T); D[1$2](P)(V,T); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG!\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG!\"$\"\"# " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "using " }{MPLTEXT 1 0 1 "D" }{TEXT -1 5 " and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff(P(V,T), V, V); \ndiff(P(V,T), V$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG!\"$\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"nG\"\"\"%\"RGF&%\"TGF&%\"VG !\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "using " }{MPLTEXT 1 0 4 "diff" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now consider " }{TEXT 330 2 " " }{XPPEDIT 331 1 "diff(P (V,T),T,T)" "-%%diffG6%-%\"PG6$%\"VG%\"TGF)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "D[2,2](P)(V,T); \nD[2$2](P)(V,T);\ndiff(P(V,T), T , T); \ndiff(P(V,T), T$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Finally, higher order " }{TEXT 332 5 "mixed" }{TEXT -1 93 " partial derivatives are obtained in a similar manner. Here for ex ample, are various ways to " }}{PARA 0 "" 0 "" {TEXT -1 11 "calculate \+ " }{TEXT 333 2 " " }{XPPEDIT 334 1 "diff(P(V,T),V,V,V,V,T)" "-%%diff G6(-%\"PG6$%\"VG%\"TGF(F(F(F(F)" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "D[1,1,1,2,1](P)(V,T); D[1$4,2](P)(V,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\"%\"RGF&%\"VG!\"&\"#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\"%\"RGF&%\"VG!\"&\"#C" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diff(P(V,T),V,V,V,T,V); d iff(P(V,T),V$4, T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\"\" %\"RGF&%\"VG!\"&\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"nG\"\" \"%\"RGF&%\"VG!\"&\"#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright a nd Author Information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 45 "Tutor1R4.mws A Maple Release 4 worksheet." }} {PARA 278 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT -1 24 "Autho r: Brian E. Blank " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 280 "" 0 "" {TEXT -1 23 "Date: 3 September 2000" }}{PARA 0 "" 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, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prio r written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 281 "" 0 "" {TEXT -1 49 "For more information, please contact \+ the author:" }}{PARA 282 "" 0 "" {TEXT -1 4 " " }}{PARA 282 "" 0 " " {TEXT -1 32 " Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Louis" }}{PARA 0 "" 0 " " {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-67 63" }}{PARA 283 "" 0 "" {TEXT -1 44 " e-mail: brian@mat h.wustl.edu" }}{PARA 284 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Brian E. Blank, All Rights Reserve d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "11 8 0" 1 }{VIEWOPTS 1 1 0 3 4 1802 }