Math 132 Online - Additional Practice Questions - Answers(1)

Math 132 - Final Exam Practice Questions - Answers 1. Which of the following represents the volume of the solid generated when the area of the region enclosed by y = 2 x and y = 2 x 2 is rotated around the line y = 3? (A) π Z 1 0 ( (3 - 2 x ) 2 - (3 - 2 x 2 ) 2 ) dx ( B ) π Z 1 0 ( (3 - 2 x 2 ) 2 - (3 - 2 x ) 2 ) dx (C) π Z 2 0 3 - y 2 2 - 3 - r y 2 2 ! dy (D) π Z 2 0 3 - r y 2 2 - 3 - y 2 2 ! dy 2. Suppose that the temperature at time t is rising with a rate of 6 t 2 + 2 degrees per minute. If the initial temperature if 0 ◦ C, what is the tem- perature after 10 minutes? (A) 602 ◦ C (B) 120 ◦ C (C) 524 ◦ C ( D )2020 ◦ C 3. O ( t ) represents the annual rate of world consumption of oil (in trillions of barrels) at time t , where t represents years since 1999. Assuming that the world rate of consumption was 80 trillion barrels consumed before 1999, which of the following expressions represents the amount of oil consumed between 1999 and 2010? (A) O 0 (11) ( B ) Z 11 0 O ( t ) dt (C) R 11 0 O 0 ( t ) dt (D) 80 + R 11 0 O ( t ) dt 1

4. Evaluate the following: d dx Z √ x 1 14 t 9 dt (A) 28 3 x 6 (B) 7 5 x 6 - 7 5 ( C ) 7 x 4 (D) 14 x 9 / 2 5. Evaluate the integral: Z 1 t 2 sin 2 t + 6 dt (A) - 1 2 cos 2 t + 6 + C ( B ) 1 2 cos 2 t + 6 + C (C) 2 cos 2 t + 6 + C (D) - cos 2 t + 6 + C 6. Recall that a series is conditionally convergent if it is convergent but not absolutely convergent. ∞ X n =1 ( - 1) n n p For which values of p is the above series conditionally convergent? (A) 0 < p < 1 (B) 0 < p ≤ 1 (C) p > 0 (D) p > 1 7. Which of the following is not an improper integral? (A) Z 1 0 1 x dx (B) Z ∞ -∞ x sin( x ) dx (C) Z 4 0 x e x - 1 dx (D) Z 2 - 1 2 x 2 + 1 dx 2

10. Which of the following are true for the Test for Divergence: I. If lim n →∞ a n does not exist, then ∞ ∑ n =0 a n diverges. II. If lim n →∞ a n exists but is not 0, then ∞ ∑ n =0 a n diverges. III. If lim n →∞ a n = 0, then ∞ ∑ n =0 a n converges. (A) I & II (B) I & III (C) I, II & III (D) II & III 11. Given that a Maclaurin series for sin( x ) is ∞ X n =0 ( - 1) n x 2 n +1 (2 n + 1)! , find the Maclaurin series for f ( x ) = x 4 sin( πx 4 ). (A) ∞ X n =0 ( - 1) n π 2 n +1 x 8 n +5 (2 n + 1)! (B) ∞ X n =0 ( - 1) n π 8 n +4 x 8 n +8 (2 n + 1)! (C) ∞ X n =0 ( - 1) n π 8 n +1 x 8 n +5 (2 n + 1)! (D) ∞ X n =0 ( - 1) n π 2 n +1 x 8 n +8 (2 n + 1)! 12. Find the interval of convergence of the power series: ∞ X n =1 ( x - 5) n n ! (A) I = { 0 } (B) I = [4 , 6] (C) I = ∅ (D) I = ( -∞ , ∞ ) 4

13. Consider the region R enclosed by curves y = x 2 + 5 and y = 2 x + 5. (a) Sketch the region R . Find and label the intersection points. (b) Find the area of the region in part (a). 4 3 (c) Find the volume of the solid obtained by rotating R around the x axis. π - 32 5 + 24 = 88 π 5 Evaluate the following integrals. 14. Z √ x sin(1 + x 3 / 2 ) dx = - 2 3 cos(1 + x 3 / 2 ) + C 15. Z 4 x 4 ln( x ) dx = 4 5 x 5 ln( x ) - 4 25 x 5 + C 16. Z 3 1 x 2 ln(3 x ) dx = 9 ln(9) - 1 3 ln(3) - 26 9 17. Z 2 x 3 √ x 2 + 4 dx = 2 3 ( x 2 + 4) 3 / 2 - 8 p x 2 + 4 + C 18. Z π/ 3 0 tan 3 ( θ ) sec 3 ( θ ) dθ = 31 5 - 7 3 = 58 15 19. Z t √ 25 t 2 - 1 dt = 1 25 p 25 t 2 - 1 + C 5

28. Determine whether the series is convergent or divergent. Clearly state which convergence test you used. ∞ X n =2 3 n (ln( n )) 2 Convergent - Integral Test 29. Determine whether the series is convergent or divergent. Clearly state which convergence test you used. ∞ X n =1 ( - 1) n 5 n 4 n + 7 Divergent - Alternating Series Test 30. Determine whether the series is convergent or divergent. Clearly state which convergence test you used. ∞ X n =1 6 n 5 n 2 + 4 n + 7 Divergent - Comparison Test/LCT 31. Determine whether the series is convergent or divergent. Clearly state which convergence test you used. ∞ X n =1 7 n 8 n + 3 Convergent - Comparison Test 32. Determine whether the series is convergent or divergent. Clearly state which convergence test you used. ∞ X n =1 n ! 20 n · 6 2 n +1 Divergent - Ratio Test 33. Using the Alternating Series Test, we can see that ∞ X n =1 ( - 1) n +1 n 2 converges. How many terms do we need to estimate the sum with error less than 0 . 001? 31 7

34. Determine whether the series is absolutely convergent, conditionally con- vergent, or divergent. Clearly state which convergence test(s) you used. ∞ X n =1 ( - 1) n n 2 2 n Absolutely Convergent - Ratio Test 35. Determine whether the series is absolutely convergent, conditionally con- vergent, or divergent. Clearly state which convergence test(s) you used. ∞ X n =2 ( - 1) n +1 n 2 n 3 - 3 Conditionally Convergent by the Alternating Series Test Not Absolutely Convergent by the Comparison Test (Ratio Test Fails) 36. Evaluate the integral. Z ∞ 1 e - √ x √ x dx Convergent to 2 e 37. Evaluate the integral. Z 1 0 1 (1 - x ) 2 dx Divergent 38. Consider the two parabolas y = x 2 - c 2 and y = c 2 - x 2 . Find the value of the constant c such that the area of the region bounded by the parabolas is 576. c = 6 39. For the following power series, find the radius and interval of conver- gence. ∞ X n =0 3 n ( n + 1) 3 (2 x - 1) n R = 1 6 , I = 2 6 , 2 3 8

45. Find the power series representation of the function and determine the radius of convergence. f ( x ) = x 9 + x 2 ∞ X n =0 ( - 1) n x 2 n +1 9 n +1 ; R = 3 46. Find the power series representation of the function and determine the radius of convergence. f ( x ) = x (1 + 4 x ) 2 ∞ X n =0 ( - 1) n 4 n ( n + 1) x n +1 ; R = 1 4 47. Eliminate the parameter to find a Cartesian equation of the curve. x = e 2 t y = t + 1 y = 1 2 ln( x ) + 1 48. Find the Cartesian coordinates of the point. a.) ( - √ 2 , 5 π/ 4) = (1 , 1) b.) (2 , - 3 π/ 4) = ( - √ 2 , - √ 2) 49. The Cartesian coordinates are given. Find two representations of this point in polar coordinates. a.) (2 , - 2) = (2 √ 2 , 7 π/ 4) , ( - 2 √ 2 , 3 π/ 4) b.) ( - 1 , √ 3) = (2 , 2 π/ 3) , ( - 2 , 5 π/ 3) 50. Find the Cartesian equation for the polar curve r = 5 cos θ x - 5 2 2 + y 2 = 25 4 51. Find the Polar equation for the curve represented by the following Carte- sian equation: y = 1 + 3 x r = 1 sin( θ ) - 3 cos( θ ) 10

52. Consider the parametric equations below. x = sin 2 t y = - cos t 0 ≤ t ≤ π (a) Draw a graph of the curve defined by this set of parametric equa- tions. Clearly label at least five ordered pairs ( x, y ) that have been plotted. Draw an arrow from the initial point to indicate which direction the curve is being traced. Points: (0 , - 1) , (1 , - √ 2 / 2) , (0 , 0) , ( - 1 , √ 2 / 2) , (0 , 1) Graph: (b) Find the equation of the line tangent to the curve at the point - √ 3 2 , 1 2 . y = - √ 3 2 x - 1 4 (c) Write an expression for the arc length of the curve as a definite integral, but do not evaluate the integral. L = Z π 0 p (sin( t )) 2 + (2 cos(2 t )) 2 dt 11