Exam 2 Key - University of Maryland Nam“ Lfl Baltimore...

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Unformatted text preview: University of Maryland, Nam“ Lfl Baltimore County Id #: Math 152-11 Exam 2 Fall 2011 Monday, November 21, 2011 [2 points] Put your name on the test above, and put a check mark next to your discussion section in the following chart: James Travis, 4:00 - 4:50 James Travis, 7:00 - 7:50 Instructions: Possible Score 0 Read each problem carefully. 0 Write legibly. 0 Show all your work on these sheets. Feel free to use the back of the page. 0 We will not accept answers without justification. o This exam has 9 pages, and 8 questions. Please make sure that all pages are in- cluded. 0 You may not use books, notes or calcu- lators. a You have 75 minutes to complete this exam. Good luck! Math 152-11 Exam 2 Fall 2011 Page 2 of 9 Question 1. [15 points extra credit, 0.5 per blank] Complete the following statements. 1 (a) The p-series Z Z; converges if a? I , and diverges if E 4— l . co 00 Q (h) The geometric series 2 411"“ or 2 ar” converges to ‘1 2." ,1 ‘ n=l n=0 if r'lé ‘ ,and diverges if Ir! 7, l . (c) The Integral Test: Let f (x) be a function that is Cam/flaws , [gets fl , and WC? . Let an = £64 . Then 2 an and l 00] 1) (WM n=l either both converge or both diverge. (d) The Comparison Test: Let 2 an and 2 bn be series with e 0 st I'M terms. 0 If an 4 bn for all n, and 2 b" converges, then 2 an M 5 . o If an a b" for all n, and 2 bn diverges, then 2 an AV—Ugé . (e) The Limit Comparison Test: Let 2 an and 2 b" be series with K as: W terms, A and let c = $3,). “6 . If c is a QQSXW and E “:1! number, then 2a,, and 2 bn either both converge or both diverge. (f) The Alternating Series Test: Let 2 an be an alternating series, i.e., an = (—1)"b,, or an = (—1)"‘1b,,, with bn > 0. If 17,, has the following two properties: bun 4’ ‘3“ , and Nfiu Ix ‘7 9 then ‘A “n" (g) The Ratio Test: Let L = “3,, l1: 0 IfL> l ,thenZan CAM) . o If L < l , then 2a,, aLss‘tsk/lq mvtrgg . OIfL= ,then 51-. rkfl'mi‘éi’ ‘3 NWLLUM __________.______—___—— LM ] (h) The Root Test is exactly like the Ratio Test, except that L = .31., n l M . Math 152-11 Exam 2 Fall 2011 Page 3 of 9 Question 2. [8 points each] 1 1 (a) Find the length of the curve g(y) = % — Zyz on the interval 1 s y S 2. (b) Find the surface area of the surface obtained by rotating the same g(y), 1 S y s 2, about the x-axis. A : 3%‘EM3/eu7 1 LFJI17{% “9J7 taffe‘é‘fly = arkng]? : Lwflj+%)-/%¢ Math 152-11 Exam 2 Fall 2011 Page 4 of 9 Question 3. [8 points each] Find the sums of the following series, or show that they diverge. °° 1 1 — — ' : 'nk t 'al . (a) ;2 (11101 + 1) Inn) (Hint Thl abou part1 sums) H; ”L 53’1"} ill -Ayker 9—- Wm L3 M A; m M Math 152—11 Exam 2 Fail 2011 Page 5 of 9 Question 4. [8 points each] Determine whether the following series converge or diverge. Justify your answers. M 00 0' m i "—3 7’ ‘__\ 5" 3 (a) Zn4+2 < v:4 2: h n=1 AM “A 00 n~3 i “bl-4’1 OWN—‘89 AV} n=2 J00 \ 3 \Jzflkx .Lxm-‘Q (kw-0 ikp is I _ \‘ S vs , in t 95¢“ .2 7({Lx3‘x’f I, {r590 £1 (15% K N it.“ :1, + ‘_\__fi _ x ‘ “M 74m“ 20.231 7 (1&1 We j’L C6030” (mu-059,» . Mil”); Math 152-11 Exam 2 Fall 2011 Page 6 of 9 (_1)n «7m Question 5. [12 points] Determine whether the series 2( is absolutely convergent, conditionally convergent, or divergent. mm») wan w L“: :57;- L“ a WSW [M r l no M M “Leo‘s 4;; i O 4 9“” C S n24 {Sad/S“ \ gran to? :43: me ”‘9‘!" Wm (”if“) Math 152-11 Exam 2 Fall 2011 Page 7 of 9 Question 6. [12 points] Find the radius of convergence and the interval of convergence for °° _ n+1 _ n the power series 2 (—D—(x—i. "=0 (n2)22n—2 B ' V‘M ‘— I‘M «74“7, L th) jal’i’ ll: m‘tlk \{[)y-[% iL.‘ Lyman K J A m ”\M MW LVN/l {L (J) 60.13) zlklril. 4‘ IL. ALL—32‘; a x-3 ”394 11 sz'h‘n \/,J/ W \l W‘%+JT) “ t,‘ / Math 152-11 Exam 2 Fall 2011 Page 8 of 9 Question 7. [10 points] Find the 3rd degree Taylor polynomial for f (x) = Vx + 6 about a=3. 4131.3 Il/lx): fi 13/(3):':: .,» V, v) pox ‘; W 973) - mg W - L, m L l i M" um)” l (3) ma 1 '3 f ,. 4, ‘ '3 Cw” '3 LU”; gummy 33 WWW/[l + My.) Math 152-11 Exam 2 Fall 2011 Page 9 of 9 ”x" '. ":0 11. Question 8. [16 points total] Recall that ex = (a) [8 points] Use the power series given above to find I e"‘4 dx. #1 h {mic gk‘fl A Z jw is J— 1‘50 :jnfiQ/ C ) t X av; 14W ICtzwfflt’C/r‘ (b) [8 points] Use the first two nonzero terms of your answer for part (a) to estimate 1 f e"‘4 dx. How accurate should you expect this estimate to be? 0 x“ " “XV : E, 0‘40 g X S— | , {\KH «v _/x:.7\ r. ——-i :1 W XOCANVXYJ09\S\5\ 7h%?5a1‘*”*“‘3,s. 2+ stu ma" 1% 3s; rm 1 a 4 ...
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