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{PARA 235 "" 0 "" {TEXT 329 19 "JG's Maple Tutorial" } {TEXT 329 0 "" }}{PARA 236 "" 0 "" {TEXT 330 29 "J. Gerlach, work in p rogress," }{TEXT 330 0 "" }}{PARA 237 "" 0 "" {TEXT 331 30 "Last Updat e: February 23, 2003" }{TEXT 331 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 212 "This is an introduction to maple intended for the use in Calculus classes. It was written for maple v ersion 8. It is work in progress, and suggestions are welcome. For m ore detailed help with maple you can use" }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 6 "- the " }{TEXT 334 15 "New User's Tour" }{TEXT 333 10 " from the " }{TEXT 334 4 "Help" }{TEXT 333 30 " icon to get mo re information," }{TEXT 333 0 "" }}{PARA 239 "" 0 "" {TEXT 333 8 "- ty pe " }{TEXT 335 8 "?command" }{TEXT 333 57 " to find out about the sp ecifics of a certain command, or" }{TEXT 333 0 "" }}{PARA 239 "" 0 "" {TEXT 333 9 "- select " }{TEXT 334 12 "Topic Search" }{TEXT 333 66 " f rom the help menu to inquire about specifics on a certain topic." } {TEXT 333 0 "" }}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 13 "Preliminarie s" }{TEXT 336 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 337 73 "All statements must end with either a semicolon (;), or with a colon (:)." }{TEXT 333 183 " Using the colon suppresses th e output, the semicolon will print an output. You can also list sever al statements and separate them by a comma(,). I will do so below to \+ save space. " }{TEXT 333 0 "" }}{PARA 238 "" 0 "" {TEXT 332 1 " " } {TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 10 "Press the " }{TEXT 337 5 "ENTER" }{TEXT 333 28 " key to execute a statement." }{TEXT 333 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }}{PARA 241 "" 0 "" {TEXT 338 146 "Note, that maple recognizes functions, or expressions etc. in the order in which they were executed, and not in the order in which they are typed!" }{TEXT 338 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }} {PARA 239 "" 0 "" {TEXT 333 33 "Use the pound symbol # to insert " } {TEXT 337 8 "comments" }{TEXT 333 11 ". Example:" }{TEXT 333 0 "" }} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "# Assignment 101" } {MPLTEXT 1 339 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 198 "You can remove the output from a worksheet using: 'Edit', and 'Remove Output', 'From Worksheet'. Conversely you can e xecute the entire worksheet using the 'Edit' , 'Execute', 'Worksheet' \+ commands." }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 123 "Make it a habit to begin with the 'restart' command. This erases all previously set definitions and lets you sta rt over. " }{TEXT 332 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 21 "Maple as a Calculator" }{TEXT 336 0 "" }}{PARA 243 "" 0 "" {TEXT 340 11 "Arithmetic:" }{TEXT 340 0 "" }}{PARA 238 "" 0 "" {TEXT 332 171 "Here are a few examples of basic operations. Notice th at * is necessary for multiplication, unlike the implied multiplicatio n, which is used on many graphing calculators." }{TEXT 332 0 "" }} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 46 "4-5.5, (-12)*(-.3), 2.5* 4, 3/.5, 2^5, 6^(-2); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 156 "Maple is a computer algebra system, it strives to pr esent you with exact expressions rather than numerical values. For ex ample, 14/49 is simplified to 2/7:" }{MPLTEXT 1 341 1 " " }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 6 "14/49;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 18 "In order to get a " }{TEXT 337 15 "numerical value" }{TEXT 333 29 ", you can e ither the command " }{TEXT 337 5 "evalf" }{TEXT 333 20 ", which stands for e" }{TEXT 334 26 "valuate in floating point " }{TEXT 333 89 "or y ou can enter the numbers with a decimal point somewhere in the express ion. Examples:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "evalf(14/49);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "14.0/49;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 244 " " 0 "" {TEXT 342 20 " Built-In Functions:" }{TEXT 342 0 "" }}{PARA 238 "" 0 "" {TEXT 332 41 "Maple has a host of built-in functions. " } {TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 4 "The " }{TEXT 337 11 "s quare root" }{TEXT 333 32 " function can be used like this:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "sqrt(x); sqr t(4); sqrt(-2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "The " }{TEXT 337 14 "absolute value" }{TEXT 333 31 " func tion is entered as follows" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 24 "abs(x); abs(4); abs(-2);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "The " }{TEXT 337 20 "exponential function" }{TEXT 333 1 " " }{XPPEDIT 18 0 "e^x;" "6#)%\"eG%\"xG" } {TEXT 333 37 " is called with exp(x). For example:" }{TEXT 333 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "exp(x); exp(1); exp(-1 /2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "Th e " }{TEXT 337 10 "constant e" }{TEXT 333 12 " is found by" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "exp(1.0);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "The " }{TEXT 337 17 "natural logarithm" }{TEXT 333 133 " can be called with ln(x) or wi th log(x). This is unlike your calculator, where log(x) is the common (base 10) logarithm. Examples:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 37 "log(x); ln(x); ln(exp(1)); log(10.0);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "The " } {TEXT 337 9 "constant " }{TEXT 333 1 " " }{XPPEDIT 18 0 "pi;" "6#%#piG " }{TEXT 333 55 " is entered as Pi, with upper case P. Check this ou t:" }{TEXT 333 0 "" }}{PARA 239 "" 0 "" {TEXT 333 1 " " }{MPLTEXT 1 341 7 "Pi; pi;" }{TEXT 333 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 5 "Both " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 333 63 "s look alike, \+ one has a numerical value, the other one doesn't:" }{TEXT 333 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "evalf(Pi); evalf(pi);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 11 "The use \+ of " }{TEXT 337 23 "trigonometric functions" }{TEXT 333 123 " is strai ght forward. Just remember to use parentheses, and of course, all tri g functions are based on radians. Examples:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 30 "sin(x); sec(5*Pi/6); tan(1.5);" } {MPLTEXT 1 339 0 "" }}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 22 "Specia l Maple Commands" }{TEXT 336 0 "" }}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 30 " Definition or the use of := " }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 104 "The command sequence := is used for definitions. For example, let's define e to be the Euler co nstant" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "e := \+ exp(1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 123 "The same := command is used to define functions or equations. For example, we may define a generic quadratic equation Q by " }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "Q := a*x^2+b *x+c;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 69 " To recall a defined variable, just enter it, followed by a semicolon." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 2 "e;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 2 "or" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 2 "Q;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 11 "I you want " } {TEXT 334 9 "undefine " }{TEXT 333 29 "the constant e you can enter " }{TEXT 334 8 "restart," }{TEXT 333 103 " in which case all other defin itions are wiped out as well, or you can reset e to the string 'e' usi ng " }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "e := 'e'; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 5 "Check" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 2 "e;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 36 "Expressions, Functions and Equations" }{TEXT 343 0 "" }} {PARA 239 "" 0 "" {TEXT 333 130 "Maple makes a strict distinction betw een expressions, functions and equations. Let's illustrate the distic tion using the example " }{XPPEDIT 18 0 "x^2-2*x-3;" "6#,(*$%\"xG\"\"# \"\"\"*&F&F'F%F'!\"\"\"\"$F)" }{TEXT 333 1 "." }{TEXT 333 0 "" }} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "e := x^2 -2* x -3;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 345 1 "e " }{TEXT 333 7 " is an " }{TEXT 337 10 "expression" }{TEXT 333 118 ". \+ It can be factored, rearranged, or evaluated for certain x values, bu t it is NOT an equation, nor is it a function." }{TEXT 333 0 "" }} {PARA 239 "" 0 "" {TEXT 333 19 "To relate it to an " }{TEXT 337 8 "equ ation" }{TEXT 333 17 ", we have to set " }{TEXT 346 1 "e" }{TEXT 333 32 " EQUAL to something, for example" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 14 "eqn := e = 12;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 50 "and once this is done, you c an solve the equation:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "solve(eqn,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 2 "A " }{TEXT 337 8 "function" }{TEXT 333 97 " assigns an output f(x) to a given input x. e can be related to a function in the following way:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 " > " 0 "" {MPLTEXT 1 339 24 "f := x -> x^2 - 2*x - 3;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 10 "Check all:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "e; eqn;f(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 198 "Notice, that e a nd f(x) are identical. Think about it this way: The function f assig ns to each input x the output f(x), and we describe the possible outpu ts by a formula, namely by the expression " }{XPPEDIT 18 0 "x^2-2*x-3; " "6#,(*$%\"xG\"\"#\"\"\"*&F&F'F%F'!\"\"\"\"$F)" }{TEXT 333 3 ". " } {TEXT 333 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}} {PARA 238 "" 0 "" {TEXT 332 0 "" }}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 11 "% and evalf" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 " " 0 "" {TEXT 333 4 "The " }{TEXT 337 1 "%" }{TEXT 333 90 " symbol is a convenient way to reactivate the expression that was last computed. \+ Example:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 7 "exp(2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 2 "%;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 12 "The command " }{TEXT 337 5 "evalf" }{TEXT 333 57 " stan ds for evaluate as a floating point number. Example" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 14 "evalf(exp(2));" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 67 "The comb ination of % and evalf is especially useful. For example: " }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "solve(x^2+x= 1,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 157 "There is another way to obtain floating point output, n amely to enter one of the numbers in floating point itself. Again, le t's illutsrate this by examples:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 " " {MPLTEXT 1 339 19 "solve(x^2+x=1.0,x);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 25 "Notice the difference in " } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "exp(2); exp(2. 0);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 194 "K eep in mind that maple is a computer algebra system, and it tries to p erform all operations in exact arithmetic. Sometimes we need to speci fically request an output as a floating point number." }{MPLTEXT 1 341 1 " " }{TEXT 333 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 19 "Maple Knows Algebra" }{TEXT 336 0 "" }}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 9 "Factoring" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 " " 0 "" {TEXT 333 32 "The factoring command is simply " }{TEXT 337 6 "f actor" }{TEXT 333 29 ". Here is how it can be used:" }{TEXT 333 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "factor(x^3 - 3*x^2 +2) ;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "x^4 - 16;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "factor(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "E := 16 + 2*x - 3*x^2;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "factor(E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 45 "The opp osite of factoring is expanding. The " }{TEXT 337 6 "expand" }{TEXT 333 39 " command will do it for you. Examples:" }{TEXT 333 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "expand((x-3)*(1-x));" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "( x-1)*(x+1)*(x^2+1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 " " {MPLTEXT 1 339 10 "expand(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 40 "E := 3 + 2*x -5*x*(x-1) - x*(x-1)*(x- 3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "expand(E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 68 "Here is a list of formulas which can be found in many al gebra books:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "expand((a+b)^2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "expand((a-b)^2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "expand((a+b)^3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "factor(a^2-b ^2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "factor(a^3-b^3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 16 "factor(a^3+b^3);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 333 88 "I worked some more problems fro m a College Algebra book below. Notice that the command " }{TEXT 337 8 "simplify" }{TEXT 333 33 " will come in handy occasionally." }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "factor(x^2-2*x-3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 35 " E := (m^2 - 4*m + 4)/(m^2 + m - 6);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 65 "E1 := (n^2 - n - 6)/(n^2 - 2*n -8); E2 := (n^2-9)/(n^2 + 7*n +12);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 6 "E1/E2;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(%); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 29 "(h+x)^3-(h-x)^3; simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 19 "Rules for Exponents" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 112 "Rules for exponents are no problem. \+ Almost that is, I had to experiment a little to obtain the desired res ults:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "a^n*a^ m; simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "a^n/a^m; simplify(%);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "1/a^n; simplify(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 4 "a^ 0;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "(a^m)^n; simplify(%,power,symbolic);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 " " 0 "" {TEXT 343 13 "Substitutions" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 66 "How can we substitute numbers for the consta nts? Here is one way:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "a := 4: a^n;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "a^(5/2);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 85 "Another way to su bstitute constant is given by the subs command. Here is an example:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "E := a*x^2 + \+ b*x + c;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 105 "Note that maple recalled my earlier definition of a. I didn't me an that, how can I undo this definition?" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 22 "One possible answer is" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "a := 'a';" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "E := a*x^2 + b*x + c;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 39 "It worke d! An alternative is to enter " }{TEXT 337 9 "restart. " }{TEXT 333 102 " But this is an emergency solution, all definitions will be wiped out, and all memory will be cleared." }{TEXT 337 1 " " }{TEXT 333 0 " " }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 83 "In order to turn E into th e expression x^2 -2x-3, I need to set a=1, b=-2, and c=3:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "subs(\{a=1, b=-2, c=-3\} , E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 61 " I can also define a new expression using this subs statement:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "e := subs(\{a=1, b=- 2, c=-3\}, E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "x:=2: e;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 " > " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 22 "Solutions of Equations" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 37 "Let's start with quadratic equations:" } {TEXT 332 0 "" }}{PARA 242 "" 0 "" {MPLTEXT 1 339 25 "E := a*x^2 + b*x + c = 0;" }{MPLTEXT 1 339 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 11 "solve(E,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "e := subs(\{a=1, b=-2, c=-3\}, E);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 11 "solve(e,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 58 "More e quations. In case of complex solutions, the symbol " }{XPPEDIT 18 0 " I;" "6#%\"IG" }{TEXT 333 44 " is used for the imaginary number such th at " }{XPPEDIT 18 0 "I*I = -1;" "6#/*&%\"IG\"\"\"F%F&,$F&!\"\"" } {TEXT 333 2 ". " }{MPLTEXT 1 341 1 " " }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 43 "solve(4/(x-3) - 8/(2*x+5) + 3/( x-3) = 0,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "solve(abs(5/(r-3)) = 10,r);" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 31 "e := x^4 +x^3 -2*x^2 + 4*x - 24;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "solve(e=0,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "factor(e);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "solve(2^(3*x)=128,x );" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "solve(10^x = 200,x);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 45 "We see \+ that this the common logarithm of 200:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "evalf(log10(200));" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 238 "" 0 "" {TEXT 332 109 "If you want the numerical sol ution only, and you do not care for the symbolic manipulations, you ca n use the " }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 14 "fsolve c ommand" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "fsolve(10^x = 200,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "fsolve(x^2 +2*x = 1,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 43 "Logarithms with base b ar e used as follows:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "log[b](x); E := b^y;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "y := solve(E=x,y);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "solve(exp(x^2 - \+ 2*x) = 1,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 134 "The solve command can also be used to solve systems of equati ons. Let's find the intersections of a circle of radius 2 with the li ne " }{XPPEDIT 18 0 "x+y = 1;" "6#/,&%\"xG\"\"\"%\"yGF&F&" }{TEXT 333 1 ":" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart; " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 39 "\nE1 := x^2 + y^2 = 4; E2 \+ := x + y = 1;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 87 "Solve two equations - listed using braces- for the two unknown s - also listed in braces" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "solve(\{E1,E2\},\{x,y\});" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 109 "### WARNING: allvalues \+ now returns a list of symbolic values instead of a sequence of lists o f numeric values" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 14 "\nallvalues (%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 54 "For numerical solutions you may use the fsolve command" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "fsolve(\{E1,E 2\},\{x,y\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 83 "By descibing the range of x and y we can for fsolve to identif y the other solution:" }{MPLTEXT 1 341 39 "fsolve(\{E1,E2\},\{x,y\},\{ x=-2..0,y=1..2\});" }{TEXT 333 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 21 "A Litt le Trigonometry" }{TEXT 343 0 "" }}{PARA 238 "" 0 "" {TEXT 332 0 "" }} {EXCHG {PARA 238 "" 0 "" {TEXT 332 44 "Maple knows the basic trigonome tric formulas" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 25 "\ne:= sin(x)^2 + c os(x)^2;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "simplify(e);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "e := sin(s+t); expand(e);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "e := sin(2*t); expa nd(e);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 42 "e := sin(t)/cos(t); e-tan(t); simplify(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 74 "Keep in mind that all com putations are done in radians! The way to enter " }{XPPEDIT 18 0 "Pi; " "6#%#PiG" }{TEXT 333 3 " is" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 38 "Pi; # The upper case P is essential!!" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "sin(5/4*Pi );" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "tan(Pi/4);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "cos(Pi)+1;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 8 "Graphing" }{TEXT 336 0 "" }}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 9 "Functions" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 " " 0 "" {TEXT 333 2 "A " }{TEXT 337 1 "f" }{TEXT 333 7 "unction" } {TEXT 337 1 " " }{TEXT 333 100 "assigns an output f(x) to a given inpu t x. This process is indicated by the -> notation. Examples:" } {TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 24 "f := x -> x^2 - 2*x - 3;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "g := x -> (cos(x)+1)/(x-Pi);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "h := x -> x*exp(-x); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 13 "Recal l that " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 333 22 " is a function, while " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 333 18 " is a n expression." }{MPLTEXT 1 341 3 " " }{TEXT 333 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 35 "Function evaluation is easily done:" } {TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "f(2); \+ g(0); h(3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 347 19 "Piecewi se Functions" }{TEXT 347 0 "" }}{PARA 238 "" 0 "" {TEXT 332 90 "Here a re some examples. For simplicity, let's plot the functions and see wha t is going on:" }{TEXT 332 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "f:= x -> piecewise(x <= -1, x^2, 1):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "p lot(f(x),x=-3..5,y=-1..4);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 " > " 0 "" {MPLTEXT 1 339 50 "f:= x -> piecewise(x<= -1,x^2,x > 2, x^2-2 *x+1,1):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "plot(f(x),x=-3..5,y=-1..4);" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 56 "f:= x -> piecewise(x<= -1,x^2,x > 2, x^2-2*x+1,x=1,2,1):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "plot(f(x),x=-3..5,y=-1..4);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 34 "It doesn 't show in the graph, but " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 20 "One more condition: " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 84 "f:= x -> piecewise(x<= -1,x^2,x > 2, x^2-2* x+1,x=1,2,x>=0 and x<1,1+ sqrt(1-x^2),1):" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "plot(f(x),x=-3..5, y=-1..4);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "f(0),f(1),f(2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 10 "Graphing I" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 85 "The plot command will always graph expressions. We can o btain graphs in several ways:" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 37 "* Enter the expression directly, or " }{TEXT 332 0 "" } }{PARA 239 "" 0 "" {TEXT 333 21 "* Define a function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 333 51 " using the -> syntax, and then plot th e expression " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 333 4 " , or" }{TEXT 333 0 "" }}{PARA 238 "" 0 "" {TEXT 332 42 "* Define an e xpression, and then plot it." }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 39 "Let's illustrate this for the function " }{XPPEDIT 18 0 "sin(x-sin(x));" "6#-%$sinG6#,&%\"xG\"\"\"-F$6#F'!