Sol_Mid2 - UNIVERSITY OF WATERLOO TEST # 2 FALL TERM 2011...

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Unformatted text preview: UNIVERSITY OF WATERLOO TEST # 2 FALL TERM 2011 Student Name (Print Legibly) (FAMILY NAME) (GIVEN NAME) Signature Student ID Number COURSE NUMBER MATH 128 COURSE TITLE Calculus 2 for the Sciences COURSE SECTION 001 DATE OF EXAM Monday, November 14, 2011 TIME PERIOD 17:30 — 18:50 DURATION OF EXAM 80 minutes NUMBER OF EXAM PAGES (Including this sheet) 8 INSTRUCTOR Koray Karabina EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED NONE (NO CALCULATORS) Notes: Marking Scheme: 1. Fill in your name, ID number, section, and sign the paper. Don’t write formulas on this page. 2. Answer all questions in the space provided. The last page is for rough work. Check that there are 8 sheets. Your grade will be in- fluenced by how clearly you express your ideas, and how well you organize your solutions. $.03 MATH 128 — Test # 2 Fall Term 2011 Page 2 of 8 1. Find the solution of the differential equation dy 2 _ i g + 21:— . y _ {1:2 that satisfies the given initial condition y(1) = 0 and x > 0. TW‘S \‘5 a 06W, Q54: (Vb/{A X x AK >< I; w“ J i I“ ’x ‘1‘ ) We, ml: 1 1’ if? t. v 2 2, [x Y 1 : C : C 1 X .. l y l fawn.) ll 5/ x‘ ‘ M 4.x‘tU‘lpt71l/x6 { 13W. 5 529/9 )l’ lam fills f" g U {Mia 06 him 2 l « l X fi i fix fj ' dv ( y’L v 4' l0 ._) « e l >2 a * l f” ’ X " C l x l f C, i— ) "M “’2”, "l y X 2" ( X C/ X X out/é. film/C {J ( D : 0 W k 1 l C O . g C W «V M 4 w.» *” ll ([3 l ( a? {We “ MATH 128 — Test # 2 Fall Term 2011 Page 3 of 8 2. Consider the logistic differential equation for population growth dP P Brian?) where t represents the time, P(t) is the size of the population at time t, k is the growth rate, and K is the carrying capacity. Find a formula for P(t) by solving this logistic equation with k = 1, K = 1000, and P(0)=100. Since 16,! 0mg If: (00:) ,i {flaw M - l” w 3> i it (Lagemfl 7g moo laws) J“; t («Rymtlfifl (NW 5/1164 Y/‘f/KD . ‘ ‘ l. M Ex :l L ~{ \ E (:50) l ‘ low l3 (“ODD ,9 > a) 5 4%“ (000 "l9 ti 6 ‘l 5A lPl - 1% llOOO"Pl : Z“ 1000’? P fit, C ~ C 0 M! (1,1. (milk; "3 c lOOOJ? a , SMALL 0W 1] / > I) ‘ :lOOJ 7) P5 MOD :7 lad) 49/ O j) W“? >,<> a ‘M :(B‘BLP rt c \m m I P l ? o/«MAffl l) MATH 128 - Test # 2 Fall Term 2011 Page 4 of 8 3. (a) Find the length of the parametric curve C': m=etsint, yzetcost, 031532. You may use (without proving) that C is traversed exactly once as t increases from 0 to 2. , 2 2 (b) Show that (if—em, —\/—_e"/4 9 2 ) is a point on the parametric curve C’: mzetsint, y=etcost, 031532, where the tangent is horizontal. . m i _‘ l v' H " r”, M t: 11 m X: e «n , C L L, ‘ MATH 128 — Test # 2 Fall Term 2011 Page 5 of 8 4. (a) Sketch the polar curve C: r=2+cos€, 030$27r by considering four portions: 1. As 6 increases from 0 to 7r/ 2, 2. As 6 increases from 7r/ 2 to 7r, 3. As 6 increases from 7r to 37r/ 2, 4. As 0 increases from 37r/ 2 to 27r, Your sketch should clearly indicate the direction of the curve C, and the order in which the portions are traced out. PM Gr" 65"?“ ',. MATH 128 - Test # 2 Fall Term 2011 Page 6 of 8 4. Let- C be the polar curve C: 7'=2+0050, 039$27r, as in part (a). Find the area enclosed by C. cos(26) + 1 9 .4 (You might find the following identity useful: cos2 9 = ) FMM 0w MD. M (a) , M 564. maul 4% Wow (4 (M? KACL‘DC’A (“w AM (S V 2T\' 2 n L i If? «I LMW £8 /\ ;; L [QQHM 20116003 > 2 <3 a we 93”” 2W x _ NW (27 + ouilglg 1p 10); 5 ) 0W - L6 / 0 I Z“ ...
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Sol_Mid2 - UNIVERSITY OF WATERLOO TEST # 2 FALL TERM 2011...

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