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exam1solns - MATH 231 FALL 2006 EXAM#1 WEDNESDAY SEPTEMBER...

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Unformatted text preview: MATH 231 FALL 2006 EXAM #1 WEDNESDAY SEPTEMBER 20‘“ Note the point values assigned to each question.- Partial credit is important — try all problems. #1 (15points): Find all equilibrium values of the following discrete-time systems. If you use a formula, state it first and then apply it. (a) Pt+1= 3 Pt (1 ' Pt) (a) fly: 3/97/7093 "‘ 3P 91" 09/?!" (:2 (b) Pen-1:2 Pt/ (Pt' 1) a g Effi'Q/fi t f¥(s?\t&) 3/02; C3 i?— CAMDJ‘: (3%”) «5) PY/fi/J 790* Q . amass/«>202: wees I #2 (lSpoints): For each of the discrete systems below, calculate a formula for the "two step" dynamics, that is, xt+2 or mm stated as a function of xt or mt. t+=3 t(1_t a z. __ (a)Xl x x) (>73 3% (/ 2%”) .9 L‘v‘f '5 3233(1—12 )[7 —3§(/_12)] (b) mt+1= V2 mx+l { = *3; (it; .31; 9’ 4 7’5; 9g 9’1? ‘7’; / / — U a; «1'2? 2 a“; O ”was; #3 (20points): Cobweb (carefully) on the figure below to determine the behavior of the dynamical system defined by the plotted updating function, as follows. First, give the values 'of all equilibria. Second, run 3 steps of the cobweb routine (by hand) starting from initial values -2, 0, and 2. From this, identify which equilibrium are stable and unstable. If you need to run the cobweb routine on more initial data points, do‘so, then determine which equilibria are stable or unstable 2 —2 -1.5 -‘l 45.5 0 0.5 1 3.5 2 5%1ZZr/lam fifty" A; /S 5%féé’ ”Wm/C 55:415— flgar' §./ /5' UUSTAfiéé N/WWJ/C’ you); flfaf’. '/./ KS 55%?! w/OSC/ccflrxw 37,2425 #4 (25points): . _ EEC/4&6 : A = A Consider the followmg dynamlcal systems. ‘5 I; I: é; 63 é b .___— __—— ____—-—- -— —_____.- (a) bt+1= 1/2 b: + 3, b0 2 0 a) a 3 .fij" 5515’ 57435-— (b) bt+1—“3 bt+3,b0—0 5) O 3 "'6 a"), ‘éo For each of these two systems, carry out the following steps. Step 1: Compute b1, b2, b3, and b4 by "running the rule", i.e., iterating the updating filnction. Step 2: Write down the exact solution bn for any linear discrete dynamical system with rate r and constant term 0 and then define each term 'in the exact solution in terms of r, c, and be (If you cannot remember the formula, I will sell it to you for 5 points). Step 3: Calculate b; for each system from the exact solution formula, and compare with Step 1 results. (59—4 if é é: éigéXi) “-9 A =g—,ZC~=$T¢25*/ .9 (A) K? 33»; 2 = 3-1563) a A? =—--.3—(a)- ‘éo / #5 (25points): JMatch solutions of? each of the following discrete- time systems starting from b0=l with one of the three plots on the next page. (a) bt+1= (1/2)bt (b) bt+1= bt(1'2bt/3) (C) by”: (1/2)bt + 1/40 Suggestion: If possible, write down the exact solution of each system, then consider what the solution would look like on linear and logarithmic graphs. Show your work. (0L) I, ' (5) W (C) i @055 .: 'Umeé g {cm were/«z ,7 :3 (c) fl” £476 £0706 7E? {WM/ffl/a/‘r?=) (4) /’ Were/50725 (g) x?” (1 10° 10‘1 1o" 10“3 1 0'5 10° 1 o“ ...
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