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-1 216 "C ourier" 1 10 255 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Helveti ca" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 232 1 {CSTYLE "" -1 -1 "Helvet ica" 1 10 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 220 "Arial Narrow" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 221 "Helvetica" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 222 "Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 223 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 224 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 233 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 225 "Helvetica" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 226 "Helvetica" 1 10 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 227 "Helvetica" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 228 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 229 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 230 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE " " -1 231 "Helvetica" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 234 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 235 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 217 "" 0 "" {TEXT 202 12 "INTRODUCTION" }}}{SECT 0 {PARA 218 "" 0 "" {TEXT 202 36 "How to get around a MAPLE worksheet: " }}{PARA 210 "" 0 "" {TEXT 232 0 "" }}{PARA 219 "" 0 "" {TEXT 201 67 "Using the UP/DOWN arrow keys, you can scroll through the worksheet." }}{PARA 220 "" 0 "" {TEXT 201 0 "" }}{PARA 221 "" 0 "" {TEXT 201 167 " Areas with RED text are INPUT REGIONS or STATEMENTS. They are in fact \+ statements of a certain programming lanuage, i.e. they must follow cer tain syntax rules. Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sin(3*Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 222 "" 0 "" {TEXT 204 0 "" }{TEXT 201 89 "To execute a statement , put the cursor on the line of the statement and press \"Enter.\"" } }{PARA 223 "" 0 "" {TEXT 201 48 "Try this now. You will see the answe r in BLUE. " }{TEXT 201 0 "" }{TEXT 201 40 "Areas with BLUE text are O UTPUT REGIONS." }}{PARA 224 "" 0 "" {TEXT 201 0 "" }}{PARA 225 "" 0 "" {TEXT 201 118 "Note that all input statements must be followed by sem icolon (or colon if you don't want the output to be displayed). " }} {PARA 226 "" 0 "" {TEXT 205 230 "Try deleting the semicolon from the p revious MAPLE statement and performing it again. You will see some dia gnostics. Type the semicolon back and perform once more. You will see \+ the diagnostic disppear and the answer appear again. " }{TEXT 201 1 " \+ " }}}{SECT 0 {PARA 227 "" 0 "" {TEXT 202 26 "The on-screen help utilit y" }}{PARA 213 "" 0 "" {TEXT 233 331 "Most users of modern software pa ckages never will buy reference books but rather entirely (or primaril y) rely on the on-screen-help. Try clicking on the on-screen help on t he right upper corner of the MAPLE window. The alternative is type ? k eyword (in an input region), and execute by hitting the ENTER key, see the example below." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?funct ion" }}}{PARA 0 "" 0 "" {TEXT 202 108 "Can you copy the examples from \+ the HELP pages into INPUT regions of the worksheet for editing and exe cuting?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 212 "" 0 "" {TEXT 234 0 "" }}}{SECT 0 {PARA 228 "" 0 "" {TEXT 202 12 "MAPLE b asics" }}{PARA 210 "" 0 "" {TEXT 232 132 "MAPLE is a package for SYMBO LIC ALGEBRA, that ALSO can do the usual numerical calculations, and pr oduce first rate graphical output." }}{SECT 0 {PARA 229 "" 0 "" {TEXT 202 13 "MAPLE algebra" }}{PARA 210 "" 0 "" {TEXT 232 362 "Execute the \+ following examples (with or without modification), and memorize the ef fects. Move the cursor to the next input area, and repeatedly hit the \+ ENTER KEY until you reach the end of this section. Each statement need s to be closed with a semi-colon (or a colon) before MAPLE will execut e it. Omitting this semi-colon is the most frequent cause for trouble. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "5*2/7-1/9+2*4;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "3+4/5-1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " exp(2)/3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 210 "" 0 "" {TEXT 232 89 "Frequently there is a need to go from exact results to a pproximations by decimal numbers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "soln:=solve(x^2+x=3);" }}}{PARA 0 "" 0 "" {TEXT 202 36 "To li st each solution separately try" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "soln[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "soln[2];" }}}{PARA 0 "" 0 "" {TEXT 202 24 "Now get a decimal answer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(soln);" }}}{PARA 210 "" 0 "" {TEXT 232 98 "The default number of digits after the dot is 10. To cha nge this number for example to 20 we write" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(soln,20);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 128 "Notice the difference between the := (assignment) and the \+ = (logical equality) in the above statement. Do not confuse these!" } }{PARA 210 "" 0 "" {TEXT 232 0 "" }}{PARA 210 "" 0 "" {TEXT 232 174 "A handy short-cut is the ditto-symbol %. This simbol means the result o f the last performed statement. Analogously, %% means the result of th e before-the-last statement, etc." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^3-5*x=2);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 52 "To ob tain a numerical approximation with 4 decimals:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "evalf(%,4);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 20 "and with 12 decimals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(%%,12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solv e(a+b*c=d,b);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 112 "Notice in th e above equation we have specified the variable with respect to which \+ we want to solve the equation." }}{PARA 210 "" 0 "" {TEXT 232 0 "" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 232 24 "MAPLE is case-sensitive:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(pi);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 84 "Multiplication has to be denoted explicitly by * , i.e. juxtap osition does not work:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x * y - xy;" }}}{PARA 210 "" 0 "" {TEXT 232 70 "Indeed in this case xy is a new variable, just like \"x1\" or \"vol\" ." }}}{SECT 0 {PARA 230 "" 0 "" {TEXT 202 27 "Defining functions in MA PLE" }}{PARA 0 "" 0 "" {TEXT 202 55 "There are different ways to defin e a function in MAPLE:" }}{PARA 203 "" 0 "" {TEXT 235 29 "The \"assign ment\" definition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:=x^2;" }}}{PARA 0 "" 0 "" {TEXT 202 155 "The value of the function at a point, say x = 2, is then obtained by evaluating the expression f for x = 2 (i.e., we subs titute x = 2 in the expression f):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=2,f);" }}}{PARA 203 "" 0 "" {TEXT 235 25 "The \+ \"arrow\" definition:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:= x->x^2; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 202 59 "is the representatio n of the function f(x) and stored as f " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 202 0 "" }}}{PARA 210 "" 0 "" {TEXT 232 36 " Then we can find \+ f(2), f(0), etc." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }} }{PARA 203 "" 0 "" {TEXT 235 29 "The \"procedure\" definition:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=proc(x) x^2; end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(2);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 139 "The procedure definition is useful when we need t o define very complicated functions whose definition requires more tha n one line commands." }}}{PARA 4 "" 0 "" {TEXT 236 0 "" }}}{SECT 0 {PARA 231 "" 0 "" {TEXT 202 19 "Plotting with MAPLE" }}{PARA 208 "" 0 "" {TEXT 237 136 "Let's look at some basic plotting. To begin enter an expression and tag it with the name f. The assignment operator is the := operator. " }}{PARA 210 "" 0 "" {TEXT 232 149 "Note that you have \+ not created a function in the mathematical sense. The notation f(2) wi ll not be recognized by Maple with the assignment made here." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "f:=x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot(f);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 78 "No domain \+ was given. Hence, no graph. The correct syntax for the plot command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(f,x=-3..3);" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 232 175 "To gain control over the verti cal scale, use a second range in the plot command. Maple interprets th e first range as the horizontal scale and the second range as the vert ical." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(f,x=-3..3,y=- 1..15);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 127 "Now suppose we wan t to examine the intersection of some simple curves and calculate the \+ points of intersection. Enter a second " }}{PARA 210 "" 0 "" {TEXT 232 28 "function g as an expression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=7+3*x-5*x^2;" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 79 "Plot both functions on one set of axes to see what to expect f or intersections." