222-wilson-05spex01 - page 1 Your Name Circle your TA’s...

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Unformatted text preview: page 1 Your Name: Circle your TA’s name: Adam Berliner John Bowman Chris Holden Eugene Tsai Dan Turetsky Mathematics 222, Spring 2005 Lecture 4 (Wilson) First Midterm Exam February 15, 2005 Write your answers to the eight problems in the spaces provided. If you must continue an answer somewhere other than immediately after the problem statement, be sure (a) to tell where to look for the answer, and (b) to label the answer wherever it winds up. In any case, be sure to make clear what is your final answer to each problem. Wherever applicable, leave your answers in exact forms (using g, x/3, cos(0.6), and similar numbers) rather than using decimal approximations. If you use a calculator to evaluate your answer be sure to Show what you were evaluating! You may refer to notes you have brought on an index card, as announced in class and on the class website. There are also some formulas given on the other side of this sheet. BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY RECEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my calculator” and “I used a formula from the book” (without more details) are not sufficient substantiation...) Problem Points Score 1 l2 2 l3 3 13 4 13 5 13 6 l2 7 l2 8 12 TOTAL 100 page 2 Some formulas, identities, and numeric values you might find useful: Values of trig functions: 9 sin6 cos6 tan6 0 0 1 0 E 1 fi fi 6 2 2 3 E Q Q 1 4 2 2 fl 2 NIH a (NI-=1 Derivative formulas: 1. d1 tanx = sec2x {r 2. % seem 2 seem tanx d - —1 _ 1 3. dac sm :3 — 1%? i —1 _ 1 4 dac tan .76 1H2 Algebra formulas: 1. ln(my) = ln(:I;) + ln(y) 2. awry = a“ ay 3. ab : 6blna Trig facts: 1. 2. tan6 = Sing c050 sec 6 = 1 cos0 . sin2 6 + cos2 6 = 1 .sec262tan26—1—1 . sin(x + y) : Sjn(x)cos(y) + cos(x) sin(y) cos(x + y) : COS($)COS(9) _ sin(x) sin(y) _ tan(x)+tan( ) tan(96 + y) — WW sm2 :3 = 5(1 — cos 2%) Integral formulas: n _ 1 n ‘ 1. fu dU—n—Hu+1+C,1fn7E—1 2. fidu=ln|u|+0 3.f\/1‘1:‘T=sin’1u+0 4. f 11:2 2 tan’1 u + C 5. fsec(u) du = in l sec(u) +tan(u)l +0 6. fudvzuv—fvdu page 3 Problem 1 (12 points) Evaluate the integral: / sin2(x) C082(5L‘) dx. page 4 Problem 2 (13 points) 3fix/x2—9d — x Evaluate the integral: / 3 96 page 5 Problem 3 (13 points) Evaluate the integral: / x2 sin(x) dx page 6 Problem 4 (13 points) dx 4 2 Evaluate the integral: / ( m + 595 + 3 m page 7 Problem 5 (13 points) 1 V1—x2. At the right is a graph of Evaluate the integral /1 dx —1 x/l — x2- page 8 Problem 6 (12 points) {11‘ 71‘ (a) Evaluate the limit 11111 e,—. 1H0 4 sm(x) —6 (b) Evaluate the limit lim 53(2/353). {L‘HOO page 9 Problem 7 (12 points) (a) Consider the sequence an 2 3 — %. Give an argument to show that {can} has a limit. You should either justify this using carefully the definition of the limit of a sequence or by appropriate use of a theorem about sequences from the book. (b) The first several terms of a different sequence {an} are 1, fi, fi, fi, . . .. (1) Find a formula giving an as a formula involving 71. (ii) This sequence does converge. What is its limit? Show all of your work. page 10 Problem 8 (12 points) For each series, tell Whether it converges or diverges. If the series converges, tell What its sum is. Be sure to show your work! 0° n—l (a) ”Egon—kl- ...
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