{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 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 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "MATH-1850 " }{TEXT -1 3 " " }{TEXT 260 18 "COMPUTER PROBLEM 4" }{TEXT -1 55 " DUE: Tuesday , September 26, 2000, in your lab class" }}{PARA 0 "" 0 "" {TEXT -1 17 "\nTeam members: " }{TEXT 268 54 "(All lines for answers below wi ll extend as you type.)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " (1) " }{TEXT 261 0 "" } {TEXT -1 0 "" }{TEXT 265 17 " SOLUTION " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " (2) " }{TEXT 262 0 "" }{TEXT -1 0 "" }{TEXT 266 9 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " (3) " }{TEXT 263 0 "" }{TEXT -1 0 "" }{TEXT 267 9 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Score: / 6 \n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 13 "INSTRUCTIONS " }{TEXT 257 20 "(Click box in front)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1192 "It is strongly suggested that th is project be worked in teams (of up to 3 persons). Approaches may be discussed among different teams, but each team must write their OWN r eport.\nREQUIREMENTS OF REPORT:\n- Mathematical contents (correctness \+ and completeness) counts most. Solve the problems and show all essenti al calculations (commands and output) and plots. Please do not submit \+ work that is not essential (e.g. checking out how the function fplot i n Problem 1 works - this may be essential to your understanding, but i t s not essential for the reader).\n- Explain the steps you carry out. Explain the approaches/techniques that you use.\n- Add explanations a nd answer questions where asked to do so. By all means, include additi onal explanations/observations/conclusions that come to your mind! (Th is could translate to extra credit.)\nHOW TO SUBMIT THE REPORT\nEach t eam submits ONE printed copy (stapled) by the due date. (Please save t he electronic version, as you will be asked to submit that, too, at a \+ future time.)\nYou can print from inside your Maple worksheet (choose \+ File/Print or click on the printer button). Set the ZOOM FACTOR to 100 % (smallest magnifying glass) before printing.\n" }{TEXT 256 62 "Pleas e remove these instructions before submitting the report." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Use the Intermediate Value Theorem (IVT) to find one " } {TEXT 269 8 "positive" }{TEXT -1 27 " solution of the equation " } {XPPEDIT 18 0 "x^2-9 = cos(2*x);" "6#/,&*$%\"xG\"\"#\"\"\"\"\"*!\"\"-% $cosG6#*&F'F(F&F(" }{TEXT -1 35 " accurate to seven decimal places." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "x^2-9 = cos(2*x);" "6#/,&*$%\"xG\"\"#\"\"\"\"\"*!\"\"-%$cosG6#*&F'F (F&F(" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "x^2-9-cos(2*x) = \+ 0;" "6#/,(*$%\"xG\"\"#\"\"\"\"\"*!\"\"-%$cosG6#*&F'F(F&F(F*\"\"!" } {TEXT -1 10 " . Let f(" }{TEXT 271 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "x^2-9-cos(2*x);" "6#,(*$%\"xG\"\"#\"\"\"\"\"*!\"\"-%$co sG6#*&F&F'F%F'F)" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> x^2-9-cos(2*x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "plot(f(x), x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 83 "From the graph, there appears to a solution (root) to the equation between 3 and 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f(3); evalf(%); f(4); evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(a) Justification for being able to apply IVT: " }{TEXT 264 68 " \+ Since f is the difference of the continuous polynomial function " } {TEXT 272 1 "y" }{TEXT 273 3 " = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\" \"#" }{TEXT 274 49 " - 9 and the continuous trigonometric function \+ " }{TEXT 275 1 "y" }{TEXT 276 8 " = cos(2" }{TEXT 277 1 "x" }{TEXT 278 126 ") , then the function f is continuous on the closed interval \+ [3, 4]. Since f(3) < 0 and f(4) > 0, then there exists a number " } {TEXT 279 1 "c" }{TEXT 280 34 " in the open interval such that f(" } {TEXT 281 1 "c" }{TEXT 282 10 ") = 0 " }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "(b) By \+ graphing the function, find an interval of length 0.00001 which contai ns a positive solution (root) to the equation. (Note: It may take se veral graphs to find the interval of length 0.00001. Show " }{TEXT 270 5 "three" }{TEXT -1 34 " graphs to get to this interval. )" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(x),x=3..4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x), x=3.1..3.2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(f(x), x=3.16..3.17);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(f(x), x=3.162..3.163) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(x),x=3.1621..3. 1622);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "(c) Starting with you r interval of length 0.00001 from Part (b) above, apply the IVT to obt ain smaller and smaller intervals, which contain the positive solution (root) to the equation, until you find an approximation to the positi ve solution (root) of the equation which is correct to seven decimal p laces." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The solution (root) is between 3.16214 and 3.16215:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f(3.162140); f(3.162145); f( 3.162150);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The solution (root) is between 3.162140 and 3.162145:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f(3.162140); f(3.162141); f (3.162142); f(3.162143); f(3.162144); f(3.162145);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The solution (roo t) is between 3.162144 and 3.162145:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f(3.1621440); f(3.1621445); f(3.1621450);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The solut ion (root) is between 3.1621440 and 3.1621445:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 83 "f(3.1621440); f(3.1621441); f(3.1621442); f(3. 1621443); f(3.1621444); f(3.1621445);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The solution (root) is be tween 3.1621441 and 3.1621442:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f(3.16214410); f(3.16214415); f(3.16214420);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The solution (r oot) is between 3.16214410 and 3.16214415:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "f(3.16214411); f(3.16214412); f(3.16214413); f(3.16 214414); f(3.16214415);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 384 "The solution (root) is between 3.1621441 1 and 3.16214412. If we pick any number in the interval [3.16214411, \+ 3.16214412] for the solution (root), then the error for our solution i s at most 3.16214412 - 3.16214411 = 0.00000001. Thus, we could take e ither 3.16214411 or 3.16214412 for our approximation to the solution ( root) of the equation which is accurate to seven decimal places." }}}} {MARK "2 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }