222-wilson-04spfin - Mathematics 222 Final Exam Lecture 1...

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Unformatted text preview: Mathematics 222 Final Exam Lecture 1 May 13, 2004 Wilson 0 Write your answers to the twelve 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 circle your final answer to each problem. 0 On the other side of this sheet there are some facts and formulas and a table for undetermined coefl'icients. o Wherever applicable, leave your answers in exact forms (using 77, e, x/g, ln(2), and similar numbers) rather than using decimal approximations. 0 You may refer to notes you have brought in, as announced in class. 0 There is scratch paper on the back of the last page. BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY RE- CEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my calculator” and “I used a formula from the book” are not sufficient substantiation...) Problem Points Score 1 14 2 l8 3 16 4 l8 5 18 6 14 7 16 8 16 9 18 10 20 ll 16 12 16 TOTAL 200 Some formulas, identities7 and numeric values you might find useful: Values of trig functions: Trig facts: 1. tan0 : (523;? 0 sinl9 cos0 tan6 2- secH : c0156 3. sin2 6 + cos2 0 = 1 0 0 1 0 4. sec2 6 = tan2 0 + 1 1 l E fl 6 2 2 3 5. sin(1; + y) : sin(1;) cos(y) + cos(m) sin(y) % V75 g 1 6. cos(1; + y) : cos(m) cos(y) — sin(m) sin(y) _ tan(ac)+tan(y) % é % fl 7 tan(m + y) — 1—tan(m) tan(y) 8. sin2 1; = a1 — cos 2x) % 1 0 7 2 1 9. cos x = 5(1 + cos 2:12) Derivative formulas: Integral formulas: 1. $tanmzsec2m 1.fu”du=n+_1u”+1+0,ifn7é—1 2. fisecmzsecxtanx 2. fidu=1n|u|+0 3. % sin’1 :1: = 11—952 3. f ldfug = sin’1 u + C 4. % tan’1 1; — Hing 4. 1122 : tanil u + C 5. % sec‘1 :1: = Mix/+74 5. fsec(u) du = In | sec(u) + tan(u)| + C 6 %lnx=% 6. fudvzuv—fvdu 7. $ 6“ = 6”” Algebra formulas: 1. ln(:1:y) = ln(:1:) + ln(y) 2. a$+y : a”: ay 3. ab = ebhl“ Terms to use in 3/10, for undetermined coefficients: For a term in f (x) which If Then use a term like is a multiple of sin(k:1;) or cos(kzm) M is not a root of the characteristic equation AcosUm) + B sin(kx) ki is a root of the characteristic equation Ax cos(km) + Bx sin(k:1:) nz n is not a root of the characteristic equation Gem” n is a single root of the characteristic equation Cm e7W n is a double root of the characteristic equation 0x2 e7W A polynomial (112 + bi; + c of degree at most 2 0 is not a root of the characteristic equation 0 is a single root of the characteristic equation 0 is a double root of the characteristic equation a polynomial Dm2 + Em + F of the same degree as (1:172 + bx + c a polynomial Dar3 + E372 + Fm of degree one more a polynomial Dar/'4 + Em3 + Fm2 of degree two more Problem 1 (14 points) Set up an integral to compute the area inside one leaf of the 5—leafed rose 7" = 3 sin(56). Be sure to make clear how you establish the limits of integration. You do not have to evaluate this integral. Problem 2 (18 points) Evaluate the integrals: (a) /e2$ sin(3x) dye Problem 3 (16 points) 9 Use Simpson7s (Parabolic) Rule to evaluate the integral / ($2 + 1) day, using 4 subintervals. 1 Problem 4 (18 points) For the differential equation 3/” — 6y’ + 10y : 0, we can express the set of all solutions as y = 6355(01 cosyc + 02 sin 1:) The first part of this problem asks you find that solution. Hence clearly you will get no credit for simply writing down the solution, credit will come from showing how to find the solution, but at the same time you can check your work and also be sure of proceeding to the last part of the problem. Part I: Find all solutions of y” — 6y’ + 10y = 0. o What is the characteristic (associated) polynomial for this differential equation? 0 What are the roots of the characteristic polynomial? o What role do the roots you found play in writing out the solution 3/ = 6330(01 cosaH—Cg sin ac)? Show where each number in the solutions came from as a part of the roots. Part 11: Find all solutions of y” — 6y’ + 10y = 46406. Part 111: Find the solution of y” — 63/ + 103/ = 46417 satisfying y(0) = 5 and y'(0) = 16. Problem 5 (18 points) Use Taylor’s theorem to estimate the error if we approximate 6% using the terms of the Maclaurin series for em through the term %. Your estimate should take into account the value % at which we are applying this. You may use the fact 6% < 2. Problem 6 (14 points) . 0° —3:1: ” . . For the ser1es E ( ) determ1ne the 1nterval of convergence (convergence set). n n21 Be sure to indicate clearly what happens at any endpoints, and give reasons for your answers. Problem 7 (16 points) The equation 25:162 + 9y2 2 225 describes a conic section. For this curve, find: (a) Where it crosses the ac—axis, if it does at all (coordinates of point(s)) (b) Where it crosses the y—axis, if it does at all (coordinates of point(s)) (c) its focus or foci (coordinates of point(s)) (d) its eccentricity (a number) Problem 8 (16 points) Let 7,? : 2?+ 2; — k and 17 : 3j+ 41$ for all parts of this problem. (a) Find a vector of unit length in the direction of 17. (b) Find cos(6) where 6 is the angle between 11 and 27. (c) Find the scalar projection of 73 upon 17. (d) Find the vector projection of 11 upon 17. Problem 9 (18 points) (a) Find an equation for the plane which passes through (2, 57 —6) and is parallel to the plane 5x — 3y + 22 = 8. (b) Find equations for the line which passes through (2, 5, —6) and is perpendicular to both of the planes in (a). You may express these in parametric or symmetric form. Problem 10 (20 points) For each series, tell whether it converges or diverges. Be sure to give reasons for your answers. 0° n (a) 7 71:1 3 1 For each series, tell whether it diverges, converges absolutely7 or converges conditionally. Be sure to give reasons for your answers. (C) ”2:; 1)n+11000n+1,000,000 (01) 303(4)"; Problem 11 (16 points) Suppose the position of an object is represented by the vector 77(15) 2 cos(2t)?— 3tj+ sin(2t)E at any time if. (a) Where is the object (coordinates of a point) at time t = g? (b) What is the velocity (vector) of the object at time t : g? (c) What is the acceleration (vector) of the object at time t : g? (d) Find equations in symmetric form for the tangent line to the path of this object at t : g. Problem 12 (16 points) 1 —d. x/l—ac2 36 1 (a) Evaluate the integral / —1 1 (b) Evaluate the integral / V1 — 362 due. —1 Hint: This is much easier if you use some geometry... WWW 222222222222? W 2222 £222} $23}: \XJYEYE RY £222 222:2 “E 222222: 2££2£2 222 222233 22‘: W55, YQ Yfixi £222? RWY $23313 i 3:23} 1 WY“ EWEL 2‘2?“ ”$23 SKY}? {.MWWWWN ”fig“ 3:13 KY? ~2 Yfifi YYRQXKRY ~~~~~~~~~~~~~~~ WW: 2222 222££ '1 .. i£222§222£§ Capgw‘ight E) 386?: Unitéd Faatuvw Sgncfiicnm, hm, Scratch Paper (not a. command!) ...
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