{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 255 255 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Define" -1 256 "Times" 1 12 0 0 0 1 1 1 2 0 0 2 0 0 0 1 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 70 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 102 117 110 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 31 138 0 0 0 2 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 12 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 257 19 "Inverse Functions, " } {XPPEDIT 260 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT 258 5 " and " } {XPPEDIT 261 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT 259 1 "\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We always begin our work with" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "" 0 "" {TEXT -1 65 "Thi s ensures that all previous variable assignments are cleared. " }} {PARA 0 "" 0 "" {TEXT -1 104 "You might find it helpful to use this co mmand to clear Maple's memory at later stages in your work also." }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 262 12 "Introduction" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Natural Logarithm and Exponential Functions" }} {PARA 0 "" 0 "" {TEXT -1 52 "The exponential function is represented i n Maple by " }{HYPERLNK 17 "exp" 2 "exp" "" }{TEXT -1 97 " (click here for help on the exponential function), and the natural logarithm can \+ be defined via " }{HYPERLNK 17 "ln" 2 "ln" "" }{TEXT -1 4 " or " } {HYPERLNK 17 "log" 2 "log" "" }{TEXT -1 29 ", i.e., Maple does not tre at " }{XPPEDIT 18 0 "log(x)" "6#-%$logG6#%\"xG" }{TEXT -1 29 " as the \+ base-10 logarithm of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 31 " as \+ is done in many text books." }}{EXCHG {PARA 257 "" 0 "" {TEXT 266 101 "We start by illustrating the relationship between the natural logarit hm and the exponential function." }}{PARA 0 "" 0 "" {TEXT -1 34 "First we define the two functions:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f \+ := x -> ln(x);\ng := x -> exp(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Next we produce a " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 54 " of this inverse pair along with the line of symmetry " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "Help on various plot options (to produce \"pretty\" plots) can be fou nd " }{HYPERLNK 17 "here" 2 "plot[options]" "" }{TEXT -1 1 "." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot([f(x), g(x), x], x=-4..4, y=- 4..4, color = [red,green,blue], legend=[\"y=ln(x)\", \"y=exp(x)\", \"y =x\"], title=\"Inverse Pairs\");" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Note that we not only specified a range for " }{XPPEDIT 18 0 "x" "6#%\"xG " }{TEXT -1 15 ", but also for " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 57 ". This is not necessary, but makes the graph look better." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We now illustrate some of Maple's \+ Calculus capabilities via our example." }}{PARA 0 "" 0 "" {TEXT -1 20 "First, we know that " }}{PARA 0 "" 0 "" {TEXT -1 12 " " } {XPPEDIT 18 0 "limit(ln(x), x=0, right) = -infinity" "6#/-%&limitG6%-% #lnG6#%\"xG/F*\"\"!%&rightG,$%)infinityG!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "Let's have Maple verify this:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Limit(f(x), x=0, right) = limit(f(x), x=0, \+ right);" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Note that the " }{HYPERLNK 17 "Limit" 2 "Limit" "" }{TEXT -1 116 " command does not actually eval uate the limit. Is is a so-called inert command (used mainly for cosme tic purposes). " }}{PARA 0 "" 0 "" {TEXT -1 26 "Also, do not confuse \+ the " }{HYPERLNK 17 "=" 2 "equation" "" }{TEXT -1 39 " operator with t he assignment operator " }{HYPERLNK 17 ":=" 2 "assignment" "" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Next, the derivative of " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT -1 22 ". What does Maple say?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The previous is Maple's way of representi ng a " }{HYPERLNK 17 "function" 2 "function" "" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 51 "To get a more familiar form we need to lo ok at the " }{HYPERLNK 17 "expression" 2 "expression" "" }{TEXT -1 44 " (the derivative function evaluated at a=x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Anoth er possibility would be to use " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " diff(f(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The difference \+ between these two commands lies in the way in which Maple interprets t he output. " }}{PARA 0 "" 0 "" {TEXT -1 19 "The first command, " } {HYPERLNK 17 "D(f)" 2 "D" "" }{TEXT -1 47 ", produces a function, wher eas the second one, " }{HYPERLNK 17 "diff(f(x), x)" 2 "diff" "" } {TEXT -1 26 ", produces an expression. " }}{PARA 0 "" 0 "" {TEXT -1 156 "This difference in representation has important consequences when working with Maple (for example, when evaluating/substituting values \+ into an expression). " }}{PARA 0 "" 0 "" {TEXT -1 27 "The preferred me thod is to " }{TEXT 265 19 "work with functions" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 43 "To produce \"pretty\" output you can use \+ the " }{HYPERLNK 17 "inert form of diff" 2 "Diff" "" }{TEXT -1 24 " to write something like" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Diff(f(x), x) = D(f)(x);\nDiff(f(x), x) = diff(f(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A similar pair of commands exists for integration. W e first use " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Int(D(f)(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "This doesn't seem to be of much help (other than for \+ pretty output). This is another inert command, and to see the actual v alue of the integral we need to use " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Note the " } {HYPERLNK 17 "shortcut %" 2 "ditto" "" }{TEXT -1 45 " which always ref ers to the previous output. " }}{PARA 0 "" 0 "" {TEXT -1 47 "Multiple \+ percentage symbols are also possible. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Another thing to note is that Maple drops the constant of integration !" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{HYPERLNK 17 "second com mand for integration" 2 "int" "" }{TEXT -1 37 " is more convenient mos t of the time:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(D(f)(x), x); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Definite integrals are comput ed similarly. For example, " }{XPPEDIT 18 0 "Int(ln(x^2), x=1..2)" "6# -%$IntG6$-%#lnG6#*$%\"xG\"\"#/F*;\"\"\"F+" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(ln(x^2), x=1..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "And a " }{HYPERLNK 17 "numerical value" 2 "evalf" "" } {TEXT -1 13 " is found by " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf( %);" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Computing Inverse Functions" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can use Maple to compute other - more complicated - inverse pairs." }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 34 "E.g., let's consider the function " }{XPPEDIT 18 0 "y = 3*ln(sqrt(x+4));" "6#/%\"yG*&\"\"$\"\"\"-%#lnG6#-%%sqrtG6#,&%\" xGF'\"\"%F'F'" }{TEXT -1 22 " and find its inverse." }}{PARA 0 "" 0 " " {TEXT -1 54 "In order to do this, we start by defining the function " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y := x -> 3*ln(sqrt(x+4));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Next we solve the equation " } {XPPEDIT 18 0 "Y=y(x)" "6#/%\"YG-%\"yG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 47 ". This will result in an exp ression describing " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 13 " in te rms of " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT -1 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x_of_Y := solve(Y = y(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "In order to get the answer in function form (as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 18 ") we first rename " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 11 " (with the " }{HYPERLNK 17 "subs" 2 " subs" "" }{TEXT -1 77 " command), and then convert the resulting expre ssion to a function using the " }{HYPERLNK 17 "unapply" 2 "unapply" " " }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 23 "Thus, as a fu nction of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 34 ", the inverse f unction is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs(Y=y, x_ of_Y);\ninv_y := unapply(%, y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Again, the graphs of the two functions are symmetric about the lin e " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 152 "plot([y(x), inv_y(x), x], x=-6..6, y=-6..6, c olor=[red,green,blue], legend=[\"y=3ln(sqrt(x+4))\", \"inv_y=-4+(exp(x /3))^2\", \"y=x\"], title=\"Inverse Pairs\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 135 "The latest release of Maple has added \+ functionality for creating plots of inverse functions. We can access t his feature by loading the " }{HYPERLNK 17 "Calculus1" 2 "Calculus1" " " }{TEXT -1 15 " module of the " }{HYPERLNK 17 "Student package" 2 "St udent" "" }{TEXT -1 32 ", and then calling the function " }{HYPERLNK 17 "InversePlot" 2 "Calculus1,InversePlot" "" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "with(Student[Calculus1]):\nInversePlot(f( x), x=-6..6);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 263 12 "Assignment 1 " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.1:" }}{PARA 0 "" 0 "" {TEXT -1 68 "Use Maple to verify the complete limiting behavior of the functions " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 87 ", i.e., c ompute the behavior of these two functions near the boundary of their \+ domains." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.2:" }}{PARA 0 "" 0 "" {TEXT -1 54 "Use Maple to verify the definition of the natural lo g." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.3:" }}{PARA 0 "" 0 "" {TEXT -1 39 "a) Display the graphs of the functions " }{XPPEDIT 18 0 " y=ln(x)" "6#/%\"yG-%#lnG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=ln (2*x)" "6#/%\"yG-%#lnG6#*&\"\"#\"\"\"%\"xGF*" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y=ln(4*x)" "6#/%\"yG-%#lnG6#*&\"\"%\"\"\"%\"xGF*" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y=ln(8*x)" "6#/%\"yG-%#lnG6#*&\"\")\" \"\"%\"xGF*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=ln(16*x)" "6#/%\"yG-%# lnG6#*&\"#;\"\"\"%\"xGF*" }{TEXT -1 26 " together in one plot for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 25 " in the interval [0,10]. " } }{PARA 0 "" 0 "" {TEXT -1 35 "Explain in words what is happening." }} {PARA 0 "" 0 "" {TEXT -1 7 "Hints: " }}{PARA 0 "" 0 "" {TEXT -1 128 "1 . You can produce separate plots by assigning each one of them to a va riable and ending the command with a colon, and then use " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 10 " from the " } {HYPERLNK 17 "plots package" 2 "plots" "" }{TEXT -1 34 ") to display t hese plots together." }}{PARA 0 "" 0 "" {TEXT -1 41 "2. You can insert text by e.g. using the " }{TEXT 264 16 "Insert-Paragraph" }{TEXT -1 6 " menu." }}{PARA 0 "" 0 "" {TEXT -1 84 "b) What is the mathematical \+ reason (formula) for the phenomenon observed in part a)?" }}{PARA 0 " " 0 "" {TEXT -1 36 "Use Maple to verify your conjecture." }}{PARA 0 " " 0 "" {TEXT -1 63 "E.g., simplify(4-2 = 8/4) returns 2=2, which is ob viously true:" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }}{PARA 0 "" 0 "" {TEXT -1 43 "a) Use Maple to determine when the function" }} {PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 18 0 "f(x) = (2*x+7)/( 3*x-17);" "6#/-%\"fG6#%\"xG*&,&*&\"\"#\"\"\"F'F,F,\"\"(F,F,,&*&\"\"$F, F'F,F,\"#