Winter2002 Midterm

Winter2002 Midterm - Math138 — W02 — Midterm Page 1...

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Unformatted text preview: Math138 — W02 — Midterm Page 1 of (pages) REPRINTED BY MATHSOC 1x1 Instructors: P. Balka, S.A. Campbell, B. Ferguson, P. Kannappan, A. Kempf, K. Rohlf, C. Struthers [8] 1. (a) Solve the differential equation (b) Find the solutions corresponding to each of the initial condtions y(0) = 2 and y(1) 2 0. [12] 2. (a) Give the precise statement of l’Hopital’s Rule. (b) Evaluate the following limits, justifying your steps. $2 i. lim ————-— w-rO ex —- 6—3 tan 5x ii. lim , w—Hr 8271 3:8 . r 111. hm xr 1:400 [12] 3. Determine Whether each of the following integrals is convergent or divergent. Justify your answers. 00 dx (8” / w[ln(w)]2 3 1 (b)/O ($_1)3dx [15] 4. (a) Define what is meant by the following statements: - 00 u 1. The sequence {an LL21 lS convergent. 00 ii. The series E ak is convergent. kzl (b) Determine whether each of the following is convergent or divergent. Justify your answers. 00 - 2n 1. {In (W) }n:1 ii- {(—1)"%}Z°=1 i1 Zn 111. ”:1 n n+1 Math138 — W02 — Midterm Page 2 of (pages) REPRINTED BY MATHSOC l><l 0° 2n+1 _ 3n IV. 471, n=1 °° 2 v.;n2+1 [6] 5. (a) State the Remainder Estimate for the Integral Test. 00 (b) You would like to estimate the sum, 3, of the convergent series 2 [6:1 1 E473 using its nth partial sum, 3". What value of 71 should you choose so that s —- 8n < 0.1? [101 6. The following differential equation dP — 2 2 — P P — 1 d, < >< ) models the size of a fish population, where t is measured in weeks and P in thousands of individuals. (a) For what values of P is the population increasing? Decreasing? (b) If the intial population is 1500, Le. P(0) : 1.5, find the population P(t) at any timet Z 0 (c) Find lim P(t). t—)oo [12] 7. State wether each of the following is True or False. If true prove it, if false give a counterexample: (a) y : 06“” is a solution of the differential equation y” + y = 56—3” for every value of C. (b) If the sequence {an }::1 is bounded and increasing, then it is convergent. (c) If lim an : 0 then 2a,, converges. n—>oo 71,21 00 00 (d) If 0 3 an 3 bn for n : 1,2,. ..and Z an is convergent, then 2 bn is convergent. 71:1 71:1 ...
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