\"\"" }{TEXT 333 43 ". First we define the respective function " }{XPPEDIT 18 0 "f;" "6#% \"fG" }{TEXT 333 35 " and the cooresponding expression ." }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 24 "f := x -> sin(x-sin(x));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "y := sin(x-sin(x));" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 48 "The respective graphs can be ob tained as follows" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "plot(sin(x-sin(x)),x=-5..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "plot(f(x),x=-5..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "plot(y,x=- 5..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 130 "Of course, all three graphs are alike. The command plot(f,x=-5..5 ) did not work, but strangely enough plot(y(x),x=-5..5) did work." } {TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 137 "The plot command com es with a host of options. Only the specification of the domain (the \+ x-values) is mandatory. Here are some options:" }{TEXT 332 0 "" }} {PARA 238 "" 0 "" {TEXT 332 41 " Exampl es:" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 39 " * viewing wi ndow view=[-5..5,-2..2]" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 41 " * color color = green" }{TEXT 332 0 "" }} {PARA 238 "" 0 "" {TEXT 332 39 " * thickness thickness = \+ 2" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 88 " * title \+ title=(`A periodic Function`) Important: Use ` and not \+ '" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 36 " * discontinuit ies discont=true" }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 22 "For more choices try " }{TEXT 337 13 "?plot,options" }{TEXT 333 36 " to see a complete list of options." }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 44 "plot(f(x),x=-5..5,color=green, thickn ess=2, " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 43 "\nview=[-5..5,-2..2] , title=`What's that?`);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 106 "Click somewhere on the graph to see a box, and the n right-click: You will see a few more graphing options." }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 333 27 " Here is the impact of the " } {TEXT 337 14 "discontinuity " }{TEXT 333 10 "statement:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "plot(tan,-3..12,-20..20); \+ " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 78 "Since plot 'connects the dots', we also obtain vertical lines at multiples \+ of " }{XPPEDIT 19 1 "pi;" "6#%#piG" }{TEXT 333 61 ". These will not a ppear if we include the discont statement:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "plot(tan,-3..12,-20..20,discont =true); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 39 "Several functions can be graphed using " }{TEXT 337 4 "plot" } {TEXT 333 94 " as well. List the functions in a bracket [], and separ ate the functions by commas. Example:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 67 "plot([f(x),sin(x)], x=-5..5, -2..2, color=[ red,blue], thickness=3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 47 "We can also DEFINE graphs, and view them later:" } {TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 80 "G1 := \+ plot(f(x),x=-5..5,-2 ..2,color=sienna, thickness=2, title=`What's that ?`):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 80 "I used the colon : to suppress the output. If I want to see G1, I can \+ just type" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 3 "G1; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 11 "Graphing II" }{TEXT 343 0 "" }}{PARA 247 "" 0 "" {TEXT 347 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 111 "There are some other fancy graphing feature s which also use up a lot of memory. If you want to use them, enter" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with( plots);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 71 "Note the list of new commands which are only available if you use \+ the " }{TEXT 337 12 "with(plots):" }{TEXT 333 85 " statement. Let's \+ illustrate the implicitplot, display and the animation features. " } {TEXT 333 0 "" }}{PARA 238 "" 0 "" {TEXT 332 229 "The graph of an equa tion consists of all points whose coordinates satisfy the said equatio n. The graph will not necessarily be that of a function, and you can' t just use plot. The classical example is the equation for a circle:" }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/ ,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT 333 35 ". To plot this equa tion we can use" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 62 "implicitplot(x^2+y^2=1,x=-1.5..2,y=-1..2,scaling=constrained);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 33 "Let's al so save this graph to G1:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 94 "G1 := implicitplot(x^2+y^2=1, x=-2..2, y=-2..2,c olor = green,thickness=2,scaling=constrained):" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 239 "" 0 "" {TEXT 333 90 "For animation we need to gene rate a family of functions. For each value of the parameter " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT 333 89 " we get a different functi on f(x). In the animation we view one function at a time - as " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT 333 24 " changes. Let's try it:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "f := \+ t -> sin(t)/t;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 64 "animate(a*f(a*x),x=-10..10,a=1..5,color=red,view=[- 7..7,-5..5]);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 141 "Click on the graph, and note the 'video options on the maple \+ toolbar. You can also save the graph, and use display to bring it bac k to life." }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 73 "G2 := animate(a*f(a*x), x=-7..7, a=1..5, color=blue, view=[-7. .7,-5..5]):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "display(G1,G2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 16 "Another example:" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "f:= t -> sin(t-sin(t)):" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 45 "a nimate(f(t-a),t=-15..15,a=0..10,color=blue);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 " " 0 "" {TEXT 343 17 "Parametric Curves" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 239 "" 0 "" {TEXT 333 40 "Parametric curves are given i n the form " }{XPPEDIT 18 0 "x(t),y(t);" "6$-%\"xG6#%\"tG-%\"yGF%" } {TEXT 333 50 ", just think of the position of an object at time " } {XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 333 99 ". Let's obtain a parameti c curve by example. First we need to define the functions x(t) and y( t):" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 30 " x := t -> exp(-0.1*t)* cos(t);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 31 "y := t -> exp(-0.1*t)*sin(2*t);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 35 "The para metric plot is obtained by " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "plot([x(t),y(t),t=0..100]);" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 146 "Note that in the brackets we have [x-function, y-function, range of the parameter]. Of course we \+ may add our variety of plot options, for example" }{TEXT 332 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 94 "plot([sin(3*t),cos(2*t), t=-Pi/2..Pi/2],view=[-3..3,-2..2],color=blue,thickness=2,title=`wow`); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 17 "Polar Coordinates" } {TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 61 "In po lar coordinates a function is described by the equation " }{XPPEDIT 18 0 "r = f(theta);" "6#/%\"rG-%\"fG6#%&thetaG" }{TEXT 333 136 ". If \+ we want to plot such a function we need to tell maple to use polar coo rdinates using a coords=polar statement. Here is an example" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 46 "plot([1/2+sin(3*t),t ,t=0..2*Pi],coords=polar);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 " " 0 "" {TEXT 332 145 "The basic syntax has the form plot([ r(t), thet a(t), t = range of t], coords=polar). Of course, we may add adidtiona l plot options, for example" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 120 "plot([3*cos(t),t,t=-Pi/4..Pi/2],-1..4,-2..2,scalin g=constrained,coords=polar,color=blue, thickness=2,title=`A circle?`); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 17 "For o ur purposes " }{XPPEDIT 18 0 "theta = t;" "6#/%&thetaG%\"tG" }{TEXT 333 42 " and we always have to stick in the extra " }{XPPEDIT 18 0 "t; " "6#%\"tG" }{TEXT 333 25 " after the definition of " }{XPPEDIT 18 0 " r(t);" "6#-%\"rG6#%\"tG" }{TEXT 333 32 ". Here is an example when bot h " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT 333 5 " and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 333 11 " depend on " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 333 2 ". " }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 61 "plot([1+t^2,exp(t),t=0..4],scaling=constrained, coords=polar);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 337 11 "Animations " }{TEXT 333 9 "require " }{TEXT 333 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 73 "Here are two examples ( click on the graph to get the animations started):" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 90 "animate([sin(4*t*a),t*a,t=0..2* Pi],a=0..1,numpoints=100,coords=polar,scaling=constrained);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 90 "a nimate([a+sin(4*t)-1,t,t=0..2*Pi],a=0..2,numpoints=100,coords=polar,sc aling=constrained);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 " " {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 11 "Plot Points" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 183 "Here we try to plot a set of points. First define the data p oints to be plotted. Each point requires a bracket, as in [x,y]. The whole set of points requires an additional bracket." }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 43 "pts := [[0,1],[1,1.6] ,[2,2],[2.5,1],[3,0]];" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 9 "Check it:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 4 "pts;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 212 "The plot is obtained by a straightforward plot co mmand with or without the usual options. However, we need to specify \+ that we want points, (style=point), else the dots would automatically be connected by lines." }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 68 "plot(pts,style=point,symbol=circle,view=[-1..4,-1. .3],color=maroon);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 107 "The same can be accomplished with the pointplot command , however, we need to call the plots package first. " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "pointplot(pt s);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 93 "If you want to generate the points by a formula you may use the followin g approach (example, " }{XPPEDIT 18 0 "y = exp(-x)*sin(x);" "6#/%\"yG* &-%$expG6#,$%\"xG!