Midterm1

Midterm1 - Alex Boisvert Math 33B, Winter ’08 February 1,...

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Unformatted text preview: Alex Boisvert Math 33B, Winter ’08 February 1, 2008, 8:00 AM Midterm 1 Name: ‘ ‘0 Student ID: I, (Y Signature: Check your section: _ 2a _ 21) (Th) TA: Neel Tiruviluamala _ 2c _ 2d (T11) TA: Eric Radke This is a closed-book exam. Do not use notes, books, papers, or electronic devices of any kind. Do all work on the sheets provided. Do not use your own paper or blue books. If you need more space for your solution, use the back of each page; you may request extra paper. Be sure to state clearly if you are continuing on a different page and label the problems well. Do all 5 problems. For full credit, you must show all your work. Do not worry about oversimplifying your answers. Please clearly indicate your final answer, for example by putting a box around it. Problem Out of Points (1) ( points) Radioactive substances decay according to the differential equation N’ = —AN where N (t) is the amount of the substance remaining at time t and A is a. constant1 called the decay constant. (a) Solve the above differential equation to find N(t). fi: —«N\) “:3 31-9 ~gflkd‘trb lhlmi‘ fiAHC l . (b) Show that after a period of T; = X, the material has decreased to 6—1 of its original value. T; is called the time constant. N CW = A e?” y" = e?" “('0le “Shea A64 '2 A08”) «HM; Fs’l‘rvfi— (C) A certain radioactive substance has half-life of 10 hours. Compute its time constant. ,_1 "10} \ F; 015‘ mg?” ' e i? "—326 \ 9/?— ==*\> —\n[2\:“'0> =~> >6: ‘55) [’0 l TX; VA: X101) (2) ( points) Only one of the following difierential equations is linear. Determine which one it is and solve it using any method from class. (a) ’=y+y2‘ (b) y,+ 93 = cost if t2 (c) y’—2ty = cosy (d) y’=t/y (3) ( points) A tank initially contains 100 gallons of pure water. Water begins entering the tank through two pipes: through pipe A at 6 gal/min and through pipe B at 4 gal/min. Simultaneously, a drain is opened at the bottom of the tank1 letting solution flow out at 10 gal/ min. (a) Supervisors quickly discover that the water coming through pipe B is contaminated, containing 0.5 lb of pollutant per gallon. If this process runs for 10 minutes1 how many pounds of pollutant are in the tank after this 10-minute period? (Call this number sup for use in part (b) of this problem) M as: itsel- atats as t. xii—,(mk M “@lri as): 9- tog-b L Qoflx __ 313; H at X — To— ‘ zo'x “’ "TB" «is , “Jr/:0 r' lo :> film—xi: We am: 'xllr\=QO—Aa { 170 =~>A=30 [X 0 =3 will? {10*306 m We, U90} ’Kllol rm (b) The supervisors shut off pipe B after this 10—minute period, allowing pipe A and the drain to function as before. How much pollutant remains after 10 minutes of this new process? (Use the notation 3,, instead of the potentially complicated number from part (a))- (no: at “05 Witt at “inhale- r .2:— it a (K : filo ‘23, *F|O'Oo_q€ :3 SQ, (ucimfihciu-TLHU X fl POD—Lat '25 \“M‘: "3‘3 inhoo-Lrtl 4c, 10. . =‘~> «smiooewfi' 10:09:: WW 1—10 «(oi : M‘OOSVHC-«p A: eye)? “)9 e xiii: «icy-105‘“: (1004M iolq we Wt mm c mp, too"°"’(to') . (4) (-points) Solve the following differential equation. Hint: Check for exactness, homogeneity. If neither holds, assume that the integrating factor is a function of either :1: alone or 3; alone. (59+ y)siuy do: + (xsiny+ cosy) dy = 0 1:; PciX +6443 P ~31; 2 (#330393 +80% n, 3%: owing 330 “BA? emcy“ Nag mi th3‘ _,_,La£’.,m @2921 a. SNCQCéup(B‘-J a): _ 2 Cori : é‘ManZH SHHHW. w-ewmrwé W chiW-Q“ Cifliazor LAWN a 29ng ciaan to oz we}. Rm): $P4x + (9(3) 2) Rag): 'L’xzir gx “9(5). (PM: iSPier= VG” SW " 3“"? (5) ( points) Consider the initial value problem 33’ 2 —:L'2cost, =1 (e) Use the existence and uniqueness theorems to prove that. a solution to this differential equation exists and is unique. rxi; fixra: F’XQCDgL‘ £(x,-E\ 1‘3 0A (b) Solve this differential equation and give the interval of existence of the solution. ...
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Midterm1 - Alex Boisvert Math 33B, Winter ’08 February 1,...

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