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222-wilson-04spex02 - Your Name Mathematics 222 Lecture 1...

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Unformatted text preview: Your Name: Mathematics 222 Lecture 1 Wilson Second Midterm Exam April 20, 2004 0 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 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 coefficients. 0 Wherever applicable, leave your answers in exact forms (using 7r, 6, 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. 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 sufl'icient substantiation...) Problem Points Score 1 12 2 12 3 13 4 l3 5 12 6 l3 7 13 8 12 TOTAL 100 Some formulas, identities7 and numeric values you might find useful: page 2 sin(:1: + y) = sin(:1:) cos(y) + cos(:1:) sin(y) cos(m + y) : cos(m) cos(y) — sin(m) sin(y) tan(w)+tan(y) litan(z) tan(y) Values of trig functions: Trig facts: 1. tan0 : Egg 0 sing cos 0 tan 0 2~ sec0 : 001s 6 ' 2 2 _ 0 0 1 0 3. sm 0+cos 0—1 4. sec2 6 = tan2 0 + 1 1 1 fi fi 6 2 2 3 5. 7r \/§ \/§ 1 T T 1 6‘ ,7 \/§ 1 7. tan(:1: + y) = 3 T 5 fl 8. sin2 x = a1 — cos 2:17) .1 1 0 7 2 9. cos2 1; = a1 + cos 21;) Derivative formulas: 1. d1 tanm : sec2 1; $ 2. % seem : secx tanm 3. % sin—1m : 117962 4. % tan—1m — H1362 5. % sec’1 1; : —m\/:2—71 6. % lnm 2% £696 : ear Algebra formulas: 1. ln(xy) = ln(x) + ln(y) 2. aw‘l'y = a9” ay 3. ab : eblna Integral formulas: 1. funduzn;+1 un+1+C, ifnyé —1 2. fiduzln|u|+0 3.f d“ —sin_1u+C Vliu2 — 4. d“ = tan‘1 u + C 1+u2 5. fsec(u) du : In | sec(u) + tan(u)| + C 6. fudvzuv—fvdu Terms to use in 3/10, for undetermined coefficients: For a term in f(x) which is a multiple of If Then use a term like sin(k:1;) or cos(k:1;) M is not a root of the characteristic equation ki is a root of the characteristic equation A cos(km) + B sin(kx) Ax cos(km) + Bx sin(k:1:) nz n is not a root of the characteristic equation n is a single root of the characteristic equation n is a double root of the characteristic equation 067741: Cm em” 0332 enw A polynomial (M2 + 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 page 3 Problem 1 (12 points) A hyperbola crosses the y—axis at (07 :12)7 and its foci are at (07 :13). (a) Find an equation for the hyperbola. (b) What is the eccentricity of this hyperbola? (c) What are the asymptotes of this hyperbola? (Give equations for the lines.) page 4 Problem 2 (12 points) Find the interval of convergence (convergence set) for the series i0: 3x" ”:1 n2” Be sure to describe the interval fully, including which endpoints (if any) are included in it. page 5 Problem 3 (13 points) Find the area of the region in the plane which is inside the Circle 7" = 3C0$6 but outside the cardioid r = 1 + cos 6. Be sure to tell how you determine the limits of integration. page 6 Problem 4 (13 points) 1 Find the Taylor series for f(x) = ln(l + 3:17), at a = g. You should show the terms through the one with x4 to get full credit: If in addition you give a correct expression for the nth power term in general you will get extra credit. page 7 Problem 5 (12 points) 3 (a) Use the Trapezoidal rule with n = 4 subintervals to approximate / (x2 — 1) dm. 1 3 (b) Calculate exactly / (:1:2 — 1) dx. 1 (c) Give an argument based on the shape of the graph of y = $2 — 1 and the nature of the trapezoidal rule to explain why your answer to (b) is larger or smaller than your answer to (a). Don’t just explain why they are different! Tell why whichever is larger had to be larger. Hint: think about concavity page 8 Problem 6 (13 points) We want to use a polynomial consisting of some beginning terms of the Maclaurin series for sin($) to approximate sin(0.2). Use the remainder term Rn($) from Taylor’s theorem to decide which terms to use if the error must be at most 0.000001 2 10’6. Write out explicitly what the polynomial should be7 i.e. which terms you would use: Don’t just tell how many or what degree. page 9 Problem 7 (13 points) Find all solutions of the differential equation y” + 63/ — 133/ 2 96’1“. page 10 Problem 8 (12 points) An ellipse is parametrized as :1: = 2 sin(t) and y = 4cos(t). Find an equation for the tangent line to this curve at the point where t = f ...
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