{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 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 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "Final Exam - SymComp - F all Semester 2002" }}{PARA 257 "" 0 "" {TEXT -1 19 "Put your name here ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 68 "Type your solutions in the spaces given. Add more prompts if ne eded." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "SAVE YOUR WORK FREQUENTLY IN CASE MAPLE CRASHES!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " } {MPLTEXT 1 0 8 "restart;" }{TEXT -1 77 " at the beginning of each prob lem unless it depends on the previous problem. " }{TEXT 261 165 " Note that it may not be necessary to construct a procedure to solve a prob lem. But, in some cases defining certain procedures may make finding the answer easier. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 125 "You may need to experiment a bit. But y ou should delete all experiments that are not required to solve the st ated problem. \n" }}{PARA 0 "" 0 "" {TEXT -1 59 "When you have finish ed, print out the exam and hand it in. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "There are 5 problems. Each is worth \+ 20 points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 10 "Problem 1." }{TEXT -1 38 " Let S be the set of postive integer s " }{XPPEDIT 18 0 "n <= 100;" "6#1%\"nG\"$+\"" }{TEXT -1 122 ". Let \+ S2 be the subset of S consisting of those integers which are a sum of \+ 2 non-zero squares, that is, are of the form " }{XPPEDIT 18 0 "a^2+b^2 ;" "6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF&F'" }{TEXT -1 238 " where a and b a re postive integers. Let S3 the the subset of S consisting of those in tegers which are a sum of three non-zeros squares and let S4 be the su bset of S consisting of those integers which are a sum of four non-zer o squares. \n" }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 267 4 "sets" }{TEXT -1 13 " S2, S3, S4.\n" }}{PARA 0 "" 0 "" {TEXT 270 5 "Hint:" }{TEXT -1 54 " To get the elements in S2 easily, just g enerate the " }{TEXT 269 3 "set" }{TEXT -1 263 " of all a^2 + b^2 wher e a goes from 1 to 10 and b goes from 1 to 10. Then remove the element s greater than 100. To get the elements of S3 you need to just add a^2 to each element of S2 for each a from 1 to 10 and remove those greate r than 100. Similarly for S4. \n" }}{PARA 0 "" 0 "" {TEXT -1 13 "(b) F ind the " }{TEXT 264 9 "number of" }{TEXT -1 39 " elements that are in S3 but not in S2." }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) Find the " } {TEXT 265 9 "number of" }{TEXT -1 39 " elements that are in S4 but not in S3." }}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Find the " }{TEXT 266 9 "n umber of" }{TEXT -1 57 " elements that are S but not in (S2 union S3 u nion S4).\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "Problem 2." }{TEXT -1 104 " Let X=\{seq(i,i=0..999)\}. Use M aple's map command to determine which of the following functions below " }{TEXT 268 35 "considered as functions from X to X" }{TEXT -1 72 " \+ are onto. DO NOT PRINT OUT THE ELEMENTS OF X IN THE PAPER YOU HAND IN ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) \+ " }{XPPEDIT 18 0 "f(x) = `mod`(37*x+23,1000);" "6#/-%\"fG6#%\"xG-%$mod G6$,&*&\"#P\"\"\"F'F.F.\"#BF.\"%+5" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "g(x) = `mod`(37*x^2+23,1000);" "6#/-%\"gG6#%\"xG-%$modG6$,&*&\"# P\"\"\"*$F'\"\"#F.F.\"#BF.\"%+5" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 9 "Problem 3" }{TEXT -1 82 ". (a) Find the determinant of t he matrix A = BC where B and C are given below. " }}{PARA 0 "" 0 "" {TEXT -1 29 " " }{XPPEDIT 18 0 "B = matrix ([[1, 1, 1, 1], [x, 5, 3, 2], [x^2, 5^2, 3^2, 2^2], [x^3, 5^3, 3^3, 2^ 3]]);" "6#/%\"BG-%'matrixG6#7&7&\"\"\"F*F*F*7&%\"xG\"\"&\"\"$\"\"#7&*$ F,F/*$F-F/*$F.F/*$F/F/7&*$F,F.*$F-F.*$F.F.*$F/F." }{TEXT -1 2 " " } {XPPEDIT 18 0 "C = matrix([[1, 2, 3, 14], [2, 3, 14, 5], [3, 14, 5, 6] , [14, 5, 6, 7]]);" "6#/%\"CG-%'matrixG6#7&7&\"\"\"\"\"#\"\"$\"#97&F+F ,F-\"\"&7&F,F-F/\"\"'7&F-F/F1\"\"(" }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "(b) Use Maple to determi ne all values of x for which the determinant is 0: (You may " }{TEXT 263 4 "know" }{TEXT -1 77 " the answer to this, but pretend you don't \+ and use Maple to find the answer.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "Problem 4. " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "(a) Solve the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2/( dx^2)*y+d/dx*y = 3*exp(x)*cos(x)+sin(x)*exp(x);" "6#/,&*(%\"dG\"\"#*$% #dxGF'!\"\"%\"yG\"\"\"F,*(F&F,F)F*F+F,F,,&*(\"\"$F,-%$expG6#%\"xGF,-%$ cosG6#F4F,F,*&-%$sinG6#F4F,-F26#F4F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "with initial conditi ons y(0) = 0 and D(y)(0) = 1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "(b) Plot the solution of the above equation for x in the interval [-2,0]. Shrink the plot to a height of 2 inches or less bef ore printing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "(c) Find t he minimum value taken by the solution when x is in the interval [-2,0 ]. Find both the exact value in terms of " }{XPPEDIT 18 0 "e;" "6#%\"e G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 73 " a nd the floating point approximation to the minimum value to 10 digits. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 10 "Problem 5." }{TEXT -1 92 " For x and y in the plane region x = -3..3, y=-1..1 the function \n\n \+ " }{XPPEDIT 18 0 "f(x,y) = (1+x/100)*sin(x+.1)^2*cos( y)^2;" "6#/-%\"fG6$%\"xG%\"yG*(,&\"\"\"F+*&%\"xGF+\"$+\"!\"\"F+F+*$-%$ sinG6#,&%\"xGF+-%&FloatG6$F+F/F+\"\"#F+-%$cosG6#%\"yGF9" }{TEXT -1 0 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "has two relative maxima, i.e., two peaks. Find the " } {TEXT 262 17 "three coordinates" }{TEXT -1 83 " of the highest peak. \+ BE CAREFUL THAT YOU DON'T GET A RELATIVE MINIMUM BY MISTAKE." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "57 0 3" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }