{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle19" -1 201 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle32" -1 204 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle29" -1 206 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle31" -1 217 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle20" -1 218 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle30" -1 219 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle21" -1 222 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle14" -1 224 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle25" -1 225 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle23" -1 231 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle4" -1 232 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle13" -1 237 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle3" -1 238 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle15" -1 240 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle27" -1 243 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle28" -1 244 "Times" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle26" -1 246 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle12" -1 247 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "_pstyle21" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle15 " -1 206 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle12" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle14" -1 215 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle 20" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle17" -1 218 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle19" -1 219 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_psty le16" -1 221 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle13" -1 223 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {PARA 211 "" 0 "" {TEXT 201 72 "General Exponential and Logari thmic Functions and Functions of the Form " }{XPPEDIT 18 0 "f(x)^g(x); " "6#)-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT 218 2 " " }{TEXT 238 0 "" }} {EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 247 8 "restart;" }}}{SECT 0 {PARA 215 "" 0 "" {TEXT 237 12 "Introduction" }}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 67 "Example 1 (Finding the intersection of the graphs o f two functions)" }}{EXCHG {PARA 221 "" 0 "" {TEXT 240 34 "We start by defining two functions" }}{PARA 218 "" 0 "" {TEXT 240 18 " \+ " }{XPPEDIT 18 0 "f(x) = 2^x;" "6#/-%\"fG6#%\"xG)\"\"#F'" } {TEXT 222 2 " " }{TEXT 240 23 " and " }{XPPEDIT 18 0 "g(x) = x^2;" "6#/-%\"gG6#%\"xG*$F'\"\"#" }{TEXT 222 2 " " }{TEXT 240 2 ". " }{TEXT 232 0 "" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 30 "f \+ := x -> 2^x; \ng := x -> x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 70 "Next you should take a couple of minutes to try to \+ solve the equation " }{XPPEDIT 18 0 "2^x = x^2;" "6#/)\"\"#%\"xG*$F&F% " }{TEXT 222 2 " " }{TEXT 240 5 " for " }{XPPEDIT 18 0 "x;" "6#%\"xG " }{TEXT 222 2 " " }{TEXT 240 10 " by hand. " }{TEXT 232 0 "" }} {PARA 221 "" 0 "" {TEXT 240 44 "You can probably guess two solutions e asily." }}{PARA 221 "" 0 "" {TEXT 240 71 "The next possiblity would be to try and solve the equation graphically." }}{PARA 221 "" 0 "" {TEXT 240 36 "Let's let Maple do this step for us:" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 104 "plot([f(x), g(x)], x=-5..5, color=[green,red] , legend=[\"y=f(x)\", \"y=g(x)\"], title=\"Graphical Solution\");" } {TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 46 "It looks as th ough there are three solutions. " }}{PARA 218 "" 0 "" {TEXT 240 37 "Th e two you have probably guessed at " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG \"\"#" }{TEXT 222 2 " " }{TEXT 240 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT 222 2 " " }{TEXT 240 61 ", as well as another \+ one which seems to be between -1 and 0. " }{TEXT 232 0 "" }}{PARA 221 "" 0 "" {TEXT 240 86 "Let's zoom in to read off a more accurate value \+ for the third solution from the graph." }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 99 "plot([f(x), g(x)], x=-1..0, color=[green,red], lege nd=[\"y=f(x)\", \"y=g(x)\"], title=\"A Closer Look\");" }{TEXT -1 0 " " }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 37 "The third solution seems t o be about " }{XPPEDIT 18 0 "x = -.77;" "6#/%\"xG,$-%&FloatG6$\"#x!\"# !\"\"" }{TEXT 222 2 " " }{TEXT 240 2 ". " }{TEXT 232 0 "" }}{PARA 221 "" 0 "" {TEXT 240 182 "(You can determine this by clicking on the \+ plot right where the two curves intersect. In the upper left-hand corn er you can read off the coordinates of the point you're clicking on.) " }}{PARA 221 "" 0 "" {TEXT 240 39 "Finally, let's see what Maple give s us:" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 24 "solve(\{f(x) = g(x)\}, x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 77 "Now we \+ have three answers, but we don't understand what the third one means. \+ " }}{PARA 218 "" 0 "" {TEXT 240 24 "(You can take a look at " } {HYPERLNK 17 "LambertW" 2 "LambertW" "" }{TEXT 240 89 " but that proba bly won't do you much good, so don't worry about the meaning of Lamber tW)." }{TEXT 232 0 "" }}{PARA 221 "" 0 "" {TEXT 240 140 "We will be co ntent with a floating point approximation of the solution which we get by typing (remember: % refers to the most recent output)" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 9 "evalf(%);" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 45 "Example 2 (Finding the extrema of a fu nction)" }}{EXCHG {PARA 218 "" 0 "" {TEXT 240 43 "As a second example, consider the function " }{XPPEDIT 18 0 "f(x) = sin(x)^x;" "6#/-%\"fG6 #%\"xG)-%$sinG6#F'F'" }{TEXT 222 2 " " }{TEXT 240 17 " on the interva l " }{XPPEDIT 18 0 "[0, Pi];" "6#7$\"\"!%#PiG" }{TEXT 222 2 " " } {TEXT 240 2 ". " }{TEXT 232 0 "" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 19 "f := x -> sin(x)^x;" }{TEXT -1 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 24 "Let's plot the graph of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 222 2 " " }{TEXT 232 0 "" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 20 "plot(f(x), x=0..Pi);" }{TEXT -1 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 16 "It seems as if " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT 222 2 " " }{TEXT 240 27 " might be approaching 1 as " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 222 2 " " }{TEXT 240 15 " approac hes 0. " }{TEXT 232 0 "" }}{PARA 221 "" 0 "" {TEXT 240 131 "Let's chec k this: (if you were doing this by hand you would have to use l'Hopita l's rule - which we will learn later this semester)" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 24 "limit(f(x), x=0, right);" }{TEXT -1 0 "" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 240 43 "Now let's think about maximum a nd minimum. " }}{PARA 221 "" 0 "" {TEXT 240 58 "From the plots above w e can already get a rough estimate. " }}{PARA 218 "" 0 "" {TEXT 240 47 "To find the locations of the extreme values of " }{XPPEDIT 18 0 "f ;" "6#%\"fG" }{TEXT 222 2 " " }{TEXT 240 40 " we need to find the cri tical points of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 222 2 " " } {TEXT 240 35 ", i.e, the zeros of the derivative." }{TEXT 232 0 "" }} {PARA 221 "" 0 "" {TEXT 240 23 "Here is the derivative:" }}{PARA 223 " > " 0 "" {MPLTEXT 1 247 15 "fprime := D(f);" }{TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 18 "and plot its graph" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 25 "plot(fprime(x), x=0..Pi);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 59 "The zeros of t he derivative seem to be around 0.4 and 1.6. " }}{PARA 221 "" 0 "" {TEXT 240 51 "To obtain an even more precise answer we use Maple." }} {PARA 221 "" 0 "" {TEXT 240 25 "Let's try solve as above:" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 22 "solve(fprime(x)=0, x);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 16 "Nothing happens." }}{PARA 221 "" 0 "" {TEXT 240 51 "Maple can't find an exact solution to this problem." }} {PARA 218 "" 0 "" {TEXT 240 25 "Therefore we need to try " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT 240 65 " instead, which attempts to g ive us an approximate numerical (or " }{TEXT 231 1 "f" }{TEXT 240 24 " loating-point) solution." }{TEXT 232 0 "" }}{PARA 221 "" 0 "" {TEXT 240 109 "Note that we have to specify an interval for Maple to look in . (Try what happens if you delete the interval.)" }}{PARA 223 "> " 0 " " {MPLTEXT 1 247 37 "x1:=fsolve(fprime(x)=0, x, 0.3..0.6);" }{TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 27 "And the second solution is " }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 37 "x2:=fsolve(fprime(x)=0, x, 1.5..1.7);" }{TEXT -1 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 29 "This last number is actually " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT 222 2 " " }{TEXT 232 0 "" }}{PARA 223 "> " 0 " " {MPLTEXT 1 247 13 "fprime(Pi/2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 240 32 "Therefore, the local maximum of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 222 2 " " }{TEXT 240 73 " is (without any \+ further tests, from the graph we can see which is which)" }{TEXT 232 0 "" }}{PARA 223 "> " 0 "" {MPLTEXT 1 247 8 "f(Pi/2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 240 19 "and the minimum is " }} {PARA 223 "> " 0 "" {MPLTEXT 1 247 6 "f(x1);" }{TEXT -1 0 "" }}}}} {SECT 0 {PARA 215 "" 0 "" {TEXT 237 12 "Assignment 2" }}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 5 "Ex.1:" }}{PARA 221 "" 0 "" {TEXT 240 71 "Usin g trial and error find all positive values of A in the interval (0," } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 240 12 ") for which \+ " }{XPPEDIT 18 0 "y = A^x;" "6#/%\"yG)%\"AG%\"xG" }{TEXT 240 20 " inte rsects y = x. \n" }}{PARA 219 "" 0 "" {TEXT 225 26 "(a) Define the fun ctions " }{XPPEDIT 2 0 "f = A^x;" "6#/%\"fG)%\"AG%\"xG" }{TEXT 225 61 " ; and G = x ( Give an initial guess to A, hint 1 < A < 2 )" }} {PARA 217 "" 0 "" {TEXT 246 0 "" }}{PARA 219 "" 0 "" {TEXT 225 20 "(b) Plot a graph of " }{XPPEDIT 18 0 "F and G;" "6#3%\"FG%\"GG" }{TEXT 225 28 " over the interval [ 1 , 5 ]" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 246 0 "" }}{PARA 219 "" 0 "" {TEXT 225 8 "(c) Use " }{TEXT 244 6 "solve " }{TEXT 225 39 " command to find all positive numbers \+ " }{TEXT 206 1 "x" }{TEXT 225 13 " such that " }{TEXT 206 1 "F" } {TEXT 225 2 " (" }{TEXT 206 2 " x" }{TEXT 225 7 " ) = " }{TEXT 206 1 "G" }{TEXT 225 3 " ( " }{TEXT 206 1 "x" }{TEXT 225 202 " ) . (If you r initial guess did not give you a intersection between these two curv es, go back to part (a). try to plug another number and update (b) and (c) till you can find at least one intersection.)" }{TEXT 219 38 "\n \n(d) Estimate the interval within (0," }{XPPEDIT 18 0 "infinity;" "6# %)infinityG" }{TEXT 240 17 ") for A on which " }{XPPEDIT 18 0 "y = A^x ;" "6#/%\"yG)%\"AG%\"xG" }{TEXT 240 16 " intersects y=x." }}}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 5 "Ex.2:" }}{PARA 218 "" 0 "" {TEXT 240 11 "a) Compute " }{XPPEDIT 18 0 "limit(x^n/exp(x), x = infinity);" "6# -%&limitG6$*&)%\"xG%\"nG\"\"\"-%$expG6#F(!\"\"/F(%)infinityG" }{TEXT 222 2 " " }{TEXT 240 37 ". What does this imply for values of " } {XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT 222 2 " " }{TEXT 240 5 " \+ and " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 222 2 " " } {TEXT 240 21 " for large values of " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT 222 2 " " }{TEXT 240 1 "?" }{TEXT 232 0 "" }}{PARA 218 "" 0 "" {TEXT 240 76 "b) Illustrate this behavior by plotting the graphs for d ifferent choices of " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 222 2 " " }{TEXT 240 1 "." }{TEXT 232 0 "" }}}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 5 "Ex.3:" }}{PARA 218 "" 0 "" {TEXT 240 18 "a) Plot the graph " } {XPPEDIT 18 0 "g(x) = 8^log[2](x);" "6#/-%\"gG6#%\"xG)\"\")-&%$logG6# \"\"#6#%\"xG" }{TEXT 222 2 " " }{TEXT 240 18 " on [0.001 , 2]. " } {TEXT 232 0 "" }}{PARA 218 "" 0 "" {TEXT 240 10 "Note that " } {XPPEDIT 18 0 "log[a](x);" "6#-&%$logG6#%\"aG6#%\"xG" }{TEXT 222 2 " \+ " }{TEXT 240 15 " is defined by " }{TEXT 231 9 "log[a](x)" }{TEXT 240 11 " in Maple. " }{TEXT 232 0 "" }{TEXT 240 0 "" }}{PARA 221 "" 0 "" {TEXT 240 176 "b) Notice that this graph appears to have the form f(x ) = x^c for some constant c. Estimate this constant by plotting f(x ) for a value of c which makes the graphs coincide." }}{PARA 218 "" 0 "" {TEXT 240 15 "c) Use Maple's " }{HYPERLNK 17 "simplify" 2 "simplify " "" }{TEXT 240 55 " command on g(x). What rule did Maple most likely use?" }{TEXT 232 0 "" }}}{SECT 1 {PARA 206 "" 0 "" {TEXT 224 5 "Ex.4: " }}{PARA 221 "" 0 "" {TEXT 240 176 "a) Define the function f(x) = cos (x)^cot(x).\nb) Plot f on the interval [-7,7].\nc) Explain the gaps in the graph. Why are they there? How wide are they?\nd) Replot the grap h on [" }{XPPEDIT 18 0 "-1/2*Pi;" "6#,$*&\"\"#!\"\"%#PiG\"\"\"F&" } {TEXT 240 1 "," }{XPPEDIT 18 0 "1/2*Pi;" "6#*&\"\"#!\"\"%#PiG\"\"\"" } {TEXT 240 212 " ].\ne) Since cot(x) is not defined for x=0 there shoul d also be a gap at x=0. Why does it not appear? (Hint: compute the on e-sided limits of f at x=0.)\nf) Find the maximum and the minimum of f in the interval [-" }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\" " }{TEXT 240 1 "," }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 240 26 " ] as in the introduction." }}}}{PARA 202 "" 0 "" {TEXT 217 0 "" }{TEXT 204 0 "" }{TEXT -1 0 "" }}}{MARK "3 4" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }