{VERSION 5 0 "IBM INTEL NT" "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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 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 " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 67 "MAT 155 Project XI \+ Fall 2002" }}{PARA 257 "" 0 "" {TEXT -1 92 " Graphi ng and Optimization " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "Do not forget to save your work reg ularly both as yourname9.mws and yourname9b.mws every 10 minutes. Rem ember to include comments explaining what you are doing." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "This lab h as four problems. For each problem you should complete all the follow ing steps:" }}{PARA 0 "" 0 "" {TEXT -1 25 "a) Define the function." }}{PARA 0 "" 0 "" {TEXT -1 90 "b) Check if the function has a vertic al asymptote by checking when the denominator is 0." }}{PARA 0 "" 0 " " {TEXT -1 41 "c) Find the derivative of the function." }}{PARA 0 " " 0 "" {TEXT -1 30 "d) Find the critical points." }}{PARA 0 "" 0 "" {TEXT -1 48 "e) Find the second derviative of the function." }} {PARA 0 "" 0 "" {TEXT -1 32 "f) Find the inflection points." }} {PARA 0 "" 0 "" {TEXT -1 88 "g) List all the important x values, cri tical numbers, inflection numbers and vertical " }}{PARA 0 "" 0 "" {TEXT -1 69 "asymptotes in order and give a good domain of x values fo r the graph." }}{PARA 0 "" 0 "" {TEXT -1 84 "h) If there is no asymp tote, find the absolute maximum and absolute minimum of the" }}{PARA 0 "" 0 "" {TEXT -1 87 "function using the critical points you've found and the end points you chose in step f." }}{PARA 0 "" 0 "" {TEXT -1 83 "i) Plot the function using the x values from step g and check th at the graph does" }}{PARA 0 "" 0 "" {TEXT -1 155 "indeed have the cor rect critical points, inflection points, global maximum and minimum, a nd asymptotes. Write a comment or correct your work if necessary." }} {PARA 0 "" 0 "" {TEXT -1 69 "j) Plot the graph again with a larger ra nge of x and write comments." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "By completing all o f these steps you will have graphed many of the important qualities of the function. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "EXAMPLE: Before doing the thre e problems complete this example here by hitting enter on each command line:" }}{PARA 0 "" 0 "" {TEXT -1 2 "a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart; f:= x -> x*(x-1)*(x+2)^2;" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "b)" }}{PARA 0 "" 0 "" {TEXT -1 60 "There is no denominator so there are no vertical asymptotes." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "c) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(D(f)(x));" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(D(f)(x)=0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "These are the three critical points and I will use evalf \+ to see their decimal approximations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(solve(D(f)(x)=0));" }{TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "e)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(D(f));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 2 "f)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(D(D(f))(x )=0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "There are inflection points at 0 and -1.5." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "g) " }}{PARA 0 "" 0 "" {TEXT -1 50 "The importa nt x values in order are approximately:" }}{PARA 0 "" 0 "" {TEXT -1 35 "-2, -1.5, -.84, 0, .59" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can choose the domain to be x in [-3,1]." }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "h)" }}{PARA 0 "" 0 "" {TEXT -1 56 "To find the absolute max and min we check the endpoints:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "f(-3);f(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and the critical points:" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(-2);f(-.84); f(.59); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "So the absolute max occurs at x=-3 and has a y value of 12." }}{PARA 0 "" 0 "" {TEXT -1 65 "The abs olute minimum occurs near x=.59 and has a y value of -1.6." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "i)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p lot(f(x), x=-3..1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "You fill in the comments for this part." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 192 "Do the following problems on yourname11.mws. Be sure to put the lett ers marking each step and to add comments just as in the example. Cut ting and pasting commands will help you work faster. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "Problem 1:" }{TEXT -1 35 " Graph f(x)= (x-50)(x-55)(x-60)^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "Problem 2:" }{TEXT -1 36 " Graph f(x)=(x-2)^2(x-5)^2(x+1)^2 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 10 "Probl em 3:" }{TEXT -1 32 " Graph f(x)= ( x^2+1 )^(1/100)" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 10 "Problem 4:" }{TEXT -1 25 " Graph f(x)=(4+4x+x^2)/x" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 12 " Exploration:" }{TEXT -1 75 " Try to graph f(x)=(4+4x+x^2)/x^2. It wi ll not work nicely. Focus on the" }}{PARA 0 "" 0 "" {TEXT -1 96 "infl ection points and critical points and discuss why there is trouble gra phing this function. " }}{PARA 0 "" 0 "" {TEXT -1 103 " \+ You may prefer to graph f(x)=tan(x) and discuss the difficulti es with this graph." }}}}{MARK "27" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }