MATH 141, 141H FINAL EXAM, SAMPLE D 1. (5 points ) Evaluate lim x 4x2 + ln(x2 + 2) . 2x3 + cos2 x 5. (5 points) Find f 0 (x) if f (x) = xsin(x) . a) f 0 (x) = cos(x)xsin(x) \u0013 \u0012 cos(x) + sin(x) ln x xsin(x) b) f 0 (x) = x a) 0 b) 4 3 cos(x) + sin(x) ln x x \u0012 \u0013 sin(x) d) f 0 (x) = + cos(x) ln x xsin(x) x c) ln 2 d) c) f 0 (x) = 5 2 e) The limit does not exist. e) f 0 (x) = sin(x)xsin(x)1 2. (5 points) Evaluate lim x sin x \u0012 \u0013 3 . x 4 Z 6. (5 points) Evaluate a) 1 b) 3 a) 8 1 3 b) 1 + 8 2 c) 1 4 2 d) 1 + 4 2 e) 1 + 8 4 c) d) 0 e) 3. (5 points) Evaluate lim x a) ln (2 + ex ) . 3x 2 3 1 Z 7. (5 points) Evaluate xex dx. 0 b) c) cos2 d. 0 1 3 a) 1 d) ln 2 b) 1 2e c) 1 e e) 1 2 4. (5 points) Evaluate lim e2/x . d) 1 x0 1 e 2 e e) 0 a) 1 b) Z c) 3 8. (5 points) Evaluate 2 d) 32 4 dx. x3 + 2x2 a) 4 ln |x3 + 2x2 | + C x 2 2 + +C b) ln x + 1 x x + 2 2 +C c) ln x x e) 0 d) ln |x + 2| 2 +C x e) 2 ln |x + 2| + 1 1 +C x MATH 141, 141H FINAL EXAM, SAMPLE D 9. (5 points) Which of the following results from an appropriate trigono- 13. (5 points) Determine if the SEQUENCE converges or diverges. If it converges find the limit. metric substitution in Z 2 \u001a \u0012 2 \u0013\u001b x 1 n +2 dx? cos(n) x3 3n2 1 Z a) Z b) sin cos d a) 0 cos d sin3 b) c) 1 Z c) sin d Z sin2 d Z sin2 d cos d) e) d) 2 e) The sequence diverges. 14. (5 points) If the series Z 10. (5 points) Determine whether the integral diverges. If it converges, evaluate it. a) 1 3 0 dx converges or 9 + x2 a) 0 b) 6 X (3)n + 2n converges, find its sum. 4n n=1 4 7 c) 2 b) 0 d) 2 d) 18 c) 10 7 e) The series diverges. 15. (5 points) Find the interval of convergence for the series e) The integral diverges. 11. (5 points) Determine if the SEQUENCE converges or diverges. If it converges find the limit. \u001a\u0012 \u0013 \u001b 5 n 1 n X (1)n (2x 3)n . 2n + 1 n=1 a) I = (1, 2] b) I = (1, 2) c) I = [1, 2) d) I = [2, 1) a) 1 e) I = (1, 1) b) 0 c) e5 16. (5 points) Find the MacLaurin series for the function f (x) = x5 cos d) e5 e) The sequence diverges. 12. (5 points) Determine if the SEQUENCE converges or diverges. If it converges find the limit. \u001a n \u001b 2 + (1)n 3n a) 0 b) X (1)n x4n+7 9n (2n + 1)! n=0 b) X (1)n 3n x4n+5 (2n)! n=0 c) X (1)n 3n x2n+5 (2n)! n=0 d) X (1)n x2n+5 9n (2n)! n=0 e) X (1)n x4n+5 9n (2n)! n=0 2 3 c) 1 d) a) 7 4 e) The sequence diverges. 2 \u0012 \u0013 x2 . 3 MATH 141, 141H FINAL EXAM, SAMPLE D 17. (5 points) Find the Taylor series expansion of f (x) = x2 centered 20. (5 points) Which integral represents the area that lies outside the cardioid r = 1 + sin and inside the circle r = 3 sin ? at a = 1. a) X (1)n (n + 1)(x 1)n n=0 b) X (1)n n(x 1)n n=0 c) X n(x 1)n1 n=0 d) X (1)n n!(x 1)n n=0 e) Z /2 a) X (n 1)x [3 sin (1 + sin )]2 d /6 n n=0 Z 2/3 \u0002 b) Z 18. (5 points) Find a power series representation for f (x) = tan1 x dx x if |x| < 1 . \u0003 9 sin2 (1 + sin )2 d /3 Z /2 \u0002 c) \u0003 9 sin2 (1 + sin )2 d /6 a) X (1)n x=0 X Z x2n+2 +C 2n + 2 X Z x=0 d) X (1)n x=0 /2 \u0002 e) n (1)n [3 sin (1 + sin )] d /6 x2n+2 b) (1) +C (2n + 2)2 x=1 c) /2 d) \u0003 (1 + sin )2 9 sin2 d /6 21. (5 points) Which of the following represents the polar graph of r = 1 + 2 cos ? x2n+1 +C 2n + 1 x2n+1 +C (2n + 1)2 a) b) c) X x2n+1 e) +C 2n +1 x=1 d) 19. (5 points) To which of the following functions does the power series X (1)n x2n converge if |x| < 1? n=0 a) 1 1 x2 b) 1 1+x c) arctan x d) 1 1 + x2 e) cos x 3 e) MATH 141, 141H FINAL EXAM, SAMPLE D 22. (5 points) Find the Cartesian equation of the curve with parametric equations x(t) = et + 1, y(t) = e2t , t > 0. a) x = y 2 + 1, 2 b) y = x + 1, b) Use Taylor's Inequality to estimate the accuracy of the ape 3e x . 2 2 x>0 x>2 d) y = (x + 1)2 , x>0 proximation f (x) T4 (x) when FINAL EXAM- SAMPLE D y<0 1. a 2. b 3. c 4. e 5. d 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. (x e) e)2 e)3 e)4 29. a) t4 (x) 2 3 2e 3e 4e4 b) 1 5 (5 points) find the slope of tangent line to curve: x sin t y cos t) at . is vertical. c) d) for problems 24 28,determine whether given series absolutely convergent, conditionally or divergent. code your answers on scantron as follows: if (1)n1 4n + 3n n (1)n+1 en (2n 1)! nn (3n 1)n (1)n ln write fourth degree taylor polynomial f ln(x) centered e.>1 c) y = (x 1)2 , e) x = y 2 1, 29. (PARTIAL CREDIT PROBLEM - WORTH 10 POINTS) sin(n + 1) n! 4