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot( \{f,g\},x=-3..3);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 183 "There ar e two intersections points to compute, and the x-coordinates of each p oint has magnitude of about one. To compute the points exactly, we can set f and g equal and solve for x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q:=solve(f=g,x);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 214 "The solve command has returned an expression sequence contain ing the two roots, listed in sequence. The ease with which teh solutio ns can be referenced and used is an essential feature of a computer al gebra system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q[1];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q[2];" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 84 "Now let's see how to obtain the y-coordinates corr esponding to the two x-coordinates" }}{PARA 210 "" 0 "" {TEXT 232 122 "just calculated. Note that mathematically you would \"plug in\" or su bstitute the x-value into the appropriate expression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y1:=subs(x=q[1],f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y2:=subs(x=q[2],f);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 77 "Obviously, there is a need for some si mplification. Two options come to mind:" }}{PARA 203 "" 0 "" {TEXT 235 14 "we can try to " }{TEXT 206 6 "expand" }{TEXT 202 20 " the expr ession for " }{TEXT 207 2 "y1" }{TEXT 235 5 " and " }{TEXT 208 2 "y2" }{TEXT 235 1 "," }}{PARA 203 "" 0 "" {TEXT 202 17 "or we can try to " }{TEXT 209 8 "simplify" }{TEXT 235 19 " these expressions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(y1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "expand(y1);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 30 "The more natural action is to " }{TEXT 210 6 "expand" } {TEXT 232 88 " the term being squared. Hence, the expand command produ ced the more appropriate result." }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 47 "Sometimes it is necessary to work numerically. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xx1:=evalf(q[1],30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=xx1,f);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 140 "Note that Maple did not transfer to the calcul ation of y the need for more digits. One way to dictate that outcome i s via the evalf command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(subs(x=xx1,f),30);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 87 "N ow let's look at another example where the choice of the correct domai n is important. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(\{ sqrt(9-x^2),-sqrt(9-x^2)\},x=-3.5..3.5,y=-3.5..3.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(\{sqrt(9-x^2),-sqrt(9-x^2)\},x=-3..3 ,y=-3..3);" }{TEXT 211 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 59 "Ca n you explain the difference between the two plots above?" }{TEXT 238 1 " " }{TEXT 213 50 "(Look near the points where the semicircles meet. )" }}{PARA 0 "" 0 "" {TEXT 238 0 "" }}{PARA 210 "" 0 "" {TEXT 232 106 "If a function is defined with an arrow or procedure definition - you \+ can still plot it, but you should use" }{TEXT 214 6 " f(x) " }{TEXT 232 17 "rather than just " }{TEXT 215 2 "f " }{TEXT 232 12 "in \"plot \"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f:=x->x^2-1; plot(f(x),x=-2. .2);" }{TEXT 216 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT 238 0 "" }}{PARA 210 "" 0 "" {TEXT 232 103 "Wit h Maple we can also do implicit plots but in order to do so we need to load the package with(plots);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "impli citplot(x^2+y^2=9,x=-3.5..3.5,y=-3.5..3.5);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 202 34 "We can also do 3-dimensional plots" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot3d(\{sqrt(9-x^2-y^2),-sqrt(9-x^2-y^2) \},x=-4..4,y=-4..4);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 202 109 "What \+ kind of solid do these equations correspond to? How would you go about filling the \"holes\" in the plot" }{TEXT 232 1 "?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot3d(-x/(1+x^2+y^2),x=-5..5,y=-3..3);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot3d(x*exp(-x^2-y^2),x =-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 202 106 "The plots p ackage also has a display command that can be used to display multiple plots on the same graph." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "curve1:=plot(x^2,x=-2..2,color=blue):\n" }{MPLTEXT 1 0 38 "curve2: =plot(x^3,x=-2..2,color=green):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 0 "" }{TEXT 202 181 "Notice that the output was suppressed when you ex ecuted these statements. What has happened is that the actual plotting commands have been stored in the variables curve1 and curve2." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "curve1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "curve2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 0 "" }{TEXT 202 78 "If you want do display these both in the same pict ure use the display command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(\{curve1,curve2\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 0 "" }{TEXT 202 56 "To get a more accurate picture, use constraine d scaling:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(\{cur ve1,curve2\},scaling=constrained);" }}}{EXCHG {PARA 232 "" 0 "" {TEXT 202 0 "" }{TEXT 220 4 "NOTE" }{TEXT 221 1 ":" }{TEXT 222 0 "" }{TEXT 202 291 " When you label plots as above with curve1 and curve2, it is \+ ESSENTIAL that you suppress the output with a colon. If you put a semi colon, MAPLE will output the list of plotting commands and it will loo k like what happens here, only perhaps lots longer when you are plotti ng complex objects.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "noc oloncurve:=plot(x^2,x=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 0 " " }{TEXT 202 192 "If this happens to you it is a sure sign that you ha ve labelled a plot but didn't suppress the printout. Remember though, \+ if you don't label a plot and use a semi colon it will draw the graph. " }{TEXT 224 0 "" }}}}{SECT 0 {PARA 233 "" 0 "" {TEXT 202 24 "Differen tiating in MAPLE" }}{PARA 210 "" 0 "" {TEXT 232 31 "If a function was \+ defined with " }{TEXT 225 14 "an assignment," }{TEXT 226 1 " " }{TEXT 232 49 "its derivative can be found by using the command " }{TEXT 227 6 "diff. " }{TEXT 232 112 "The process of computing a derivative is ca lled differentiation, and that is where the notation diff comes from." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=3*x^2-5*x+2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 75 "Let's construct, at the point (2,4), the line tangent to the g raph of f(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=diff(f, x);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 69 "The slope of this tange nt line is the value of the derivative at x=2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "m:=subs(x=2,d);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 45 "The equation of the tangent line is therefore" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y:=m*(x-2)+4;" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 65 "A graph of the tangent line and of the function f is created via " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(\{f,y\},x=0..3);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 0 "" }}{PARA 210 "" 0 "" {TEXT 232 116 "We see that the line through (2,4) \+ with slope 7, as calculated from the derivative, is tangent to the gra ph of f(x)." }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 67 "Let's explore t he calculation of a derivative for another function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=sin(x);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 76 "The difference quotient (which represents the slopes of \+ the secant lines) is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m:= (subs(x=x+h,f)-f)/h;" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 64 "The li mit of the difference quotient is found by the computation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(m,h=0);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 82 "Finally, we compare this result with a direct ca ll to the differentiation command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 228 6 "REMARK" }{TEXT 232 96 ": A common mistake results when a name has been assigned a value earlie r. See the example below:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(x^2,x);" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 232 355 "Consequently requests like dif ferentiating x^2 with respect to x don't make any sense at all: you ca n't differentiate 4 with respect to 2. Everyone runs into this troubl e -- there is simply no way a person can keep in mind all the previous assignments. There are two ways: First we may selectively reset x to \+ simply mean x (the letter), and nothing else:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " diff(x^2,x);" }}}{PARA 210 "" 0 "" {TEXT 232 141 "More radically, we m ay restart MAPLE, thus wiping out all prior assignments, loaded packag es, calculations etc. (but the worksheet is safe!)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:=x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(y,x) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:=x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(y,x);" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 0 "" }}{PARA 0 "" 0 "" {TEXT 238 6 "Again:" }{TEXT 202 60 " if your func tion was defined with an arrow definition, put " }{TEXT 238 5 "f(x) " }{TEXT 202 3 "or " }{TEXT 238 4 "y(x)" }{TEXT 202 18 " etc. in place o f " }{TEXT 238 2 "f " }{TEXT 202 4 "and " }{TEXT 238 1 "y" }{TEXT 202 32 " above. Of course, in this case " }{TEXT 238 5 "subs " }{TEXT 202 34 "is not needed: you may just write " }{TEXT 238 13 "d(2), f(x+h)," }{TEXT 202 5 " etc." }}{PARA 0 "" 0 "" {TEXT 238 0 "" }}{PARA 0 "" 0 " " {TEXT 238 0 "" }{TEXT 202 103 "For functions defined with arrow defi nition you may also find their derivatives using another command: " } {TEXT 238 2 "D." }{TEXT 202 48 " The syntax is different: see the exam ple below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->3*x^2-5 *x+3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(2);" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT 239 1 "I" }{TEXT 229 19 "ntegrating in Maple" }}{PARA 0 "" 0 " " {TEXT 238 0 "" }{TEXT 202 61 "To write an integral, either definite \+ or indefinite, use Int:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "I nt(sin(x),x);" }{TEXT 238 0 "" }}{PARA 0 "" 0 "" {TEXT 238 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Int(sin(x),x=1..2);" }}} {PARA 0 "" 0 "" {TEXT 238 44 "To calculate the integral, use lowcase i nt: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(sin(x),x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(sin(x),x=0..Pi/2); " } {TEXT 238 0 "" }}{PARA 0 "" 0 "" {TEXT 202 84 "Remember the difference between pi (just a greek letter) and Pi (the number 3.14...)" }}}}} {EXCHG {PARA 212 "" 0 "" {TEXT 234 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 239 1 " " }{TEXT 230 9 "EXERCISES" }}{PARA 210 "" 0 "" {TEXT 232 146 "Work through the following problems. If you need extra Maple \+ prompts in order to write and execute more commands, click on the butt on with symbol " }{TEXT 231 2 "[>" }{TEXT 232 9 " above. " }}{SECT 0 {PARA 234 "" 0 "" {TEXT 202 8 "CALCULUS" }}{EXCHG {PARA 210 "" 0 "" {TEXT 232 55 "C1: Plot the graph of y=cos(4*arcsin(x)) from -1 to 1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 209 "C2: Find the fourth derivative of arctan(x) with respec t to x. Write the result as a single fraction (combine,simplify,....) \+ (do a help on diff to find out how to compute second, third, fourth,.. . derivatives)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 210 "" 0 "" {TEXT 232 155 "C3: Write the difference quoti ent for the function sin(x). Compute the limit for h->0 and compare wi th the value of the derivative computed by using diff." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 232 37 "C4: Find an antiderivative of sin(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 212 "" 0 "" {TEXT 234 0 "" }}}{SECT 0 {PARA 235 "" 0 "" {TEXT 239 0 "" }{TEXT 202 7 "PLOTING" }}{PARA 0 "" 0 "" {TEXT 238 0 "" }{TEXT 202 43 "P1: On the same graph, plot the fun ctions " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mn(\"2\"), Typ esetting:-mrow(Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \+ \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mverbatim(\"-%$sinG6#%\"xG\"), Typesetting:-mi(\"\")), Typesetti ng:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6%-I#mnGF$6#Q\"2F'-F#6%-I#moGF$63Q1⁢F'/%%formGQ&infixF' /%&fenceGQ&falseF'/%*separatorGF:/%'lspaceGQ$0emF'/%'rspaceGF?/%)stret chyGF:/%*symmetricGF:/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeo pGF:/%.movablelimitsGF:/%'accentGF:/%0font_style_nameGQ+2D~CommentF'/% %sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-I *mverbatimGF$6#Q.-%$sinG6#%\"xGF'-I#miGF$6#Q!F'F\\o" }{TEXT 238 2 ", " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -mrow(Typesetting:-mverbatim(\"%$cosG\"), Typesetting:-mo(\"&ApplyFunc tion;\", form = \"infix\", fence = \"false\", separator = \"false\", l space = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \" prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmath space\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn(\"3\"), Typesett ing:-mrow(Typesetting:-mo(\"⁢\", form = \"infix\", fenc e = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mverbatim(\"%\"xG\")), Typesetting:-mi(\"\")), Typesetting:-mo(\")\", \+ form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \+ \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\" , symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", font_s tyle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", b ackground = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:- mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6#Q!F'-F#6&-I*mverbatimGF$6#Q'%$cosGF'-I#moGF$63Q0&ApplyFunction ;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF>/%'lspaceGQ$0emF' /%'rspaceGFC/%)stretchyGF>/%*symmetricGF>/%(maxsizeGQ)infinityF'/%(min sizeGQ\"1F'/%(largeopGF>/%.movablelimitsGF>/%'accentGF>/%0font_style_n ameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+background GQ.[255,255,255]F'-F#6%-F663Q\"(F'/F:Q'prefixF'/F=Q%trueF'F?/FBQ.thinm athspaceF'/FEFfo/FGFdoFHFJFMFPFRFTFVFYFfnFin-F#6%-I#mnGF$6#Q\"3F'-F#6$ -F663Q1⁢F'F9F " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }