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2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {PARA 207 "" 0 "" {TEXT 206 0 "" }{TEXT 200 0 "" }{TEXT 200 9 "Maple Lab" }}{PARA 208 "" 0 "" {TEXT 206 6 "Week 2" }}{PARA 209 "" 0 "" {TEXT 206 0 "" }{TEXT 200 0 "" }{TEXT 200 12 "Introduction" }} {PARA 0 "" 0 "" {TEXT 207 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 208 38 " Arithmetic, elements of the worksheet" }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 122 "What you are seeing is a \"worksheet\". We'll be using \+ them in class, and you'll be using them to help you with homeworks." } }{PARA 0 "" 0 "" {TEXT 207 46 "A worksheet contains text and Maple com mands. " }{TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 31 "Let's look at a few examples.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "2^1000;" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 11 "int(x^2 ,x);" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 17 "diff(x*exp(x),x); " }}}{EXCHG {PARA 205 "" 0 "" {TEXT 209 2 "\n" }{TEXT 209 112 "These a re Maple commands. You can execute them in any order. You can also mod ify any command and re-execute it. " }}{PARA 205 "" 0 "" {TEXT 209 58 "For instance, here is another copy of the third command:\n" }}{PARA 205 "> " 0 "" {MPLTEXT 1 0 11 "int(x^2,x);" }}}{EXCHG {PARA 205 "" 0 " " {TEXT 209 20 "Change the variable " }{MPLTEXT 1 204 1 "x" }{TEXT 209 20 " after the comma to " }{MPLTEXT 1 205 1 "X" }{TEXT 209 1 "." } {TEXT 209 107 " Then you can re-execute the command and immediately se e the result. Do it now; does the result make sense?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 139 "Generally, t he worksheets are completely modifiable. Everything, including this te xt can be changed. Right now we are in a \"text region\"." }}{PARA 0 " " 0 "" {TEXT 207 140 "Although it is possible to place executable comm ands in the text region it is better to enter them in what are called \+ \"input regions\" and " }}{PARA 0 "" 0 "" {TEXT 207 49 "follow the \"> \" prompt as in the examples above." }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 210 "" 0 "" {TEXT 207 99 "The sleek design of the latest versi on of Maple allows you to input commands without the prompt [>." } {TEXT 207 1 " " }}{PARA 211 "" 0 "" {TEXT 207 101 "This often leads to \"funny\" behavior from Maple (and sometimes to extreme annoyance of \+ the users). " }}{PARA 212 "" 0 "" {TEXT 207 75 "Please use the prompt \+ button \"[>\" in the menu before entering a command. " }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 145 "It is a good idea t o spend some time to check the menus and toolbar buttons to see what i s available. An important thing is the \"save\" command " }}{PARA 0 "" 0 "" {TEXT 207 137 "under the File menu. It will allow you to save yo ur current worksheet. Note also the \"Interrupt\" button on the toolba r. This will stop " }}{PARA 0 "" 0 "" {TEXT 207 136 "the Maple kernel \+ in the middle of whatever it is doing (an infinite loop in particular! ) and return you to the worksheet. You can learn " }}{PARA 0 "" 0 "" {TEXT 207 87 "the meaning of the toolbar buttons by moving the mouse p ointer over a puzzling button. " }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 41 " Certain constants are built into Maple. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Pi;\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 17 "Note the capital " }{MPLTEXT 1 0 2 "Pi" }{TEXT 207 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "P i-pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(Pi, 200);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(pi,200);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 137 "Maple prints the symbols the sam e way, but the meaning is completely different. This is one of the har d-to-catch mistakes. Don't make it!" }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 208 14 " Basic algebra" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "(x+y)^10;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 12 "Compare with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x + y^10;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expand( ( x+ y)^10);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 62 " The expand command does pretty \+ much what you would expect. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(x+y)^2;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "expand( %);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 13 "Other things:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^2+3*x+2,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^2+3*x-3,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2+3.0*x- 3,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 86 "The equations are the s ame, but the answers are in different forms. Can you guess why?" }} {PARA 0 "" 0 "" {TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 94 " Real arithmetic has a precision of 10 digits by default. This can \+ lead to strange results. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(x^2+2, real);\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 208 14 " Some calculus" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:= (x+1)^ 3 * cos(x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 9 "The sign " } {MPLTEXT 1 0 2 ":=" }{TEXT 207 48 " means \"assign\"; from now on, you can type in " }{MPLTEXT 1 0 1 "f" }{TEXT 207 63 " instead of the whol e expression. It is important not to forget" }}{PARA 0 "" 0 "" {TEXT 207 47 "the colon; a simple equality sign would not do!" }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 27 "You can compute de rivatives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f, x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 37 " How about the second derivat ive of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 207 2 " ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(f,x,x);" }{TEXT 207 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 52 "Symbolic manipulation is one of t he main advantages:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff (c^x, x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(c^x, c) ;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 80 "At the same time, the exa mple shows why you need to specify the variable in the " }{MPLTEXT 1 0 5 "diff " }{TEXT 207 8 "command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(f, x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:= exp(x) * sin(x ^2 + 1);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 24 "Let's see if this \+ works:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(g, x);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 6 "The \"" }{MPLTEXT 1 0 3 "erf" } {TEXT 207 122 "\" stands for the error function; Maple has a lot of bu ilt-in functions. What if you need to find the intgral from 1 to 3?" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(g,x=1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 22 "So that's not helpful." }}{PARA 0 "" 0 " " {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 37 "If you want to get a \+ number, use the " }{MPLTEXT 1 0 5 "evalf" }{TEXT 207 33 ". Let's try d ifferent approaches:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eva lf(int(g,x=1..3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 8 "What is " } {XPPEDIT 18 0 "I;" "6#%\"IG" }{TEXT 207 116 "? This is a complex numbe r, one of the square roots of -1. The answer is of course a real numbe r, that's why there's" }}{PARA 0 "" 0 "" {TEXT 207 9 "0 before " } {XPPEDIT 18 0 "I;" "6#%\"IG" }{TEXT 207 110 ". What happens is Maple f irst computes the integral, using a fancy formula, and then \"evaluate s\" the result." }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 63 " For numeric computation of integrals, it's better to us e this:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(Int(g,x=1. .3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 112 "Through some magic, th is instructs Maple to compute the integral numerically. Need better pr ecision? No problem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eva lf(Int(g,x=1..3),100);" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 210 18 " A bi t of practice" }}{PARA 0 "" 0 "" {TEXT 207 8 "Compute " }{XPPEDIT 18 0 "int(h(x), x = 0 .. 2);" "6#-%$intG6$-%\"hG6#%\"xG/F);\"\"!\"\"#" } {TEXT 207 9 " , where " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT 207 4 " i s " }{XPPEDIT 18 0 "x^2*exp(x^2);" "6#*&)%\"xG\"\"#\"\"\"-%$expG6#*$F$ F'F'" }{TEXT 207 13 "get a number." }}{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 3 "" 0 "" {TEXT 208 9 " Graphics " }}{EXCHG {PARA 0 "" 0 "" {TEXT 207 42 " Maple has good facilities \+ for plotting " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f, x= -2..2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "h:= x + 2 * x^ 2/(x^2-1);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 57 " A glance at h shows it will behave badly at +1 and -1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(h, x=-2..2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 113 " The last graph is not so informative so I restrict the val ues of the variable y and increase the x-range a bit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(h, x=-3..3, y=-6..6);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 118 " The vertical lines are there be cause of discontinuity. To remove them, use an appropriate option in t he plot function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(h, x=-3..3, y=-6..6,discont=true);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot3d( x^2 -y^2, x=-2..2, y=-2..2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 98 "The nice thing is that you can play with the graph; rotate it arou nd; put in/remove the axes, etc." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(plots);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 50 " Thi s means you have loaded a \"package\" called " }{MPLTEXT 1 0 5 "plots" }{TEXT 207 50 ". Next time, terminate the instruction with colon." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "animate3d( cos(t) * (x^2 - \+ y^2), x=-2..2, y=-2..2, t=0..2*Pi, frames=10);\n" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 164 "Click on the graph, select \" play\" button from the animation menu. You can choose to run the anima tion faster or slower; frame-by-frame, one time, or continuously." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }