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3 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bulle t Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Maple essentials tutorial " }} {PARA 0 "" 0 "" {TEXT -1 367 "This Maple tutorial is based on one from the Mathematics Department at Seattle Central Community College. It i s designed to give you a comprehensive self-paced hands-on introductio n to the basic commands that you will need to use Maple effectively. \+ \nThe tutorial will step you through each topic with many opportunitie s for you to practice and experiment with Maple.\n" }}{PARA 0 "" 0 "" {TEXT -1 52 "The original tutorial was downloaded from Maplesoft " } {URLLINK 17 "Maple Tutorial" 4 "http://www.mapleapps.com/categories/wh atsnew/html/SCCCmapletutorial.shtml" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Version 15.2.2004" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "I ntroduction to the tutorial " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 264 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "This tutorial has been designed to give you a" }{TEXT 263 1 " " } {TEXT 259 22 "comprehensive hands-on" }{TEXT -1 218 " introduction to \+ the basic commands that you will need to use Maple effectively. The si x sections of the tutorial will step you through each topic with many \+ opportunities for you to practice and experiment with Maple. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 207 ": This tutorial was developed to provide a thorough and \+ efficient introduction to Maple for students about to enter a Calculus course. Therefore the tutorial only assumes familiarity with mathemat ics at the " }{TEXT 259 11 "precalculus" }{TEXT -1 9 " level. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Each sect ion of the tutorial has the same basic outline: " }}{PARA 15 "" 0 "" {TEXT 261 8 "Examples" }{TEXT -1 89 ": a sequence of short, completely worked out examples that illustrate each new procedure." }}{PARA 15 " " 0 "" {TEXT 261 9 "Exercises" }{TEXT -1 89 ": short practice problems based on the material from the section. For your convenience a " } {TEXT 261 17 "Student Workspace" }{TEXT -1 87 " is provided after each exercise so that you have a separate place to enter your work. " } {TEXT 261 7 "Answers" }{TEXT -1 75 " are supplied for all of the exerc ises so that you can check your progress." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "When you have worked through " } {TEXT 261 12 "Sections 1-6" }{TEXT -1 21 " continue on to the \"" } {TEXT 261 17 "Practice Problems" }{TEXT -1 233 "\". These problems wil l give you an opportunity to try out the full range of Maple commands \+ that you have learned in the Tutorial. By the time you complete these \+ problems you will be ready to use Maple effectively in your math class . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "At \+ the end of the tutorial you will find a section entitled \"" }{TEXT 261 26 "Maple Quick Reference Card" }{TEXT -1 182 "\". You may find it convenient to print out the contents of this section so that you can \+ have a convenient reference to the syntax of the commands that you wil l learn in the tutorial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 89 "The Sections 1 - 6 of the tutorial focus on essent ial Maple commands. For details on the " }{TEXT 259 19 "worksheet inte rface" }{TEXT 634 1 " " }{TEXT -1 22 "see the last section \"" }{TEXT 261 32 "Notes on the Worksheet Interface" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Section \+ 1: Numerical calculations " }}{PARA 0 "" 0 "" {TEXT -1 203 "In this se ction you will use Maple to do some standard numerical calculations. M aple's ability to produce exact answers in addition to numerical appro ximations gives you more options in solving problems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Doing exact arithmetic with Maple " }}{PARA 0 "" 0 "" {TEXT -1 126 "Using Maple to do numerical computat ions is very straightforward. Just enter the numerical expression and \+ end the line with a " }{TEXT 280 0 "" }{TEXT 281 0 "" }{TEXT 259 9 "se micolon" }{TEXT -1 1 " " }{TEXT 0 1 ";" }{TEXT -1 11 ". Pressing " } {TEXT 279 7 "[Enter]" }{TEXT -1 97 " will then execute the line and th e result will be displayed in blue in the center of the screen." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 278 9 "Example 1" }{TEXT -1 0 "" }{TEXT 635 0 "" }{TEXT -1 2 " : " }{TEXT 0 3 "+,*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 97 "A simple calculation has been entered on the next line. Click anywhere in the red line and press " }{TEXT 282 7 "[Enter]" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+4;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "12*34567890;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The asterisk " }{TEXT 0 1 "*" }{TEXT -1 40 " is required to indicate multiplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Each red input \+ line is \"live\" and can be modified at any time. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Change the \"4\" in the l ine above to an \"8\" and press " }{TEXT 283 7 "[Enter]" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Not ice how the blue output is automatically updated to display the new re sult." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 284 11 "Example 2 : " }{TEXT 0 1 "^" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "For o ur next example let's calculate " }{XPPEDIT 18 0 "134^39" "6#*$\"$M\" \"#R" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "134^39;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Unlike your calculator, Maple g ives you the exact answer to this problem, all 83 digits worth! " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 285 9 "Example 3" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Mapl e can calculate with fractions without converting to decimals:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "3/5 + 5/9 + 7/12;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 286 9 "Example 4" }{TEXT -1 2 ": " }{TEXT 0 4 "sqrt" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "T o enter the square root of a number use " }{TEXT 260 4 "sqrt" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sqrt(24);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Notice that Maple has simplified " }{XPPEDIT 18 0 "sqrt(24)" "6#-%%sqrtG6#\"#C" }{TEXT -1 134 " but has left the answe r in exact radical form. In the next section you will learn how to get a decimal approximation for this number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 287 9 "Example 5" }{TEXT -1 2 ": " }{TEXT 0 2 "Pi" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Maple has all of the important mathematical constants bui lt in. To enter " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 8 " type \+ " }{TEXT 0 2 "Pi" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "4*(3+Pi);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The asterisk " }{TEXT 0 1 "*" }{TEXT -1 26 " indicates multiplication." }}{PARA 0 "" 0 "" {TEXT -1 77 "Again Maple carries out the calculation but leaves the an swer in exact form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 288 11 "Example 6: " }{TEXT 0 11 "sin,cos,tan" }{TEXT 636 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Un like your calculator, Maple gives you the exact answer when applying t rigonometric functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sin(5*Pi/3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "cos(Pi/4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tan(3*Pi/4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "To get the inverse sine of a number use the " }{TEXT 0 6 "arcsin" }{TEXT -1 10 " function:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "arcsin(-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "If you ask Maple \+ to calculate a value that is " }{TEXT 259 9 "undefined" }{TEXT -1 39 " it will respond with an error message:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tan(Pi/2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 289 11 "Example 7: " }{TEXT 0 3 "exp" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "To enter the natural exp onential function " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" } {TEXT -1 19 " in Maple, type: " }{TEXT 260 6 "exp(x)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "To get the number " }{XPPEDIT 18 0 "exp(1);" "6#-%$expG6# \"\"\"" }{TEXT -1 18 " by itself, type: " }{TEXT 260 6 "exp(1)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 290 11 "Example 8: " }{TEXT 0 3 "abs" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "To enter the ab solute value function " }{XPPEDIT 18 0 "abs(x)" "6#-%$absG6#%\"xG" } {TEXT -1 16 " in Maple type: " }{TEXT 312 6 "abs(x)" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "abs(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "For example, this function at " }{XPPEDIT 18 0 "x = -3;" "6#/%\"xG,$\"\"$!\"\"" }{TEXT -1 8 " yields:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "abs(-3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Note that Maple gives the correct, exact answer for the following since " }{XPPEDIT 18 0 "exp(1)-Pi < 0" "6#2,&-%$expG6#\"\"\"F(%#PiG!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "abs(exp(1)-Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 291 11 "Example 9: " }{TEXT 0 7 "ifactor" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "Maple ha s many special purpose procedures for working with numbers. You will l earn these as you need them in your math course. Here is one last exam ple for now. If we have an integer and want to factor it into primes w e can use the Maple procedure " }{TEXT 0 7 "ifactor" }{TEXT -1 51 ". F eel free to experiment by changing the number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ifactor(3172 2722304);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 292 10 "Example 10" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "There may be times when you want to enter more than one \+ command on a single line. You can do this in Maple, but just be sure t o end " }{TEXT 259 4 "each" }{TEXT -1 92 " command with a semicolon. I t also helps to put spaces between the commands. When you press " } {TEXT 293 7 "[Enter]" }{TEXT -1 100 " all of the expressions are execu ted and the results are listed, in order, in a single output field." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sin(Pi/3); cos(Pi/3); tan(Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"\"$#\"\"\"F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$ #\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "To display the commands in a single execution group but o n separate lines, press " }{TEXT 303 17 "[Shift] + [Enter]" }{TEXT -1 284 ", to obtain a new line within the execution group. The result wil l be the same. It is the semi-colon that signifies the end of a line o f Maple code and the outputs for successive lines of code are printed \+ on successive lines whether or not the lines of code appear on separat e lines. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sin(Pi/3);\ncos(Pi/3);\ntan(Pi/3);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 294 12 "Example 11: " }{TEXT 0 3 "seq" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "To calculate and display a sequence of numbers use the procedur e " }{TEXT 0 3 "seq" }{TEXT -1 65 ". Here we calculate the squares of \+ the first 100 natural numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(k^2,k=1..100);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "evalf( )" {TEXT -1 31 "Numerical approximations using \+ " }{TEXT 0 5 "evalf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 250 "Recall that in the previous section we asked Maple to \+ add three fractions and the result was also displayed as a fraction. T his sort of exact arithmetic is very useful but there are times when w e prefer an answer in decimal form. The Maple procedure " }{TEXT 0 5 " evalf" }{TEXT -1 28 " performs this task for us. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 295 9 "Example 1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Compare the results \+ of the next two lines." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "3/5+5/9+7/12;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(3/5+5/ 9+7/12);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 296 22 "Example 2 : assignment " }{TEXT 0 2 ":=" }{TEXT 640 1 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Assignin g a name to the result of a calculation makes it easier to use that re sult in a subsequent calculation." }}{PARA 0 "" 0 "" {TEXT -1 26 "To a ssign a name we use a " }{TEXT 259 5 "colon" }{TEXT -1 1 " " }{TEXT 0 1 ":" }{TEXT -1 34 " followed by an equal sign ( i.e. " }{TEXT 260 16 "name := result ;" }{TEXT -1 50 " ). On the next line we have assigne d the letter " }{TEXT 260 1 "k" }{TEXT -1 45 " to the original output \+ above. Then we apply " }{TEXT 0 5 "evalf" }{TEXT -1 4 " to " }{TEXT 260 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k := 3/5+5/9+7/12;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(k) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 638 14 "Important Note" }{TEXT -1 11 ": Maple is " }{TEXT 259 14 "case sensitive" }{TEXT -1 34 " so, for example, Maple considers " }{TEXT 260 1 "k" }{TEXT -1 5 " and " }{TEXT 260 1 "K" }{TEXT -1 27 " t o be different variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "K := 2/15*111/264;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "k;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "K;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "You can also use \"words\" as variable names. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rhubarb := 2 ^5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sqrt(rhubarb);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 297 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 133 "If we want fewer or more digits of accuracy than \+ the default number which is 10 digits we can add an extra argument to \+ the procedure " }{TEXT 0 5 "evalf" }{TEXT -1 17 " as shown below. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "w := 4*(3+Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "To see four digits of " }{TEXT 260 1 "w " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "evalf(w,4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "To see forty-five digits of " }{TEXT 260 1 "w" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(w,45);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 298 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "If you enter numbers with a decim al point Maple automatically gives decimal results. Compare the result s of the two lines below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sqrt(34);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sqrt(34.0);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Here is a nother example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "4-1/3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "4.0-1/3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 299 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We can apply the pro cedure " }{TEXT 0 5 "evalf" }{TEXT -1 156 " to a sequence of numbers. \+ Below we first generate the exact square roots of the first 10 natura l numbers, then apply evalf to get decimal approximations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "re sult := seq(sqrt(k),k=1..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(result);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT 300 16 "Maple shortcut: " }{TEXT 0 1 "% " }{TEXT 637 57 " gives a quick reference to the last chronological ou tput" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 189 "There will be many times when using Maple that you wil l string together a sequence of computations. Rather than giving a nam e to each result as you go along, you can use the percent sign ( " } {TEXT 260 1 "%" }{TEXT -1 19 " ) to refer to the " }{TEXT 304 16 "chro nologically " }{TEXT 301 34 "last expression computed by Maple." } {TEXT -1 97 " Note, however, that this device can lead to great conf usion, and is not generally recommended." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Here are some examples of how it w orks. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "3/5+5/9+7/12;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Pi;\nevalf(%);\n%+ 5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Fo r more on using the symbol " }{TEXT 260 1 "%" }{TEXT -1 5 " see " } {TEXT 302 12 "Exercise 1.4" }{TEXT -1 9 " below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 526 7 "Warning" }{TEXT -1 9 ": U se of " }{TEXT 260 1 "%" }{TEXT -1 80 " should be avoided as much as p ossible, as it may make the worksheet unreadable." }}{PARA 0 "" 0 "" {TEXT -1 125 "Its use is probably best restricted to a sequence of Map le commands which are executed together as in the two examples above. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Even use in expressions like " } {TEXT 260 3 "%+5" }{TEXT -1 55 ", as in the second example, should pro bably be avoided." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 265 12 "Ex ercise 1.1" }}{PARA 0 "" 0 "" {TEXT 305 35 "Use Maple to calculate the number " }{XPPEDIT 18 0 "37^43" "6#*$\"#P\"#V" }{TEXT 306 2 " ." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 266 21 "Student Workspace 1.1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 20 "" 0 "" {TEXT 267 10 "Answer 1.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "37^43;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"_o`9LWHELk31&4WLJQl\\(ei*Q&H)4/1\")f 1&)e\"3F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 268 12 "Exercise \+ 1.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 10 "C alculate " }{XPPEDIT 18 0 "sqrt(34)" "6#-%%sqrtG6#\"#M" }{TEXT 308 16 " to 18 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 269 21 "Student Workspace 1.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 270 10 "Answer 1.2" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "m := sqrt( 34);\nevalf(m,18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG*$\"#M#\" \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3Z+`%[*=&4$e!#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 271 12 "Exercise 1.3" }} {PARA 0 "" 0 "" {TEXT 309 54 "Find a numerical approximation for the e xpression : " }{XPPEDIT 18 0 "(3+Pi)/(7-sqrt(13))" "6#*&,&\"\"$\"\" \"%#PiGF&F&,&\"\"(F&-%%sqrtG6#\"#8!\"\"F." }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 272 21 "Student \+ Workspace 1.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 273 10 "Answer 1.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "answer := (3+Pi)/(7-sqrt(13));\neva lf(answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG*&,&\"\"$\"\" \"%#PiGF(F(,&\"\"(F(*$\"#8#F(\"\"#!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$)[I4=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 274 12 "Exercise 1.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "The percent sign ( " }{TEXT 260 1 "%" } {TEXT -1 80 ") is a handy shortcut but it can occasionally lead to so me unexpected results. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Here is an example. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 96 "First execute each of the next three \+ lines. You should be able to predict the result in advance." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "4+ Pi;\nevalf(%);\n%+10;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Now go back and re-execute the last line (i.e., " } {TEXT 260 5 "%+10;" }{TEXT -1 37 "). Note that the output changes from " }{XPPMATH 20 "6#$\"+l#fTr\"!\")" }{TEXT -1 4 " to " }{XPPMATH 20 "6 #$\"+l#fTr#!\")" }}{PARA 0 "" 0 "" {TEXT -1 20 "Can you explain why?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 275 21 "Student Workspace 1.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 276 10 "Answer 1.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The percent symbol (" }{TEXT 260 1 "%" }{TEXT -1 18 ") represents the " }{TEXT 277 47 "chronologically last number calculated by Maple" }{TEXT -1 45 ". So after you execute d the first three lines" }}{PARA 0 "" 0 "" {TEXT 260 1 "%" }{TEXT -1 3 " = " }{XPPMATH 20 "6#$\"+l#fTr\"!\")" }{TEXT -1 41 ". The second t ime you executed the line " }{TEXT 260 5 "%+10;" }{TEXT -1 19 " Maple \+ added 10 to " }{XPPMATH 20 "6#$\"+l#fTr\"!\")" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 60 "To avoid this sort of confusion assign na mes to each output:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a := 4+Pi;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b := evalf(a);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b+10;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "Assigning new values to variabl es and clearing variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 42 "Assigning new values to exi sting variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "Once you have defined a variable, Maple will remember it s value during your entire working session. If you want to overwrite t he variable with a new value, you can simply make a new assignment." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "For exam ple each assignment below redefines the value of the variable " } {XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 99 ": to check the current value for a variable just type it on a command line followed by a semicolon." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Assign " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 14 " the value 56." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "h := \+ 56;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "N otice the value of " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 27 ", 56, after the assignment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "To overwrite " }{XPPEDIT 18 0 "h;" "6 #%\"hG" }{TEXT -1 47 " with a new value, make a new assignment using \+ " }{TEXT 0 2 ":=" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h := sqrt(Pi);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "You'll notice " } {XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 36 " now has the new value you \+ assigned." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 310 86 "The unevaluated form of a variable, clearing one variable at a time a nd the procedure " }{TEXT 0 8 "unassign" }{TEXT 311 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Som etimes you will want to \"clear\" a variable in memory so that you can use it in a new situation. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Here is \+ an example. First we assign " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 13 " the value 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a := 5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 91 "Now assume that we start a new problem an d want to enter the general algebraic expression " }{XPPEDIT 18 0 "a^ 2-4*a+7;" "6#,(*$%\"aG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"\"\"(F'" }{TEXT -1 24 " and assign it the name " }{TEXT 260 1 "b" }{TEXT -1 80 ". If we j ust enter this, Maple automatically substitutes the previous value for " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b := a^2-4*a +7;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "I n order to get " }{TEXT 527 1 "a" }{TEXT -1 134 " to be a general vari able (or unassigned variable) again we must first \"clear\" (i.e. eras e from Maple's memory) our earlier value for " }{TEXT 639 1 "a" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 109 "This can be accomplis hed by asigning the varable to the variable written with single quotes around it ( e.g, " }{TEXT 260 8 "a := 'a'" }{TEXT -1 4 " ). " }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 10 ": Use the " }{TEXT 259 21 "vertical single quote" }{TEXT -1 1 " " }{TEXT 0 1 "'" }{TEXT -1 25 " given by the key on the " }{TEXT 259 10 "right side" }{TEXT -1 21 " of the keyboard and " }{TEXT 259 3 "not" }{TEXT -1 20 " the sl anting quote " }{TEXT 0 1 "`" }{TEXT -1 55 " given by the key at the t op left side of the keyboard." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Execute the next two lines to see how thi s works." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a := 'a';\nb := a^2-4*a+7;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Putting single quotes around th e name of a variable on a command line constructs the " }{TEXT 259 16 "unevaluated form" }{TEXT -1 17 " of the variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a := 6;\nb : = a^2-4*a+7;\n'a';\n'b';" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 94 "Putting single quotes around an expression constru cts the unevaluated form of the expression. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "'sin(Pi/sqrt(16))' ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%#PiG\"\"\"-%%sqrtG6# \"#;!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Putting single quotes around the name of a function causes the \+ function to remain unevaluated. " }}{PARA 0 "" 0 "" {TEXT -1 61 "In th e following, just the sine function remains unevaluated." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "'sin' (Pi/sqrt(16));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,$*&\"\"%! \"\"%#PiG\"\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Putting quotes around " }{TEXT 260 4 "sqrt" }{TEXT -1 35 " means the Maple does not evaluate " }{TEXT 260 4 "sqrt" }{TEXT -1 24 ", and then the function " }{TEXT 260 3 "sin" }{TEXT -1 37 " cannot recognise that the result is " }{XPPEDIT 18 0 "sin(Pi/4)=sqrt(2)/2" " 6#/-%$sinG6#*&%#PiG\"\"\"\"\"%!\"\"*&-%%sqrtG6#\"\"#F)F0F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sin(Pi/'sqrt'(16));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$sinG6#*&%#PiG\"\"\"-%%sqrtG6#\"#;!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Without any quotes we do get th e expected result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "sin(Pi/sqrt(16));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"F%#\"\"\"F%F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "una ssign" }{TEXT -1 80 " can also be used to clear variables. Quotes must be used in this case as well. " }}{PARA 0 "" 0 "" {TEXT -1 80 "This m ethod is useful if you have a number of variables which you wish to cl ear." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "c := 3;\nd := a^2+b^2;\nunassign('a','b','c','d','x') ;\na*x^3+b*x^2+c*x+d;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 30 44 "Clearing all variables at once: the command " }{TEXT 0 7 "restart" }{TEXT 30 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 0 7 "restart" }{TEXT -1 176 " command \+ will clear Maple's memory of all definitions that you have made. It is like starting a new Maple session. If you are starting a completely n ew problem you can use the " }{TEXT 0 7 "restart" }{TEXT -1 154 " comm and to guarantee that there are no leftover definitions from your earl ier work. Before you execute the second line below, quickly predict th e output." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := 1; b := 2; c := 3; d := a+b+c;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a; \nb; \+ \nc;\nd;\nh;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "If you have worked through the previous subsections befor e executing this, the variable " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 35 " will have been assigned the value " }{XPPEDIT 18 0 "sqrt(Pi)" "6#-%%sqrtG6#%#PiG" }{TEXT -1 71 ". You may not have remembered this. \+ That's why it's a good idea to use " }{TEXT 0 7 "restart" }{TEXT -1 126 " to remove all definitions at once. (As you work through this tut orial you will notice that we start most new sections with a " }{TEXT 0 7 "restart" }{TEXT -1 11 " command.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Now check the values of variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a; \nb; \nc;\nd;\nh;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Section 2: Algebraic calculations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 337 12 "Maple is a \"" }{TEXT 528 5 "C.A.S" }{TEXT 648 13 "\", i.e., a " }{TEXT 645 1 "C" }{TEXT 339 8 "omputer " } {TEXT 646 1 "A" }{TEXT 340 7 "lgebra " }{TEXT 647 1 "S" }{TEXT 341 262 "ystem. This means that Maple knows every rule of algebra that you know. Maple also has essential symbolic operations built into its lar ge set of procedures to handle problems from calculus, to solve differ ential equations and to work with vectors and matrices. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 204 "In this section \+ you will learn how to enter an algebraic expression and substitute val ues in for the variables. Then you will learn the commands that allow \+ you to expand, factor and simplify expressions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "s ubs( )" {TEXT -1 4 "The " }{TEXT 0 4 "eval" }{TEXT -1 5 " and " } {TEXT 0 4 "subs" }{TEXT -1 9 " commands" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Example 1: \+ " }{TEXT 0 15 "eval,Eval,value" }}{PARA 0 "" 0 "" {TEXT 342 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "For our first example let's start with th e expression " }{XPPEDIT 18 0 "3*x^2+8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\" #F&F&\"\")F&" }{TEXT -1 25 " and assign it the name " }{TEXT 260 4 "e xpr" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expr := 3*x^2+8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Suppose now that you want to evaluate the expression " }{XPPEDIT 18 0 "3x^2+8" "6#,&*&\"\"$\"\" \"*$%\"xG\"\"#F&F&\"\")F&" }{TEXT -1 7 " where " }{TEXT 529 1 "x" } {TEXT -1 76 " has the value 4. The quickest way to do this is to use t he Maple procedure " }{TEXT 0 4 "eval" }{TEXT -1 28 ". Here's what it \+ looks like:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "eval(3*x^2+8,x=4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "Alternatively, you can apply the proc edure " }{TEXT 0 4 "eval" }{TEXT -1 4 " to " }{TEXT 260 4 "expr" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(expr,x=4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Note that the unevaluated form " } {TEXT 260 20 "'eval(3*x^2+8,x=4)';" }{TEXT -1 16 " is displayed as" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%evalG6$,&*&\"\"$\"\"\")%\"xG\"\"#F) F)\"\")F)/F+\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 55 "in the corresponding Maple output. This ties up with a " }{TEXT 259 30 "standard mathemati cal notation" }{TEXT -1 30 " used to indicate evaluation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "'eva l(3*x^2+8,x=4)';" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Alternatively, you can use the " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Can you predict the output for \+ each of the following commands before you execute it? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "'eval(e xpr,x=4)';" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "'eval'(expr,x=4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "'eval'('expr',x=4);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The speci al name " }{TEXT 0 4 "Eval" }{TEXT -1 21 " (with an upper case " } {TEXT 260 1 "E" }{TEXT -1 43 ") can be used for the unevaluated functi on " }{TEXT 0 6 "'eval'" }{TEXT -1 2 ". " }{TEXT 0 4 "Eval" }{TEXT -1 18 " is the so-called " }{TEXT 259 5 "inert" }{TEXT -1 23 " form of th e procedure " }{TEXT 0 4 "eval" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "The inert form of a procedure can be evaluated by using t he procedure " }{TEXT 0 5 "value" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "A good way to perform an evaluation so that you can demonstrate what you are doing via the cor responding Maple output is to use both " }{TEXT 0 4 "Eval" }{TEXT -1 5 " and " }{TEXT 0 5 "value" }{TEXT -1 24 " in a pair of commands. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Eval(expr,x=4);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "eval" }{TEXT -1 41 " wor ks equally well with symbolic values:" }}{PARA 0 "" 0 "" {TEXT -1 11 " To replace " }{TEXT 530 1 "x" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "5+2*u " "6#,&\"\"&\"\"\"*&\"\"#F%%\"uGF%F%" }{TEXT -1 19 " in the expression " }{XPPEDIT 18 0 "3*x^2+8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\")F& " }{TEXT -1 62 " execute the following line. In this case we label the result " }{TEXT 260 5 "expr2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expr := 3*x^ 2+8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "expr2 := eval(expr,x=5+2*u);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "And now to get Maple to \+ \"multiply out\" this expression we use the procedure " }{TEXT 0 6 "ex pand" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "expand(expr2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 343 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 4 "eval" }{TEXT -1 119 " procedure is very versatile. You can use it to evaluate expressions \+ involving more than one variable. Here we replace " }{XPPEDIT 18 0 "x " "6#%\"xG" }{TEXT -1 10 " by 7 and " }{TEXT 350 1 "y" }{TEXT -1 27 " \+ by 12 in the expression " }{XPPEDIT 18 0 " U=2/5*x^2+3*y" "6#/%\"UG, &*(\"\"#\"\"\"\"\"&!\"\"%\"xGF'F(*&\"\"$F(%\"yGF(F(" }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "U := (2/5)*x^2+3*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG, &*(\"\"#\"\"\"\"\"&!\"\"%\"xGF'F(*&\"\"$F(%\"yGF(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 260 1 "U " }{TEXT -1 167 " has two variables, the evaluating point then needs t o be in a set, indicated by curley brackets \{ \}, or a list indicate d by square brackets [ ] separated by commas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q := eval(U, \{x=7,y=12\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following command has the same effect. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q := \+ eval(U,[x=7,y=12]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "As a floating-point (decimal) number, we have:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(q);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "You can also use the procedure " }{TEXT 0 4 "eval" }{TEXT -1 221 " to sub stitute a value into an equation. This is the sort of thing you might \+ want to do to test whether a particular value \"satisfies\" the equati on. In the next few lines we substitute different values into the equa tion " }{XPPEDIT 18 0 "x^3-5*x^2+7*x-12=0" "6#/,**$%\"xG\"\"$\"\"\"*& \"\"&F(*$F&\"\"#F(!\"\"*&\"\"(F(F&F(F(\"#7F-\"\"!" }{TEXT -1 55 " . A re any of these values a solution to the equation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Note we use \" " }{TEXT 0 2 ":=" }{TEXT -1 32 " \" to assign the name and just \"" }{TEXT 260 3 " = " }{TEXT -1 26 "\" for the equation itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn := x^3-5 *x^2+7*x-12=0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "To see if 3 is a solution for eqn, execute:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval( eqn,x=3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Alternatively, the pair of commands " }{TEXT 0 10 "Eval,value" }{TEXT -1 14 " may be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Eval(eqn,x=3);\nvalue(%);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Similar ly, to see if 4 is a solution for eqn, execute:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(eqn,x=4 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "As you can see, since -9 is not equal to 0, 3 is not a solution to " } {XPPEDIT 18 0 "x^3-5*x^2+7*x-12=0" "6#/,**$%\"xG\"\"$\"\"\"*&\"\"&F(*$ F&\"\"#F(!\"\"*&\"\"(F(F&F(F(\"#7F-\"\"!" }{TEXT -1 82 ", and 0 = 0 in dicates that 4 is in fact a solution. Again, 5 is not a solution as " }{XPPEDIT 18 0 "x^3-5*x^2+7*x-12 = 23;" "6#/,**$%\"xG\"\"$\"\"\"*&\"\" &F(*$F&\"\"#F(!\"\"*&\"\"(F(F&F(F(\"#7F-\"#B" }{TEXT -1 5 " when" } {XPPEDIT 18 0 "x = 5;" "6#/%\"xG\"\"&" }{TEXT -1 28 ", which the follo wing shows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "eval(eqn,x=5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 11 "Example 5: " }{TEXT 0 4 "subs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 4 "subs" }{TEXT -1 27 " works in a similar way to " }{TEXT 0 4 "eval" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "When using " }{TEXT 0 4 "subs" }{TEXT -1 107 " the equation or set of equations specifying t he substitution or substitutions to be made are given as the " }{TEXT 259 14 "first argument" }{TEXT -1 56 " rather than as the second argum ent as is the case with " }{TEXT 0 4 "eval" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Examples: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=4,3*x^2+8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#c" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(u=exp(x),u/(u+u^(-1))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#%\"xG\"\"\",&F$F(*&F(F( F$!\"\"F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "There is no special notation for the output associated with the unevaluated form of the procedure " }{TEXT 0 4 "subs" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "'subs'(x=4,3*x^2+8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sub sG6$/%\"xG\"\"%,&*&\"\"$\"\"\")F'\"\"#F,F,\"\")F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "subs" }{TEXT -1 92 " can replace a sub-expression appearing in a la rger expression by an alternative expression." }}{PARA 0 "" 0 "" {TEXT -1 50 "The following command replaces each occurrence of " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 19 " in the expres sion " }{XPPEDIT 18 0 "sin(x)*cos(x)/(3+2*sin(x)-cos(x));" "6#*(-%$sin G6#%\"xG\"\"\"-%$cosG6#F'F(,(\"\"$F(*&\"\"#F(-F%6#F'F(F(-F*6#F'!\"\"F4 " }{TEXT -1 4 " by " }{TEXT 704 1 "s" }{TEXT -1 24 " and each occurren ce of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 4 " by " }{TEXT 705 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "subs(\{sin(x)=s,cos(x)=c\},s in(x)*cos(x)/(3+2*sin(x)-cos(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *(%\"sG\"\"\"%\"cGF%,(\"\"$F%*&\"\"#F%F$F%F%F&!\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Here is another exam ple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "(a+b+c)^3;\nsubs(a+b+c=-A-B-C,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$),(%\"aG\"\"\"%\"bGF'%\"cGF'\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),(%\"AG!\"\"%\"BGF'%\"CGF'\"\"$\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "There are limitations on this behaviour. For more details see the help page " } {HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 325 12 "Exercise 2.1" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 531 4 "Let " }{XPPEDIT 18 0 "P=a *x^3+b*x^2+c*x+d" "6#/%\"PG,**&%\"aG\"\"\"*$%\"xG\"\"$F(F(*&%\"bGF(*$F *\"\"#F(F(*&%\"cGF(F*F(F(%\"dGF(" }{TEXT 539 6 ". Find" }{TEXT 533 1 " " }{TEXT 534 4 "P if" }{TEXT 532 1 " " }{XPPEDIT 18 0 "x = 0" "6#/%\" xG\"\"!" }{TEXT 538 6 ".01 , " }{XPPEDIT 18 0 "a =-1/5" "6#/%\"aG,$*& \"\"\"F'\"\"&!\"\"F)" }{TEXT 535 3 " , " }{XPPEDIT 18 0 "b=2/5" "6#/% \"bG*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 537 5 " and " }{XPPEDIT 18 0 "d=13 /15" "6#/%\"dG*&\"#8\"\"\"\"#:!\"\"" }{TEXT 536 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 326 21 "Student Work space 2.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 327 10 "Answer 2.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P := a*x^3+b*x^2+c*x+d;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Answer using " }{TEXT 0 4 "eval" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eval(P,\{x=0.01,a=-1/5,b=2/5 ,c=0,d=13/15\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Answer using " }{TEXT 0 4 "subs" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs(\{x=0.01,a=-1/5,b=2/5,c=0,d=13/15\},P);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 330 12 "Exercise 2.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Use" }{TEXT 349 1 " " }{TEXT 0 4 "eval" }{TEXT -1 4 " or " }{TEXT 0 4 "subs" }{TEXT -1 73 " to check \+ if any of the numbers: 1, 2 or 3 is a solution of the equation:" }} {PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3-16*x^2+51*x-36=0 " "6#/,**$%\"xG\"\"$\"\"\"*&\"#;F(*$F&\"\"#F(!\"\"*&\"#^F(F&F(F(\"#OF- \"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 328 21 "Student Workspace 2.2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 329 10 "Answer 2.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn := x^3-16*x^2+51*x-36=0;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(eqn,x=1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(eqn,x=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(eqn,x=3);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Therefor e " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 97 " are solutions of the equation. (In Section 5 you will learn how to solve equations usi ng Maple.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The following three commands use " }{TEXT 0 4 "subs" }{TEXT -1 13 " in place of " }{TEXT 0 4 "eval" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=1,e qn);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=2,eqn);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=3,eqn);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "expand( )" {TEXT -1 14 "The procedure " }{TEXT 0 6 "expand" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The principal use of the procedure " }{TEXT 0 6 "expand" }{TEXT -1 136 " is to \"multiply out\" products of polynomial expressions. It can also be used to expand trigonometric and other more general funct ions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {TEXT -1 18 "Use the procedure " }{TEXT 0 6 "expand" }{TEXT -1 19 " to multiply out " }{XPPEDIT 18 0 "(x+2)^2*(3x-3)*(x+5 )" "6#*(,&%\"xG\"\"\"\"\"#F&F',&*&\"\"$F&F%F&F&F*!\"\"F&,&F%F&\"\"&F&F &" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(x+2)^2*(3*x-3)*(x+5);\nexpand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Maple applies some famili ar trigonometric identities to expand " }{XPPEDIT 18 0 "sin*2*x;" "6#* (%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*2 *x;" "6#*(%$cosG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 14 "Note that all " }{TEXT 259 19 "function evaluation" } {TEXT -1 44 " must be indicated explicitly in Maple with " }{TEXT 259 14 "round brackets" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "We must use the forms " }{TEXT 260 8 "sin(2*x)" }{TEXT -1 5 " and " } {TEXT 260 8 "cos(2*x)" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sin*2*x;" " 6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "co s*2*x;" "6#*(%$cosG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 33 " respectively i n Maple commands. " }}{PARA 0 "" 0 "" {TEXT -1 72 "It is not incorrect (but perhaps just a little clumsy) to use the forms " }{XPPEDIT 18 0 "sin(2*x)" "6#-%$sinG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cos(2*x)" "6#-%$cosG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 139 " in the mathematical notation, but it may be preferable to use th ese forms in any discussion related to Maple commands to avoid confusi on. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(sin(2*x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(cos(2*x));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Try expan ding the sine and cosine of some other integer multiples of " }{TEXT 351 1 "x" }{TEXT -1 15 ". For example: " }{XPPEDIT 18 0 "sin(3*x)" "6# -%$sinG6#*&\"\"$\"\"\"%\"xGF(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(6* x)" "6#-%$cosG6#*&\"\"'\"\"\"%\"xGF(" }{TEXT -1 7 " , etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 67 "Here is a final example. Have Maple multiply out the expr ession: " }{XPPEDIT 18 0 "x^(1/2)*(x^(3/2)+x^(-1/2));" "6#*&)%\"xG*& \"\"\"F'\"\"#!\"\"F',&)F%*&\"\"$F'F(F)F')F%,$*&F'F'F(F)F)F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x^(1/2)*(x^(3/2)+x^(-1/2));\nexpand(%);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT 706 12 "Exercise 2.3" }}{PARA 0 "" 0 " " {TEXT -1 16 "Assign the name " }{TEXT 713 1 "k" }{TEXT -1 19 " to th e expression " }{XPPEDIT 18 0 "x^2+4*x-3" "6#,(*$%\"xG\"\"#\"\"\"*&\" \"%F'F%F'F'\"\"$!\"\"" }{TEXT -1 24 " . Then assign the name " }{TEXT 714 1 "M" }{TEXT -1 20 " to the expression " }{XPPEDIT 18 0 "k^2-9" " 6#,&*$%\"kG\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Finally have Maple calcul ate " }{XPPEDIT 18 0 "3*M+6" "6#,&*&\"\"$\"\"\"%\"MGF&F&\"\"'F&" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 64 ": To get Maple to multiply the expres sion out use the procedure " }{TEXT 0 6 "expand" }{TEXT -1 18 ". That \+ is enter: " }{MPLTEXT 1 0 14 "expand(3*M+6);" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 707 21 "Student Workspace 2.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 708 10 "Answer 2.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k \+ := x^2+4*x-3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "M := k^2-9;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "3*M+6;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "expand(3*M+6 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 709 12 "Exercise 2.4" }}{PARA 0 "" 0 "" {TEXT -1 8 "Expand " }{XPPEDIT 18 0 "( 1+x)^4 " "6#*$,&\"\"\"F%%\"xGF%\"\"%" }{TEXT -1 21 " using the procedure " } {TEXT 0 6 "expand" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 20 "" 0 "" {TEXT 710 21 "Student Workspace 2.4" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 711 10 "Answer 2.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w := (1+x)^4;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " . . . or we can do this all in one step with: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((1+x)^4);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 322 12 "Exercise 2.5" }} {PARA 0 "" 0 "" {TEXT 345 8 "Expand " }{XPPEDIT 18 0 "( x +1 )^n" "6# ),&%\"xG\"\"\"F&F&%\"nG" }{TEXT 346 6 " for " }{XPPEDIT 18 0 "n =2" " 6#/%\"nG\"\"#" }{TEXT 641 11 ", 3 and 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 323 21 "Student Workspace 2.5" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 324 10 "Answer 2.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((x+1)^2);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((x+ 1)^3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((x+1)^4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "factor( )" {TEXT -1 14 "The procedure " }{TEXT 0 6 "factor" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 23 "Factor the expression: " }{XPPEDIT 18 0 "3*x^2-10*x-8" "6#,(*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*& \"#5F&F(F&!\"\"\"\")F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := 3*x^2-10*x-8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor( w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Y ou can do it all on one line . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "factor(3*x^2-10*x-8);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 " . . . or use two lines executed together with a " }{TEXT 0 1 "%" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "3*x^2-10*x-8;\nfactor(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "First enter in the expression " } {XPPEDIT 18 0 "2*(x-2)*(2*x^2+5*x+2)*(x+4)" "6#**\"\"#\"\"\",&%\"xGF%F $!\"\"F%,(*&F$F%*$F'F$F%F%*&\"\"&F%F'F%F%F$F%F%,&F'F%\"\"%F%F%" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "H := 2*(x-2)*(2*x^2+5*x+2)*(x+4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Expand " }{TEXT 716 1 "H" }{TEXT -1 21 " using the procedure " }{TEXT 0 6 "expand" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ans := expand(H);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Then apply the procedure " }{TEXT 0 6 "factor" }{TEXT -1 16 " to the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(ans);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Can you e xplain why the final result looks different than the original expressi on?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Th e calculation can also be performed with a group of commands using " } {TEXT 0 1 "%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "2*(x-2)*(2*x^2+5*x+2)*(x+4); \nexpand(%);\nfactor(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 89 "Maple can factor expre ssions with more than one variable. For example, the expression " } {XPPEDIT 18 0 "x^2y+2xy+y" "6#,(*&%\"xG\"\"#%\"yG\"\"\"F(*&F&F(%#xyGF( F(F'F(" }{TEXT -1 17 ", that is, . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h := x^2*y+2*x*y+y;" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " . . . f actors to . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "factor(h);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 125 "If Maple can't factor an expression using rationa l numbers (i.e. integers and fractions) then it returns the input unch anged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "factor(3*x^2-10*x-9);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "factor" }{TEXT -1 69 " is not limited to polynomials. It can be used to factor other for ms." }}{PARA 0 "" 0 "" {TEXT -1 26 "For example we can factor " } {XPPEDIT 18 0 "cos^2*x-sin^2*x:" "6#,&*&%$cosG\"\"#%\"xG\"\"\"F(*&%$si nGF&F'F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 73 "Note tha t Maple does not use the common shorthand mathematical notation: " } {XPPEDIT 18 0 "sin^2*x,cos^2*x;" "6$*&%$sinG\"\"#%\"xG\"\"\"*&%$cosGF% F&F'" }{TEXT -1 76 ". Also function evaluation must be indicated expli citly with round brackets." }}{PARA 0 "" 0 "" {TEXT -1 39 "The Maple e xpression corresponding to " }{XPPEDIT 18 0 "cos^2*x-sin^2*x:" "6#,&* &%$cosG\"\"#%\"xG\"\"\"F(*&%$sinGF&F'F(!\"\"" }{TEXT -1 4 " is " } {TEXT 260 17 "cos(x)^2-sin(x)^2" }{TEXT -1 40 ", and the corresponding Maple output is " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)-%$cosG6#%\" xG\"\"#\"\"\"F+*$)-%$sinGF(F*F+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(cos(x)^2-sin(x)^2); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We c ould type two lines of code using " }{TEXT 0 1 "%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cos(x)^2-sin(x)^2;\nfactor(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 17 "If the procedure " }{TEXT 0 6 "factor" }{TEXT -1 50 " is used with a rational expression such as . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A := (x ^3-7*x^2+15*x-9)/(x^2+4*x+4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 " . . . the numerator and denominator are \+ each factored, as we see from . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(A);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The common factors i n the expression . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B := (x^3-7*x^2+15*x-9)/(x^2-4*x+3) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " . \+ . . are cancelled to simplify the expression, as we see from . . . " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(B);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The next example allows you to see the factored form with out cancellation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Exa mple 7: " }{TEXT 0 11 "numer,denom" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 5 "numer" }{TEXT -1 5 " and " }{TEXT 0 5 "denom" }{TEXT 347 1 " " }{TEXT -1 81 "commands allo w you to isolate either the numerator or denominator of a fraction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "For exa mple, consider " }{XPPEDIT 18 0 "(x^3-7*x^2+15*x-9)/(x^2-4*x+3);" "6#* &,**$%\"xG\"\"$\"\"\"*&\"\"(F(*$F&\"\"#F(!\"\"*&\"#:F(F&F(F(\"\"*F-F(, (*$F&F,F(*&\"\"%F(F&F(F-F'F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B := (x^3-7* x^2+15*x-9)/(x^2-4*x+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG*&,* *$)%\"xG\"\"$\"\"\"F+*&\"\"(F+)F)\"\"#F+!\"\"*&\"#:F+F)F+F+\"\"*F0F+,( *$F.F+F+*&\"\"%F+F)F+F0F*F+F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "Here we use " }{TEXT 0 5 "numer" }{TEXT -1 5 " and " }{TEXT 0 5 "denom" }{TEXT -1 77 " commands to examine the factors of the numerator and denominator separately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "factor(nu mer(B));\nfactor(denom(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\" xG\"\"\"F&!\"\"F&),&F%F&\"\"$F'\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"F&!\"\"F&,&F%F&\"\"$F'F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 313 12 "Exercise 2.6" }}{PARA 0 "" 0 "" {TEXT -1 22 "Factor the expression " }{XPPEDIT 18 0 "3x^4-2x^3+22x^2-18x-45 \+ " "6#,,*&\"\"$\"\"\"*$%\"xG\"\"%F&F&*&\"\"#F&*$F(F%F&!\"\"*&\"#AF&*$F( F+F&F&*&\"#=F&F(F&F-\"#XF-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 314 18 "Student Workspace " } {TEXT 331 3 "2.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 332 10 "Answer 2.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "factor(3*x^4-2*x^3+22 *x^2-18*x-45);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 315 12 "E xercise 2.7" }}{PARA 0 "" 0 "" {TEXT -1 25 "Factor the expression \+ " }{XPPEDIT 18 0 "sqrt(x)-x^(3/2);" "6#,&-%%sqrtG6#%\"xG\"\"\")F'*&\" \"$F(\"\"#!\"\"F-" }{TEXT -1 33 " and then use the procedure " } {TEXT 0 6 "expand" }{TEXT -1 22 " to check the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 316 18 "Student Wo rkspace " }{TEXT 333 3 "2.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 334 10 "Answer 2.7" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sqrt(x)-x^ (3/2);\nfactor(%);\nexpand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "simplify( )" {TEXT -1 14 "The procedure " }{TEXT 0 8 "simplify" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 8 "simplify" }{TEXT -1 108 " can apply identities to simplify many lengthy mathematical expre ssions, such as trigonometric expressions. " }}{PARA 0 "" 0 "" {TEXT -1 14 "For example, c" }{TEXT 348 24 "onsider the expression: " } {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^5 *x+sin^4*x+2*cos^2*x-2*sin^2*x-cos*2*x" "6#,,*&%$cosG\"\"&%\"xG\"\"\"F (*&%$sinG\"\"%F'F(F(*(\"\"#F(*$F%F-F(F'F(F(*(F-F(*$F*F-F(F'F(!\"\"*(F% F(F-F(F'F(F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "The following form of this expression, in whic h all function evaluations are indicated explicitly with round bracket s, should be used to obtain the corresponding Maple expression." }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2* sin(x)^2 - cos(2*x)" "6#,,*$-%$cosG6#%\"xG\"\"&\"\"\"*$-%$sinG6#F(\"\" %F**&\"\"#F**$-F&6#F(F1F*F**&F1F**$-F-6#F(F1F*!\"\"-F&6#*&F1F*F(F*F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "cos(x)^5+sin(x)^4+2*cos(x)^2-2*sin(x)^2-cos(2* x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)-%$cosG6#%\" xG\"\"&\"\"\"F+*$)-%$sinGF(\"\"%F+F+*&\"\"#F+)F&F2F+F+*&F2F+)F.F2F+!\" \"-F'6#,$*&F2F+F)F+F+F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)-%$cosG6 #%\"xG\"\"%\"\"\",&F%F*F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 "cos(2*x)=co s(x)^2-sin(x)^2" "6#/-%$cosG6#*&\"\"#\"\"\"%\"xGF),&*$-F%6#F*F(F)*$-%$ sinG6#F*F(!\"\"" }{TEXT -1 38 " has been used to eliminate the terms \+ " }{XPPEDIT 18 0 "2*cos(x)^2-2*sin(x)^2-cos(2*x)" "6#,(*&\"\"#\"\"\"*$ -%$cosG6#%\"xGF%F&F&*&F%F&*$-%$sinG6#F+F%F&!\"\"-F)6#*&F%F&F+F&F1" } {TEXT -1 36 ", and then the remaining expression " }{XPPEDIT 18 0 "cos (x)^5+sin(x)^4" "6#,&*$-%$cosG6#%\"xG\"\"&\"\"\"*$-%$sinG6#F(\"\"%F*" }{TEXT -1 21 " is factored to give " }{XPPEDIT 18 0 "cos(x)^4*(cos(x)+ 1)" "6#*&-%$cosG6#%\"xG\"\"%,&-F%6#F'\"\"\"F,F,F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Trigonometric expre ssions with arguments in multiples of some angle will be simplified to trig functions in the single angle if possible:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sin(5*t)+sin (3*t);\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "simplify" }{TEXT -1 42 " can be use d to add rational expressions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "The sum " }{XPPEDIT 18 0 "1/(x+1)+x/(x-1) " "6#,&*&\"\"\"F%,&%\"xGF%F%F%!\"\"F%*&F'F%,&F'F%F%F(F(F%" }{TEXT -1 47 ", is simplified to a single algebraic fraction." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(1/(x+1))+ (x/(x-1));\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 317 12 "Exercise 2.8" }}{PARA 0 "" 0 "" {TEXT -1 8 "Express " } {XPPEDIT 18 0 "7/(x+2)+(3*x)/(x+2)^2" "6#,&*&\"\"(\"\"\",&%\"xGF&\"\"# F&!\"\"F&*(\"\"$F&F(F&*$,&F(F&F)F&F)F*F&" }{TEXT -1 32 " as a single a lgebraic fraction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 321 18 "Student Workspace " }{TEXT 335 3 "2.8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 336 10 "A nswer 2.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "7/(x+2)+(3*x)/(x+2)^2;\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 318 12 "Exercise 2.9" }}{PARA 0 "" 0 "" {TEXT -1 24 "How does Maple simplify " }{XPPEDIT 18 0 "sin(3*t)-sin(7* t)" "6#,&-%$sinG6#*&\"\"$\"\"\"%\"tGF)F)-F%6#*&\"\"(F)F*F)!\"\"" } {TEXT -1 101 "? Whether or not this \"simplified\" form is of use to \+ you will depend on what you plan to do with it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 319 21 "Student Workspa ce 2.9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 320 10 "Answer 2.9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sin(3*t)-sin(7*t);\nsimplify(%);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Section 3: Plotting graphs " }{TEXT 352 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "In this section y ou will learn how to plot the graph of a function defined by an expres sion. Other topics covered include: combining the graphs of several ex pressions into a single plot, plotting points, and combining different plot structures into a single picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "plot( ) " {TEXT -1 29 "Plotting an expression using " }{TEXT 0 4 "plot" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 11 "We use the " } {TEXT 0 4 "plot" }{TEXT -1 33 " procedure to plot the graph of " } {XPPEDIT 18 0 "3*x^2-8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\")!\"\"" }{TEXT -1 5 " for" }{TEXT 360 2 " x" }{TEXT -1 9 " between " } {XPPEDIT 18 0 "-5" "6#,$\"\"&!\"\"" }{TEXT -1 7 " and 5." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot( 3*x^2-8,x=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Notice that Maple scales the " }{TEXT 361 1 "y" }{TEXT -1 32 "-axis automatically, choosing a " }{TEXT 362 1 "y" }{TEXT -1 74 "-range that shows the entire graph corresponding to the specified \+ domain. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "You can override automatic " }{TEXT 391 1 "y" }{TEXT -1 35 "-scali ng by specifying a range for " }{TEXT 389 1 "y" }{TEXT -1 12 " as well as " }{TEXT 390 1 "x" }{TEXT -1 39 ". On the next line we have limite d the " }{TEXT 392 1 "y" }{TEXT -1 23 "-range to the interval " } {XPPEDIT 18 0 "[-20,40]" "6#7$,$\"#?!\"\"\"#S" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(3*x^2-8,x=-5..5,y=-20..40);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 "If you click on a graph with the \+ left mouse button, the graph is selected and the bottom toolbar (calle d the Context Bar) options are changed. Now when you click on the grap h, the point coordinates of its location are shown. The 1:1 button mak es the " }{TEXT 363 1 "x" }{TEXT -1 11 "-scale and " }{TEXT 364 1 "y" }{TEXT -1 15 "-scale equal. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 109 "Scroll back up to the previous graph and experiment with these features. Try the other graph options as well. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Automatic scaling is a useful feature but there are times when \+ you may want to set the " }{TEXT 365 1 "y" }{TEXT -1 102 " range manua lly. For example automatic scaling isn't appropriate for graphs with v ertical asymptotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "Compare the next two graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(x/(x-2),x=-5. .5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " Notice how we have set the limits for " }{TEXT 573 1 "y" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[-20,20]" "6#7$,$\"#?!\"\"F%" } {TEXT -1 28 " in the following procedure " }{TEXT 0 4 "plot" }{TEXT -1 2 ". " }{TEXT 366 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(x/(x-2),x=-5..5,y=-20..20);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 " " 0 "" {TEXT -1 20 "Plot the graph of " }{XPPEDIT 18 0 "y=x^3+1-exp( x)" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"F)F)-%$expG6#F'!\"\"" }{TEXT -1 17 " \+ over the domain " }{XPPEDIT 18 0 "[-8, 8]" "6#7$,$\"\")!\"\"F%" } {TEXT -1 12 ". Choose a " }{TEXT 394 1 "y" }{TEXT -1 39 "-range that \+ allows you to see the four " }{TEXT 393 1 "x" }{TEXT -1 12 "-intercept s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Fir st let's take a look at the plot with automatic scaling of " }{TEXT 395 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(x^3+1-exp(x),x=-8..8);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The large negative values for " }{TEXT 396 1 "y" }{TEXT -1 66 " near 8 have for ced the vertical scale to be too large to see the " }{TEXT 397 1 "x" } {TEXT -1 21 "-intercepts clearly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "A better view is achieved by setting limi ts on the " }{TEXT 398 1 "y" }{TEXT -1 8 "-range. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(x^3+1-e xp(x),x=-8..8,y=-5..15);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 355 12 "Exercise 3.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Plot " }{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$si nG6#%\"xG" }{TEXT -1 6 " over " }{TEXT 259 3 "two" }{TEXT -1 18 " comp lete periods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 3.1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Answer 3.1" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot( sin(x),x=-2*Pi..2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(sin(x), x=0..4*Pi);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 356 12 "Exercise 3.2" }}{PARA 0 "" 0 "" {TEXT -1 6 "Plot " }{XPPEDIT 18 0 "y = 3*x^4-6*x^2" "6#/%\"yG,&*&\"\" $\"\"\"*$%\"xG\"\"%F(F(*&\"\"'F(*$F*\"\"#F(!\"\"" }{TEXT -1 17 " over \+ the domain " }{XPPEDIT 18 0 "[-10,10]" "6#7$,$\"#5!\"\"F%" }{TEXT -1 16 " with automatic " }{TEXT 373 1 "y" }{TEXT -1 11 "-scaling. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "After obs erving the graph, edit the domain and range so that you can see the " }{TEXT 381 1 "x" }{TEXT -1 35 "-intercepts clearly. Estimate the " } {TEXT 374 1 "x" }{TEXT -1 34 "-intercepts with the mouse cursor." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "S tudent Workspace 3.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Answer 3.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(3*x^4-6 *x^2,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Notice how large " }{TEXT 367 1 "y" }{TEXT -1 15 " becomes when \+ " }{XPPEDIT 18 0 "x= -10" "6#/%\"xG,$\"#5!\"\"" }{TEXT -1 5 " or " } {XPPEDIT 18 0 "x= 10" "6#/%\"xG\"#5" }{TEXT -1 18 ", with automatic \+ " }{TEXT 368 1 "y" }{TEXT -1 62 ", scaling it is difficult to see how \+ the function behaves for " }{TEXT 371 1 "x" }{TEXT -1 9 " between " } {XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT -1 42 " and 2. In the next plot we restrict the " }{TEXT 369 1 "y" }{TEXT -1 51 "-scale in order to focus on the behavior for small " }{TEXT 370 1 "y" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(3*x^4-6*x^2,x=-3..3,y=-5..15);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 372 1 "x" }{TEXT -1 22 "-intercepts are about " }{XPPEDIT 18 0 "-1.4" "6#,$-%&FloatG6$ \"#9!\"\"F(" }{TEXT -1 12 ", 1.4 and 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "P lotting several expressions - the \"" }{TEXT 0 5 "color" }{TEXT -1 7 " \" and \"" }{TEXT 0 9 "thickness" }{TEXT -1 15 "\" optionds for " } {TEXT 0 4 "plot" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 102 "To show more than one graph in the same \+ picture list them in square brackets [ ] separated by commas." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot([cos(x),x^2],x=-1..4,y=-4..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "Notice that each of the graphs is displayed using a different colour. You can specify the colours for e ach function by adding a colour option at the end of the command. The \+ colours are assigned in the same order as the functions. Note that the colours must also be listed in a square bracket [ ] . " }}{PARA 0 " " 0 "" {TEXT -1 76 "Either of the spellings \"color\" or \"colour\" ma y be used. Here is an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([cos(x),x^2],x=-1..5,y= -4..4,color=[coral,blue]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Here are the colours available in Maple. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 243 " aqu amarine black blue navy coral cyan \n brown go ld green gray grey khaki \n magenta maroon orang e pink plum red \n sienna tan turquoise viole t wheat white \n yellow" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 115 "It is also possible to specify the color for a plot by specifying the red, green and blue components of the co lour." }}{PARA 0 "" 0 "" {TEXT -1 71 "The thickness of the plot can be specified with the \"thickness\" option." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(exp(x),x=-2..2,c olour=COLOR(RGB,.6,.1,.9),thickness=2);" }}}{PARA 0 "" 0 "" {TEXT 574 2 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT 357 12 "Exercise 3.3" }}{PARA 0 "" 0 "" {TEXT -1 20 "Graph the functions " }{XPPEDIT 18 0 "y = x^2-5 *x+6" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&\"\"&F)F'F)!\"\"\"\"'F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y= 1/(x-2)^2" "6#/%\"yG*&\"\"\"F&*$,&% \"xGF&\"\"#!\"\"F*F+" }{TEXT -1 37 " together. Experiment with differe nt " }{TEXT 572 1 "y" }{TEXT -1 59 "-ranges so that complete pictures \+ of both graphs are shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student Workspace 3.3" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 3.3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot([x^2-5*x+6,1/(x-2)^2],x=-3..8,y=-1..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "Plotting points - the \"" }{TEXT 0 5 "style" }{TEXT -1 7 "\" and \"" }{TEXT 0 6 "symbol" }{TEXT -1 14 "\" options for " } {TEXT 0 4 "plot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 4 "plot" }{TEXT -1 64 " procedure can also \+ be used to plot one or more isolated points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "Plot the point" }{XPPEDIT 18 0 " ``(2 , 3) " "6#-%!G6$\"\"#\"\"$" }{TEXT -1 70 ". Note in the following lin e that we use two sets of square brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([ [2,3] ],sty le=point);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We ca n control the size of the " }{TEXT 382 2 "x-" }{TEXT -1 5 " and " } {TEXT 383 2 "y-" }{TEXT -1 64 "ranges shown by adding these to the com mand as in the next line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot([ [2,3] ],x=-7..7,y=-7..7,styl e=point);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To gr aph more than one point list them in the " }{TEXT 0 4 "plot" }{TEXT -1 122 " command. Note the commas. Remember square brackets for each p oint and an extra pair of square brackets surround the list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "pl ot([ [2,3],[-2,5],[1,-4] ],x=-7..7,y=-7..7,style=point);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Changing style to \"line \" connects the points in the order listed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([ [2,3],[-2,5 ],[1,-4] ],x=-7..7,y=-7..7,style=line);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Note that if the style option is o mitted then the \"line\" style is used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([ [2,3],[-2,5],[ 1,-4] ],x=-7..7,y=-7..7);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Extra plot options can be used to specify the symbol a nd colour to be used for the point. " }}{PARA 0 "" 0 "" {TEXT -1 129 " The symbol options are \"symbol=diamond\", which is the default, \"sym bol=circle\", \"symbol=cross\", \"symbol=box\" and \"symbol=point\". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([[3,2],[-2,3],[2,-1]],style=point,color=blue,symbol=circl e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "U pper case letters may also be used when specifying the \"symbol\" opti on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([[3,2],[-2,3],[2,-1]],style=point,color=magenta, symbol=BOX);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 359 12 "Exer cise 3.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Plot the following points using the color red and the diamond symb ol: " }{XPPEDIT 18 0 "[1, 4] , [-2, -3], [4, -5]" "6%7$\"\"\"\"\"%7$, $\"\"#!\"\",$\"\"$F)7$F%,$\"\"&F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[-6, 5] " "6#7$,$\"\"'!\"\"\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Then connect the points w ith lines in a separate " }{TEXT 0 4 "plot" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 642 21 "Student Workspace 3.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 20 "" 0 "" {TEXT 643 10 "Answer 3.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([[1,4], [-2,-3],[4,-5],[-6,5]],style=point,color=red,symbol=diamond);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([[1,4],[-2,-3],[4,-5],[-6,5]],style=line,color=red,symbol=dia mond);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "with(plots)" {TEXT -1 49 "Combining graphs of \+ expressions and points using " }{TEXT 0 7 "display" }{TEXT -1 10 " fro m the " }{TEXT 0 5 "plots" }{TEXT -1 8 " package" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "A special plotting packag e called \"" }{TEXT 0 5 "plots" }{TEXT -1 129 "\" contains many additi onal special graphing commands. To use these commands, you need to exe cute the following line which loads \"" }{TEXT 0 5 "plots" }{TEXT -1 3 "\". " }{TEXT 379 0 "" }{TEXT -1 149 "Recall, the colon at the end o f the statement allows this line to be executed without displaying any distracting output. To see the contents of the \"" }{TEXT 0 5 "plots " }{TEXT -1 50 "\" package you can change the colon to a semicolon." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 7 "display" }{TEXT -1 138 " allows you to combine grap hs of expressions and points in the same picture. The first step is to name the individual picture components. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 575 9 "IMPORTANT" }{TEXT -1 19 ": Be sure to use a " }{TEXT 259 5 "colon" }{TEXT -1 176 " at the end of the lin e to suppress output (see first three lines below), otherwise, on exec ution, Maple will output the complete \"PLOT\" structure, which may be extensive. The " }{TEXT 0 7 "display" }{TEXT -1 75 " command is then \+ used to do the actual plot (this ends with a semicolon). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pict 1 := plot([-3*x+5,9-x^2],x=-3..5,color=[green,red]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "pict2 := \+ plot([[-1,8],[4,-7]],style=point,color=blue,symbol=circle):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "di splay([pict1,pict2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Alternatively we can list these three related " } {TEXT 0 4 "plot" }{TEXT -1 48 " commands in a single execution group b y typing " }{TEXT 377 17 "[Shift] + [Enter]" }{TEXT -1 21 " at end of \+ each line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 149 "pict1 := plot([-3*x+5,9-x^2],x=-3..5,color=[green, red]):\npict2 := plot([[-1,8],[4,-7]],style=point,color=blue,symbol=ci rcle):\ndisplay([pict1,pict2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "For more on this see \"" }{TEXT 261 43 "E xecution groups with more than one command" }{TEXT 378 18 "\" in the s ection \"" }{TEXT 261 38 "Notes on the Maple worksheet interface" } {TEXT -1 30 "\" at the end of this tutorial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 358 12 "Exercise 3.5" }}{PARA 0 "" 0 "" {TEXT -1 47 "Display a graph that contains both the function" }{MPLTEXT 1 0 1 " " }{XPPEDIT 18 0 "y = x^2+x-6" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"F'F)\" \"'!\"\"" }{TEXT -1 10 " and its " }{TEXT 353 2 "x-" }{TEXT -1 5 " an d " }{TEXT 354 2 "y-" }{TEXT -1 32 "intercepts, marked with circles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student Workspace 3.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 3.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "pict4 := pl ot(x^2+x-6,x=-5..4,y=-8..8):\npict5 := plot([[0,-6],[-3,0],[2,0]],styl e=point,symbol=circle,color=blue):\ndisplay([pict4,pict5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 20 "Interactive graphing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "In this section, you will see that Maple has tools you can use to plot graphs. Using \"" }{TEXT 0 9 "sma rtplot" }{TEXT -1 6 "\" or \"" }{TEXT 0 11 "interactive" }{TEXT -1 70 "\" tools, you can plot easily without having to specify details in th e " }{TEXT 0 4 "plot" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Using the " }{TEXT 0 9 "smartplot" } {TEXT -1 6 " tools" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 380 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "smartplot" } {TEXT -1 93 " generates an initial plot which can be further tailored \+ through use of interactive controls." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Here is an example. To plot " } {XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 19 ", type and execute:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "smartplot(sin(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "By right-clicking on the graph, you can now adjust the details as you want. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "For example, to ch ange the domain to -4..4, right-click anywhere on the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Select " }{TEXT 576 4 "Axes" }{TEXT -1 6 " then " }{TEXT 577 5 "Range" }{TEXT -1 39 ". The Axes - Ranges window will pop up." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 107 "Under X, select the second option to customize the values and replace -10.4 with -4 and 10.4 with 4. Click " }{TEXT 578 2 "OK" }{TEXT -1 49 ", and the graph will update itself \+ automatically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Other changes can be made in very similar ways. Try and c hange the line and axes styles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "Another useful feature of the " }{TEXT 0 9 "smartplot" }{TEXT -1 27 " is drag-and-drop plotting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Suppose a first " } {TEXT 384 9 "smartplot" }{TEXT -1 5 " (of " }{XPPEDIT 18 0 "sin(x);" " 6#-%$sinG6#%\"xG" }{TEXT -1 36 ") has been generated. Then, execute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "cos(x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Select the blue output of this command then drag it \+ and drop it onto the " }{TEXT 385 9 "smartplot" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 14 ". A graph of " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 21 " will app rear in the " }{TEXT 386 9 "smartplot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 387 9 "sma rtplot" }{TEXT -1 87 " can also be launched interactively, via the con text-sensitive menu. Enter and execute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "Select all the blue output and right-click on it. The c ontext-sensitive pop-up menu gives the option Plots which leads to 2-D Plot, which, if selected, launches a " }{TEXT 388 9 "smartplot" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Using the \+ " }{TEXT 0 11 "interactive" }{TEXT -1 6 " tools" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "interactive" }{TEXT -1 17 " is part of the \"" }{TEXT 0 5 "plots" }{TEXT -1 104 "\" package. It allows you to build plots interactively \+ by opening a window where you can specify details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, to graph " } {XPPEDIT 18 0 "f(x) = sin(x)+1;" "6#/-%\"fG6#%\"xG,&-%$sinG6#F'\"\"\"F ,F," }{TEXT -1 21 " using the procedure " }{TEXT 0 11 "interactive" } {TEXT -1 19 ", type and execute:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "interactive(sin(x)+1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Select " }{TEXT 580 8 "2-D Plot" }{TEXT -1 11 " and click " }{TEXT 579 4 "Next" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "This will open the 2 -D Plot window. After entering in your preferences, click " }{TEXT 581 4 "Plot" }{TEXT -1 33 ". This will display the graph of " } {XPPEDIT 18 0 "sin(x)+1;" "6#,&-%$sinG6#%\"xG\"\"\"F(F(" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Try different options for line style, axes, or color as well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The interactive p lot builder can also be launched interactively, by typing and executin g . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " . . . then selecting, and right-clicking on the blue output. Choose " } {TEXT 582 5 "Plots" }{TEXT -1 5 " and " }{TEXT 583 12 "Plot Builder" } {TEXT -1 86 " from the pop-up context-sensitive menu, and the interact ive plot-builder will launch." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Section 4: Solving equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In this section you will learn how to apply the Maple procedure " }{TEXT 0 5 "solve" }{TEXT -1 13 " to find the " }{TEXT 259 5 "exact" }{TEXT -1 163 " solutions of equations (when this is possible). In many cases it is not possible to find exact solutions to equations and so we rely o n numerical solvers to find " }{TEXT 259 11 "approximate" }{TEXT -1 67 " solutions. Later in this section you will use the Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 114 " to find decimal approximations \+ for solutions. The solution to linear systems of equations will also b e discussed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Entering and manipulatin g equations: the " }{TEXT 0 3 "lhs" }{TEXT -1 5 " and " }{TEXT 0 3 "rh s" }{TEXT -1 9 " commands" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Recall that we can give a name to an entire equation just as we have done for expressions." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "On the n ext line we enter the equation " }{XPPEDIT 18 0 "x^3-5*x^2+23=2*x^2+4* x-8 " "6#/,(*$%\"xG\"\"$\"\"\"*&\"\"&F(*$F&\"\"#F(!\"\"\"#BF(,(*&F,F(* $F&F,F(F(*&\"\"%F(F&F(F(\"\")F-" }{TEXT -1 24 " and give it the name \+ \"" }{TEXT 260 4 "eqn1" }{TEXT -1 3 "\" ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn1 := x^3-5*x^2+23= 2*x^2+4*x-8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Exampl e 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "We can isolate the left-hand and right-hand sides of the equation by usi ng the " }{TEXT 0 3 "lhs" }{TEXT -1 5 " and " }{TEXT 0 3 "rhs" }{TEXT -1 11 " commands. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "lhs(eqn1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rhs(eqn1);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Here we use the " }{TEXT 0 3 "lhs" }{TEXT -1 5 " and " }{TEXT 0 3 "rhs" }{TEXT -1 75 " commands to find an equation that is equivalent to the original equation \"" } {TEXT 260 4 "eqn1" }{TEXT -1 70 "\", but has zero on the right-hand si de. The new equation is labelled \"" }{TEXT 260 4 "eqn2" }{TEXT -1 2 " \"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn2 := lhs(eqn1)-rhs(eqn1)=0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " solve( )" {TEXT -1 30 "Finding exact solutions using " }{TEXT 0 5 "so lve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We first consider " }{TEXT 259 10 "polynomial" }{TEXT -1 49 " equations. Algorithms exist for calculating the " }{TEXT 259 5 "exact" }{TEXT -1 15 " solutions for " }{TEXT 584 10 "polynomial" }{TEXT -1 17 " equa tions up to " }{TEXT 585 8 "degree 4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "For example, the quadratic equation " }{XPPEDIT 18 0 " a*x^2+b*x+c=0" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'%\"cG F'\"\"!" }{TEXT -1 20 " has the solutions: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*& \"\"#F(%\"aGF(!\"\"F," }{TEXT -1 1 " " }{TEXT 586 1 "+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*a);" "6#*&-%%sqrtG6#,&*$%\"bG\"\" #\"\"\"*(\"\"%F+%\"aGF+%\"cGF+!\"\"F+*&F*F+F.F+F0" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Maple's \+ procedure " }{TEXT 0 5 "solve" }{TEXT -1 30 " implements these algorit hms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {TEXT -1 56 "To find the exact solutions to the polyn omial equation " }{XPPEDIT 18 0 "3*x^3-4*x^2-43*x+84=0" "6#/,**&\"\"$ \"\"\"*$%\"xGF&F'F'*&\"\"%F'*$F)\"\"#F'!\"\"*&\"#VF'F)F'F.\"#%)F'\"\"! " }{TEXT -1 19 " use the procedure " }{TEXT 0 5 "solve" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Not e that the second argument of the command tells Maple that " }{TEXT 587 1 "x" }{TEXT -1 51 " is the unknown variable that we are solving f or. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(3*x^3-4*x^2-43*x+84=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Here Maple has found all \+ three solutions and listed them for you. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "Sometimes you will want to select one so lution from the list of solutions and use it in another computation. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "You c an do this by first assigning a name (we use the name \"sol\"" }{TEXT 412 1 " " }{TEXT -1 45 "in this case) to the output of the procedure \+ " }{TEXT 0 5 "solve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol := solve(x^2-5*x+3=0,x ); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "T hen \"" }{TEXT 260 6 "sol[1]" }{TEXT -1 50 "\" is the first number in \+ the expression sequence, " }{TEXT 433 1 "\"" }{TEXT 260 6 "sol[2]" } {TEXT 588 1 "\"" }{TEXT -1 34 " is the second number and so on. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol[2];" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Note that square brackets are used for indexing a variable in Maple input." }} {PARA 0 "" 0 "" {TEXT -1 20 "If the the variable " }{TEXT 589 1 "t" } {TEXT -1 73 " has no assigned value, executing the following command g ives the output " }{XPPEDIT 18 0 "t[3];" "6#&%\"tG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t[3];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The order of the solutions in such a sequence is " } {TEXT 259 22 "not necessarily unique" }{TEXT -1 50 ". It might change \+ if the command is re-executed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 84 "An alternative way to access the separate solutions of an equation as determined by " }{TEXT 0 5 "solve" } {TEXT -1 14 " is to make a " }{TEXT 259 19 "multiple assignment" } {TEXT -1 54 ". This method can be used when the equation solved by " } {TEXT 0 5 "solve" }{TEXT -1 87 " has more than one solution, but you n eed to know exactly how many solutions there are." }}{PARA 0 "" 0 "" {TEXT -1 64 "The following command assigns the two solutions of the eq uation " }{XPPEDIT 18 0 "x^2-5*x+3=0" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"&F (F&F(!\"\"\"\"$F(\"\"!" }{TEXT -1 33 " simultaneously to the two names " }{TEXT 260 2 "x1" }{TEXT -1 5 " and " }{TEXT 260 2 "x2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x1,x2 := solve(x^2-5*x+3=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "When working with the procedure " } {TEXT 0 5 "solve" }{TEXT -1 67 " it is often convenient to begin by gi ving a name to the equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Note we use " }{TEXT 0 2 ":=" }{TEXT -1 32 " to assign the name and just \"" }{TEXT 260 1 "=" }{TEXT -1 26 " \" for the equation itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn1 := 7*x^3-11*x^2-27*x-9=0;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Next we s olve the equation using the procedure" }{TEXT 414 1 " " }{TEXT 0 5 "so lve" }{TEXT -1 36 " and assign the output to the name \"" }{TEXT 260 4 "soln" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "soln := solve(eqn1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "For practice let 's check that each of these values satisfies the equation. This is eas y to do using the procedure " }{TEXT 0 4 "eval" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eval(eqn1,x=soln[1]);\neval(eqn1,x=soln[2]);\neval(eqn1,x=soln[3]) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Ch ecking the solutions can also be achieved by referencing the individua l solutions with different names via a " }{TEXT 259 19 "multiple assig nment" }{TEXT -1 13 ". We may use " }{TEXT 0 4 "subs" }{TEXT -1 12 " i nstead of " }{TEXT 0 4 "eval" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x1,x2,x3 := \+ solve(eqn1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "subs(x=x1,eqn1);\nsubs(x=x2,eqn1);\nsubs(x=x3, eqn1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 718 7 "WARNING" }{TEXT -1 30 ": The solutions obtained from " }{TEXT 0 5 " solve" }{TEXT -1 31 " may not be in a unique order. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Sometimes the \"exact\" \+ solutions are too cumbersome to be of much use. In the next two lines \+ we solve the equation " }{XPPEDIT 18 0 "x^3-34*x^2+4=0" "6#/,(*$%\"xG \"\"$\"\"\"*&\"#MF(*$F&\"\"#F(!\"\"\"\"%F(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqn1 := x^3-34*x^2+4=0;\nsol := solve(eqn1,x);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "As you can see, reading these exact solutions is quite a challenge! Note that the " }{TEXT 415 1 "I" }{TEXT -1 13 " stands for " }{XPPEDIT 18 0 "sqrt(-1)" "6#-% %sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 129 ". When a solution is this compli cated it is more useful to look at the approximate numerical (decimal) solutions using procedure " }{TEXT 0 5 "evalf" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "num_sol := evalf(sol);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The small imaginary part attached to each real \+ root can be removed with the procedure " }{TEXT 0 7 "fnormal" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 0 7 "fnorm al" }{TEXT -1 77 " is applied to a list of the solutions formed by put ting the square brackets " }{TEXT 0 3 "[ ]" }{TEXT -1 22 " around the \+ variable \"" }{TEXT 260 7 "num_sol" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fnormal([n um_sol]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Finally, the imaginary unit, " }{XPPEDIT 18 0 "i = sqrt(-1);" " 6#/%\"iG-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 22 ", can be removed with \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(fnormal([num_sol]));" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "A good alternative to the procedure \+ " }{TEXT 0 5 "solve" }{TEXT -1 43 " in a situation like this is the pr ocedure " }{TEXT 0 6 "fsolve" }{TEXT -1 46 " which will be discussed i n the next section. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 5 "solve" }{TEXT -1 50 " can also be used to find the exact so lutions for " }{TEXT 259 14 "non-polynomial" }{TEXT -1 253 " equations . Some simple examples are listed below. However if the equations are \+ at all complicated, for example combining exponential, polynomial and \+ trigonometric expressions, then an exact solution will typically not b e available. Again the procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 20 " \+ is an alternative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 50 "For this equation, the appropriate syntax would be" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(eqn1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 57 " will be discussed in greater detail in the next section." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "solve" }{TEXT -1 52 " can work with exponential and logarithm funct ions. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 " 5*exp(x/4) = 43" "6#/*&\"\"&\"\"\"-%$expG6#*&%\"xGF&\"\"%!\"\"F&\"#V" }{TEXT -1 27 " can be solved as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(5*exp(x/4)=43,x );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Sometim es Maple does not display " }{TEXT 259 3 "all" }{TEXT -1 112 " of the \+ solutions. How would you use the result below to write down the entire set of solutions to the equation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(sin(x)=1/2,x);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 20 "environment variable" }{TEXT -1 2 " \"" }{TEXT 260 16 "_ EnvAllSolutions" }{TEXT -1 243 "\", if set to \"true\", will force all inverse transcendental functions to return the entire set of solution s. This usually requires additional, system created, variables, which \+ take integer values. Normally such variables are named with prefix \" " }{TEXT 260 2 "_Z" }{TEXT -1 23 "\" for integer values, \"" }{TEXT 260 3 "_NN" }{TEXT -1 71 "\" for non-negative integer values and \"_B \" for binary values (0 and 1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "If you are a programmer, you will know wh at is meant by a " }{TEXT 259 15 "global variable" }{TEXT -1 222 ". Ma ple environment variables behave like global variables except that if \+ an assignment is made to an environment variable in the body of a proc edure, the assignment will automatically be undone on exit from the pr ocedure." }}{PARA 0 "" 0 "" {TEXT -1 41 "For more information click th e hyperlink " }{HYPERLNK 17 "envvar" 2 "envvar" "" }{TEXT -1 57 " to o btain the help page concerning environment variables" }{HYPERLNK 17 " " 2 "envar" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "For exam ple, the general solution of the equation " }{XPPEDIT 18 0 "sin(x)=1/2 " "6#/-%$sinG6#%\"xG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 25 " can be found \+ as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "_EnvAllSolutions := true:\nsolve(sin(x)=1/2,x);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The output is " }{XPPEDIT 18 0 "Pi/6+2*Pi/3*_B1+2*Pi*_Z1;" "6#,(*&%#PiG\"\"\"\"\"'!\"\"F&**\"\"#F&F%F &\"\"$F(%$_B1GF&F&*(F*F&F%F&%$_Z1GF&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Pi/6+2*Pi/3*_B1" "6#,&*&%#PiG \"\"\"\"\"'!\"\"F&**\"\"#F&F%F&\"\"$F(%$_B1GF&F&" }{TEXT -1 8 " can be " }{XPPEDIT 18 0 "Pi/6+0" "6#,&*&%#PiG\"\"\"\"\"'!\"\"F&\"\"!F&" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "Pi/6+2*Pi/3=5*Pi/6" "6#/,&*&%#PiG\" \"\"\"\"'!\"\"F'*(\"\"#F'F&F'\"\"$F)F'*(\"\"&F'F&F'F(F)" }{TEXT -1 18 ", this shows that " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 10 " whenever " }{TEXT 428 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*Pi*n;" "6#*(\"\"#\"\"\"%#PiGF%%\"nGF%" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "5*Pi/6;" "6#*(\"\"&\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*Pi*n;" "6#*(\"\"#\"\"\"%#PiGF%%\"nGF%" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 15 " is a n integer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 215 ": If a value is assigned to an environment variable which is unassigned by default, or if a change is made to th e default value, then it is good practice to return the variable to it s default state before proceding." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "Here, we have to obtain the general solut ion of just the one equation, so execute . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "_EnvAllSolut ions := '_EnvAllSolutions';" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 399 12 "Exercise 4.1" }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the equation " }{XPPEDIT 18 0 "x^3-11*x^2+7*x+147 = 0." "6#/,**$%\"xG\"\" $\"\"\"*&\"#6F(*$F&\"\"#F(!\"\"*&\"\"(F(F&F(F(\"$Z\"F(-%&FloatG6$\"\"! F4" }{TEXT -1 114 " Why does Maple produce only two distinct solution s for this cubic equation? Why is one of them written twice? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 259 4 "HINT" }{TEXT -1 46 ": Factor the left hand side of the equatio n.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student Wo rkspace 4.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 407 10 "Answer 4.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(x^3-11*x^2+7*x+ 147=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "factor(x^3-11*x^2+7*x+147);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Because " }{XPPEDIT 18 0 " x-7;" "6#,&%\"xG\"\"\"\"\"(!\"\"" }{TEXT -1 6 " is a " }{TEXT 259 15 " repeated factor" }{TEXT -1 59 ", the cubic equation has only two disti nct roots, namely, " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 9 " and 7. " }}{PARA 0 "" 0 "" {TEXT -1 25 "We say that root 7 has a \+ " }{TEXT 259 17 "multiplicity of 2" }{TEXT -1 25 ", meaning that the f actor" }{XPPEDIT 18 0 "``(x-7);" "6#-%!G6#,&%\"xG\"\"\"\"\"(!\"\"" } {TEXT -1 9 " appears " }{TEXT 259 5 "twice" }{TEXT -1 28 " in the fact ored polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "fsolve( )" {TEXT -1 36 "Finding approximate solutions using \+ " }{TEXT 0 6 "fsolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 20 "The Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 65 " can be used to find approximate solutions for any equation. For " } {TEXT 259 10 "polynomial" }{TEXT -1 11 " equations " }{TEXT 0 6 "fsolv e" }{TEXT -1 79 " finds all the real solutions in one step (see Exampl e 1). For other equations " }{TEXT 0 6 "fsolve" }{TEXT -1 20 " can be \+ used to get " }{TEXT 259 22 "one solution at a time" }{TEXT -1 25 " (s ee Examples 2 and 3). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "fsolve" }{TEXT -1 94 " procedure wil l compute a numerical approximation for each solution of a polynomial \+ equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "To obtain approximate solutions for the equation " }{XPPEDIT 18 0 "x^4-x^3-17*x^2-6*x+2=0" "6#/,,*$%\"xG\"\"%\"\"\"*$F&\"\"$!\"\"*& \"# " 0 " " {MPLTEXT 1 0 30 "eqn := x^4-x^3-17*x^2-6*x+2=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,,*$)%\"xG\"\"%\"\"\"F+*$)F)\"\"$F+!\"\"*&\"# " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!+iN@9M!\"*$!+wV'y&e!#5$\"+D:7( 3#F($\"+ZyG\"z%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The four solutions listed above provide us with a comple te collection of the solutions to the given polynomial equation." }} {PARA 0 "" 0 "" {TEXT -1 145 "The solutions are given in the form of a n expression sequence. To form a genuine Maple list of solutions put s quare brackets around the command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 95 "The following sequence of commands draws a picture to show how the solutions coorespond to the " }{TEXT 605 1 "x" }{TEXT -1 43 "-intercepts of the graph of the polynomial " } {XPPEDIT 18 0 "x^4-x^3-17*x^2-6*x+2" "6#,,*$%\"xG\"\"%\"\"\"*$F%\"\"$! \"\"*&\"# " 0 "" {MPLTEXT 1 0 158 "xvals := [fsolve(eqn,x)];\nx_intercepts := [seq([xvals[i],0],i=1. .4)];\nplot([lhs(eqn),x_intercepts],x=-4..5,style=[line,point],color=[ red,blue],symbol=circle);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 18 "For the equation " }{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$exp G6#F&!\"\"\"\"!" }{TEXT -1 21 ", that is, for . . . " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn := x^3 +1-exp(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(*$)%\"xG\" \"$\"\"\"F+F+F+-%$expG6#F)!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " . . . " }{TEXT 0 6 "fsolve" }{TEXT 434 26 " yields just the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Maple returns " }{TEXT 259 3 "one" } {TEXT -1 205 " real solution. This time Maple has not given us the who le story. Are there any other solutions? How do we find them? A syste matic procedure for finding the remaining solutions is presented in Ex ample 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 51 "We find the other real solutions for th e equation " }{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/,(*$%\"xG\"\"$\"\"\" F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 416 103 "The first step in finding the other solutions is to plot a graph of the left-hand side of the equat ion." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 6 ": The " }{TEXT 603 1 "x" }{TEXT 259 11 "-intercepts" }{TEXT 602 1 " \+ " }{TEXT -1 4 "of " }{XPPEDIT 18 0 "y=x^3+1-exp(x)" "6#/%\"yG,(*$%\"x G\"\"$\"\"\"F)F)-%$expG6#F'!\"\"" }{TEXT -1 54 " correspond exactly to the solutions of the equation " }{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/ ,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(x^3+1-exp(x),x=-3..5,y=-5..15);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The graph shows " }{TEXT 259 4 "fo ur" }{TEXT -1 1 " " }{TEXT 435 1 "x" }{TEXT -1 88 "-intercepts. One of these corresponds to the solution we found in Example 2. Which one? \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 63 " solution is als o easy to spot. How do we find the other three?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We can extend the procedu re " }{TEXT 0 6 "fsolve" }{TEXT -1 136 " to look for a solution in a p articular interval. For example to find the negative solution we ask M aple to search on the interval from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG, $\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=-0" "6#/%\"xG,$\"\" !!\"\"" }{TEXT -1 107 ".2, since we can see from the graph that there \+ definitely is one (and only one) solution on that interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 27 ": It is sufficient to give " }{TEXT 0 6 "fsolve" }{TEXT -1 21 " just the expression " }{TEXT 260 12 "x^3+1-exp(x)" }{TEXT -1 34 " on \+ the left side of the equation " }{TEXT 260 16 "x^3+1-exp(x) = 0" } {TEXT -1 24 " as its first argument. " }{TEXT 0 6 "fsolve" }{TEXT -1 47 " \"assumes\" that you want to find the values of " }{TEXT 609 1 "x " }{TEXT -1 26 " for which the expression " }{XPPEDIT 18 0 "x^3+1-exp( x)" "6#,(*$%\"xG\"\"$\"\"\"F'F'-%$expG6#F%!\"\"" }{TEXT -1 20 " has th e value 0. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(x^3+1-exp(x),x=-1..-0.2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "To find the other tw o solutions we use " }{TEXT 0 6 "fsolve" }{TEXT -1 39 " again, this ti me with search interval " }{XPPEDIT 18 0 "[1, 2]" "6#7$\"\"\"\"\"#" } {TEXT -1 25 " and then with interval " }{XPPEDIT 18 0 "[4, 5]" "6#7$ \"\"%\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "fsolve(x^3+1-exp(x),x=1..2);\nfsolv e(x^3+1-exp(x),x=4..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 96 "What happens if you ask Maple to search for a solu tion on an interval where no solution exists? " }}{PARA 0 "" 0 "" {TEXT -1 63 "Let's try it out. From the graph it is clear that there a re no " }{TEXT 604 1 "x" }{TEXT -1 57 "-intercepts (and therefore no s olutions) between 2 and 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(x^3+1-exp(x),x=2..4);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Notice t hat Maple simply returns the original input line unchanged when it can not find a solution on the given interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 "Are there any other solutions? For example, are there any solutions larger than 5? We can check th is out by expanding the interval over which the graph is plotted. On t he next line we expand the interval to " }{XPPEDIT 18 0 "[-3, 50]" "6# 7$,$\"\"$!\"\"\"#]" }{TEXT -1 11 ". No other " }{TEXT 436 1 "x" } {TEXT -1 190 "-intercepts appear. The graph confirms what we should ex pect by looking at the terms of the expression, namely the exponential term dominates and causes the graph to go down in the long run." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(x^3+1-exp(x),x=-3..50,y=-10..15);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Alternatively we can use the pr ocedure " }{TEXT 0 6 "fsolve" }{TEXT -1 42 ", now searching over this \+ larger interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "fsolve(x^3+1-exp(x),x=5..50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "As expected no solut ions are found by Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 52 "In a similar way, we can check for solutions to th e " }{TEXT 259 4 "left" }{TEXT -1 49 ". Here we search for solutions o ver the interval " }{XPPEDIT 18 0 "[-50, -1] " "6#7$,$\"#]!\"\",$\"\" \"F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(x^3+1-exp(x),x=-50..-1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "None ther e either !" }}{PARA 0 "" 0 "" {TEXT -1 58 "We have now calculated all \+ four solutions of the equation " }{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/, (*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 59 "They are: -0.8251554597, 0, 1.545007279 \+ and 4.567036837. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "To be sure that the values given for the non-zero soluti ons are correct to the number of digits finally displayed you can perf orm the numerical solution with " }{TEXT 0 6 "fsolve" }{TEXT -1 29 " u sing a few extra so-called " }{TEXT 259 12 "guard digits" }{TEXT -1 204 ", and then perform suitable rounding. Maple often does this inter nally. The following calculation should convince you that the original numerical solutions obtained above are in fact correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 29 "Note that, instead of giving " } {TEXT 0 6 "fsolve" }{TEXT -1 125 " an interval in which to search for \+ a solution, a single starting approximation for each of the non-zero s olutions is given. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(fsolve(x^3+1-exp(x),x=-0.82),15);\nx1 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0c&)opa:D)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+(pa:D)!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(fsol ve(x^3+1-exp(x),x=1.545),15);\nx2 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0Et!zs+X:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x 2G$\"+zs+X:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "evalf(fsolve(x^3+1-exp(x),x=4.567),15);\nx3 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(f%p$o.nX!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+Po.nX!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 7 "We use " }{TEXT 0 6 "fsolve" }{TEXT -1 52 " to find the ap proximate solutions of the equation " }{XPPEDIT 18 0 "x^2/20-10*x=15* cos(x+15)" "6#/,&*&%\"xG\"\"#\"#?!\"\"\"\"\"*&\"#5F*F&F*F)*&\"#:F*-%$c osG6#,&F&F*F.F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eq := x^2/20-10*x-15*cos(x+1 5)=0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 "Just as in the last example we will use a graph to help us determ ine the number and approximate location of the solutions. Our task is \+ simplified if we start by converting the given equation to an equivale nt one that has zero on the right-hand side. " }}{PARA 0 "" 0 "" {TEXT -1 43 "So we shall solve the equivalent equation " }{XPPEDIT 18 0 "x^2/20-10*x-15*cos(x+15)=0" "6#/,(*&%\"xG\"\"#\"#?!\"\"\"\"\"*& \"#5F*F&F*F)*&\"#:F*-%$cosG6#,&F&F*F.F*F*F)\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "If we now graph the " }{TEXT 259 14 "left-hand side" }{TEXT -1 68 " of this equ ation, we once again will find solutions at each of the " }{TEXT 437 1 "x" }{TEXT -1 13 "-intercepts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(lhs(eq),x=-10..10);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "From the graph it appears that there is a solution on the interval " } {XPPEDIT 18 0 "[1, 2]" "6#7$\"\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We now direct Mapl e to search for a solution on this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fsolve(eq,x=1..2); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "Hav e we found all of the solutions to this equation? In fact there is an other solution! To find it, expand the interval over which the graph i s drawn, then use " }{TEXT 0 6 "fsolve" }{TEXT -1 92 " to find a numer ical approximation for this second solution. (This is left as an exer cise.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 409 12 "Exercise 4 .2" }}{PARA 0 "" 0 "" {TEXT -1 39 "Find all the solutions to the equat ion " }{XPPEDIT 18 0 "x^5-4*x^3+3*x^2+7*x-1=0" "6#/,,*$%\"xG\"\"&\"\" \"*&\"\"%F(*$F&\"\"$F(!\"\"*&F,F(*$F&\"\"#F(F(*&\"\"(F(F&F(F(F(F-\"\"! " }{TEXT -1 41 ". Begin by looking at a relevant graph. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student W orkspace 4.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 4.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We begin by graphing the \+ left-hand side of the equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := x^5-4*x^3+3*x^2+7*x-1 =0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(lhs(eqn),x=-5..5,y=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This picture indicates th at there are solutions near " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" } {TEXT -1 3 " , " }{XPPEDIT 18 0 "-1.5" "6#,$-%&FloatG6$\"#:!\"\"F(" } {TEXT -1 9 ", and 0. " }}{PARA 0 "" 0 "" {TEXT -1 29 "We next try the \+ unrestricted " }{TEXT 0 6 "fsolve" }{TEXT -1 46 " command to see which solution(s) Maple finds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Since this is a polynomia l equation " }{TEXT 0 6 "fsolve" }{TEXT -1 48 " gives us a complete li st of the real solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 408 12 "Exercise 4.3" }}{PARA 0 "" 0 "" {TEXT -1 39 "Find all th e solutions to the equation " }{XPPEDIT 18 0 "x^2 - 2 = ln(x+5)" "6#/, &*$%\"xG\"\"#\"\"\"F'!\"\"-%#lnG6#,&F&F(\"\"&F(" }{TEXT -1 20 ". Use \+ the graph of " }{TEXT 259 3 "one" }{TEXT -1 38 " expression to locate \+ the solutions. " }}{PARA 0 "" 0 "" {TEXT -1 69 "Check each solution b y substituting it back in the original equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student Workspac e 4.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 4.3" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "First we put the equati on in \"standard form\" , i.e. with zero on one side. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := \+ x^2-2-ln(x+5)=0;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Now we can graph the right-hand side equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pl ot(lhs(eqn),x=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 43 "There appear to be two solutions: one near " } {XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT -1 23 " and the other near \+ 2. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 0 6 "fsolve" }{TEXT -1 83 " with a restricted domain to find the two solutions you've loca ted more precisely. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "soln1 := fsolve(eqn,x=-5..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "soln2 := fsolve(eqn,x=1..3);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Check by \+ subsituting back into the original equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(eval(eqn,x=s oln1));\nevalf(eval(eqn,x=soln2));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 177 "Notice that the original equation is on ly \"approximately\" satisfied by each of our solutions. The slight di screpancy is a result of round-off error in the approximate solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Check that there are no additional solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 410 12 "Exercise 4.4" }}{PARA 0 "" 0 "" {TEXT -1 15 "T he graphs of " }{XPPEDIT 18 0 "y=10-x^2" "6#/%\"yG,&\"#5\"\"\"*$%\"xG \"\"#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y=4*sin(2*x)+5" "6#/ %\"yG,&*&\"\"%\"\"\"-%$sinG6#*&\"\"#F(%\"xGF(F(F(\"\"&F(" }{TEXT -1 33 " intersect twice on the interval " }{XPPEDIT 18 0 "[-5, 5]" "6#7$, $\"\"&!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 91 "(a) Graph the two equations together and \+ estimate the intersection points using the mouse. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) Write an equation tha t can be solved to find the " }{TEXT 607 1 "x" }{TEXT -1 41 "-coordina tes of the intersection points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "(c) Use " }{TEXT 0 6 "fsolve" }{TEXT -1 24 " to solve this equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(d) Use the results from part (c) to estimate t he " }{TEXT 606 1 "y" }{TEXT -1 40 "-coordinates of the intersection p oints." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "(e) It appears that the curves may intersect at a third point near" } {XPPEDIT 18 0 " ``(1, 9)" "6#-%!G6$\"\"\"\"\"*" }{TEXT -1 6 ". Use " } {TEXT 0 6 "fsolve" }{TEXT -1 94 " and/or a relevant graph to demonstra te that there is no intersection point at that location. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student W orkspace 4.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 4.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "First we enter \+ the two expressions " }{XPPEDIT 18 0 "10-x^2" "6#,&\"#5\"\"\"*$%\"xG\" \"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "4*sin(2*x)+5" "6#,&*&\" \"%\"\"\"-%$sinG6#*&\"\"#F&%\"xGF&F&F&\"\"&F&" }{TEXT -1 23 " giving t hem the names " }{TEXT 260 2 "y1" }{TEXT -1 5 " and " }{TEXT 260 2 "y2 " }{TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y1 := 10-x^2;\ny2 := 4*sin(2 *x)+5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Here is a plot of the two equations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([y1,y2],x=-5..5) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Int ersection points are located approximately at" }{XPPEDIT 18 0 "``(-1.8 , 6.6)" "6#-%!G6$,$-%&FloatG6$\"#=!\"\"F+-F(6$\"#mF+" }{TEXT -1 4 " an d" }{XPPEDIT 18 0 " ``( 2.75, 2) " "6#-%!G6$-%&FloatG6$\"$v#!\"#\"\"# " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) The equation to solve is " }{XPPEDIT 18 0 "y[1] = y[2 ];" "6#/&%\"yG6#\"\"\"&F%6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eqn := y1=y2 ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "(c) We now find the two solutions using " }{TEXT 0 6 "fsolve" }{TEXT -1 1 "." }{TEXT 644 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "x_soln1 := fsolve(y1=y2, x=-4..0);\nx_soln2 := fsolve(y1=y2,x=0..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(d) We can use the procedure " }{TEXT 0 4 "eval" }{TEXT -1 27 " to find the corresponding " }{TEXT 438 1 "y" }{TEXT -1 13 "-coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y_soln1 := eval(y1,x=x_s oln1);\ny_soln2 := eval(y1,x=x_soln2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "So the points of intersection are" } {XPPEDIT 18 0 " ``(-1.800,6.763) " "6#-%!G6$,$-%&FloatG6$\"%+=!\"$!\" \"-F(6$\"%jnF+" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(2.773,2.311)" " 6#-%!G6$-%&FloatG6$\"%tF!\"$-F'6$\"%6BF*" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "(e) Here is a clos er look at what is happening near " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG \"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot([y1,y2],x=.5..1.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Alternatively w e can use " }{TEXT 0 6 "fsolve" }{TEXT -1 43 " to confirm that there i s no solution near " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(y1=y2,x=.5..1.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Solving literal equations" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Often Maple can solve literal equations for any one of the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "Suppose we want to solve for the variable g in the equation " }{XPPEDIT 18 0 "4-v=2*T-k*g" "6#/,&\"\"%\"\"\"%\" vG!\"\",&*&\"\"#F&%\"TGF&F&*&%\"kGF&%\"gGF&F(" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 5 "solve" }{TEXT -1 19 " works well here. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "so lve(4-v=2*T-k*g,g);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "Here is a little nicer way of displaying the same resul t:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g = solve(4-v=2*T-k*g,g);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 401 12 "Exercise 4.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Edit the last command to solve \+ for each of the other letters " }{TEXT 417 4 "T, k" }{TEXT 598 5 " and " }{TEXT 599 1 "v" }{TEXT 600 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 402 21 "Student Workspace 4.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 403 10 "Answer 4. 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "T = solve(4-v=2*T-k*g,T);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "k = solve(4-v=2*T- k*g,k);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v = solve(4-v=2*T-k*g,v);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 404 12 "Exercise 4.6" }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the e quation " }{XPPEDIT 18 0 "x^2+y^2=9" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF' F(\"\"*" }{TEXT -1 5 " for " }{TEXT 418 1 "y" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 67 "Assign the expression sequence of solutio ns to a variable named S. " }}{PARA 0 "" 0 "" {TEXT -1 50 "How are the two solutions S[1] and S[2] related ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 " " 0 "" {TEXT 405 21 "Student Workspace 4.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 406 10 "Answer 4. 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "S := solve(x^2+y^2=25,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S[2];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The solutions are the negatives of each other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 36 "Solving a linear system of equations" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 51 "Please execute th e next two lines before proceeding" }{TEXT 590 1 ":" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 5 "solve" }{TEXT -1 39 " can also be used to solv e a system of " }{TEXT 425 1 "m" }{TEXT -1 21 " linear equations in " }{TEXT 419 1 "n" }{TEXT -1 26 " variables. We call these " }{TEXT 591 1 "m" }{TEXT 259 1 " " }{TEXT 593 1 "x" }{TEXT 259 1 " " }{TEXT 592 1 "n" }{TEXT 259 15 " linear systems" }{TEXT -1 11 " for short." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Consider the 2 " } {TEXT 594 1 "x" }{TEXT -1 46 " 2 linear system consisting of the equat ions: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x+2*y = 3 " "6#/,&*&\"\"$\"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'F'F&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "x -y=-4" "6#/,&%\"xG\"\"\"%\"yG!\"\",$\"\"%F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "First we set up the two equations in Maple." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn1 := 3*x+2*y=3;\neqn2 := x-y=-4;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " The solution can be found using \+ " }{TEXT 0 5 "solve" }{TEXT -1 25 " by typing and executing:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "so ln := solve(\{eqn1,eqn2\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the procedure " }{TEXT 0 4 "eval" } {TEXT -1 24 " to directly select the " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT -1 6 "- and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 49 "-coord inates from the expression returned by the " }{TEXT 0 5 "solve" } {TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For example, for the " }{TEXT 429 1 "x" }{TEXT -1 36 " -coordinate, type and execute . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(x,soln);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " . . . and for th e " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 36 "-coordinate, type and \+ execute . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(y,soln);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "In the first of these usages, we are as king Maple to evaluate the symbol " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT -1 44 " using the information it finds in the set \"" }{TEXT 260 4 "soln" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "soln;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Similarly, by evaluating \+ the symbol " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 48 " with the inf ormation in this set, the value of " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 13 " is selected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The solution can be expressed as a point in Maple via the usage . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "pt := eval([x,y],soln);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Again, note the use of the " }{TEXT 0 4 "eval" }{TEXT -1 24 " construct. The symbol " } {XPPEDIT 18 0 "[x, y];" "6#7$%\"xG%\"yG" }{TEXT -1 47 " is evaluated w ith the information in the set \"" }{TEXT 260 3 "sol" }{TEXT -1 28 "\" , and results in the point " }{XPPEDIT 18 0 "[-1, 3];" "6#7$,$\"\"\"! \"\"\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 124 "A graph of the two lines represented by the two e quations can be generated if first, each equation is solved explicitly for " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "y1 := solv e(3*x+2*y=3,y);\ny2 := solve(x-y=-4,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "Now we construct a picture made u p of two parts: \"p1\" contains the graphs the two equations and \"p2 \" plots the solution point that we found. This point should be the in tersection point of the two lines. Is it ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "p1 := plot([y1,y2 ],x=-5..5,color=[red,blue]): \np2 := plot([[-1,3]],style=point,color =black,symbol=circle):\ndisplay([p1,p2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Alternatively, since the equat ions of the lines were given implicitly, the desired picture could be \+ obtained with Maple's " }{TEXT 0 12 "implicitplot" }{TEXT -1 12 " proc edure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "p1 := implicitplot([eqn1,eqn2],x=-5..5,y=-6..8,color =[red,blue]):\np2 := plot([soln],style=point,symbol=circle,color=black ):\ndisplay([p1,p2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Here is an example of a " }{TEXT 420 2 "3 " }{TEXT 595 1 "x" }{TEXT 596 2 " 3" }{TEXT -1 23 " system with variables " } {TEXT 430 1 "x" }{TEXT -1 2 ", " }{TEXT 431 1 "y" }{TEXT -1 6 ", and \+ " }{TEXT 432 1 "z" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 259 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([x+y+z = 1 , ``],[3*x+y = 3, ``],[x-2*y-z = 0, ``]);" "6#-%*PIECEWISEG6%7$/,(%\"x G\"\"\"%\"yGF*%\"zGF*F*%!G7$/,&*&\"\"$F*F)F*F*F+F*F2F-7$/,(F)F**&\"\"# F*F+F*!\"\"F,F8\"\"!F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "solve" } {TEXT -1 67 " gives the solution in the form of a set of equations of \+ the form: " }{XPPEDIT 18 0 "\{ x=` . . . `,y=` . . . `,z=` . . . `\}" "6#<%/%\"xG%(~.~.~.~G/%\"yGF&/%\"zGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "soln := s olve(\{x+y+z=1,3*x+y=3,x-2*y-z=0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%solnG<%/%\"xG#\"\"%\"\"&/%\"zG#!\"#F*/%\"yG#\"\"$F*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The solution can b e expressed as a point via the command . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eval([x,y,z] ,soln);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 400 12 "Exercise \+ 4.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Fi nd the solution to the 2 " }{TEXT 597 1 "x" }{TEXT -1 39 " 2 system co nsisting of the equations: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "4*x+3*y = 12;" "6#/,&*&\"\"%\"\"\"%\"xGF'F'*&\"\"$F'%\" yGF'F'\"#7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5*x-7*y = 35" "6#/,&* &\"\"&\"\"\"%\"xGF'F'*&\"\"(F'%\"yGF'!\"\"\"#N" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Check by \+ evaluating both equations in the system at the solution pair." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 " Student Workspace 4.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 4.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eq ns := \{4*x+3*y=12,5*x-7*y=35\};" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "Solve the system using the procedure " } {TEXT 0 5 "solve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ans := solve(eqns);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "And to ch eck if the solution " }{XPPEDIT 18 0 "x = 189/43,y = -80/43;" "6$/%\"x G*&\"$*=\"\"\"\"#V!\"\"/%\"yG,$*&\"#!)F'F(F)F)" }{TEXT -1 42 " is a co rrect solution, use the procedure " }{TEXT 0 4 "eval" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(eqns,ans);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Linear sys tems with an infinite number of solutions (optional)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "When a system has more \+ " }{TEXT 259 9 "variables" }{TEXT -1 6 " than " }{TEXT 259 9 "equation s" }{TEXT -1 59 " we often get not one, but an infinite number of solu tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Here is an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 " Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We consider the solution of the system consisting of the two equat ions: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+y+z = 1; " "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 4 "and " }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "3*x+y=3" "6#/,& *&\"\"$\"\"\"%\"xGF'F'%\"yGF'F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solns := sol ve(\{x+y+z=1,3*x+y=3\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 68 "Notice this time we do not get a single set of num erical values for " }{TEXT 610 1 "x" }{TEXT -1 2 ", " }{TEXT 611 1 "y " }{TEXT -1 5 " and " }{TEXT 612 1 "z" }{TEXT -1 43 ". Instead Maple t ells us how the values of " }{TEXT 613 1 "x" }{TEXT -1 2 ", " }{TEXT 614 1 "y" }{TEXT -1 5 " and " }{TEXT 615 1 "z" }{TEXT -1 49 " must be \+ related to construct a typical solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "In particular the expression " } {XPPEDIT 18 0 "x=x" "6#/%\"xGF$" }{TEXT 424 1 " " }{TEXT -1 35 "in the output above indicates that " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 8 " can be " }{TEXT 259 3 "any" }{TEXT -1 76 " number. We refer to it \+ as the \"free\" variable in the solution. To find any " }{TEXT 259 10 "particular" }{TEXT -1 65 " solution (among the infinte number possibl e) pick any value for " }{TEXT 439 1 "x" }{TEXT -1 56 " and use this t o calculate the corresponding values for " }{TEXT 616 1 "y" }{TEXT -1 5 " and " }{TEXT 617 1 "z" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "For example let " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eval(solns,x=4); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "So one solution is: " }{XPPEDIT 18 0 "x = 4,y = -9,z = 6;" "6%/%\"xG\"\"%/%\"yG,$\"\"*!\"\"/%\"zG\"\"' " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Take a minute and check by hand that these three numbers \+ do in fact satisfy our original equations " }{XPPEDIT 18 0 " x+y+z=1" "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "3*x+y=3" "6#/,&*&\"\"$\"\"\"%\"xGF'F'%\"yGF'F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Now let's look at the solution that is generated when we take " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(solns,x=2);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "So two of the infinitely many solutions are:" }{XPPEDIT 18 0 "``(x,y,z) = ``(4, -9,6);" "6#/-%!G6%%\"xG%\"yG%\"zG-F%6%\"\"%,$\"\"*!\"\"\"\"'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(2,-3,2);" "6#-%!G6%\"\"#,$\"\"$!\"\"F& " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "Each of the two equations each represent a plane in 3 di menional space. The two planes intersect in an infinite straight line. " }}{PARA 0 "" 0 "" {TEXT -1 61 "The two planes can be drawn by solvin g the two equations for " }{TEXT 618 1 "y" }{TEXT -1 9 " to give " } {XPPEDIT 18 0 "y=1-x-z" "6#/%\"yG,(\"\"\"F&%\"xG!\"\"%\"zGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3-3*x" "6#/%\"yG,&\"\"$\"\"\"*&F&F'%\"x GF'!\"\"" }{TEXT -1 65 ", and then constructing a 3-dimensional plot u sing the procedure " }{TEXT 0 6 "plot3d" }{TEXT -1 57 " with suitable \+ ranges of values chosen for the variables " }{TEXT 619 1 "x" }{TEXT -1 5 " and " }{TEXT 620 1 "z" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The point given by " }{XPPEDIT 18 0 "x = 4,y = -9,z = 6; " "6%/%\"xG\"\"%/%\"yG,$\"\"*!\"\"/%\"zG\"\"'" }{TEXT -1 67 " can be s een to lie on the line of intersection of the two planes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot 3d([1-x-z,3-3*x],x=-4..4,z=-4..6,axes=framed,color=[red,blue],labels=[ `x`,`z`,`y`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 421 12 "Exe rcise 4.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Solve the system " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x+2*y+z = 2,``], [3*x+y = 1,``])" "6#-%*PIECEWISEG6$7 $/,(%\"xG\"\"\"*&\"\"#F*%\"yGF*F*%\"zGF*F,%!G7$/,&*&\"\"$F*F)F*F*F-F*F *F/" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 45 "and find at least \+ three particular solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 422 21 "Student Workspace 4.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 423 10 "Answer 4.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqns := \{x+2*y+z=2,3*x+y=1\};" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "soln := solve(eqns);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "Here are the solutions for " }{XPPEDIT 18 0 "x= 1 " "6#/%\"xG\"\"\"" }{TEXT -1 14 ", 2, 3, and 4:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(soln,x= 1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(soln,x=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(soln,x=3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval (soln,x=4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "Section 5: Defining, evaluating and graphing func tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "In this section you will learn how to define a function in Maple. The remainder of the section covers evaluating functions, solving equatio ns with functions, and graphing functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Defining a function in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Maple requires special notation when defining a " }{TEXT 259 8 "function" }{TEXT -1 28 ", as opposed \+ to defining an " }{TEXT 259 10 "expression" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For example, th e function " }{XPPEDIT 18 0 "f(x) = cos(Pi*x)+3;" "6#/-%\"fG6#%\"xG,&- %$cosG6#*&%#PiG\"\"\"F'F.F.\"\"$F." }{TEXT -1 15 " is defined by:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> cos(Pi*x)+3; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "Take note of the syntax here. The \"arrow \" " }{TEXT 0 2 "->" }{TEXT -1 110 " is necessary, and is made by typ ing (with no intervening space) a \"minus sign\" and a \"greater than \" symbol. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 443 7 "WARNING" }{TEXT -1 2 ": " }{TEXT 442 51 "Maple will not define \+ a function if you type either" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT -1 1 " " }{TEXT 445 21 "f(x) := cos(Pi*x)+3; \+ " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 259 "" 0 "" {TEXT 260 17 " f := cos(Pi*x)+3;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The first input fails to define the desir ed function because of the assignment to the symbol " }{XPPEDIT 18 0 " f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "In general, this kind of assignment should be avoided in Maple." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The seco nd input creates an " }{TEXT 259 10 "expression" }{TEXT -1 8 ", not a \+ " }{TEXT 259 8 "function" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "Below is a comparison of an expre ssion and a function. Note the difference in syntax and how Maple retu rns the output for each." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "This first command defines an expression . . . " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y := (x+2)/(x^3+5*x+2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 ". . . and this one defines a function . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> (x+2)/(x^3+5*x+2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Standard function notatio n can now be used to evluate the function f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(1/2);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#?\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0aS0a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0aS0a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"\"\"#F&F&,(*$)F% \"\"$F&F&*&\"\"&F&F%F&F&F'F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f(exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#%\"xG\"\"\"\"\"#F)F),(*$)F%\"\"$F)F)*& \"\"&F)F%F)F)F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "Gen erally, functions require an arrow when typed in from the keyboard. Wh en created this way, the corresponding Maple output which echos the Ma ple input command should also have an arrow. " }{TEXT 259 23 "Always c heck the output" }{TEXT -1 9 " for the " }{TEXT 259 5 "arrow" }{TEXT -1 53 " to confirm that you have in fact defined a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "If you select ( highlight with the mouse) the output" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"\"\"#F/F/,(*$)F .\"\"$F/F/*&\"\"&F/F.F/F/F0F/!\"\"F(F(F(" }}{PARA 0 "" 0 "" {TEXT -1 99 "you will see the following expression in the window in the context bar at the top of the worksheet." }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{TEXT 608 65 "f := proc (x) options operator, arrow; (x+2)/(x^3+5*x+2 ) end proc" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Maple has s et up a " }{TEXT 259 9 "procedure" }{TEXT -1 70 " to handle evaluation of the function f as indicated by the key-words " }{TEXT 0 4 "proc" } {TEXT -1 7 " . . . " }{TEXT 0 8 "end proc" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This code can be \+ used on a command line to set up the function f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f := proc(x) options operator,arrow; (x+2)/(x^3+5*x+2) end proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\" \"\"\"#F/F/,(*$)F.\"\"$F/F/*&\"\"&F/F.F/F/F0F/!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "If the " }{TEXT 260 23 "options operator,arrow;" }{TEXT -1 226 " statement is omitted \+ from the body of the procedure. This prevents Maple from using the arr ow notation in the output form but otherwise the procedure still works in the same way. I'll use a g instead of f to illustrate this. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g := proc(x) (x+2)/(x^3+5*x+2) end proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"F(F(*&,&9$\"\"\"\"\"#F,F,,(*$)F+\"\"$ F,F,*&\"\"&F,F+F,F,F-F,!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g(1/2);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#?\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0aS0a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0aS0a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"\"\"#F&F&,(*$)F%\"\"$F&F&*&\"\"&F&F%F&F& F'F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "g(exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&- %$expG6#%\"xG\"\"\"\"\"#F)F),(*$)F%\"\"$F)F)*&\"\"&F)F%F)F)F*F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 440 12 "Exercise 5.1" }} {PARA 0 "" 0 "" {TEXT -1 65 "(a) In the workspace below, set up the Ma ple function h given by " }{XPPEDIT 18 0 "h(x) = x^3*sin(2*x);" "6#/-% \"hG6#%\"xG*&F'\"\"$-%$sinG6#*&\"\"#\"\"\"F'F/F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 63 "(b) Use the Maple function h to find a sy mbolic expression for " }{XPPEDIT 18 0 "h(Pi/3)" "6#-%\"hG6#*&%#PiG\" \"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "(c) Fi nd an approximate decimal value for " }{XPPEDIT 18 0 "h(Pi/3)" "6#-%\" hG6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 5.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 10 "Answer 5.1" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h := x -> x^3*sin(2*x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&)9$\"\"$\"\"\"-%$sinG6#,$*&\"\"#F0F.F0F0F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "(b) and (c) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h(Pi/3);\nevalf(%);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#a!\"\"%#PiG\"\"$F(#\"\"\"\"\"#F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+))yEX**!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "Once you have defined a function, Maple will remember that function during you r entire working session. If you want to overwrite the function with \+ a new definition, you simply retype the definition. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "For example, if you wan t to change the definition of the function f so that " }{XPPEDIT 18 0 "f(x) = ln(cos(5*x));" "6#/-%\"fG6#%\"xG-%#lnG6#-%$cosG6#*&\"\"&\"\"\" F'F0" }{TEXT -1 7 ", type:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x -> ln(cos(5*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%# lnG6#-%$cosG6#,$*&\"\"&\"\"\"9$F5F5F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can confirm the current defi nition for the function f by typing " }{TEXT 260 5 "f(x);" }{TEXT -1 19 " on a command line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$cosG6#,$*&\"\"&\"\"\"%\"xGF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Alternatively, the defini tion of f can be obtained by using " }{TEXT 0 4 "eval" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)ope ratorG%&arrowGF&-%#lnG6#-%$cosG6#,$*&\"\"&\"\"\"9$F3F3F&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Note that simply typing " }{TEXT 260 2 "f;" }{TEXT -1 23 "on a command line doe s " }{TEXT 259 10 "not recall" }{TEXT -1 16 " the definition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"fG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "If you want to clear the \+ function f without redefining it, type:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := 'f';" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGF$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "A number of functions, say, f, g and h, c an cleared at the same time using the procedure " }{TEXT 0 8 "unassign " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "unassign('f','g','h');" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "In order to save memory \+ it is always a good idea to clear your functions, as well as any other variables, when you start a new problem. Alternatively, you can use \+ " }{TEXT 446 8 "restart;" }{TEXT -1 10 " to clear " }{TEXT 259 10 "eve rything" }{TEXT -1 13 " from memory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 449 12 "Exercise 5.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "If the following three commands " }} {PARA 0 "" 0 "" {TEXT 450 3 " " }{TEXT 260 8 "restart;" }}{PARA 0 " " 0 "" {TEXT 451 3 " " }{TEXT 260 12 "f(x) := x^2;" }}{PARA 0 "" 0 " " {TEXT 452 3 " " }{TEXT 260 5 "f(3);" }}{PARA 0 "" 0 "" {TEXT -1 78 "are entered in the workspace below, the output will not be the exp ected . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "f(3) = 3^2;" "6#/-%\"fG6#\"\"$*$F'\"\"#" }{XPPEDIT 18 0 " ``= 9" "6#/%!G\"\"*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Revise this sequence of three comman ds so the function " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F '\"\"#" }{TEXT -1 39 " is correctly defined and evaluated at " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 5.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Answer 5.2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The M aple commands: " }}{PARA 0 "" 0 "" {TEXT 458 2 " " }{TEXT 260 8 "rest art;" }}{PARA 0 "" 0 "" {TEXT 459 2 " " }{TEXT 260 12 "f(x) := x^2;" }}{PARA 0 "" 0 "" {TEXT -1 85 "do not create a function. In fact, it \+ is generally not wise to assign to the symbol " }{XPPEDIT 18 0 "f(x); " "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 58 "The correct way to enter this function and evaluate it at " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart;\n f := x -> x^2;\nf(3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 453 12 "Exercise \+ 5.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "If the following three commands . . . " }}{PARA 0 "" 0 "" {TEXT 454 3 " \+ " }{TEXT 260 8 "restart;" }}{PARA 0 "" 0 "" {TEXT 455 3 " " } {TEXT 260 9 "g := x^3;" }}{PARA 0 "" 0 "" {TEXT 456 3 " " }{TEXT 260 5 "g(2);" }}{PARA 0 "" 0 "" {TEXT -1 71 "are entered in the worksp ace below, the output will not be the expected" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "f(2) = 2^3;" "6#/-%\"fG6#\"\"#*$F'\"\"$" }{XPPEDIT 18 0 " ``= 8" "6#/%!G\"\")" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "Revise this sequence of three commands so the function " } {XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 39 " is correctly defined and evaluated at " }{XPPEDIT 18 0 "x = 2;" "6#/% \"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 5.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Answer 5.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple commands" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 260 8 "restart;" }}{PARA 0 "" 0 "" {TEXT 460 3 " " }{TEXT 260 9 "g := x^2;" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "create an expression, not a function. To \+ create the function g and evaluate it at " }{XPPEDIT 18 0 "x = 3;" "6# /%\"xG\"\"$" }{TEXT -1 23 ", use the syntax . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart;\ng \+ := x -> x^3;\nf(2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Evaluating a funct ion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We start by defining a function f with the mathematical formulation " } {XPPEDIT 18 0 "f(x)=3*x+x^2" "6#/-%\"fG6#%\"xG,&*&\"\"$\"\"\"F'F+F+*$F '\"\"#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> 3*x+x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"$ \"\"\"9$F/F/*$)F0\"\"#F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Typing " }{TEXT 447 5 "f(a);" }{TEXT -1 44 " will give you the value of f at the point " }{XPPEDIT 18 0 "x = \+ a;" "6#/%\"xG%\"aG" }{TEXT -1 15 ". For example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "Maple will leave expressions involvin g " }{XPPEDIT 18 0 "sqrt(``)" "6#-%%sqrtG6#%!G" }{TEXT -1 37 " in sym bolic form, as shown below. \n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "val := f(3+sqrt(20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$va lG,(\"\"*\"\"\"*&\"\"'F'\"\"&#F'\"\"#F'*$),&\"\"$F'*&F,F'F*F+F'F,F'F' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "To o btain a floating-point (numerical) value, use the procedure " }{TEXT 0 5 "evalf" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(val);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+eB#\\#y!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "The value of " }{TEXT 444 1 "x" }{TEXT -1 57 " does not have to be numerically computable. For example:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expr := f(x+4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,(*&\" \"$\"\"\"%\"xGF(F(\"#7F(*$),&F)F(\"\"%F(\"\"#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "To simplify this expressi on, use the procedure " }{TEXT 0 8 "simplify" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#6\"\"\"%\"xGF&F&\"#GF&*$)F'\"\"#F&F&" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Furtherm ore, you can manipulate the results of " }{TEXT 448 5 "f(x);" }{TEXT -1 60 " directly. For example, the Newton (or difference) quotient " } {XPPEDIT 18 0 "(f(x+h)-f(x))/h" "6#*&,&-%\"fG6#,&%\"xG\"\"\"%\"hGF*F*- F&6#F)!\"\"F*F+F." }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "NQ := (f(x+h)-f(x) )/h;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " which can be simplified, via the procedure " }{TEXT 0 8 "simplify" } {TEXT -1 4 ", to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "simplify(NQ);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "Given the second function . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g := x -> cos(x)+1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 " . . . the composition " }{XPPEDIT 18 0 "f(g(x)); " "6#-%\"fG6#-%\"gG6#%\"xG" }{TEXT -1 30 " is obtained with the notati on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f(g(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "To assign the composite function " }{XPPEDIT 18 0 "f(g(x));" "6#-%\"fG6#-%\"gG6#%\"xG" }{TEXT -1 10 " the name " } {XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 5 ", use" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "h := x -> f(g(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The specific value " }{XPPEDIT 18 0 "f(g(Pi/3));" "6#-%\" fG6#-%\"gG6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 15 " is then . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(g(Pi/3));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "h(Pi/3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 441 12 "Exercise 5.4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "De fine the function " }{XPPEDIT 18 0 "s(t)= (3+t^2)/sqrt(3*t+1)" "6#/-% \"sG6#%\"tG*&,&\"\"$\"\"\"*$F'\"\"#F+F+-%%sqrtG6#,&*&F*F+F'F+F+F+F+!\" \"" }{TEXT -1 28 " then have Maple calculate " }{XPPEDIT 18 0 "s(2),s (t-3);" "6$-%\"sG6#\"\"#-F$6#,&%\"tG\"\"\"\"\"$!\"\"" }{TEXT -1 6 ", a nd " }{XPPEDIT 18 0 "s(t)-s(3);" "6#,&-%\"sG6#%\"tG\"\"\"-F%6#\"\"$!\" \"" }{TEXT -1 28 " and simplify your results. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Don't forget the arrow no tation!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Student Workspace 5.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Answer 5.4" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "s := t -> (3 + t^2)/(sqrt(3*t+1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "s(2);" "6#-%\"sG6#\"\"#" }{TEXT -1 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "s(t-3);" "6#-%\"sG6# ,&%\"tG\"\"\"\"\"$!\"\"" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "s(t - 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "To simplify the result: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(s(t-3));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "s(t)-s(3);" "6#,&-%\"sG6#%\"tG\"\"\" -F%6#\"\"$!\"\"" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s(t) - s(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "To simplify the result:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(s(t)-s(3));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Notice that if you define a fun ction, there is no need to use the procedure " }{TEXT 0 4 "eval" } {TEXT -1 32 " like you do with expressions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Solving equations involving functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Once y our function is defined, you can solve equations containing this funct ion either exactly or approximately." }}{PARA 0 "" 0 "" {TEXT -1 63 "T he following command sets up in Maple the function g given by " } {XPPEDIT 18 0 "g(t)=t^3-6*t^2+6*t+8" "6#/-%\"gG6#%\"tG,**$F'\"\"$\"\" \"*&\"\"'F+*$F'\"\"#F+!\"\"*&F-F+F'F+F+\"\")F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := t -> t^3-6*t^2+6*t+8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The exact solutions of the equation " } {XPPEDIT 18 0 "g(t) = 0;" "6#/-%\"gG6#%\"tG\"\"!" }{TEXT -1 25 " are o btained from . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(t)=0,t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " . . . whereas an approximate ( numerical) solutions are obtained from . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(g(t)= 0,t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Graphing a function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 4 "plot" }{TEXT -1 92 " function works the same for \+ functions, provided the function is evaluated at its argument. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "For examp le, consider the function " }{XPPEDIT 18 0 "g(x) = x*exp(-x);" "6#/-% \"gG6#%\"xG*&F'\"\"\"-%$expG6#,$F'!\"\"F)" }{TEXT -1 26 ", given in Ma ple by . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "unassign('g','y','x'); \ng := x -> x*exp(-x); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " The curve " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 24 " is generated via . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(g(x),x=-1..4,y=-2..1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Several \+ functions can be graphed simultaneously just at we did for expressions , again, provided that the functions are evaluated at their arguments. " }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f(x) =4/(x^2+1)" "6#/-%\"fG6#%\"xG*&\"\"%\"\"\",&*$F'\"\"#F*F*F*!\" \"" }{TEXT -1 30 ", entered into Maple as . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> 2/ (x^2+1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Below we graph this function along with the horizontal shifts " } {XPPEDIT 18 0 "f(x+1)" "6#-%\"fG6#,&%\"xG\"\"\"F(F(" }{TEXT -1 3 " , \+ " }{XPPEDIT 18 0 "f(x-3) " "6#-%\"fG6#,&%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1 4 "and " }{XPPEDIT 18 0 "f(x-6)" "6#-%\"fG6#,&%\"xG\"\"\"\"\"'!\"\" " }{TEXT -1 25 " . Can you identify each?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([f(x),f(x+1),f(x -3),f(x-6)],x=-5..10,y=-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 159 "By not assigning unique and recognizable colors to the functions plotted, we have created a real puzzle that c ould have been avoided with syntax such as . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot([f(x),f (x+1),f(x-3),f(x-6)],x=-5..10,y=-1..3, color=[black,red,green,blue]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The \+ order of the functions corresponds to the order of the colors!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 649 12 "Exercise 5.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 658 21 "Define the fun ction " }{XPPEDIT 18 0 "f(x)=2*x-abs(x^2-5)" "6#/-%\"fG6#%\"xG,&*&\" \"#\"\"\"F'F+F+-%$absG6#,&*$F'F*F+\"\"&!\"\"F2" }{TEXT -1 50 " in Mapl e and then answer the following questions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) Find the value of f(6.5) \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) S implify the expression " }{XPPEDIT 18 0 "f(z-4);" "6#-%\"fG6#,&%\"zG\" \"\"\"\"%!\"\"" }{TEXT -1 7 " where " }{TEXT 664 1 "z" }{TEXT -1 14 " \+ is a variable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot a graph of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#% \"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 "(d) Find all values of " }{TEXT 671 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 650 21 "Stud ent Workspace 5.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 651 10 "A nswer 5.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "First define the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"x G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> 2*x-abs(x^2-5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"#\" \"\"9$F/F/-%$absG6#,&*$)F0F.F/F/\"\"&!\"\"F8F(F(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) To find the value o f f at " }{TEXT 665 1 "x" }{TEXT -1 7 " = 6.5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(6.5);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "(b) First evaluate " }{TEXT 666 1 "f" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = z- 4;" "6#/%\"xG,&%\"zG\"\"\"\"\"%!\"\"" }{TEXT -1 50 " then simplify the expression using the procedure " }{TEXT 0 8 "simplify" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(z-4);\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "(c) Use the " }{TEXT 0 4 "plot" }{TEXT -1 10 " command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-6..8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(d) We can use the proced ure " }{TEXT 0 6 "fsolve" }{TEXT -1 40 " to find the solutions in the \+ intervals " }{XPPEDIT 18 0 "[0, 2]" "6#7$\"\"!\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[3, 4]" "6#7$\"\"$\"\"%" }{TEXT -1 1 "." }{TEXT 659 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fsolve(f(x)=0,x=0..2);\nfsolve(f(x)=0,x=3..4);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 652 12 "Exercise 5.6" }}{PARA 0 "" 0 "" {TEXT 660 21 "Define the functions " }{XPPEDIT 18 0 "g(x)= 5*exp(-0.5*x)" "6 #/-%\"gG6#%\"xG*&\"\"&\"\"\"-%$expG6#,$*&-%&FloatG6$F)!\"\"F*F'F*F3F* " }{TEXT 661 7 " and " }{XPPEDIT 18 0 "h(x)=x+1" "6#/-%\"hG6#%\"xG,& F'\"\"\"F)F)" }{TEXT 662 24 " then do the following." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "(a) Plot a graph that \+ shows both functions " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 43 ". Experiment with different values for the " }{TEXT 672 1 "x" } {TEXT -1 5 " and " }{TEXT 673 1 "y" }{TEXT -1 8 "-ranges." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "(b) Estimate the \+ coordinates of the point of intersection of these two graphs by using \+ left mouse-button click." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "(c) Use " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to sol ve the equation " }{XPPEDIT 18 0 "g(x) = h(x);" "6#/-%\"gG6#%\"xG-%\"h G6#F'" }{TEXT -1 77 ". How does the solution of this equation relate t o your answer to part (b)?. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 " " {TEXT 653 21 "Student Workspace 5.6" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 20 "" 0 "" {TEXT 654 10 "Answer 5.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "First set up the function s in Maple using the arrow notation. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g := x -> 5*exp(-0.5*x); \nh := x -> x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,$*&\"\"&\"\"\"-%$expG6#,$*&$F.!\"\"F/9$F/F6F/F /F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,&9$\"\"\"F.F.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 104 "(a) To plot a graph that shows both fun ctions, simply put the functions in a list using square brackets " } {TEXT 0 3 "[ ]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Start by taking fairly wide ranges of val ues for " }{TEXT 674 1 "x" }{TEXT -1 5 " and " }{TEXT 675 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([g(x),h(x)],x=-5..5,y=-20..20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Now \"zoom in \" towards the point of intersection. This will be useful for checking the answer to part (c). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([g(x),h(x)],x=1..2,y=1..4);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "(c) To \+ solve the equation " }{XPPEDIT 18 0 "g(x) = h(x);" "6#/-%\"gG6#%\"xG- %\"hG6#F'" }{TEXT -1 7 " using " }{TEXT 0 6 "fsolve" }{TEXT -1 20 ", t ype and execute: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xval := fsolve(g(x)=h(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The solution to " } {XPPEDIT 18 0 "g(x) = h(x);" "6#/-%\"gG6#%\"xG-%\"hG6#F'" }{TEXT -1 8 " is the " }{TEXT 667 1 "x" }{TEXT -1 44 "-coordinate of the point of \+ intersection of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 28 ". To \+ find the corresponding " }{TEXT 668 1 "y" }{TEXT -1 76 "-coordinate of the intersection point evaluate either g or h at this value. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g(xval);\nh(xval);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "The coordinates can be checked at least roughly by \+ means of a mouse click applied to the intersection point of the two gr aphs in the second of the two plots given in the answer to part (a). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 655 12 "Exercise 5.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 663 21 "Define the function " } {XPPEDIT 18 0 "k(x)=x+3*sin(2*x)" "6#/-%\"kG6#%\"xG,&F'\"\"\"*&\"\"$F) -%$sinG6#*&\"\"#F)F'F)F)F)" }{TEXT -1 24 ", then do the following:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(a) Plot \+ the graph of this function on the domain " }{XPPEDIT 18 0 "[-1, 8] " " 6#7$,$\"\"\"!\"\"\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "(b) Modify your plot from part (a) t o include the horizontal line " }{XPPEDIT 18 0 "y = 4;" "6#/%\"yG\"\"% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 73 " Use this new pl ot to estimate the number and approximate values for " }{TEXT 676 1 "x " }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "k(x) = 4;" "6#/-%\"kG6#% \"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 9 "(c) What " }{TEXT 259 6 "single" }{TEXT -1 83 " fun ction could you graph that would give you the same information as in p art (b) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "(d) Use the Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 46 " t o approximate all solutions to the equation " }{XPPEDIT 18 0 "k(x) = 4 ;" "6#/-%\"kG6#%\"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 656 21 "Student Workspace 5.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 657 10 "Answer 5.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "(a) First set up the function k v ia . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "k := x -> x+3*sin(2*x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 " . . . then plot the graph of this function using the " }{TEXT 0 4 "plot" }{TEXT -1 15 " command . . . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(k(x),x=-1..8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "(b) As it can be seen from the following, ther e appear to be three intersection points located approximately where \+ " }{XPPEDIT 18 0 "x = 3.25 " "6#/%\"xG-%&FloatG6$\"$D$!\"#" }{TEXT -1 17 ", 4.85 and 5.95. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot([k(x),4],x=-1..8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) We could graph " } {XPPEDIT 18 0 "k(x)-4;" "6#,&-%\"kG6#%\"xG\"\"\"\"\"%!\"\"" }{TEXT -1 14 " and look for " }{TEXT 669 1 "x" }{TEXT -1 38 "-intercepts. These \+ will correspond to " }{TEXT 670 1 "x" }{TEXT -1 26 "-values found in p art (b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(k(x)-4,x=-1..8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Here are the solutions obtained by g iving " }{TEXT 0 6 "fsolve" }{TEXT -1 54 " suitable intervals in which to search for a solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "fsolve(k(x)=4,x=2..3.5);\nfsolve(k( x)=4,x=3.5..5);\nfsolve(k(x)=4,x=5..7);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Giving " }{TEXT 0 6 "fsolve" } {TEXT -1 66 " the approximatevalues given above as starting values als o works. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "fsolve(k(x)=4,x=3.25);\nfsolve(k(x)=4,x=4.85);\nfsolv e(k(x)=4,x=5.95);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Converting expressions to functions using " }{TEXT 0 7 "unapply" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "If an expression such as . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expr_in_x := x*sin(x)-exp(2*x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 88 " . . . already appears in Maple, it can be made into a \+ function by use of the procedure " }{TEXT 0 7 "unapply" }{TEXT -1 24 " , whose syntax is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := unapply(expr_in_x,x);" }}} {PARA 0 "" 0 "" {TEXT -1 35 "The expression is now the function " } {XPPEDIT 18 0 "f(x) = x*sin(x)-exp(2*x);" "6#/-%\"fG6#%\"xG,&*&F'\"\" \"-%$sinG6#F'F*F*-%$expG6#*&\"\"#F*F'F*!\"\"" }{TEXT -1 38 ", so that \+ evaluation at, for example, " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\" " }{TEXT -1 29 ", is accomplished with . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Let f be the functio n given by " }{XPPEDIT 18 0 "f(x)=2/(x^2+1)" "6#/-%\"fG6#%\"xG*&\"\"# \"\"\",&*$F'F)F*F*F*!\"\"" }{TEXT -1 50 " and which was considered in \+ the previous section." }}{PARA 0 "" 0 "" {TEXT -1 24 "The function g g iven by " }{XPPEDIT 18 0 "g(x)=f(x)+f(x+1)+f(x-3)+f(x-6)" "6#/-%\"gG6# %\"xG,*-%\"fG6#F'\"\"\"-F*6#,&F'F,F,F,F,-F*6#,&F'F,\"\"$!\"\"F,-F*6#,& F'F,\"\"'F4F," }{TEXT -1 24 " is a rational function." }}{PARA 0 "" 0 "" {TEXT -1 70 "The following commands construct a simplified form for the expression " }{XPPEDIT 18 0 "f(x)+f(x+1)+f(x-3)+f(x-6)" "6#,*-%\" fG6#%\"xG\"\"\"-F%6#,&F'F(F(F(F(-F%6#,&F'F(\"\"$!\"\"F(-F%6#,&F'F(\"\" 'F0F(" }{TEXT -1 32 " as a single algebraic fraction." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{TEXT 0 7 "unapply" }{TEXT -1 67 " is used to d efine the function g using this rational expression. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "f := x \+ -> 1/(x^2+1);\nf(x)+f(x+1)+f(x-3)+f(x-6);\nsimplify(%);\ng := unapply( %,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We can now plot the graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "plot(g(x),x=-7..11);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 677 12 "Exercise 5.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "If the following four commands " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 260 8 "restart;" }}{PARA 0 "" 0 "" {TEXT 680 3 " " }{TEXT 260 13 "k := x^2 + 5;" }}{PARA 0 " " 0 "" {TEXT 678 3 " " }{TEXT 260 12 "f := x -> k;" }}{PARA 0 "" 0 " " {TEXT 679 3 " " }{TEXT 260 5 "f(2);" }}{PARA 0 "" 0 "" {TEXT -1 58 "are entered in the workspace below, the output will not be" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "f(2) = 2^2+5;" "6#/-%\"fG6#\"\"#,&*$F 'F'\"\"\"\"\"&F*" }{TEXT -1 5 " = 9 " }}{PARA 0 "" 0 "" {TEXT -1 21 "b ecause the function " }{XPPEDIT 18 0 "f(x) = x^2+5;" "6#/-%\"fG6#%\"xG ,&*$F'\"\"#\"\"\"\"\"&F+" }{TEXT -1 50 " will not have been correctly \+ formulated in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "Show the proper way to convert the expression " } {XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 15 " to a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " so the value of " } {XPPEDIT 18 0 "f(2);" "6#-%\"fG6#\"\"#" }{TEXT -1 23 " is correctly co mputed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 "Student Workspace 5.8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 5.8" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Entering . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;\nk := x^2+5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " . . . creates an " }{TEXT 259 10 "expression" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "An expressions that " }{TEXT 259 13 "already exist" }{TEXT -1 111 " in Maple cannot be made \+ into function with an arrow. Hence the following commands fail to set \+ up the function " }{XPPEDIT 18 0 "f(x)=x^2+5" "6#/-%\"fG6#%\"xG,&*$F' \"\"#\"\"\"\"\"&F+" }{TEXT -1 19 " and evauate it at " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> k;\nf(2);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "However t he procedure " }{TEXT 0 7 "unapply" }{TEXT -1 13 " may be used." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The corre ct syntax for this is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart;\nk := x^2+5;\nf := \+ unapply(k,x);\nf(2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 457 12 "Exercise 5.9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Let f be the function defined by " }{XPPEDIT 18 0 "f(x)=8 *x/(5*x^2+1)" "6#/-%\"fG6#%\"xG*(\"\")\"\"\"F'F*,&*&\"\"&F**$F'\"\"#F* F*F*F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "(a) Constr uct a single rational expression for " }{XPPEDIT 18 0 "f(x-2)+f(x)+f(x +2)" "6#,(-%\"fG6#,&%\"xG\"\"\"\"\"#!\"\"F)-F%6#F(F)-F%6#,&F(F)F*F)F) " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "(b) Define a function g such that " }{XPPEDIT 18 0 "g(x)=f(x-2)+f(x)+f(x+2)" "6#/-%\"gG6#% \"xG,(-%\"fG6#,&F'\"\"\"\"\"#!\"\"F--F*6#F'F--F*6#,&F'F-F.F-F-" } {TEXT -1 47 " by using the rational expression found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 22 "(c) Plot the graph of " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(d) F ind approximate values for the non-zero solutions of the equation " } {XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 21 " Student Workspace 5.9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Answer 5.9" }}{PARA 0 "" 0 "" {TEXT -1 56 "First we se t up the function f using the arrow notation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> 8* x/(5*x^2+1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) The rational expression for " }{XPPEDIT 18 0 "f(x-2)+ f(x)+f(x+2)" "6#,(-%\"fG6#,&%\"xG\"\"\"\"\"#!\"\"F)-F%6#F(F)-F%6#,&F(F )F*F)F)" }{TEXT -1 33 " can be constructed using . . . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f(x-2 )+f(x)+f(x+2);\nrat_expr := simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "(b) The function g can then be def ined using " }{TEXT 0 7 "unapply" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g := unapply (rat_expr,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) The graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 20 " may now be plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(g(x),x=-5..5);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "(d) Using approximate starting values for " }{TEXT 0 6 "fsolve" }{TEXT -1 67 " \+ found by clicking on the graph at the approximate location of the " } {TEXT 681 1 "x" }{TEXT -1 63 " intercepts we obtain four non-zero solu tions for the equation " }{XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG\"\" !" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(g(x),x=1.8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(g(x), x=1.3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(g(x),x=-1.3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(g(x),x=-1.8);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Note tha t the second two solutions are the negatives of the first two. Why is \+ this?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Section 6: More on graphing" }{TEXT 461 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Pa rametric equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 68 " ca n also be used to graph curves described by parametric equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "To graph \+ the parametric curve corresponding to the pair of parametric equations : " }{XPPEDIT 18 0 "x = f(t);" "6#/%\"xG-%\"fG6#%\"tG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "y = g(t);" "6#/%\"yG-%\"gG6#%\"tG" }{TEXT -1 27 " on the parameter interval " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG " }{TEXT -1 17 " use the command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 0 "" {TEXT 260 61 "plot([f(t), g(t), t = a..b], x = xmin. .xmax, y = ymin..ymax);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "There are two things to take carefu l note of here. First note that there are three entries in the square \+ brackets: the two parametric expressions for " }{TEXT 476 1 "x" } {TEXT -1 5 " and " }{TEXT 477 1 "y" }{TEXT -1 1 " " }{TEXT 259 3 "and " }{TEXT -1 101 " the parameter domain. Also note that the viewing win dow for the plot is separately specified by the " }{TEXT 478 2 "x-" } {TEXT -1 5 " and " }{TEXT 479 1 "y" }{TEXT -1 14 "-ranges (i.e.," } {TEXT 260 31 " x = xmin..xmax, y = ymin..ymax" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 41 "Plot the parametric curve determined by " } {XPPEDIT 18 0 "x=t^2-t" "6#/%\"xG,&*$%\"tG\"\"#\"\"\"F'!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y=2*t-t^3" "6#/%\"yG,&*&\"\"#\"\"\"%\" tGF(F(*$F)\"\"$!\"\"" }{TEXT -1 10 " over the " }{TEXT 480 1 "t" } {TEXT -1 15 "-interval from " }{XPPEDIT 18 0 "t=-2" "6#/%\"tG,$\"\"#! \"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([t^2-t,2*t-t^3,t=-2..2],x=-2..5,y=-5..5);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 462 12 "Exercise 6.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 473 34 "Plot the param etric curve defined " }{XPPEDIT 18 0 "x=sin(3*t)" "6#/%\"xG-%$sinG6#*& \"\"$\"\"\"%\"tGF*" }{TEXT 474 5 " and " }{XPPEDIT 18 0 "y=sin(4*t)" " 6#/%\"yG-%$sinG6#*&\"\"%\"\"\"%\"tGF*" }{TEXT 475 10 " over the " } {TEXT 481 1 "t" }{TEXT 482 15 "-interval from " }{XPPEDIT 18 0 "t=0" " 6#/%\"tG\"\"!" }{TEXT 621 4 " to " }{XPPEDIT 18 0 "t=2*Pi" "6#/%\"tG*& \"\"#\"\"\"%#PiGF'" }{TEXT 622 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "F or a viewing window let " }{TEXT 483 1 "x" }{TEXT -1 5 " and " }{TEXT 484 1 "y" }{TEXT -1 26 " range between -2 and 2 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 463 21 "Student Workspa ce 6.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 464 10 "Answer 6.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([sin(3*t),sin(4*t),t=0..2*Pi], x=-2..2,y=-2..2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "implicitplot( )" {TEXT -1 22 "Implic it plots: using " }{TEXT 0 12 "implicitplot" }{TEXT -1 10 " from the \+ " }{TEXT 0 5 "plots" }{TEXT -1 9 " package " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 12 "implicitplot" }{TEXT -1 10 " from the " }{TEXT 0 5 "plots" }{TEXT -1 85 " package can plot curves that are implicitly defined by an equa tion in the variables " }{TEXT 485 1 "x" }{TEXT -1 5 " and " }{TEXT 486 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 58 "To plot the graph of t he hyperbola given by the equation: " }{XPPEDIT 18 0 "x^2/4 - y^2/9 = \+ 1" "6#/,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yGF'\"\"*F)F)F*" }{TEXT -1 19 " use the procedure " }{TEXT 0 12 "implicitplot" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "implicitplo t" }{TEXT -1 12 " is in the \"" }{TEXT 260 5 "plots" }{TEXT -1 35 "\" \+ package along with the procedure " }{TEXT 0 7 "display" }{TEXT -1 24 " discussed in section 3." }}{PARA 0 "" 0 "" {TEXT -1 50 "The plots pac kage can be loaded using the command " }{TEXT 260 12 "with(plots);" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 132 "Our Maple code can be m ade to be independent of such a loading command by referring directly \+ to the required procedure via the name " }{TEXT 0 19 "plots[implicitpl ot]" }{TEXT -1 124 " in which the name of the individual procedure app ears in square brackets after the name of the package in which it resi des." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 48 "The required hyperbola can be drawn as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plots[implicitplot](x^2/4-y^2/4=1,x=-5..5,y=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 468 39 "We can plot the graph of the equation " }{XPPEDIT 18 0 "x^2/25+y^2/9=1" "6#/,&*&%\"xG\"\"#\"#D!\"\"\"\"\"*&%\"yGF'\"\"*F)F*F* " }{TEXT 469 21 " using the procedure " }{TEXT 0 12 "implicitplot" } {TEXT 470 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "This is the equation of an ellipse with the lengths of m ajor and minor axes equal to 10 and 6 respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Our first attempt at g etting the expected graph comes up short ! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plots[implicitplot ](x^2/25+y^2/9=1,x=-5..5,y=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 46 "Why did we get a circle instead of an el lipse?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The problem here is that the scaling along the " }{TEXT 487 2 "x-" } {TEXT -1 5 " and " }{TEXT 488 1 "y" }{TEXT -1 99 "-axes is not the sam e. To force equal scaling in the two coordinate directions include the option \"" }{TEXT 260 19 "scaling=constrained" }{TEXT -1 90 "\", or c lick on the graph to expose the graphing toolbar, and select the butto n marked 1:1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plots[implicitplot](x^2/25+y^2/9=1,x=-5..5,y=-5. .5,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 465 12 "Exercise 6.2" }}{PARA 0 "" 0 "" {TEXT 471 19 "Graph the \+ equation " }{XPPEDIT 18 0 "x^2+4*y^2=4" "6#/,&*$%\"xG\"\"#\"\"\"*&\"\" %F(*$%\"yGF'F(F(F*" }{TEXT 472 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 20 "" 0 "" {TEXT 466 21 "Student Workspace 6.2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT 467 10 "Answer 6.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plots[implicitplot](x ^2+4*y^2=4,x=-3..3,y=-2..2,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "Polar graphs (optional)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "Graphs of polar equations " } {XPPEDIT 18 0 "r = f(theta)" "6#/%\"rG-%\"fG6#%&thetaG" }{TEXT -1 30 " are handled by the procedure " }{TEXT 0 9 "polarplot" }{TEXT -1 24 ", which is part of the \"" }{TEXT 0 5 "plots" }{TEXT -1 25 "\" package \+ accessed using " }{TEXT 260 12 "with(plots);" }{TEXT -1 31 " or indepe ndently via the name " }{TEXT 260 16 "plots[polarplot]" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Here \+ are some examples. Note that we include the option \"" }{TEXT 260 19 " scaling=constrained" }{TEXT -1 31 "\" to get geometric perspective." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plots[polarplot](1+cos(theta),theta=-Pi..Pi,scaling=constrained) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "polarplot(sin(3*theta),theta=-Pi..Pi,scaling=constrai ned);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Another way of graph polar graphs is to use the plot option \"" } {TEXT 260 12 "coords=polar" }{TEXT -1 85 "\" and graph the curve using parametric equations. The general form of the command is:" }}{PARA 259 "" 0 "" {TEXT 489 46 "plot([r(s), theta(s), s=a..b], coords=polar) ; " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "If the parameter " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 23 " is actually the angle " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 21 ", the command become s" }}{PARA 259 "" 0 "" {TEXT 490 51 "plot([r(theta), theta, theta=a..b ], coords=polar); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, to graph " }{XPPEDIT 18 0 "r = 1+cos(theta); " "6#/%\"rG,&\"\"\"F&-%$cosG6#%&thetaGF&" }{TEXT -1 42 " in polar coor dinates using the procedure " }{TEXT 0 4 "plot" }{TEXT -1 8 ", type: \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([1+cos(theta),theta,theta=-Pi..Pi],coords=polar);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "The \"" } {TEXT 260 12 "coords=polar" }{TEXT -1 30 "\" option can be applied to \+ an " }{TEXT 0 12 "implicitplot" }{TEXT -1 17 " command as well." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "For examp le, the following command graphs the lemniscate " }{XPPEDIT 18 0 "r^2 \+ = 4*cos(2*theta);" "6#/*$%\"rG\"\"#*&\"\"%\"\"\"-%$cosG6#*&F&F)%&theta GF)F)" }{TEXT -1 10 " over the " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 15 " interval from " }{XPPEDIT 18 0 "theta=-Pi" "6#/%&thetaG ,$%#PiG!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "theta=Pi" "6#/%&theta G%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 13 ": By default " }{TEXT 0 12 "impl icitplot" }{TEXT -1 106 " uses a grid 25 by 25 grid consisting of 625 \+ points in order to construct the plot. Including the option \"" } {TEXT 260 12 "grid=[50,50]" }{TEXT -1 13 "\" will cause " }{TEXT 0 12 "implicitplot" }{TEXT -1 102 " to user a finer 50 by 50 grid of points to try to find pairs of neighbouring points along the curve. " }} {PARA 0 "" 0 "" {TEXT -1 192 "About 415 line separate line segments ar e drawn to construct the plot instead of about 196 obtained with a 25 \+ by 25 grid. This means that the memory requirements for this plot are \+ quite large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plots[implicitplot](r^2=4*cos(2*theta),r=0..2,the ta=0..2*Pi,coords=polar, scaling=constrained,grid=[50,50]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Plot options" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "There are many options available when you use the procedure " }{TEXT 0 4 "plot " }{TEXT -1 30 ". To see a list go to Maple's " }{TEXT 624 9 "Help Pag e" }{TEXT -1 41 " on this command. Clicking the hyperlink " } {HYPERLNK 17 "plot[options]" 2 "plot[options]" "" }{TEXT -1 110 " does this as does executing the following command (no semi-colon is needed ). You can also select the \"word\" " }{TEXT 258 13 "plot[options]" } {TEXT -1 49 " with the mouse and access the help page via the " } {TEXT 623 4 "Help" }{TEXT -1 7 " menu. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "?plot[options]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Notes on the Maple worksheet interface" }{TEXT 523 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 27 "Creating a new command line" }}{PARA 0 "" 0 "" {TEXT -1 34 "There are several ways to do this." }}{PARA 0 "" 0 "" {TEXT -1 29 "a quick keystroke : Ctrl-J , " }}{PARA 0 "" 0 "" {TEXT -1 27 "or click on the button: " }{TEXT 493 2 "[>" }{TEXT -1 44 " \+ in the toolbar at the top of the screen ," }}{PARA 0 "" 0 "" {TEXT -1 28 "or use the Menu selection: " }{TEXT 494 6 "Insert" }{TEXT 499 3 " / " }{TEXT 495 15 "Execution Group" }{TEXT 500 3 " > " }{TEXT 496 12 "After Cursor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 26 "Creating a new text region" }}{PARA 0 "" 0 "" {TEXT -1 121 "The easiest way to do this is to first create a new comm and line as described above and then convert it to a text region." }} {PARA 0 "" 0 "" {TEXT -1 56 "To convert to a text region do any one of the following:" }}{PARA 0 "" 0 "" {TEXT -1 29 "a quick keystroke : Ct rl-T " }}{PARA 0 "" 0 "" {TEXT -1 27 "or click on the button: " } {TEXT 492 1 "T" }{TEXT -1 47 " in the toolbar at the top of the scre en. " }}{PARA 0 "" 0 "" {TEXT -1 28 "or use the Menu selection: " }{TEXT 497 6 "Insert" }{TEXT -1 3 " / " }{TEXT 498 10 "Text Input" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "You can \+ also type text between command lines by creating a space before or aft er a command line as follows." }}{PARA 0 "" 0 "" {TEXT -1 118 "First c lick with the mouse close to the [ part of the command line indicator \+ [> so that it appears as a double line. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Then use the Menu selection: " }{TEXT 625 6 "Insert" }{TEXT -1 3 " / " }{TEXT 627 9 "Paragraph" }{TEXT 632 3 " > " }{TEXT 626 12 "After Cursor" }}{PARA 0 "" 0 "" {TEXT -1 4 "or: " }}{PARA 0 "" 0 "" {TEXT -1 46 " " }{TEXT 628 6 "I nsert" }{TEXT -1 3 " / " }{TEXT 629 9 "Paragraph" }{TEXT 631 3 " > " } {TEXT 630 13 "Before Cursor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 20 "Semicolon vs. Colon " }}{PARA 0 "" 0 " " {TEXT -1 37 "End a command line with a semicolon (" }{TEXT 260 1 "; " }{TEXT -1 150 ") to execute the line and display the result as blue \+ output or a graph. Alternatively suppress the display of output by end ing the line with a colon (" }{TEXT 260 1 ":" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 " Opening and closing sections" }}{PARA 0 "" 0 "" {TEXT -1 98 "Click on \+ + button to open a section. The button changes to - . Click on - to close a section. \n" }{TEXT 259 26 "Quick close with space bar" } {TEXT -1 184 ": To close a section if the button has scrolled off the \+ screen without having to scroll back up to find it, click on section l ine at left to make it a double line then press Space Bar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 27 "Moving t hrough a worksheet " }}{PARA 0 "" 0 "" {TEXT -1 205 "Use the vertical \+ scrollbar on the right or up/down arrow keys. If you press [Enter] whe n you are on a command line Maple executes the line and then the curso r automatically moves to the next command line. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 39 "How to remove ou tput from the worksheet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "To remove the output from a single command line, or group of commands, select (highlight with the mouse) the appropriate \+ command or execution group." }}{PARA 0 "" 0 "" {TEXT -1 30 "Then use t he Menu selection: " }{TEXT 682 37 "Edit / Remove Output > From Selec tion" }}{PARA 0 "" 0 "" {TEXT -1 80 "Note that it is actually sufficie nt to select only part of the command or group." }}{PARA 0 "" 0 "" {TEXT -1 111 "The output from a number of consecutive execution groups can be removed at the same time by selecting them all." }}{PARA 0 "" 0 "" {TEXT -1 68 "(It helps to hold down the shift key when selecting \+ a large region.)" }}{PARA 0 "" 0 "" {TEXT -1 97 "This cannot be done w ithout also selecting any intervening text, but this will remain unaff ected." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "To remove " }{TEXT 259 10 "all output" }{TEXT -1 26 " use the Menu selection: " }{TEXT 501 37 "Edit / Remove Output > From Worksheet" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "Expanding or Collapsing all Sections" }}{PARA 0 "" 0 "" {TEXT -1 20 " Use Menu selection: " }{TEXT 502 53 "View / Expand All Sections (or Co llapse All Sections)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 61 "Creating a New File/ Saving a File / Open ing an Existing File" }}{PARA 0 "" 0 "" {TEXT -1 21 "Use Menu selectio n : " }{TEXT 503 34 "File / New , Open , Save , Save As" }{TEXT -1 37 " (standard Windows file management)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Cutting and pasting" }} {PARA 0 "" 0 "" {TEXT -1 81 "Standard Windows editing: cut, copy and p aste available from the Menu selection: " }{TEXT 504 21 "Edit / Copy o r Paste " }}{PARA 0 "" 0 "" {TEXT -1 26 "or use buttons on Toolbar." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 22 "Using the Help system " }}{PARA 0 "" 0 "" {TEXT 259 110 "To get quick help on a particular procedure: select (highlight) the name of the pr ocedure and press the F1 key" }{TEXT -1 45 ". A Help page for the proc edure will appear. " }}{PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 507 5 " close" }{TEXT -1 70 " the Help page when you have finished reading it \+ use the Menu choice: " }{TEXT 506 22 "File, Close Help Topic" }{TEXT -1 22 ", or simply click the " }{TEXT 633 1 "X" }{TEXT -1 43 " mark at the top right side of the window. " }}{PARA 0 "" 0 "" {TEXT -1 58 "To look up the Help page for a particular procedure: type " }{TEXT 260 22 "?name-of-the-procedure" }{TEXT -1 61 " and press enter. A Help pag e for that procedure will appear." }}{PARA 0 "" 0 "" {TEXT -1 85 "Mapl e's Help system is quite elaborate. To learn more about it from the Me nu choose: " }{TEXT 505 17 "Help / Using Help" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 508 43 " Execution groups with more than one command" }}{PARA 0 "" 0 "" {TEXT -1 65 "To include more than one command in a single execution group us e:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 259 68 "[SHIFT] + [ENTER] \+ instead of [ENTER] at the end of each command line" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 50 "Instead of executing one command at at t ime . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "a:=4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b :=7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=8;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 48 ". . . all three can be executed toget her . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "a:=4;\nb:=7;\nc:=8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Qu ick reference for Maple commands" }{TEXT 524 0 "" }}{PARA 3 "" 0 "" {TEXT 514 0 "" }}{PARA 0 "" 0 "" {TEXT 0 8 "denom( )" }{TEXT -1 82 " \+ Selects the denominator of a fraction." }}{PARA 0 "" 0 "" {TEXT 0 10 "display( )" }{TEXT 548 5 " \+ " }{TEXT 517 74 " Combines graphs \+ of functions and points." }}{PARA 0 "" 0 "" {TEXT 553 74 " \+ Note: the " } {TEXT 260 5 "plots" }{TEXT 554 30 " package must be loaded using " } {TEXT 260 13 "with(plots); " }{TEXT 693 23 "unless the longer form " } {TEXT 260 14 "plots[display]" }{TEXT 694 10 " is used.\n" }{TEXT 0 14 "eval(expr,x=v)" }{TEXT 549 29 " " }{TEXT 522 38 "Evaluates the expression at the point " }{XPPEDIT 18 0 "x = v " "6#/%\"xG%\"vG" }{TEXT 701 1 "." }}{PARA 0 "" 0 "" {TEXT 0 14 "Eval( expr,x=v)" }{TEXT 683 45 " \"Inert\" form \+ of " }{TEXT 260 4 "eval" }{TEXT 700 31 ": it constructs the expression " }{XPPEDIT 18 0 "eval(expr,x=v)" "6#-%%evalG6$%%exprG/%\"xG%\"vG" } {TEXT -1 47 " without actually performing the evaluation. " }}{PARA 0 "" 0 "" {TEXT -1 76 " \+ (see also: " }{TEXT 260 8 "value( )" }{TEXT -1 2 ".) " }}{PARA 0 "" 0 "" {TEXT 0 11 "evalf(expr)" }{TEXT 509 63 " \+ Numercially evaluates the " }{TEXT -1 16 "g iven expression" }{TEXT 510 33 " to the default number of digits." }} {PARA 0 "" 0 "" {TEXT 0 13 "evalf(expr,n)" }{TEXT 511 58 " \+ Numercially evaluates the " }{TEXT -1 16 "given ex pression" }{TEXT 512 4 " to " }{TEXT 695 1 "n" }{TEXT 696 8 " digits. " }}{PARA 0 "" 0 "" {TEXT 0 12 "expand(expr)" }{TEXT 551 1 " " }{TEXT -1 63 " Expands the given expression. " }}{PARA 0 "" 0 "" {TEXT 0 12 "factor(expr)" }{TEXT -1 65 " \+ Factors the given expression. " }}{PARA 0 "" 0 "" {TEXT 0 11 "fsolve(eqn)" }{TEXT -1 90 " \+ Finds numerical (approximate) solutions to equations." }} {PARA 0 "" 0 "" {TEXT -1 75 " \+ Example: " }{TEXT 260 28 "fsolve(x^2=cos(x)+4 ,x=0..5);" }}{PARA 0 "" 0 "" {TEXT 0 10 "ifactor(n)" }{TEXT 552 1 " " }{TEXT 515 94 " Gives prime inte ger factorization for a given integer. " }}{PARA 0 "" 0 "" {TEXT 0 8 " lhs(eqn)" }{TEXT -1 87 " S elects the left hand side of an equation." }}{PARA 0 "" 0 "" {TEXT 0 15 "implicitplot( )" }{TEXT -1 62 " Plots im plicitly defined functions." }}{PARA 0 "" 0 "" {TEXT 555 75 " \+ Note: the " } {TEXT 260 5 "plots" }{TEXT 556 30 " package must be loaded using " } {TEXT 260 13 "with(plots); " }{TEXT 691 23 "unless the longer form " } {TEXT 260 19 "plots[implicitplot]" }{TEXT 692 9 " is used." }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ Example: " }{TEXT 260 72 "implicitplot(x^2/25 +y^2/9=1 ,x=-5 . . 5,y=-5 . . 5,scaling=constrained);" }}{PARA 0 "" 0 "" {TEXT 0 8 "numer( )" }{TEXT -1 81 " \+ Selects the numerator of a fraction." }}{PARA 0 "" 0 "" {TEXT 0 7 "plot( )" }{TEXT 550 10 " " }{TEXT 516 89 " \+ Plots functions defined by an algebraic expression: " }}{PARA 0 "" 0 "" {TEXT -1 29 " " } {TEXT 520 4 " " }{TEXT 260 32 " " } {TEXT -1 10 "Example: " }{TEXT 260 40 "plot(3*x^2-8 ,x=-5 . . 5 ,y=-2 0 . . 40);" }}{PARA 0 "" 0 "" {TEXT -1 104 " \+ Plots more than one function at a time:" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ " }{TEXT 521 36 " " }{TEXT -1 9 "Ex ample: " }{TEXT 260 58 "plot( [3*x^2-8 ,sin(x) ,cos(x) ],x=-5 . . 5 ,y =-20 .. 40);" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ Plots points:" }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ Example: " }{TEXT 260 47 "plot([[2, 3], [-2, 5], [1, -4]],s tyle = point);" }}{PARA 0 "" 0 "" {TEXT -1 28 " \+ " }{TEXT 519 65 " Plots param etric equations :" }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ Example: " }{TEXT 260 53 "plot( [cos(t),sin(t),t=0 . . 2*Pi],x=-2..2, y=-2..2);" }}{PARA 0 "" 0 "" {TEXT 0 17 "rationalize(expr)" }{TEXT 547 1 " " }{TEXT 513 74 " \+ Rationalizes the denominator of the given expressio n." }}{PARA 0 "" 0 "" {TEXT 0 7 "restart" }{TEXT -1 88 " \+ Clears Maple's memory of all definiti ons." }}{PARA 0 "" 0 "" {TEXT 0 8 "rhs(eqn)" }{TEXT -1 88 " \+ Selects the right hand side of an eq uation." }}{PARA 0 "" 0 "" {TEXT 0 14 "simplify(expr)" }{TEXT -1 62 " \+ Simplifies the given expression." }} {PARA 0 "" 0 "" {TEXT 0 10 "solve(eqn)" }{TEXT -1 123 " \+ Finds exact solutions to equations, including literal equations and linear systems." }}{PARA 0 "" 0 "" {TEXT 0 14 " subs(x=v,expr)" }{TEXT 544 1 " " }{TEXT -1 51 " \+ Substitutes the value " }{TEXT 702 1 "v" }{TEXT -1 5 " for " } {TEXT 703 1 "x" }{TEXT -1 19 " in the expression." }}{PARA 0 "" 0 "" {TEXT -1 75 " \+ Example: " }{TEXT 260 18 "subs(x=4,3*x^2+8);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 15 "unapply(expr,x)" }{TEXT 545 8 " \+ " }{TEXT 30 80 " Constructs the function x -> expr \+ from the given expression." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 24 "unassign('var1','var2') " }{TEXT 546 5 " " }{TEXT 30 35 "Clear s the variables var1 and var2." }}{PARA 0 "" 0 "" {TEXT -1 82 " \+ The shorter \+ form " }{TEXT 260 14 "var1 := 'var1'" }{TEXT -1 41 " can be used to cl ear a single variable. " }}{PARA 0 "" 0 "" {TEXT 0 8 "value( )" } {TEXT -1 99 " Completes th e evaluation of an unevaluated expression." }}{PARA 0 "" 0 "" {TEXT -1 76 " \+ (see also: " }{TEXT 260 7 "Eval( )" }{TEXT -1 2 " )" }}{PARA 0 "" 0 "" {TEXT 0 7 "with( )" }{TEXT -1 90 " \+ Loads a package of additional procedures. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The standard mathe matical " }{TEXT 259 9 "constants" }{TEXT -1 5 " are:" }}{PARA 0 "" 0 "" {TEXT 0 2 "Pi" }{TEXT -1 60 " \+ " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 8 " \+ " }{TEXT 543 7 "Caution" }{TEXT -1 51 ": Do not use \"pi\", capit al \"P\" is required. " }}{PARA 0 "" 0 "" {TEXT 0 6 "exp(1)" } {TEXT -1 50 " " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT 0 1 "I" }{TEXT -1 61 " \+ " }{XPPEDIT 18 0 "sqrt(-1)" "6#-% %sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 28 " . . . the imaginary unit. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The names of the standard mathematical " }{TEXT 259 9 "functions" }{TEXT -1 5 " are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 7 "sq rt(x)" }{TEXT -1 46 " " } {XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6#%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 0 6 "abs(x)" }{TEXT -1 49 " \+ " }{XPPEDIT 18 0 "abs(x)" "6#-%$absG6#%\"xG" }} {PARA 0 "" 0 "" {TEXT 0 6 "exp(x)" }{TEXT -1 49 " \+ " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"x G" }}{PARA 0 "" 0 "" {TEXT 0 5 "ln(x)" }{TEXT -1 76 " \+ the natural logarithm of " }{TEXT 684 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 0 6 "log(x)" } {TEXT 540 11 " " }{TEXT 518 63 " \+ the natural logarithm of " }{TEXT 685 1 "x" }{TEXT 686 10 ", same as " }{TEXT 0 5 "ln(x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 9 "log[n](x)" }{TEXT 541 1 " " }{TEXT -1 62 " \+ the logarithm to base " }{TEXT 687 1 " n" }{TEXT -1 4 " of " }{TEXT 688 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 0 21 "sin(x),cos(x),tan(x)," }}{PARA 0 "" 0 "" {TEXT 0 20 " cot(x),sec(x),csc(x)" }{TEXT -1 27 " the sine of " } {TEXT 689 1 "x" }{TEXT -1 6 ", etc." }}{PARA 0 "" 0 "" {TEXT 0 29 "arc sin(x),arccos(x),arctan(x)" }{TEXT 542 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 " \+ the inverse sine of " }{TEXT 690 1 "x" }{TEXT -1 73 " etc. (some stud ents may be more familiar with the mathematical notation " }{XPPEDIT 18 0 "sin^(-1)*x" "6#*&)%$sinG,$\"\"\"!\"\"F'%\"xGF'" }{TEXT -1 2 ".) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT 525 0 "" }{TEXT -1 17 "Practice problems " }{TEXT 697 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 1" }}{PARA 0 "" 0 "" {TEXT -1 16 "Assign the name " }{TEXT 715 1 "w" }{TEXT -1 16 " to the \+ number " }{XPPEDIT 18 0 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" } {TEXT -1 20 " and then find the " }{TEXT 259 11 "exact value" }{TEXT -1 7 " and a " }{TEXT 259 21 "decimal approximation" }{TEXT -1 20 " fo r the following: " }{XPPEDIT 18 0 "w^2, sqrt(w)" "6$*$%\"wG\"\"#-%%sqr tG6#F$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(w);" "6#-%$sinG6#%\"wG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 491 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 13 "Problem No. 2" }}{PARA 0 "" 0 "" {TEXT -1 8 "Factor " }{XPPEDIT 18 0 "x^8-2*x^4+1" "6#,(*$%\"xG\"\")\"\"\"*&\"\"#F'*$F%\"\"% F'!\"\"F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 559 18 "Student Workspace " }}{PARA 0 "" 0 " " {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 3" }}{PARA 0 "" 0 "" {TEXT -1 19 "E nter the function " }{XPPEDIT 18 0 "f(x)=20*x+30*x^2-sqrt(46-x^2)" "6# /-%\"fG6#%\"xG,(*&\"#?\"\"\"F'F+F+*&\"#IF+*$F'\"\"#F+F+-%%sqrtG6#,&\"# YF+*$F'F/!\"\"F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "Then \+ find the approximate value of " }{XPPEDIT 18 0 "f(3.29) + f(-3.1) " " 6#,&-%\"fG6#-%&FloatG6$\"$H$!\"#\"\"\"-F%6#,$-F(6$\"#J!\"\"F3F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 1 {PARA 20 "" 0 "" {TEXT 560 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 " __________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 4" }}{PARA 0 "" 0 "" {TEXT -1 50 "Find the approxim ate value of T in the formula " }{XPPEDIT 18 0 "T = sqrt((2*a-3*b^2 )/(c-20))" "6#/%\"TG-%%sqrtG6#*&,&*&\"\"#\"\"\"%\"aGF,F,*&\"\"$F,*$%\" bGF+F,!\"\"F,,&%\"cGF,\"#?F2F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "a=4.6 " "6#/%\"aG-%&FloatG6$\"#Y! \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b= -3.8" "6#/%\"bG,$-%&FloatG6$ \"#Q!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c=2.9" "6#/%\"cG-%&Fl oatG6$\"#H!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 20 "" 0 "" {TEXT 561 18 "Student Workspace " }}{PARA 0 " " 0 "" {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 5" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "f(x)=x^2-2*x+3" "6#/-%\"fG6#%\"xG,(*$F'\" \"#\"\"\"*&F*F+F'F+!\"\"\"\"$F+" }{TEXT -1 23 ", find and simplify, \+ " }{XPPEDIT 18 0 "f(3*t+2)" "6#-%\"fG6#,&*&\"\"$\"\"\"%\"tGF)F)\"\"#F) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 562 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 13 "Problem No. 6" }}{PARA 0 "" 0 "" {TEXT -1 35 "When mult iplied out the expression " }{XPPEDIT 18 0 "(x-4)^2*(x+1)^3" "6#*&,&% \"xG\"\"\"\"\"%!\"\"\"\"#,&F%F&F&F&\"\"$" }{TEXT -1 32 " is a fifth-d egree polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 31 "What is the coeffic ient of the " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 6 " term ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 563 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "___________ _______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "__ ________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Pro blem No. 7" }}{PARA 0 "" 0 "" {TEXT -1 21 "Plot the expressions " } {XPPEDIT 18 0 "cos(x) " "6#-%$cosG6#%\"xG" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "cos(x)*sin(10*x)" "6#*&-%$cosG6#%\"xG\"\"\"-%$sinG6#*& \"#5F(F'F(F(" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,2*Pi ]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 51 ". Then plot these sam e expressions on the interval " }{XPPEDIT 18 0 "[0,4*Pi]" "6#7$\"\"!*& \"\"%\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 20 "" 0 "" {TEXT 564 18 "Student Workspace " }}{PARA 0 " " 0 "" {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Plot the function " }{XPPEDIT 18 0 "f(x)=sec(x) +4" "6#/-%\"fG6#%\"xG,&-%$secG6#F'\"\"\"\"\"%F," } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*& \"\"#\"\"\"%#PiGF'" }{TEXT -1 65 ". Automatic scaling does not produce a useful picture. Specify a " }{TEXT 557 1 "y" }{TEXT -1 65 "-range t hat gives a good view of this function on this interval. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 565 18 "Student \+ Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________________ ______" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Problem No. 9" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Recall th at if a " }{TEXT 259 15 "rational number" }{TEXT -1 151 " has an infin ite decimal expansion then somewhere in the expansion the digits must \+ repeat. A familiar example is the decimal expansion of the fraction " }{XPPEDIT 18 0 "1/3 = 0;" "6#/*&\"\"\"F%\"\"$!\"\"\"\"!" }{TEXT -1 48 " .33333... where we have the digit 3 repeated. " }}{PARA 0 "" 0 "" {TEXT -1 53 "A bit more interesting is the decimal expansion for " } {XPPEDIT 18 0 "33/14 = 2;" "6#/*&\"#L\"\"\"\"#9!\"\"\"\"#" }{TEXT -1 101 ".3571428571428... with the repeating digits 571428. Now look at a decimal expansion of the fraction " }{XPPEDIT 18 0 "2/19" "6#*&\"\"# \"\"\"\"#>!\"\"" }{TEXT -1 101 ". Can you identify the repeating seque nce. Check yourself by looking at one thousand decimal places. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 566 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________ ______________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Problem No. 1 0" }}{PARA 0 "" 0 "" {TEXT -1 42 "(a) Plot the following points on a g raph: " }}{PARA 0 "" 0 "" {TEXT -1 67 " (1, 0.53), (1.5, 0.65), (2 , 0.91), (2.5 , 0.95) and (3, 1.10 )" }}{PARA 0 "" 0 "" {TEXT -1 97 "( b) Create a single picture that has the points above together with the graphs of the functions: " }{XPPEDIT 18 0 "f(x)=sin(x/2)" "6#/-%\"fG6 #%\"xG-%$sinG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x)=x^2/5" "6#/-%\"gG6#%\"xG*&F'\"\"#\"\"&!\"\"" }{TEXT -1 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 94 "Use your picture to decide which of \+ these two functions most closely fits this set of points? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 567 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________________ ______" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Problem No. 11" }} {PARA 0 "" 0 "" {TEXT -1 49 "Approximate all real solutions of the equ ation " }{XPPEDIT 18 0 "x^4-4*x^3+3*x-12=0" "6#/,**$%\"xG\"\"%\"\"\" *&F'F(*$F&\"\"$F(!\"\"*&F+F(F&F(F(\"#7F,\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 568 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________ ______________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Problem No. 1 2" }}{PARA 0 "" 0 "" {TEXT -1 49 "Approximate all real solutions of th e equation " }{XPPEDIT 18 0 "x^4-4*x^3=cos(3*x)+3" "6#/,&*$%\"xG\"\" %\"\"\"*&F'F(*$F&\"\"$F(!\"\",&-%$cosG6#*&F+F(F&F(F(F+F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 569 18 "Student Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "___________ _______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "__ ________________________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Pro blem No. 13" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " } {XPPEDIT 18 0 "f(x)=20-x" "6#/-%\"fG6#%\"xG,&\"#?\"\"\"F'!\"\"" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "h(x)=1.012^x" "6#/-%\"hG6#%\"xG) -%&FloatG6$\"%75!\"$F'" }{TEXT -1 26 " intersect at one point. " }} {PARA 0 "" 0 "" {TEXT -1 107 "Use the numerical solving capabilities o f Maple to approximate the coordinates of this intersection point. " } }{PARA 0 "" 0 "" {TEXT -1 52 "Start by entering an appropriate equatio n to solve. " }}{PARA 0 "" 0 "" {TEXT -1 76 "Check your answer by crea ting a picture that shows the graphs intersecting. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 570 18 "Student Works pace " }}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________________ ______" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Problem No. 14" }} {PARA 0 "" 0 "" {TEXT -1 10 "Solve for " }{TEXT 558 1 "r" }{TEXT -1 19 " in the equation: " }{XPPEDIT 18 0 "r*(p*k-18*m)=32*(2-p*r*m)/m^2 " "6#/*&%\"rG\"\"\",&*&%\"pGF&%\"kGF&F&*&\"#=F&%\"mGF&!\"\"F&*(\"#KF&, &\"\"#F&*(F)F&F%F&F-F&F.F&*$F-F2F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 96 "Be sure that you have entered the equation correctly. I n particular check that you have used an " }{TEXT 0 1 "*" }{TEXT -1 5 " for " }{TEXT 259 5 "every" }{TEXT -1 16 " mulitplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT 571 18 "Stude nt Workspace " }}{PARA 0 "" 0 "" {TEXT -1 26 "________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "____________ ______________" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Problem No. 1 5" }}{PARA 0 "" 0 "" {TEXT -1 36 "(a) Use Maple to solve the equation \+ " }{XPPEDIT 18 0 "ln(4-x^2)=0" "6#/-%#lnG6#,&\"\"%\"\"\"*$%\"xG\"\"#! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) Illustr ate the solutions of this equation as the " }{TEXT 698 1 "x" }{TEXT -1 41 "-intercepts of the graph of the equation " }{XPPEDIT 18 0 "y=ln (4-x^2)" "6#/%\"yG-%#lnG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "(c) Use Maple to solve the equation " }{XPPEDIT 18 0 "ln(2+x)+ln(2-x)=0" "6#/,&-%#lnG6#,&\"\"#\"\"\"%\"xG F*F*-F&6#,&F)F*F+!\"\"F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 306 "(d) Using the multiplication property of logarithms you \+ can see that the equation in part (c) is equivalent to the equation in part (a) so you would expect Maple to give the same solutions as befo re. This does not happen (with Maple 9). Are the solutions obtained in part (c) correct? Justify your answer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Hint" }{TEXT -1 16 ": the procedur e " }{TEXT 0 11 "rationalize" }{TEXT -1 111 " can be used to rationali se the denominator of a numerical fraction which contains radicals in \+ the denominator." }}{PARA 0 "" 0 "" {TEXT -1 19 "For example, . . . " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "1/(1+sqrt(2));\nrationalize(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 699 18 "Student Workspace " }}{PARA 0 "" 0 " " {TEXT -1 26 "__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "__________________________" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0" 17 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }