{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {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 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2280-1" }}{PARA 257 "" 0 " " {TEXT -1 15 "Maple Project 1" }}{PARA 265 "" 0 "" {TEXT 257 30 "Popu lation models, section 2.1" }}{PARA 259 "" 0 "" {TEXT 256 35 "Numerica l Methods, sections 2.4-2.6" }}{PARA 258 "" 0 "" {TEXT -1 30 "Problems due Friday February 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Create a Maple document, perhaps by modifying this \+ one, in which you combine text and Maple commands to answer (and expla in) the following problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "(0) Be sure to include your name(s) at the to p of your document. You may work individually or in groups of up to t hree people, and each group should hand in only one document." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 12 "Populati ons:" }}{PARA 0 "" 0 "" {TEXT -1 290 "(1) Complete Investigation C on \+ page 89 of the text. This involves finding parameters for a logistic \+ model of world population from Figure 2.1.10 on page 90, and then pred icting world population in 2025. We carried out an analgous study for U.S. populations, and the Maple file located at" }}{PARA 266 "" 0 "" {TEXT -1 62 "http://www.math.utah.edu/~korevaar/2280spring06/jan20mapl e.mws" }}{PARA 0 "" 0 "" {TEXT -1 42 "has useful commands to cut, past e, modify." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 21 "Numerical techniques:" }}{PARA 0 "" 0 "" {TEXT -1 67 "(2) Cons ider the same initial value problem as in the class handout" }}{PARA 267 "" 0 "" {TEXT -1 65 "http://www.math.utah.edu/~korevaar/2280spring 06/numerical1.mws , " }}{PARA 0 "" 0 "" {TEXT -1 7 "namely " }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "dy/dx = y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"y G" }{TEXT -1 0 "" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "y(0) = 1;" "6#/-% \"yG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 418 "Use the improved Euler algorithm, with n= 100, on the x-interval [0,1], \+ to approximate the solution. Print out your approximations at x value s which are multiples of 0.1. Compare the accuracy of the n=100 impr oved Euler approximation to e, versus the results in the handout for u nimproved Euler. Note, that on this problem and the subsequent ones, \+ the class handout has useful routines to copy, paste, and modify." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "(3) Use the Runge-Kutta algorithm for the same initial value problem and x-i nterval as in (2) above, with n=20. Compare the accuracy of your resu lts to the work you did there using improved Euler n=100. There shoul d be a huge improvement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 339 "(4) Do problem # 29 in Section 2.4, page 120. (This problem involves work done by hand in part 29a. You may staple this work to your Maple file if you don' t want to bother typing it. But use Maple for 29bc) This problem illu strates how a numerical algorithm may accidently pass right through an x-value where the real solution blows up." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "(5) Do problem #5 in sections 2.4, 2.5, 2.6 homework. This is the same initial value problem in ea ch case, studied successively with Euler, improved Euler, and Runge-Ku tta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 263 " (6) Do problem #25 in section 2.5. Additionally, compare your numeri cal answers to the exact answers obtained by solving this initial valu e problem by hand. (As in problem 4 above, you may either staple your hand-work to the Maple file or type it directly in.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 368 "(7) Go to example 4 \+ on page 138-139. Work through and understand what the text has to say on these two pages. Recreate the data in the last two columns of Fi gure 2.6.9 on page 138 using the Runge-Kutta routine from the class ha ndout. That is, recompute the the Runge-Kutta approximations with h=0 .05 and compare to the actual solution values, at multiples of 0.4." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Understa nd (for yourself) how the form of the general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "y(x) = exp(-x)+C *exp(5*x);" "6#/-%\"yG6#%\"xG,&-%$expG6#,$F'!\"\"\"\"\"*&%\"CGF.-F*6#, $*&\"\"&F.F'F.F.F.F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "me ans that small variations in initial values " }{XPPEDIT 18 0 "y[0];" " 6#&%\"yG6#\"\"!" }{TEXT -1 403 " lead to huge variations in y(x) for x >>0. This sort of instability in actual solution behavior leads to nu merical instability like that illustrated in this important example, e ven when you're using a ``good'' numerical technique like Runge Kutta \+ and a \"small\" time step. Thus people who create numerical solutions to problems must also be well-grounded in the mathematical theory beh ind the problems." }}}{MARK "37 2" 403 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }