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2 2 2 0 0 0 1 }{CSTYLE "_cstyle8 " -1 222 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{PSTYLE "_pstyl e5" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 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 "_cstyle9" -1 223 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle10" -1 224 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 207 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 202 "" 0 "" {TEXT 215 19 "Inverse Functions, " } {XPPEDIT 209 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT 216 1 " " }{TEXT 215 5 " and " }{XPPEDIT 208 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT 216 1 " " }{TEXT 217 1 "\n" }}{EXCHG {PARA 203 "" 0 "" {TEXT 218 29 "We always \+ begin our work with" }}{PARA 203 "" 0 "" }{PARA 203 "> " 0 "" {MPLTEXT 1 219 8 "restart;" }}{PARA 203 "" 0 "" {TEXT 218 65 "This ens ures that all previous variable assignments are cleared. " }}{PARA 203 "" 0 "" {TEXT 218 104 "You might find it helpful to use this comma nd to clear Maple's memory at later stages in your work also." }}} {SECT 1 {PARA 204 "" 0 "" {TEXT 220 12 "Introduction" }}{SECT 0 {PARA 205 "" 0 "" {TEXT 221 43 "Natural Logarithm and Exponential Functions" }}{PARA 203 "" 0 "" {TEXT 218 52 "The exponential function is represe nted in Maple by " }{HYPERLNK 222 "exp" 2 "exp" "" }{TEXT 218 97 " (cl ick here for help on the exponential function), and the natural logari thm can be defined via " }{HYPERLNK 222 "ln" 2 "ln" "" }{TEXT 218 4 " \+ or " }{HYPERLNK 222 "log" 2 "log" "" }{TEXT 218 29 ", i.e., Maple does not treat " }{XPPEDIT 18 0 "log(x)" "6#-%$logG6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 29 " as the base-10 logarithm of " }{XPPEDIT 18 0 "x" "6 #%\"xG" }{TEXT 218 1 " " }{TEXT 218 31 " as is done in many text books ." }}{EXCHG {PARA 206 "" 0 "" {TEXT 223 101 "We start by illustrating \+ the relationship between the natural logarithm and the exponential fun ction." }}{PARA 203 "" 0 "" {TEXT 218 34 "First we define the two func tions:" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 16 "f := x -> ln(x);" } {MPLTEXT 1 219 18 "\ng := x -> exp(x);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 18 "Next we produce a " }{HYPERLNK 222 "plot" 2 "plot" "" } {TEXT 218 54 " of this inverse pair along with the line of symmetry " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT 218 1 " " }{TEXT 218 1 "." }}{PARA 203 "" 0 "" {TEXT 218 70 "Help on various plot options (to pr oduce \"pretty\" plots) can be found " }{HYPERLNK 222 "here" 2 "plot[o ptions]" "" }{TEXT 218 1 "." }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 128 "plot([f(x), g(x), x], x=-4..4, y=-4..4, color = [red,green,blue], leg end=[\"y=ln(x)\", \"y=exp(x)\", \"y=x\"], title=\"Inverse Pairs\");" } }}{PARA 203 "" 0 "" {TEXT 218 44 "Note that we not only specified a ra nge for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 15 ", but also for " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 218 1 " " } {TEXT 218 57 ". This is not necessary, but makes the graph look better ." }}{EXCHG {PARA 203 "" 0 "" {TEXT 218 72 "We now illustrate some of \+ Maple's Calculus capabilities via our example." }}{PARA 203 "" 0 "" {TEXT 218 20 "First, we know that " }}{PARA 203 "" 0 "" {TEXT 218 12 " " }{XPPEDIT 18 0 "limit(ln(x), x=0, right) = -infinity" "6 #/-%&limitG6%-%#lnG6#%\"xG/F*\"\"!%&rightG,$%)infinityG!\"\"" }{TEXT 218 1 " " }{TEXT 218 2 ". " }}{PARA 203 "" 0 "" {TEXT 218 29 "Let's ha ve Maple verify this:" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 50 "Limit( f(x), x=0, right) = limit(f(x), x=0, right);" }}}{PARA 203 "" 0 "" {TEXT 218 14 "Note that the " }{HYPERLNK 222 "Limit" 2 "Limit" "" } {TEXT 218 116 " command does not actually evaluate the limit. Is is a \+ so-called inert command (used mainly for cosmetic purposes). " }} {PARA 203 "" 0 "" {TEXT 218 26 "Also, do not confuse the " } {HYPERLNK 222 "=" 2 "equation" "" }{TEXT 218 39 " operator with the as signment operator " }{HYPERLNK 222 ":=" 2 "assignment" "" }{TEXT 218 1 "." }}{EXCHG {PARA 203 "" 0 "" {TEXT 218 24 "Next, the derivative of " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 4 " is " }{XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT 218 1 " " }{TEXT 218 22 ". What does Maple say?" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 5 "D(f);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 46 "The previous is Maple's way of representing a " }{HYPERLNK 222 "function" 2 "function" "" }{TEXT 218 1 "." }}{PARA 203 "" 0 "" {TEXT 218 51 "To get a more familiar form we need to look at the " }{HYPERLNK 222 "exp ression" 2 "expression" "" }{TEXT 218 44 " (the derivative function e valuated at a=x)" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 8 "D(f)(x);" }} }{EXCHG {PARA 203 "" 0 "" {TEXT 218 36 "Another possibility would be t o use " }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 14 "diff(f(x), x);" }}} {EXCHG {PARA 203 "" 0 "" {TEXT 218 96 "The difference between these tw o commands lies in the way in which Maple interprets the output. " }} {PARA 203 "" 0 "" {TEXT 218 19 "The first command, " }{HYPERLNK 222 "D (f)" 2 "D" "" }{TEXT 218 47 ", produces a function, whereas the second one, " }{HYPERLNK 222 "diff(f(x), x)" 2 "diff" "" }{TEXT 218 26 ", pr oduces an expression. " }}{PARA 203 "" 0 "" {TEXT 218 156 "This differ ence in representation has important consequences when working with Ma ple (for example, when evaluating/substituting values into an expressi on). " }}{PARA 203 "" 0 "" {TEXT 218 27 "The preferred method is to " }{TEXT 224 19 "work with functions" }{TEXT 218 2 ". " }}{PARA 203 "" 0 "" {TEXT 218 43 "To produce \"pretty\" output you can use the " } {HYPERLNK 222 "inert form of diff" 2 "Diff" "" }{TEXT 218 24 " to writ e something like" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 24 "Diff(f(x), \+ x) = D(f)(x);" }{MPLTEXT 1 219 31 "\nDiff(f(x), x) = diff(f(x), x);" } }}{EXCHG {PARA 203 "" 0 "" {TEXT 218 64 "A similar pair of commands ex ists for integration. We first use " }{HYPERLNK 222 "Int" 2 "Int" "" } {TEXT 218 1 ":" }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 16 "Int(D(f)(x), \+ x);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 159 "This doesn't seem to b e of much help (other than for pretty output). This is another inert c ommand, and to see the actual value of the integral we need to use " } }{PARA 203 "> " 0 "" {MPLTEXT 1 219 9 "value(%);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 9 "Note the " }{HYPERLNK 222 "shortcut %" 2 "ditto" "" }{TEXT 218 45 " which always refers to the previous output. " }} {PARA 203 "" 0 "" {TEXT 218 47 "Multiple percentage symbols are also p ossible. " }}{PARA 203 "" 0 "" {TEXT 218 70 "Another thing to note is \+ that Maple drops the constant of integration!" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 2 "A " }{HYPERLNK 222 "second command for integration" 2 "int" "" }{TEXT 218 37 " is more convenient most of the time:" }} {PARA 203 "> " 0 "" {MPLTEXT 1 219 16 "int(D(f)(x), x);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 56 "Definite integrals are computed simila rly. For example, " }{XPPEDIT 18 0 "Int(ln(x^2), x=1..2)" "6#-%$IntG6$ -%#lnG6#*$%\"xG\"\"#/F*;\"\"\"F+" }{TEXT 218 1 " " }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 21 "int(ln(x^2), x=1..2);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 6 "And a " }{HYPERLNK 222 "numerical value" 2 "evalf" "" }{TEXT 218 13 " is found by " }}{PARA 203 "" 0 "" }{PARA 203 "> " 0 "" {MPLTEXT 1 219 29 "evalf(%);Int(ln(x^2),x=1..2);" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 221 27 "Computing Inverse Functions" }}{PARA 203 "" 0 "" {TEXT 218 69 "We can use Maple to compute other - more complicate d - inverse pairs." }}{EXCHG {PARA 203 "" 0 "" {TEXT 218 34 "E.g., let 's consider the function " }{XPPEDIT 18 0 "y = 3*ln(sqrt(x+4));" "6#/% \"yG*&\"\"$\"\"\"-%#lnG6#-%%sqrtG6#,&%\"xGF'\"\"%F'F'" }{TEXT 218 1 " \+ " }{TEXT 218 22 " and find its inverse." }}{PARA 203 "" 0 "" {TEXT 218 54 "In order to do this, we start by defining the function" }} {PARA 203 "> " 0 "" {MPLTEXT 1 219 26 "y := x -> 3*ln(sqrt(x+4));" }}} {EXCHG {PARA 203 "" 0 "" {TEXT 218 27 "Next we solve the equation " } {XPPEDIT 18 0 "Y=y(x)" "6#/%\"YG-%\"yG6#%\"xG" }{TEXT 218 1 " " } {TEXT 218 5 " for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 218 1 " " } {TEXT 218 47 ". This will result in an expression describing " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 13 " in terms \+ of " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 218 1 " " }{TEXT 218 2 ". " } }{PARA 203 "> " 0 "" {MPLTEXT 1 219 29 "x_of_Y := solve(Y = y(x), x);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 62 "In order to get the answer \+ in function form (as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT 218 1 " " }{TEXT 218 18 ") we first rename " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 218 1 " " }{TEXT 218 4 " to " }{XPPEDIT 18 0 "x" "6#% \"xG" }{TEXT 218 1 " " }{TEXT 218 11 " (with the " }{HYPERLNK 222 "sub s" 2 "subs" "" }{TEXT 218 77 " command), and then convert the resultin g expression to a function using the " }{HYPERLNK 222 "unapply" 2 "una pply" "" }{TEXT 218 9 " command." }}{PARA 203 "" 0 "" {TEXT 218 23 "Th us, as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 218 1 " " } {TEXT 218 34 ", the inverse function is given by" }}{PARA 203 "> " 0 " " {MPLTEXT 1 219 18 "subs(Y=y, x_of_Y);" }{MPLTEXT 1 219 24 "\ninv_y : = unapply(%, y);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 218 68 "Again, the graphs of the two functions are symmetric about the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT 218 1 " " }{TEXT 218 1 "." }}{PARA 203 "> " 0 "" {MPLTEXT 1 219 152 "plot([y(x), inv_y(x), x], x=-6..6, y =-6..6, color=[red,green,blue], legend=[\"y=3ln(sqrt(x+4))\", \"inv_y= -4+(exp(x/3))^2\", \"y=x\"], title=\"Inverse Pairs\");" }}}{EXCHG {PARA 203 "> " 0 "" }}{PARA 203 "" 0 "" }}}{SECT 0 {PARA 204 "" 0 "" {TEXT 220 12 "Assignment 1" }}{SECT 1 {PARA 205 "" 0 "" {TEXT 221 5 "E x.1:" }}{PARA 203 "" 0 "" {TEXT 218 68 "Use Maple to verify the comple te limiting behavior of the functions " }{XPPEDIT 18 0 "ln(x)" "6#-%#l nG6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 5 " and " }{XPPEDIT 18 0 "exp(x )" "6#-%$expG6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 87 ", i.e., compute \+ the behavior of these two functions near the boundary of their domains ." }}}{SECT 1 {PARA 205 "" 0 "" {TEXT 221 5 "Ex.2:" }}{PARA 203 "" 0 " " {TEXT 218 39 "a) Display the graphs of the functions " }{XPPEDIT 18 0 "y=ln(x)" "6#/%\"yG-%#lnG6#%\"xG" }{TEXT 218 1 " " }{TEXT 218 2 ", " }{XPPEDIT 18 0 "y=ln(2*x)" "6#/%\"yG-%#lnG6#*&\"\"#\"\"\"%\"xGF*" } {TEXT 218 1 " " }{TEXT 218 2 ", " }{XPPEDIT 18 0 "y=ln(4*x)" "6#/%\"yG -%#lnG6#*&\"\"%\"\"\"%\"xGF*" }{TEXT 218 1 " " }{TEXT 218 2 ", " } {XPPEDIT 18 0 "y=ln(8*x)" "6#/%\"yG-%#lnG6#*&\"\")\"\"\"%\"xGF*" } {TEXT 218 1 " " }{TEXT 218 2 ", " }{XPPEDIT 18 0 "y=ln(16*x)" "6#/%\"y G-%#lnG6#*&\"#;\"\"\"%\"xGF*" }{TEXT 218 1 " " }{TEXT 218 26 " togethe r in one plot for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 218 1 " " } {TEXT 218 25 " in the interval [0,10]. " }}{PARA 203 "" 0 "" {TEXT 218 35 "Explain in words what is happening." }}{PARA 203 "" 0 "" {TEXT 218 7 "Hints: " }}{PARA 203 "" 0 "" {TEXT 218 128 "1. You can pr oduce separate plots by assigning each one of them to a variable and e nding the command with a colon, and then use " }{HYPERLNK 222 "display " 2 "plots,display" "" }{TEXT 218 10 " from the " }{HYPERLNK 222 "plot s package" 2 "plots" "" }{TEXT 218 34 ") to display these plots togeth er." }}{PARA 203 "" 0 "" {TEXT 218 41 "2. You can insert text by e.g. \+ using the " }{TEXT 224 16 "Insert-Paragraph" }{TEXT 218 6 " menu." }} {PARA 203 "" 0 "" {TEXT 218 84 "b) What is the mathematical reason (fo rmula) for the phenomenon observed in part a)?" }}{PARA 203 "" 0 "" {TEXT 218 36 "Use Maple to verify your conjecture." }}{PARA 203 "" 0 " " {TEXT 218 63 "E.g., simplify(4-2 = 8/4) returns 2=2, which is obviou sly true:" }}}{SECT 1 {PARA 205 "" 0 "" {TEXT 221 5 "Ex.3:" }}{PARA 203 "" 0 "" {TEXT 218 43 "a) Use Maple to determine when the function" }}{PARA 203 "" 0 "" {TEXT 218 8 " " }{XPPEDIT 18 0 "f(x) = (2* x+7)/(3*x-17);" "6#/-%\"fG6#%\"xG*&,&*&\"\"#\"\"\"F'F,F,\"\"(F,F,,&*& \"\"$F,F'F,F,\"#