162E3-F2007 - i 4 MATH 162 — FALL 2007 — THIRD EXAM —...

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Unformatted text preview: i 4 MATH 162 — FALL 2007 — THIRD EXAM NOVEMBER 15, 2007 — l STUDENT NAME———————————— STUDENT ID INSTRUCTOR RECITATION INSTRUCTOR RECITATION TIME————+——— INSTRUCTIONS 1. Verify that you have 7 pages. 2. The exam has twelve questions, each worth 8.3 points. 3. Fill in the blank spaces above. ' 4. Use a number 2 pencil to write on your mark-sense sheet. I 5. On your mark sense sheet, write your name, your student ID number, the division and section numbers of your recitation, and fill the corresponding circles. _ 6. Mark the letter of your response for each question on this booklet and on the mark- _. sense sheet. ‘ 7. Work only on the spaces provided or on the backside of the pages. 8. No books, notes or calculators may be used. 4A 2-5 as 4.5 sic g. C 7./l\ 78C OLA l0!) (:3 11-0 2 1) (8.3 points) The series Z;I(—1)nn2:1 converges absolutely A) True B) False 2) (8.3 points) Which of the following series cOnverge? 00 1 oo 2 n 00 3 TI. 1):? II); (3) , III): (5) 71:1 71:1 n—1 A) only I" B) only 11 C) only III D) only I and 11 E) only II and III 3 3) (8.3 points) Suppose we know that an 2 2—1713 n = 1, 2, 3, Which statement below must be true? ” 22:1 afl converges B) 22:1 an diverges C) 22:1 an converges, provided Iimnnoo an = 0 _D) 22:1 an converges, provided an 2 an+1 for all n E) None of the statements above is necessarily true 4) (8.3 points) Suppose we want to. approximate the sum of the series Sill—1 "£3 1 by the sum 3m = Em (-1)";‘% of the first m terms. By the theory of alternating series, n=1 the error will be less than 10‘3 provided m = A) 4 B) 5 C) 6 D) 7 E) 10 4 5) (8.3 points) The series i 5%" . -——_ k IS =1 (2k 1)2 A) convergent because 11m;MOO fly; = 0 B) divergent because lim;woo $35,; is not equal to zero C) convergent by the root test D) divergent by the root test E) divergent by the ratio test 6) (8.3 points) The series is A) convergent by the root test B) convergent by the integral test C) divergent by the ratio test D) convergent by the ratio test E) none of the alternatives above is correct 7) (8.3 points) Which of the following series converge? °° 1 °° ‘ 1 0 {Emma I” 33mm A) only I B) only II C) neither D) bOth E) I converges conditionally and II converges absolutely 8 )(8.3 points) The interval of convergence of the series 00 n($ —- 1)n - 2(4) 477. 3n ls n=1 A) [—2,zl] B) (—3, 3) C) (—2, 4] D) [—2, 4) E) (——3, 3} 6 9) (8.3 points) Which of the following is a power series representation of the function 1 .— =———? f(x) x2—2a:+2 A) Z;o(*1)"($ ~ 1)2" B) 22:0(13 “ DZ“ 0) E” (wig—rt n=0 3n+1 WZ%PW%¥ E) Z?=o(-1)"($_- 1)” 10) (8.3 points) Let f be the fimction which is represented by the power series. I rm=2emfififi The third derivative of the function f at a: = 4 is equal to .A)1/9 B) 42/3 C) 1/27 D) —1/9 E) 1/25 7 11)(8.3 points) Let f(:z:) be a flmction such that f’ = :32 C031: and that. f(0) = 0. The Maclaurin series of f is ‘ ’ A) Z;o(—1)”(ZTJ§§3—(2n—). B) E;o(-1)"(—27% 0) ZZo(-1)”(—2,£TT(Z;)7 D) 2%(-1)"(§;—$ 3n+1 E) 210(4)"?3mfi 12)(8.3 points) The first three terms of the binomial series expahsion of ’ f(a:) = (1 + 2.7:)‘i are A) 1—%x+§a:2 1—ix+§5§az2 C)1—%a;+-§-x2 D) 1~§m+§mz E) 1-%x+§33x2 ...
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