MATH 141, 141E, 141H FINAL EXAM 1. If f is invertible, f (3) = 5, f (4) = 3, f 0 (3) = 7 and f 0 (4) = 11, find (f 1 )0 (3). a) 1/7 b) 1/11 SAMPLE A Z 1 dx. x2 + 3x 4 5. Evaluate the integral a) 1 x + 4 ln +C 5 x 1 b) 1 ln |(x 1)(x + 4)4 | + C 5 c) 1/5 1 ln |(x 1)4 (x + 4)| + C 5 1 x 1 d) ln +C 5 x + 4 c) d) 1/3 e) Cannot be determined e) ln |x2 + 3x 4| + C 2 2. Find the limit lim e x3 . x3 /2 Z a) 0 6. Evaluate the integral 3 sin2 cos3 d. 0 b) c) e2/3 a) 2/5 d) e b) 2/15 e) e2/3 c) 2/5 d) 0 x 3. Find the derivative of arcsin(e ). a) e) The integral is divergent 1 7. Evaluate the integral d) e) Z 4. Evaluate x ln xdx. 1 ex b) 1 e2x c) 2 Z 1 e2x a) ln 2 1 1 b) ln 16 3 e2x 1 c) ln 2 ex 1 ex d) ln 4 3 2 e) ln 4 3 4 ex 1 e2x 1 dx. x2 x2 9 8. For the series 3 +C x x b) +C 2 x 9 + X n=1 a) (1)n determine which of the following staten2 + 1 ments is/are true? (i) The series diverges by the test for divergence. + X 1 (ii) The series diverges by comparison with . n n=1 (iii) The series converges by the alternating series test. x2 + 1 c) +C x2 9 x2 9 d) +C 9x x2 9 e) +C 3x a) Only (i) is true b) Only (ii) is true c) Only (iii) is true d) Only (i) and (ii) are true e) None of (i), (ii), or (iii) is true 1 MATH 141, 141E, 141H FINAL EXAM 9. Find the Taylor series of f (x) = ex centered at a = 3. a) X en (x 3)n n! n=0 b) X e3 (x 3)n n! n=0 13. Find dy/dx for the parametric curve y = tan1 (t2 ), x = ln t. X xn c) n! n=0 d) X (x 3)n n! n=0 e) X exn n! n=0 10. Express f (x) = x as a power series. x+3 a) X (1)n xn+1 3n+1 n=0 b) X (1)n+1 xn+1 3n n=0 c) X (1)n xn 3n n=0 d) X (1)n+1 xn+1 3n1 n=0 e) X (1)n xn+1 3n1 n=0 (R = 3) (R = 3) (R = 3) (R = 3) (R = 3) 11. Write the Maclaurin series of cos 2x. a) 1 x4 x6 x2 + + ... 2! 4! 6! b) 2 2x2 2x4 2x6 + + ... 2! 4! 6! c) 1 16x4 64x6 4x2 + + ... 2! 4! 6! d) 2x 32x5 128x7 8x3 + + ... 3! 5! 7! e) 2x 2x3 2x5 2x7 + + ... 3! 5! 7! 12. Find the polar equation for the parabola y = 1 2 x . 2 a) r = 2 tan b) r = 2 tan sec c) r = 1 cot csc 2 d) r = 1 tan cos 2 SAMPLE A e) r = 2 sin 2 a) t 1 + t2 b) t 1 + t4 c) 2t t + t5 d) 1 t + t3 e) 2t2 1 + t4 MATH 141, 141E, 141H FINAL EXAM SAMPLE A 14. Identify the curve described by the equation r = 1cos , (0, 2]. 15. If lim an = 0, then n + X an is convergent. n=1 a) a) True b) False 16. If + X an is a convergent series with positive terms, then n=1 + X n=1 1 an is divergent. a) True b) b) False 17. lim (sin x)1/x is an indeterminate form. x0+ a) True b) False c) Z 18. The improper integral 2 1 dx converges. x(ln x)2 a) True b) False 19. If the radius of convergence of the power series + X cn (x 2)n is 1, n=0 d) then the series converges for x = 0. a) True b) False For Problems 20 - 24, determine whether each series is absolutely convergent, conditionally convergent, or divergent. Code on your scantron sheet: e) A - if the series is Absolutely convergent, C - if the series is Conditionally Convergent, D - if the series is Divergent. + X (1)n 20. 1+ n n=1 21. + X (1)n+1 n=1 For Problems 15-19, mark (a) or (b) on your scantron sheet. 22. + X n=2 23. + X 3 (1)n1 (ln n)n (1)n+1 n=1 n 2 + 3n 3n n 6 n! MATH 141, 141E, 141H FINAL EXAM Evaluate the following limits. Choose each answer from options a) through e). Note that an answer choice can be used more than once. Be sure to bubble your answer choice on the scantron. Each limit is worth 3 points. a) 1 b) 1 c) 0 d) + e) 24. lim x 2 arctan x SAMPLE A FINAL EXAM 1. B 2. A 3. E 4. D 5. D 6. A 7. E 8. C 9. B 10. A 11. C 12. B 13. E 14. D 15. B 16. A 17. B 18. A 19. B 20. C 21. D 22. A 23. A 24. A 25. B 26. C 27. E 28. R = 2; (4, 8] 29. a)ex = 2 b)ex = 25. sin x 1 cos x lim x0 SAMPLE A Z c) X xn x2 x3 =1+x+ + + , (, ) n! 2! 3! n=0 X x2n x4 x6 = 1 + x2 + + + , (, ) n! 2! 3! n=0 2 ex dx = C + X n=0 x2n+1 x3 x5 x7 = C +x+ + + + , (2n + 1)n! 3 5 2! 7 3! (, ) 26. lim t0+ 30. a) 2 circles, both through pole, 1 symmetric to y-axis, 1 sym metric to x-axis. Intersection points: ( 3, ) and pole. 6 t ln t 6 Z b) A = 0 27. lim t0+ ln t t 28. For the power series X (1)n (x 6)n , n2n n=1 a) (2pts) find the radius of convergence. b) (6pts) Find the interval of convergence. 29. a) (2 pts) Write the Maclaurin series for f (x) = ex in both summation notation and term by term (4 terms). 2 b) (2 pts) Write the Maclaurin series for g(x) = ex in both summation notation and term by term (4 terms). Z 2 c) (5 pts) Express ex dx as a power series in both summation notation and term by term (4 terms). 30. a) (4 pts) Sketch the curves r = 2 cos and r = 2 3 sin on one pair of axes, labeling all points of intersection. b) (6 pts) Set up - but do not evaluate - an integral expression for the area of the intersection of the regions contained inside both curves. 4 1 (2 3 sin )2 d + 2 Z 2 6 1 (2 cos )2 d 2