\"\"\"\"\"-%$sinG6#F*F," }{TEXT 333 2 ")." }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "N:= 10: dx := 1/4:" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 24 "\nfor k from 1 to N/dx do" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 30 "\na[k] := sin(k*dx)*exp(-k*dx) :" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 28 "\np(k) := [k*dx , a[k]] : \+ od:" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 75 "Th e seq command lets us place the points p(k) into one object, called p ts:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "pts := s eq(p(k),k=1..N/dx):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 " " {MPLTEXT 1 339 26 "plot([pts],style = point);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 6 "Limits" }{TEXT 336 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{PARA 239 "" 0 "" {TEXT 333 71 "Computations of limits are done using the limit \+ command. Example, let " }{XPPEDIT 18 0 "f(x) = sin(x)/x;" "6#/-%\"fG6 #%\"xG*&-%$sinGF&\"\"\"F'!\"\"" }{TEXT 333 41 " , and take the limit a s x tends to zero:" }{TEXT 333 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "limit(sin(x)/x,x=0);" }{MPLTEXT 1 339 0 "" }}} {PARA 238 "" 0 "" {TEXT 332 68 "We could also define a function f(x) f irst, and then take the limit:" }{TEXT 332 0 "" }}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 19 "f := x -> sin(x)/x;" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "limit(f(x),x=0);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 62 "It is a \+ good idea to use the inert from of this command first:" }{TEXT 332 0 " " }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "Limit(f(x),x=0);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 147 "Using t he limit command with an upper case L just displays the limit in quest ion, nothing is being calculated. In order to evaluate the limit, use " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 125 "We shal l use this technique below. By displaying the limit before we compute it we can catch type-o's before it is too late." }{TEXT 332 0 "" }}} {PARA 238 "" 0 "" {TEXT 332 26 "One-sided limits, example:" }{TEXT 332 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 24 "f := x -> sin (x)/abs(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "Limit(f(x),x=0);value(%);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 30 "Limit(f(x),x=0,left);val ue(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 31 "Limit(f(x),x=0,right);value(%);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 57 "plot(f(x),x=-4..4, y=-1. 5..1.5, color=red, discont=true);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(0);" }{MPLTEXT 1 339 0 "" }}} {PARA 238 "" 0 "" {TEXT 332 30 "Limits at infinity. Examples:" } {TEXT 332 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "f(x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "L imit(f(x),x=infinity);value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "g := x -> (exp(x)-1)/(exp(x)+1);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "L imit(g(x),x=infinity);value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 33 "Limit(g(x),x=-infinity);value(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 31 "p lot(g(x),x=-25..25,color=red);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 15 "Differentiation" }{TEXT 336 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 111 "It will be important to distinguish bewteen the derivatives of functions vers us the derivatives of expressions." }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 43 "y := x^2-2*x; # y is an expr ession" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 63 "f := x -> x^2- 2*x; # f is a function, f(x) is an expressio n" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 54 "Firs t let's look at the D operator for differentiation" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "D(f);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "D(f)(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 36 "Now let's use diff \+ for the same task" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "diff(x^2-2*x,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 10 "diff(y,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "diff(f(x),x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 27 "The following do not wo rk: " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "D(y);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 11 "D (x^2-2*x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "D(y)(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 " > " 0 "" {MPLTEXT 1 339 10 "diff(f,x);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 23 "What can we conclude? " } {TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 337 4 "diff" }{TEXT 333 28 " \+ takes the derivative of an " }{TEXT 337 10 "expression" }{TEXT 333 90 ". This expression was communicated by either typing it in explicitly, or by just entering " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 333 4 " or " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 333 39 ". In all c ases diff had to deal with " }{XPPEDIT 18 0 "x^2-2*x;" "6#,&*$%\"xG\" \"#\"\"\"*&F&F'F%F'!\"\"" }{TEXT 333 21 " and the answer was " } {XPPEDIT 18 0 "2*x-2;" "6#,&*&\"\"#\"\"\"%\"xGF&F&F%!\"\"" }{TEXT 333 3 ". " }{TEXT 333 0 "" }}{PARA 239 "" 0 "" {TEXT 337 1 "D" }{TEXT 333 29 " expects that the input is a " }{TEXT 337 8 "function" }{TEXT 333 53 ", and it returns a function, namely the derivative. " } {XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT 333 23 " is the derivat ive of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 333 12 ". Think of " } {XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT 333 4 " as " }{TEXT 334 2 "f'" }{TEXT 333 26 ", and if you want to form " }{TEXT 334 6 "f' (x)," }{TEXT 333 12 " just enter " }{XPPEDIT 18 0 "D(f)(x);" "6#--%\"D G6#%\"fG6#%\"xG" }{TEXT 334 1 "." }{TEXT 333 1 " " }{TEXT 333 0 "" }} {PARA 239 "" 0 "" {TEXT 333 125 "It takes some knowledge of Calculus, \+ and a positive attitude towards maple to interpret the results from D( y)or from D(y)(x)." }{MPLTEXT 1 341 2 " " }{TEXT 333 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 74 "Higher order derivatives can be conven iently computed with the D operator:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 33 "restart; f:= x -> exp(-x)*cos(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "D(f)(x); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 60 "For the second derivative we can apply the D operator twice:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 11 "D(D(f))(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 101 "This would be arkw ard if we tried to find the eigth derivative. Fortunately, there is a simpler way:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 41 "(D@@2)(f)(x); # for the second derivative" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 40 "(D@@8)(f)(x); # for the eigth derivative" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 84 "Evaluation of a derivative at a given x value is equally simple with the D operator:" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 13 "f'(2) becomes" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "D(f)(2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 " " 0 "" {TEXT 333 22 "The 8th derivative at " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT 333 9 " becomes" }{TEXT 333 0 "" }}{PARA 242 " > " 0 "" {MPLTEXT 1 339 14 "(D@@8)(f)(Pi);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 79 "Now let us define a function, a nd plot this function along with its derivative." }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 46 "restart; f := t -> exp(-0.25*t) *sin(t-sin(t));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "g := D(f):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 50 "plot([f(t),g(t)],t=-2..5,-1..1, color =[blue,red]);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 119 "Let's conclude this section with the graph of a function toge ther with a sketch of a tangent line at a specified point." }{TEXT 332 0 "" }}{PARA 239 "" 0 "" {TEXT 337 8 "Problem:" }{TEXT 333 5 " Let " }{XPPEDIT 18 0 "f(x) = (exp(x)-1)/(exp(x)+1);" "6#/-%\"fG6#%\"xG*&, &-%$expGF&\"\"\"F,!\"\"F,,&F*F,F,F,F-" }{TEXT 333 68 ", sketch the ta ngent line to the graph of f at the point where x=1." }{TEXT 333 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 337 9 "Solution:" }{TEXT 333 81 " Fi rst define f(x), and select the x value where the tangent line is to b e taken." }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "rest art;" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 33 "\nf := x -> (exp(x)-1)/ (exp(x)+1);" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 8 "\na := 1;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 61 "Determin e the slope of the tangent line at the selected point" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "m := D(f)(a):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 23 "Define the tangent \+ line" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "y := m* (x-a)+f(a):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 16 "Sketch the graph" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 66 "plot([f(x),y],x=-.5..5,color=[blue,black],view=[-0. 5..5,-1..1.5]);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 55 "In case you were curiuous about m and y, here they are:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "m; y; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 18 "Implicit Functions" } {TEXT 336 0 "" }}{PARA 239 "" 0 "" {TEXT 333 28 "Let us look at an exa mple: " }{XPPEDIT 18 0 "x^3+y^3 = 4*xy;" "6#/,&*$%\"xG\"\"$\"\"\"*$% \"yGF'F(*&\"\"%F(%#xyGF(" }{TEXT 333 48 ". It is non-trivial to solve this equation for " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 333 39 ", an d to obtain a function of the form " }{XPPEDIT 18 0 "y = f(x);" "6#/% \"yG-%\"fG6#%\"xG" }{TEXT 333 40 " . Let us first define this equatio n: " }{TEXT 333 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "r estart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "E := x^3 + y^3 = 4*x*y;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 32 "Solving for y yields the result:" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 11 "solve(E,y);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 95 "This exp ression is way too messy. It may be better to deal with the function \+ in implicit form." }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 121 " Let's look at a graph first. When we wish to employ the command 'impl icitplot', we need to load the plots-package first." }{TEXT 332 0 "" } }{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 32 "implicitplot(E,x =-3..3,y=-3..3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 270 "There are several things we can do to improve the looks of this graph: Use more points and get a better resolution, increase the thickness so that the paper output is easier to read, and set the coordinates so that the spacing on the x-axis and the y-axis are the \+ same." }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 114 "implicitplot(E,x=-3..3,y=-3..3,view = [-3..3,-3..3], numpoints=20 00,thickness=2,scaling = constrained,color=blue);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 62 "What about slopes? Let's \+ find y' by implicit differentiation:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "m := implicitdiff(E,y,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 38 "What is the slope at the \+ point P(2,2)?" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "subs(x=2,y=2,m);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 79 "At what points does the graph have horizontal tangent l ines? We need to solve " }{XPPEDIT 18 0 "m = 0;" "6#/%\"mG\"\"!" } {TEXT 333 2 ": " }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "H := solve(m=0);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 258 "These are all values for which m vanishes. We only ne ed the points that are also on the curve, that is we need to solve m=0 along with the equation E. We obtain a system of two equations and t wo variables. The maple solution can be implemented as follows:" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "solve(\{E,m=0 \}, \{x,y\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 61 "The numerical values are more useful, let's redo the problem:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "fsolve(\{E,m =0\}, \{x,y\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 104 "We missed the origin, probably because m is undefined a t the origin. Where are the points of slope m=1?" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "P := fsolve(\{E,m=1\},\{x,y\}); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 31 "Let's identify the other point:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 43 "Q := fsolve(\{E,m=1\},\{x,y\},\{x=1..2,y=0..1\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 22 "Finally another graph." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 119 "G := implicitplot(E,x=-3..3,y=-3..3,view = [-3..3,-3..3], numpoin ts=2000,thickness=2,scaling = constrained,color=blue):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 43 "With cut-and-paste \+ we get the tangent lines" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 44 "line1 := x -> x - .7005338460 + 1.608867231:" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "G 1 := plot(line1(x),x=-3..3,y=-3..3): " }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 44 "line2 := x -> x - 1.6088 67231 + .7005338460:" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "G2 := plot(line2(x),x=-3..3,y=-3..3): " } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "d isplay(G,G1,G2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 11 "Integr ation" }{TEXT 336 0 "" }}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 43 "Antid erivatives and the Indefinite Integral" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 239 "" 0 "" {TEXT 333 36 "Antiderivatives are found usi ng the " }{TEXT 335 4 "int " }{TEXT 333 74 "command. This is equivale nt to finding an indefinite integral. To apply " }{TEXT 337 3 "int" } {TEXT 333 51 " you need enter the integrand, i.e. the expression " } {TEXT 337 4 "f(x)" }{TEXT 333 56 ", and the name of the variable. Thi s is similar to the " }{TEXT 334 4 "diff" }{TEXT 333 87 " command. he re are some examples, note that the constant of integration is suppres sed:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "int(x^3 - 3*x^2 + 6,x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 19 "int(1/x + 1/x^2,x);" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 14 "int(exp(x),x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 56 "You can also define a functi on first using the standard " }{TEXT 334 15 "f := x -> expr " }{TEXT 333 101 " procedure, and then compute its antiderivative, or even defi ne a new function as an anti-derivative:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "f:= x -> x*ln(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "int(f(x),x); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "F := x -> int(f(x),x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 " > " 0 "" {MPLTEXT 1 339 5 "F(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 18 "Check the result: " }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 13 "diff(F(x),x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 12 "The command " }{TEXT 348 3 "Int" }{TEXT 333 122 " is the inert version of the integration c ommand. It simply displays the integral, but it doesn't compute anyth ing. The " }{TEXT 348 5 "value" }{TEXT 333 184 " command can be used \+ to evaluate the integral. It is highly recommended to use this proces s, because the integral is displayed and it makes it easier to catch e rrors. (The definition " }{XPPEDIT 18 0 "f(x) = ln(x);" "6#/-%\"fG6#% \"xG-%#lnGF&" }{TEXT 333 20 " is still in effect)" }{TEXT 333 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "Int(f(x),x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 21 "The Definite Integral" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 92 "For the definite integral use int again, but this time we specify the limits of integration." }{TEXT 332 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "int(x^2-2*x, x=-1..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "f := x -> x*ln(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "int(f(x),x=1..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 7 "Again, " }{TEXT 334 3 "Int" }{TEXT 333 42 " can be used to display the integral, and " } {TEXT 334 5 "value" }{TEXT 333 29 " will perform the evaluation:" } {TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 37 "Int(f(x),x=1.. 5); value(%); evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 143 "The is another way to obtain an anti-derivative, ba sed on the Fundamental Theorem of Calculus (the definition f(x) =ln(x) is still in effect): " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "F := x -> int(f(t),t=1..x);" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "F(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 53 "The Numerical Approximations of the De finite Integral" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 29 "The student package constains" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 88 "* graphics to visualize the Left Sum, the Right Sum an d the Midpoint Sum approximations" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 85 "* numerical evaluations of these sums, plus a routine \+ to evaluate the trapezoid sum." }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 128 "In all cases you need to communicate the function, the \+ endpoints, and the number of subintervals (default = 4). Here is a de mo:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "r estart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 14 "with(student);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 27 "f:= x-> x^4 - 7*x^2 +8*x+1;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 48 "leftbox(f(x) ,x=0..2,10,color=red,shading=blue); " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 51 "\nrightbox(f(x),x=0..2,10,color=blue,shading=green);" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 54 "\nmiddlebox(f(x),x=0..2,10,col or=brown,shading=yellow);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 73 "Here are the numerical approximations. We compute the exact value first." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "True := int(f(x),x=0..2.0);" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 29 "L := leftsum(f(x),x=0. .2,10);" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 31 "\nR := rightsum(f(x) ,x=0..2,10);" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 32 "\nM := middlesu m(f(x),x=0..2,10);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 112 "The display of the sums does not appear to be too helpf ul. The floating point values contain more information:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 51 "True; L := evalf(L); R : = evalf(R); M := evalf(M); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 41 "T := trapezoid(f(x),x=0..2,10): evalf(T); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 134 "It a ppears that midpoint sum yields the best approximation in this example . Note that the trapezoid result is the average of L and R:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "evalf((R + L )/2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 97 " Simpson's rule can be obtained as a weighted average of the trapezoid \+ rule and the midpoint rule:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "S := evalf((2*M+T)/3);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 41 "It's approximation is the close st so far:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "S \+ - True;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 39 "Maple's built-in Simpson routine yields" }{MPLTEXT 1 341 0 "" } {MPLTEXT 1 341 42 "\nSm := simpson(f(x),x=0..2,10); evalf(Sm);" } {TEXT 333 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 103 "In order to \+ duplicate the value which we obtained by weighted averaging, we need t o use 20 subintervals" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 20 "(note the rounding):" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 41 "Sm := simpson(f(x),x=0..2,20): evalf(Sm);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 22 "Differential Equations" }{TEXT 336 0 "" }} {SECT 1 {PARA 245 "" 0 "" {TEXT 343 6 "Basics" }{TEXT 343 0 "" }} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 129 "We shall illustrat e the concepts using y' = x-y as the standard example. We define t he differential equation by the statement" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "ODE := D(y)(x) = x - y(x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 203 "Note, t hat you have to write y'(x) and y(x) explicitly , that is, you have to include the x. A simple y' or y won't do the trick, you need to comm unicate to maple that y is a function of the variable x." }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 82 "The dsolve command lets you fin d the general solution of the differential equation" }{TEXT 332 0 "" } }{PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "dsolve(ODE,y(x));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 65 "and a slight modi fication can be used for initial value problems." }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "dsolve(\{ODE,y(0)=1\},y(x));" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 57 "Another \+ problem: Solve the initial value problem y' = " }{XPPEDIT 18 0 "1-y ^2;" "6#,&\"\"\"F$*$%\"yG\"\"#!\"\"" }{TEXT 333 10 "; y(0)=0." } {TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "D E := D(y)(x) = 1-y(x)^2;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> \+ " 0 "" {MPLTEXT 1 339 25 "dsolve(\{DE,y(0)=0\},y(x));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 17 "Graphical Methods" }{TEXT 343 0 "" } }{EXCHG {PARA 238 "" 0 "" {TEXT 332 50 "Most graphics commands require that we use DEtools" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "restart; with(DEtools):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 37 "Let's use the example y' = x-y again." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "DE := D(y)(x ) = x - y(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 248 "" 0 "" {TEXT 349 16 "Direction field:" }{TEXT 349 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 37 "dfieldplot(DE,y(x), x=-2..2,y=-2..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 249 "" 0 "" {TEXT 350 33 "Direction field wit h some extras:" }{TEXT 350 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 78 "dfieldplot(DE,y(x),x=-2..2,y=-2..2,color = (x-y),title = `A Direct ion Field`);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 241 "" 0 "" {TEXT 338 46 "Direction field with selected solution curves:" }{TEXT 338 0 " " }}{PARA 238 "" 0 "" {TEXT 332 102 "DE plot uses a numerical method. \+ Occasionally you may have to reset the stepsize to refine the graph." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 69 "DEplot(DE, y (x), x = -2..2,[[y(0)=0]],color=black,linecolor=[blue]); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 30 "Let us look at an other example" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 30 "DE := D(y)(x) = x^2 +y(x)^2-1;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 39 "The general solution looks complicated :" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "dsolve(\{D E\},y(x));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 105 "DEplot(DE, y(x), x = -2..2, [[y(0)=0],[y(0)=1], [y (0)=-1]], view=[-2..2,-2..2],linecolor=blue,color=red);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 109 "I don't know where the arrows went, and I had to insert the view statement to avoid floa ting point overflows." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 1 " " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 39 "Elasti c Spring Equation (2nd order ODE)" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 150 "This is a demo for a second order differential \+ equation. We model the motion of a spring of mass m, with spring cons tant k, and friction parameter r." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 48 "ODE := m*(D@@2)(x)(t) + r* D(x)(t) + \+ k*x(t) = 0;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 105 "We want ot study the impact of changing the friction paramete r, and for simplicity we set m and k to 1. " }{TEXT 332 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "m := 1: k := 1: ODE;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 142 "Further we shall always use the initial conditions y(0)=1, and y'(0) = 0 , i.e. we start at position x=1, with initial velocity x'(0)=0." } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "dsolve(\{ODE,x (0)=1,D(x)(0)=0\},x(t));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 136 "We see that the solution depends on r, and r=2 app ears to be an interesting limiting case. First we shall look at small values for r: " }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 52 "r := 0: sln1 := dsolve(\{ODE,x(0)=1,D(x)(0)=0\},x(t ));" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 56 "\nr := 0.05: sln2 := dso lve(\{ODE,x(0)=1,D(x)(0)=0\},x(t));" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 55 "\nr := 0.1: sln3 := dsolve(\{ODE,x(0)=1,D(x)(0)=0\},x(t));" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 60 "We shall now obtain a graph of all three solution functions." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 41 "\nG1 := plot(rhs(sln1),t=0..25,color=red) :" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 42 "\nG2 := plot(rhs(sln2),t=0 ..25,color=blue):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 43 "\nG3 := pl ot(rhs(sln3),t=0..25,color=black):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 19 "\ndisplay(G1,G2,G3);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 79 "We see that the blue and the black curve slo wly lose amplitude due to damping. " }{TEXT 332 0 "" }{TEXT 332 57 "\n What happens near r=2? Again, let's look at examples: " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 54 "r := 1.5: sln1 := dsolve (\{ODE,x(0)=1,D(x)(0)=0\},x(t));" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 53 "\nr := 2: sln2 := dsolve(\{ODE,x(0)=1,D(x)(0)=0\},x(t));" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 56 "\nr := 2.1: sln3 := dsolve(\{O DE,x(0)=1,D(x)(0)=0\},x(t)); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "factor(rhs(sln1));factor(rhs(sln2));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 153 "We se e that for r < 2, oscillations are still present (with strong dampin g though), while for r >= 2, the solutions approach zero without oscil lations." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "G1 \+ := plot(rhs(sln1),t=0..8,color=red):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 41 "\nG2 := plot(rhs(sln2),t=0..8,color=blue):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 42 "\nG3 := plot(rhs(sln3),t=0..8,color=black):" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 19 "\ndisplay(G1,G2,G3);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 351 8 "r: =0.1; " }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 15 "\nwith(DEtools):" } {MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 77 "\nDEplot(ODE,x(t),t=0..30,[[x( 0)=1, D(x)(0)=0]],linecolor=blue, thickness=1); " }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 85 "This is an example where it is advisable to reset the stepsize for better \+ resolution:" }{TEXT 332 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 351 90 " DEplot(ODE,x(t),t=0..30,[[x(0)=1, D(x)(0)=0]],linecolor=blue, thicknes s=1, stepsize=0.1); " }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}} {EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 246 "> \+ " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 36 "Predator/Prey (Systems of Equations)" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 200 "A coupled system of differential equations \+ is usually harder to solve than a single differential equation. We sh all look at examples for predator/prey system (Compare SEction 7.4 of \+ our maple book). " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "restart;with(DEtools):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 74 "First define some coefficients for the syste m, and then the system itself:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 29 "a:=1: b:=1/2: k:=1: r := 1/3:" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 41 "\nSYS := [D(x)(t) = k*x(t) - a*x(t)*y(t)," } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 42 "\n D(y)(t) = -r*y(t) + \+ b*x(t)*y(t)];" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 81 "Notice that the system does not contain the variable t. Such \+ systems are called " }{TEXT 334 11 "autonomous." }{TEXT 333 56 " Nex t we look at the direction field in the xy-plane: " }{TEXT 333 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 49 "dfieldplot(SYS,[x(t),y(t)],t=0. .1,x=0..2,y=0..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 45 "This is the equivalent of the DEplot command:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 121 "phaseportrait(SYS,[ x(t),y(t)],t=0..12,[[x(0)=0.5,y(0)=0.8],[x(0)=0.8,y(0)=1.0]],linecolor =blue,thickness=2,stepsize=0.1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 261 "A general solution of the system via \+ the dsolve command did not work for me. As a substitute one can find \+ numerical solutions and plot these individually as functions of t, or \+ the phase portrait in the xy-plane. We use initial conditions x(0)= 0 .8, y(0)=1.0. " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 40 "SYS := [D(x)(t) = k*x(t) - a*x(t)*y(t)," }{MPLTEXT 1 339 0 "" } {MPLTEXT 1 339 41 "\n D(y)(t) = -r*y(t) + b*x(t)*y(t)," } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 29 "\n x(0)=0.8, y(0)=1.0]; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 44 "sln := dsolve(SYS,[x(t),y(t)],type=numeric);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "o deplot(sln,[[t,x(t)],[t,y(t)]],0..25);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 31 "odeplot(sln,[x(t),y(t)], 0..13);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}} {SECT 1 {PARA 245 "" 0 "" {TEXT 343 14 "Euler's Method" }{TEXT 343 0 " " }}{EXCHG {PARA 242 "" 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 48 "This is a routine w hich executes Euler's method." }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 21 "The input consists of" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 58 "f the right hand side of the differential equ ation" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 58 "a, y0 the co ordinates of the starting point, i.e. y(a)=y0" }{TEXT 332 0 "" }} {PARA 238 "" 0 "" {TEXT 332 26 "b the final x-value" }{TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 22 "h the step size" } {TEXT 332 0 "" }}{PARA 238 "" 0 "" {TEXT 332 83 "The output is an arra y of points, containing the coordinates of the approximations." } {TEXT 332 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 351 25 "Euler := proc( f,a,y0,b,h)" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 27 "\nlocal N, x,y,j ,pt,dy, pts;" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 19 "\nN := round(b/ h)+1;" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 17 "\nx := a; y:= y0;" } {MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 16 "\npt(1) := [x,y];" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 21 "\nfor j from 2 to N do" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 16 "\ndy := h*f(x,y);" }{MPLTEXT 1 351 0 "" } {MPLTEXT 1 351 24 "\nx := x+h; y := y + dy;" }{MPLTEXT 1 351 0 "" } {MPLTEXT 1 351 16 "\npt(j) := [x,y];" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 8 "\nend do;" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 28 "\npts := \+ [seq(pt(j),j=1..N)];" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 10 "\nend p roc:" }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 337 8 "Example:" }{TEXT 333 38 " Solve the initial valu e problem " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 333 4 "' = " } {XPPEDIT 18 0 "1-x*y^2;" "6#,&\"\"\"F$*&%\"xGF$*$%\"yG\"\"#F$!\"\"" } {TEXT 333 3 " " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT 333 157 " on the interval [0,2.5] with stepsize h=0.1, and generate a plot of the solution. NOTE: The x is omitted in the y-te rm below, it is y^2, not y(x)^2.." }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 21 "f:= (x,y) -> 1-x*y^2;" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 27 "pts := Euler(f,0,1,2.5,. 1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "plot(pts,style=point);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 87 "Let's get a plot of the exact solution along with the approximation via Euler's method:" }{TEXT 332 0 "" }}} {EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 351 12 "with(plots):" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 29 "\nG1 := plot(pts,style=point):" } {MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 48 "\nY := dsolve(\{D(y)(x) = f(x, y(x)),y(0)=1\},y(x)):" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 13 "\nY := rhs(Y):" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 24 "\nG2 := plot(Y,x=0. .2.5):" }{MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 16 "\ndisplay(G1,G2);" } {MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 38 "Sequences, Sums, Serie s, Taylor Series" }{TEXT 336 0 "" }}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 9 "Sequences" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 " " 0 "" {TEXT 333 69 "Let us illustrate the concepts by studying the se quence defined by " }{XPPEDIT 18 0 "a[n] = (2*n^2+(-1)^n*n*cos(n))/( n^2+1);" "6#/&%\"aG6#%\"nG*&,&*&\"\"#\"\"\"*$F'F+F,F,*(),$F,!\"\"F'F,F 'F,-%$cosGF&F,F,F,,&F-F,F,F,F1" }{TEXT 333 0 "" }{TEXT 333 62 "\nFirst we define the sequence as a function of the variable n:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 40 "a := n -> (2*n^2+(-1)^n* cos(n))/(n^2+1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 45 "Here are the first, tenth and hundredth term " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 20 "a(1), a(10), a(100); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 51 "Their floating point values can be found with evalf" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 73 "The seq command proves helpf ul. To list the first five terms, just enter" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "seq(a(n),n=1..5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 145 "The seq proves useful for plots as well. First use seq to define the points, and then use plots for the display. Note the style=point option.. " }{MPLTEXT 1 341 0 "" }{MPLTEXT 1 341 32 "\npts := seq([n, a(n)], n=1.. 20):" }{MPLTEXT 1 341 0 "" }{MPLTEXT 1 341 25 "\nplot([pts],style=poin t);" }{TEXT 333 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 60 "For the limit of a sequence, use the limit command as usual." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "limit(a(n),n=infinity);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 37 "Of cours e, limits don't always exist:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 35 "Limit(cos(n),n=infinity); value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 42 "Limit((-1)^n *n/(n+1),n=infinity);value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 44 "Limit((-1)^n*n^2/(n+1),n=infinity);va lue(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 61 "Now let \+ us look at iterative sequences, and use the example " }{XPPEDIT 18 0 "a[k+1] := 1/(1+a[k]);" "6#>&%\"aG6#,&%\"kG\"\"\"F)F)*&F)F),&F)F)&F%6# F(F)!\"\"" }{TEXT 333 7 " with " }{XPPEDIT 18 0 "a[1] = 1;" "6#/&%\"a G6#\"\"\"F'" }{TEXT 333 76 ". The the sake of definiteness let us tak e the first ten terms only and set" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "N := 10:" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 32 "Now we define the first N terms " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "a[1] := 1:" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 25 "\nfor k from 1 to (N-1) do" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 22 "\na[k+1] := 1/(1+a[k]):" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 4 "\nod:" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 29 "Use seq to display the values" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "seq(a[k],k=1. .N);evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 35 "A plot can be generated as follows:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "pts := seq([k,a[k]],k=1..N):" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 25 "\nplot([pts],style=point);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 115 "Clearly , there appears to be a limit, but the limit command cannot be use, si nce we only defined the first N terms:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "limit(a[k],k=infinity);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 56 "Our limit is a FIXED POIN T. The sequence is defined by " }{XPPEDIT 18 0 "a[k+1] = f(a[k]);" "6 #/&%\"aG6#,&%\"kG\"\"\"F)F)-%\"fG6#&F%6#F(" }{MPLTEXT 1 341 1 " " } {TEXT 333 6 "where " }{XPPEDIT 18 0 "f(x) = 1/(1+x);" "6#/-%\"fG6#%\"x G*&\"\"\"F),&F)F)F'F)!\"\"" }{TEXT 333 28 ". The limit L will satisfy " }{XPPEDIT 18 0 "L = f(L);" "6#/%\"LG-%\"fG6#F$" }{TEXT 333 83 ", wh ich makes it a so-called fixed point of the function. Let's compute t he limit:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "f \+ := x -> 1/(1+x);" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 17 "\nsolve(x=f (x),x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 34 "The numerical value in question is" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "evalf(%[1]);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 4 "Sums" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 53 "Sums, fi nite or infinite are can be computed via the " }{TEXT 334 4 "sum " } {TEXT 333 8 "command." }{TEXT 333 0 "" }}{PARA 238 "" 0 "" {TEXT 332 34 "Let S = 1+2+3+ ..+9+1000. We find" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "S := sum(k,k=1..1000);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 108 "Sometim es it is helpful to view the expression for the sum before it is evalu ated. In this case we use the " }{TEXT 334 3 "Sum" }{TEXT 333 24 " co mmand along with the " }{TEXT 334 5 "value" }{TEXT 333 19 " command. \+ Example:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 14 "Sum(k,k=1..N);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 58 "Let's factor the answer to obtain a mo re familiar formula:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "factor(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 18 "Some more examples" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 22 " S := Sum(x^k,k=0..N);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(S);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "factor(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 " S := Sum(m^2,m=1..10);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(S);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 6 "Series" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 " " 0 "" {TEXT 332 106 "Series are infinite sums, and all we need to do, is to make the upper limit of the sum infinite. Example:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "S := Sum(x^k ,k=0..infinity);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(S);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 38 "Looks like maple doesn't care whether " } {XPPEDIT 18 0 "abs(x) < 1;" "6#2-%$absG6#%\"xG\"\"\"" }{TEXT 333 27 " \+ or not. If we substitute " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" } {TEXT 333 10 " we obtain" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "S2 := subs(x=2,S);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "value(S2);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 38 "This was to be expected, h owever, for " }{XPPEDIT 18 0 "x = -2;" "6#/%\"xG,$\"\"#!\"\"" }{TEXT 333 10 " we obtain" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "S2 := subs(x=-2,S);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 163 "If you use maple 5, you get a wrong r esult. It looks like a maple programmer forgot to put an absolute val ue in its appropriate place. In maple 7 this got fixed." }{MPLTEXT 1 341 2 " " }{TEXT 333 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 98 "S ometimes maple uses special functions to express the result of an infi nte summation. For example" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "S := Sum(2*k/(k^3+1),k=1..infinity);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 63 "A floati ng point approximation may be sufficient for our needs:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 87 "If a calculation become s too difficult for maple, it responds by restating the problem:" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 34 "sum((2+sin(k)) /k^2,k=1..infinity);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 134 "In this case evalf does not help, but using a partial \+ sum may yield a satisfactory approximation (use calculus to estimate t he error) " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "ev alf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "sum((2+sin(k))/k^2,k=1..10);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "evalf(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 13 "Taylor Series" }{TEXT 343 0 "" }} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 4 "The " }{TEXT 337 6 " series" }{TEXT 333 8 " or the " }{TEXT 337 6 "taylor" }{TEXT 333 226 " command are our tool to find series expansions. According to the map le help, the series command is the more general command, and the talor command is a mere restriction of the series command. Let's look at a few expansions:" }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "series(exp(x),x=0);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 18 "series(1/x,x=1,5);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 63 "The synt ax for the series command is series[f(x),x=a,n], where " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 333 176 " indicates the function, a =x stands for the point of the series expansion. The parameter n is o ptional, if used, it is the degree of the Taylor polynomial -1. More \+ examples: " }{TEXT 333 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "taylor(sin(x)/x,x=0,5);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "taylor(sin(x)/x,x=0,6);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "taylor(sin(x )/x,x=0,7);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 22 "taylor(arctan(x),x=0);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 76 "The next example shows the diff erence between the taylor and series command:" }{TEXT 332 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "f := x -> cos(x)/x;" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "t aylor(f(x),x=0);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 59 "A plot shows why the Taylor expansion of f is bound to f ail" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 38 "plot(f(x ),x=-3..3,-5..5,discont=true);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 31 "The function is not defined at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 333 104 " and a Taylor series (or Maclaurin series) does not exist. The series command returns an answ er anyhow:" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "s eries(f(x),x=0);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 42 "and we see that the function behaves like " }{XPPEDIT 18 0 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT 333 6 " near " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 333 1 "." }{TEXT 333 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 63 "plot([f(x),1/x],x=-2..2, -10..10,discont=true,color=[red,blue]);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 337 19 "Taylor polynomials: " }{TEXT 333 256 " Although the series command displays the series we \+ have to do more to define an approximating polynomial. As the first s tep we can use coeftayl to get the Taylor series coefficient. Again w e must indicate the function, the point and the order. Example:" } {TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "f := x -> sin( x)/x;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 65 "coeftayl(f(x), x=0,0);coeftayl(f(x),x=0,1);coeftayl(f(x), x=0, 2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 55 "No w we use this command to build the Taylor polynomial." }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "restart; " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 42 "\nf := x -> sin(x)/x: # define the function" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 52 "\na := 0: # define the point of expansion" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 40 "\nN \+ := 5: # define the order " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 21 "\nfor j from 0 to N do" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 34 "\nc[j] := coeftayl(f(x),x=a,j): od:" }{MPLTEXT 1 339 0 "" } {MPLTEXT 1 339 36 "\np := x -> sum(c[k]*(x-a)^k,k=0..N):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "p(x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 49 "p lot([f(x),p(x)],x=-8..8,-1..2,color=[blue,red]);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 51 "Use cut and paste to find o ther Taylor polynomials:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "restart; " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 42 " \nf := x -> sqrt(x): # define the function" }{MPLTEXT 1 339 0 "" } {MPLTEXT 1 339 52 "\na := 4: # define the point of expansi on" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 40 "\nN := 3: # d efine the order " }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 21 "\nfor j fro m 0 to N do" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 34 "\nc[j] := coefta yl(f(x),x=a,j): od:" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 36 "\np := x -> sum(c[k]*(x-a)^k,k=0..N):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 5 "p(x);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "simplify(p(x));" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 50 "plot([f(x),p (x)],x=-2..12,-3..5,color=[blue,red]);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}}{SECT 1 {PARA 240 "" 0 "" {TEXT 336 17 "Several Variables" }{TEXT 336 0 "" }}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 9 "Functions" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" } }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 82 "Functions of two or three var iables are quite easily defined. See examples below:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "f := (x,y) -> x^2-2*x*y-y^2; " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 33 "g := (x,y,z) -> z*sin(x^2+2*y^2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 245 "" 0 "" {TEXT 343 6 "Graphs" }{TEXT 343 0 "" }}{EXCHG {PARA 238 "" 0 "" {TEXT 332 108 "The plot3d command is suited to graph functions of variables. One can either enter the expression directly," }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 38 "plot3d(x^2-2*x*y-y^2,x=-2..2,y= -2..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 60 "or, define a function f(x,y) first, and then plot z =f(x,y):" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "f := (x,y) -> \+ x^2-2*x*y-y^2;" }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 32 "\nplot3d(f(x, y),x=-2..2,y=-2..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 43 "You may apply the usual bells and whistles:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 62 "plot3d(f(x,y),x=-2.. 2,y=-2..2,axes=normal,title=`A Surface`); " }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 238 "" 0 "" {TEXT 332 116 "For several graphs in the same plot, list the functions within parentheses, and separate the functio ns with a comma:" }{TEXT 332 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 351 55 "plot3d(\{f(x,y),-1+2*x-2*y\},x=-2..2,y=-2..2,axes=normal," } {MPLTEXT 1 351 0 "" }{MPLTEXT 1 351 42 "\ntitle=`A Function with a Tan gent Plane`);" }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 80 "Contours, or level curves, require the plots-package. So we need to first enter" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 12 "with(plots):" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 238 "" 0 "" {TEXT 332 25 "Here is the contour plot:" } {TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 36 "contourplot(f( x,y),x=-2..2,y=-2..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 96 "If we want to select the levels ourselves we need to \+ add a contour=... statement to the options:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 63 "contourplot(f(x,y),x=-2..2,y=-2..2,co ntours=[-2, -1 , 0, 1, 2]," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 23 " \nscaling=constrained); " }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> \+ " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 180 " Graphs of functions with three (or more) variables are impossible to g raph in 3-space. But we can look at contours, and I found that implic itplot3d works (with(plots) required). " }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 39 "g:= (x,y,z) -> z^2-2*x*z+2*x*y-y^2-z-y;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 66 " implicitplot3d(x^2-2*y^2-z=0,x=-2..2,y=-2..2,z=-2..2,axes=normal);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 34 "Here are several contour surfaces:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 80 "c := 0: G0 := implicitplot3d(x^2-2*y^2-z=c,x=-2..2, y=-2..2,z=-2..2,axes=normal):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 80 "c := 2: G2 := implicitplot3d(x^2-2*y^ 2-z=c,x=-2..2,y=-2..2,z=-2..2,axes=normal):" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 82 "c := -2: G2a := implicit plot3d(x^2-2*y^2-z=c,x=-2..2,y=-2..2,z=-2..2,axes=normal):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "display(G0 ,G2,G2a);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 15 "Differ entiation" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 "restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 33 "Let's define the functions first:" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 28 "f := (x,y) -> x^2-2*x*y-y^2:" } {MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 34 "\ng := (x,y,z) -> z*sin(x^2+2* y^2):" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 57 " The diff command can be used to find partial derivarives:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "diff(f(x,y),x);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "d iff(g(x,y,z),y);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 17 "diff(g(x,y,z),z);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 212 "It's that easy. It should also be cl ear now why we had to throw in the extra x in the diff-command - unlik e differentiation with the D operator. The use of the D operator with partial derivatives is as follows:" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 16 "D[1](f);D[2](f);" }{MPLTEXT 1 339 0 " " }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 15 "D[1](g)(x,y,z);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 60 "Higher o rder mixed partial derivatives can also be computed:" }{TEXT 332 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "D[1,1](f);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 10 "D[1,3](g);" } {MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 19 "D [1,2,3](g)(x,y,z);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 125 "Let's \+ find stationary points using solve. We need to set the partial deriva tives to zero and solve for x and y. Here we go:" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 49 "solve(\{diff(f(x,y),x)=0,diff(f (x,y),y)=0\},\{x,y\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 45 "Let's try it for a more complicated function:" } {TEXT 332 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 351 40 "g := (x,y,z) - > z^2-2*x*z+2*x*y-y^2-z-y;" }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n " }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "solve(\{diff(g(x,y, z),x)=0," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 27 "\n diff(g(x,y ,z),y)=0," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 28 "\n diff(g(x, y,z),z)=0\}," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 17 "\n \{x,y, z\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(%);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 127 "Ooops? Is it possible that the function does not have \+ a stationary point? Let's modify the function and repeat the procedur e." }{TEXT 332 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 351 40 "g := (x,y ,z) -> z^2-2*x*z+2*x*y+y^2-z-y;" }{MPLTEXT 1 352 0 "" }{MPLTEXT 1 352 1 "\n" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 26 "solve(\{diff(g (x,y,z),x)=0," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 27 "\n diff( g(x,y,z),y)=0," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 28 "\n diff (g(x,y,z),z)=0\}," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 17 "\n \+ \{x,y,z\});" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}}{SECT 1 {PARA 245 "" 0 "" {TEXT 343 11 "Integr ation" }{TEXT 343 0 "" }}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 8 " restart;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 333 106 "Find the value of a double or triple integral, apply the int comm and repeatedly. Let's work the integral " }{XPPEDIT 18 0 "int(int(2-x ^2*y,y = 0 .. 1),x = 0 .. 1);" "6#-%$intG6$-F$6$,&\"\"#\"\"\"*&%\"xGF) %\"yGF*!\"\"/F-;\"\"!F*/F,F0" }{TEXT 333 45 " from the book (p. 710). \+ First define f(x,y)" }{TEXT 333 0 "" }}{PARA 242 "> " 0 "" {MPLTEXT 1 339 23 "f:= (x,y) -> 2 - x^2*y;" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 13 "Now integrate" }{TEXT 332 0 "" }} {PARA 242 "> " 0 "" {MPLTEXT 1 339 45 "A := Int(Int(2-x^2*y,y = 0 .. 1 ),x = 0 .. 1);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(A);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 40 "plot3d(f(x,y),x=0..1,y=0..1,axes=boxed);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 238 "" 0 "" {TEXT 332 99 "Now le t us calculate the volume of a paraboloid. First we need to define a \+ suited function f(x,y):" }{TEXT 332 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 25 "f:= (x,y) -> 4 - x^2-y^2;" }{MPLTEXT 1 339 0 "" }} }{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 54 "plot3d(f(x,y),x=-2..2,y =-2..2,view=[-2..2,-2..2,0..4]," }{MPLTEXT 1 339 0 "" }{MPLTEXT 1 339 33 "\nscaling=constrained,axes=boxed);" }{MPLTEXT 1 339 0 "" }}} {EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 58 "V := Int(Int(f(x,y),y=-s qrt(4-x^2)..sqrt(4-x^2)),x=-2..2);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 242 "> " 0 "" {MPLTEXT 1 339 9 "value(V);" }{MPLTEXT 1 339 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 344 0 "" }}}{PARA 238 "" 0 "" {TEXT 332 0 "" }}}}{PARA 251 "" 0 "" {TEXT 225 0 "" }}{PARA 251 "" 0 "" {TEXT 225 0 "" }}{PARA 252 "" 0 "" {TEXT 315 0 "" }}{PARA 253 "" 0 "" {TEXT 353 0 "" }}{PARA 254 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }