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" }{TEXT 233 0 "" }}{PARA 212 " " 0 "" {TEXT 233 104 "You might find it helpful to use this command to clear Maple's memory at later stages in your work also." }{TEXT 233 0 "" }}{SECT 1 {PARA 213 "" 0 "" {TEXT 234 12 "Introduction" }{TEXT 234 0 "" }}{SECT 1 {PARA 214 "" 0 "" {TEXT 235 43 "Natural Logarithm a nd Exponential Functions" }{TEXT 235 0 "" }}{PARA 209 "" 0 "" {TEXT 230 52 "The exponential function is represented in Maple by " } {HYPERLNK 17 "exp" 2 "exp" "" }{TEXT 230 97 " (click here for help on \+ the exponential function), and the natural logarithm can be defined vi a " }{HYPERLNK 17 "ln" 2 "ln" "" }{TEXT 230 4 " or " }{HYPERLNK 17 "lo g" 2 "log" "" }{TEXT 230 29 ", i.e., Maple does not treat " }{XPPEDIT 18 0 "log(x)" "6#-%$logG6#%\"xG" }{TEXT 230 32 " as the base-10 log arithm of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 34 " as is done in many text books." }{TEXT 230 0 "" }}{EXCHG {PARA 215 "" 0 "" {TEXT 237 101 "We start by illustrating the relationship between the n atural logarithm and the exponential function." }{TEXT 237 0 "" }} {PARA 212 "" 0 "" {TEXT 233 34 "First we define the two functions:" } {TEXT 233 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 16 "f := x -> ln(x );" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 18 "\ng := x -> exp(x);" } {MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 18 "Next we \+ produce a " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT 230 54 " of this i nverse pair along with the line of symmetry " }{XPPEDIT 18 0 "y=x" "6# /%\"yG%\"xG" }{TEXT 230 4 " ." }{TEXT 230 0 "" }}{PARA 209 "" 0 "" {TEXT 230 70 "Help on various plot options (to produce \"pretty\" plot s) can be found " }{HYPERLNK 17 "here" 2 "plot[options]" "" }{TEXT 230 1 "." }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 128 "pl ot([f(x), g(x), x], x=-4..4, y=-4..4, color = [red,green,blue], legend =[\"y=ln(x)\", \"y=exp(x)\", \"y=x\"], title=\"Inverse Pairs\");" } {MPLTEXT 1 232 0 "" }}}{PARA 209 "" 0 "" {TEXT 230 44 "Note that we no t only specified a range for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 18 " , but also for " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 230 60 " . This is not necessary, but makes the graph look better." } {TEXT 230 0 "" }}{EXCHG {PARA 212 "" 0 "" {TEXT 233 72 "We now illustr ate some of Maple's Calculus capabilities via our example." }{TEXT 233 0 "" }}{PARA 212 "" 0 "" {TEXT 233 20 "First, we know that " } {TEXT 233 0 "" }}{PARA 209 "" 0 "" {TEXT 230 4 " " }{XPPEDIT 18 0 " limit(ln(x), x=0, right) = -infinity" "6#/-%&limitG6%-%#lnG6#%\"xG/F* \"\"!%&rightG,$%)infinityG!\"\"" }{TEXT 230 5 " . " }{TEXT 230 0 "" }}{PARA 212 "" 0 "" {TEXT 233 29 "Let's have Maple verify this:" } {TEXT 233 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 50 "Limit(f(x), x= 0, right) = limit(f(x), x=0, right);" }{MPLTEXT 1 232 0 "" }}}{PARA 209 "" 0 "" {TEXT 230 14 "Note that the " }{HYPERLNK 17 "Limit" 2 "Lim it" "" }{TEXT 230 116 " command does not actually evaluate the limit. \+ Is is a so-called inert command (used mainly for cosmetic purposes). \+ " }{TEXT 230 0 "" }}{PARA 209 "" 0 "" {TEXT 230 26 "Also, do not conf use the " }{HYPERLNK 17 "=" 2 "equation" "" }{TEXT 230 39 " operator w ith the assignment operator " }{HYPERLNK 17 ":=" 2 "assignment" "" } {TEXT 230 1 "." }{TEXT 230 0 "" }}{EXCHG {PARA 209 "" 0 "" {TEXT 230 24 "Next, the derivative of " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG " }{TEXT 230 7 " is " }{XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\"xG!\"\" " }{TEXT 230 25 " . What does Maple say?" }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 5 "D(f);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 46 "The previous is Maple's way of represe nting a " }{HYPERLNK 17 "function" 2 "function" "" }{TEXT 230 1 "." } {TEXT 230 0 "" }}{PARA 209 "" 0 "" {TEXT 230 51 "To get a more familia r form we need to look at the " }{HYPERLNK 17 "expression" 2 "expressi on" "" }{TEXT 230 44 " (the derivative function evaluated at a=x)" } {TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 8 "D(f)(x);" } {MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 233 36 "Another \+ possibility would be to use " }{TEXT 233 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 14 "diff(f(x), x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 233 96 "The difference between these two comma nds lies in the way in which Maple interprets the output. " }{TEXT 233 0 "" }}{PARA 209 "" 0 "" {TEXT 230 19 "The first command, " } {HYPERLNK 17 "D(f)" 2 "D" "" }{TEXT 230 47 ", produces a function, whe reas the second one, " }{HYPERLNK 17 "diff(f(x), x)" 2 "diff" "" } {TEXT 230 26 ", produces an expression. " }{TEXT 230 0 "" }}{PARA 212 "" 0 "" {TEXT 233 156 "This difference in representation has important consequences when working with Maple (for example, when evaluating/su bstituting values into an expression). " }{TEXT 233 0 "" }}{PARA 209 " " 0 "" {TEXT 230 27 "The preferred method is to " }{TEXT 238 19 "work \+ with functions" }{TEXT 230 2 ". " }{TEXT 230 0 "" }}{PARA 209 "" 0 "" {TEXT 230 43 "To produce \"pretty\" output you can use the " } {HYPERLNK 17 "inert form of diff" 2 "Diff" "" }{TEXT 230 24 " to write something like" }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 24 "Diff(f(x), x) = D(f)(x);" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 31 "\nDiff(f(x), x) = diff(f(x), x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 64 "A similar pair of commands exists for \+ integration. We first use " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT 230 1 ":" }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 16 "Int(D(f )(x), x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 233 159 "This doesn't seem to be of much help (other than for pretty outpu t). This is another inert command, and to see the actual value of the \+ integral we need to use " }{TEXT 233 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 9 "value(%);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 9 "Note the " }{HYPERLNK 17 "shortcut %" 2 "ditto" " " }{TEXT 230 45 " which always refers to the previous output. " } {TEXT 230 0 "" }}{PARA 212 "" 0 "" {TEXT 233 47 "Multiple percentage s ymbols are also possible. " }{TEXT 233 0 "" }}{PARA 212 "" 0 "" {TEXT 233 70 "Another thing to note is that Maple drops the constant of inte gration!" }{TEXT 233 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 2 "A \+ " }{HYPERLNK 17 "second command for integration" 2 "int" "" }{TEXT 230 37 " is more convenient most of the time:" }{TEXT 230 0 "" }} {PARA 211 "> " 0 "" {MPLTEXT 1 232 16 "int(D(f)(x), x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 56 "Definite integrals \+ are computed similarly. For example, " }{XPPEDIT 18 0 "Int(ln(x^2), x= 1..2)" "6#-%$IntG6$-%#lnG6#*$%\"xG\"\"#/F*;\"\"\"F+" }{TEXT 230 3 " \+ " }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 21 "int(ln(x^2) , x=1..2);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 6 "And a " }{HYPERLNK 17 "numerical value" 2 "evalf" "" }{TEXT 230 13 " is found by " }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 9 "evalf(%);" }{MPLTEXT 1 232 0 "" }}}}{SECT 1 {PARA 214 "" 0 " " {TEXT 235 27 "Computing Inverse Functions" }{TEXT 235 0 "" }}{PARA 212 "" 0 "" {TEXT 233 69 "We can use Maple to compute other - more com plicated - inverse pairs." }{TEXT 233 0 "" }}{EXCHG {PARA 209 "" 0 "" {TEXT 230 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 230 25 " and find its inverse." }{TEXT 230 0 "" }} {PARA 212 "" 0 "" {TEXT 233 54 "In order to do this, we start by defin ing the function" }{TEXT 233 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 26 "y := x -> 3*ln(sqrt(x+4));" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 27 "Next we solve the equation " } {XPPEDIT 18 0 "Y=y(x)" "6#/%\"YG-%\"yG6#%\"xG" }{TEXT 230 8 " for \+ " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 50 " . This will result in an expression describing " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 16 " in terms of " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 230 5 " . \+ " }{TEXT 230 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 232 29 "x_of_Y := s olve(Y = y(x), x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 62 "In order to get the answer in function form (as a functi on of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 21 " ) we first rena me " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 230 7 " to " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 14 " (with the " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 230 77 " command), and then convert the resulting exp ression to a function using the " }{HYPERLNK 17 "unapply" 2 "unapply" "" }{TEXT 230 9 " command." }{TEXT 230 0 "" }}{PARA 209 "" 0 "" {TEXT 230 23 "Thus, as a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 230 37 " , the inverse function is given by" }{TEXT 230 0 "" }} {PARA 211 "> " 0 "" {MPLTEXT 1 232 18 "subs(Y=x, x_of_Y);" }{MPLTEXT 1 232 0 "" }{MPLTEXT 1 232 24 "\ninv_y := unapply(%, x);" }{MPLTEXT 1 232 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 230 68 "Again, the graphs o f the two functions are symmetric about the line " }{XPPEDIT 18 0 "y=x " "6#/%\"yG%\"xG" }{TEXT 230 4 " ." }{TEXT 230 0 "" }}{PARA 211 "> \+ " 0 "" {MPLTEXT 1 232 152 "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\");" }{MPLTEXT 1 232 0 "" }}}}}{SECT 1 {PARA 213 "" 0 "" {TEXT 234 11 "Assignment " }{TEXT 234 0 "" }}{SECT 1 {PARA 218 "" 0 "" {TEXT 241 74 "Ex.1: Use Maple to veri fy the complete limiting behavior of the functions " }{XPPEDIT 18 0 "l n(x)" "6#-%#lnG6#%\"xG" }{TEXT 241 8 " and " }{XPPEDIT 18 0 "exp(x) " "6#-%$expG6#%\"xG" }{TEXT 241 90 " , i.e., compute the behavior of these two functions near the boundary of their domains." }{TEXT 241 0 "" }}{PARA 219 "" 0 "" {TEXT 242 24 "(a) Define the function " } {XPPEDIT 18 0 "f = ln(x);" "6#/%\"fG-%#lnG6#%\"xG" }{TEXT 242 14 " \+ and plot " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 242 54 " in valid do main (the x-interval(s)) with diagonal " }{XPPEDIT 18 0 "y=x" "6#/% \"yG%\"xG" }{TEXT 242 4 " " }{TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 9 "(b) Use " }{HYPERLNK 17 "Limit" 2 "Limit" "" }{TEXT 242 46 " command to verify the boundary of function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 242 29 " you \+ get from the graph " }{TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 24 "(c) Define the function " } {XPPEDIT 18 0 "h = exp(x);" "6#/%\"hG-%$expG6#%\"xG" }{TEXT 242 14 " \+ and plot " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT 242 19 " with di agonal " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT 242 3 " " } {TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 9 "(d) Use " }{HYPERLNK 17 "Limit" 2 "Limit" "" }{TEXT 242 45 " command to verify the boundary of function " }{XPPEDIT 18 0 "h; " "6#%\"hG" }{TEXT 242 4 " " }{TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 10 "(e) plot " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 242 9 " and " }{XPPEDIT 18 0 " h;" "6#%\"hG" }{TEXT 242 38 " together and again with diagonal " } {XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT 242 79 " . Please incl ude a \"Legend\" to label each curve and a \"Title\" for the graph." } {TEXT 242 0 "" }}}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{SECT 1 {PARA 218 "" 0 "" {TEXT 241 5 "Ex.2:" }{TEXT 244 1 " " }{TEXT 241 113 "Integ rate 1/x between 2 and 5, 20 and 50, 200 and 500 , 2000 and 5000 . Wha t property of logs does this describe?" }{TEXT 241 0 "" }}}{SECT 1 {PARA 218 "" 0 "" {TEXT 241 64 "Ex.3: Use Maple to determine the doma in on which the function " }{XPPEDIT 218 0 "f(x) = (2*x-3)/(x+3)" "6# /-%\"fG6#%\"xG*&,&*&\"\"#\"\"\"F'F,F,\"\"$!\"\"F,,&F'F,F-F,F." }{TEXT 244 3 " " }{TEXT 241 27 " is a one-to-one function." }{TEXT 241 0 " " }}{PARA 219 "" 0 "" {TEXT 242 25 "(a) Define the function " } {XPPEDIT 18 0 "y = (2*x-3)/(x+3)" "6#/%\"yG*&,&*&\"\"#\"\"\"%\"xGF)F) \"\"$!\"\"F),&F*F)F+F)F," }{TEXT 242 26 " and plot the function. " } {TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 33 "(b) Find the inverse function of " }{XPPEDIT 18 0 "f" "6 #%\"fG" }{TEXT 242 37 " using the method outlined below. " }{TEXT 242 0 "" }}{PARA 219 "" 0 "" {TEXT 242 32 " Step 1: solve the equa tion " }{XPPEDIT 18 0 "Y=y(x)" "6#/%\"YG-%\"yG6#%\"xG" }{TEXT 242 8 " \+ for " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 242 50 " . This will re sult in an expression describing " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT 242 16 " in terms of " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 242 5 " . " }{TEXT 242 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }} {PARA 209 "" 0 "" {TEXT 230 3 " " }{TEXT 238 72 " Step 2: In order \+ to get the answer in function form (as a function of " }{XPPEDIT 217 0 "x" "6#%\"xG" }{TEXT 238 21 " ) we first rename " }{XPPEDIT 216 0 "Y" "6#%\"YG" }{TEXT 238 7 " to " }{XPPEDIT 215 0 "x" "6#%\"xG" } {TEXT 238 14 " (with the " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 238 98 " command), and then convert the resulting expression to a function using the " }{HYPERLNK 17 "unapply" 2 "unapp ly" "" }{TEXT 238 33 " command. Thus, as a function of " }{XPPEDIT 202 0 "x" "6#%\"xG" }{TEXT 238 37 " , the inverse function is given \+ by" }{TEXT 230 0 "" }}{PARA 217 "" 0 "" {TEXT 240 0 "" }}{PARA 219 "" 0 "" {TEXT 242 50 "(c) Plot the inverse pair along with the diagonal \+ " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT 242 31 " on an approp riate interval." }{TEXT 242 0 "" }}}}{PARA 220 "" 0 "" {TEXT 245 0 "" }}{PARA 221 "" 0 "" {TEXT 246 0 "" }}{PARA 222 "" 0 "" {TEXT -1 0 "" } }}{MARK "7 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }