hw1s - MAT 1330 Fall 2011 Assignment 1 Due September 28 at...

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Unformatted text preview: MAT 1330, Fall 2011 Assignment 1 Due September 28 at the beginning of class. Late assignments will not be accepted; nor will unstapled assignments. Instructor (circle one): Robert Smith? Jason Levy Olga Vassilieva Catalin Rada DGD (circle one): 1 . 2 3 4 Student Name Student Number By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. Signature QUESTION 1. A patient’s metabolism will break down morphine to 60% of its original amount every day. The patient is given 8 units of morphine to raise to concentration of the medication at the end each day. Let Mt denote the concentration of morphine at time t. (a) Write down the associated DTDS and indentify the updating function. (b) If the patient body contains Mt = 5 units of morphine at the start of the day, write down the amount of the medication at the end of the day and at the end of the next day. End of the day: End of the next day: (c) Compose the updating function with itself and find the two step DTDS. fofUz/t) =f{§(Me))= ff0.6/%—+¢v): 0.6-(O.6M£ +2) +8 = a 36 m. mm ((1) Find the backwards DTDS (the inverse of the updating function) and use it to find the - value at the previous time (t = 5), knowing that Ms : 11. Value at t = 5: Mi,” : +8 €0.6/lfg» =Méf/ “J /% = wng—f : f "my; 4:52;; (6) Find the equilibrium point of the dynamical system. Equilibrium point: D (f) Find the general solution algebraicall and use the formula to find the concentration of E? 0i medication at t = 1, 2, 3, 4 starting at o = . f it «new, man) _ _ General solution, MI: 0-5/Vo *3 = 0‘60 +4? =3 M2 = +59 : OvéL‘Ofi/‘h *J’J‘Fc?‘ (0,6)‘2/‘10 +ag.§+ (P M5: 0.6M; +39: aéféléia/L/o *a6'<?*c?,]+g =‘ = C0. 03/910 749.048 Hag.ng ,' A 2.4-4) , l .4940”: Me = (M1740 «DAYS/‘0” f4”) 4"” + ' + 3‘96 : (QtéysMa 7L 8£@‘6t—I£+ (0’6) 2.1.24. .. / - 2" 2 gagfyo + 9[ $3979.]: W0 *JOE/‘@'6)j Mas), M4 539 [/—(0,6)2.7 flow” M3 :- gzo£ /[email protected],6)3.7’ may } /L/¢:go[/{aa)"]=1% woo” (g) Draw the solution of the DTDS with the four iterations obtained in (i) Draw the cobweb diagram of the DTDS starting With Mo = 0 (four iterations are enough). {3M Mada/"(71% £LLDL¢téO M (£16m about {-9 blewJ. (j) Determine the stability of the equilibrium point using the cobweb diagram. QUESTION 2. For a bacteria colony that takes 3 hours to double in size, it took the colony 12 hours to reach the size of 2,000,000 bacteria. How long did it take to reach 1,000,000? Answer: b£+I=Vbt bt=Fibo abo=rsbo3 ail; =r57r’: = ’2. a e t l ‘ Qz‘bo‘g9b" §> fig“) = 33-50 15:15: 5 [DO a 521,0 ala’M‘OOO 6 z (gig-[)0 =QE-Ic7rooo 252% 22> £=9£ow€ QUESTION 3. Consider an experiment where salt crystals are grown in a supersaturated solution. The mass of a. crystal grows over 24 hours based on the updating function mt+1 Z = Given that the crystal originally has a mass of 10 grams, express the solution to this system, both in a table (four calculations are enough), graphically and as a formula. Mo=lO m2: /,5~.m,: /,$.IS‘=23.S“ (Vb = Lymg = L? 22:23: 33, 7A? mit: QfS—yv’flo. 3bn QUESTION 4. A population grows according to the formula I) . 1 + 0.2bn n+1 :2 (a) Find all the equilibrium points of the DTDS. % , 96 - Equilibrium points: 13 1’0 [9340 (b) Assume that we start with b0 2 100. What will be the size of the population after the 1st, 2nd, and 3rd year. .. 3bo ‘ 3.100 51 ~ Haabc ‘ [+0.3-IOO '37"! 132: 3b! 5-H . ——-———— e: News, “ Hus-W IL” 2’0 6+MWPVL¥L¢6P cot/pau'téi (e) Sketch the solution curves obtained in (b), (c), (d) in the same tbt—plane. Give short comments about the above results (three sentences are enough). ‘HLL (WWI/thou gl’ze 4Qch I giuu L‘i‘S {la/(Hal size W ! abwvf 25M earrgd‘uép capacnloc, ‘— Hm cw Pig/{Hg au'w‘g, , .. . [Am T {+4 \Dolxumuflou $126 [mm/@494, (flue 2,5211) IMHOLHtar l 'HUL (NDMIGDEIOQ MM be/oa/ 2+9 CMV‘fz‘u/afi aapau‘tfi . ...
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