Name: Class: Math 7B Date:

Transcription

1 1. Match the given differential equations to their families of solutions. 2. Match the given differential equations and the graphs of their solutions. PAGE 1

2 3. Match the differential equation with its direction field (labeled 1 4). I II III IV PAGE 2

3 4. Evaluate the definite integral: Select the correct answer. a b c d e f Use the substitution and the identity to evaluate Select the correct answer. a. b. c. d. e. PAGE 3

4 6. Evaluate the integral. Select the correct answer. a. b. c. d. 7. Estimate the area of the shaded region in the graph by using the Trapezoidal Rule with n=4. Select the correct answer. a b. 7.4 c d. 6.9 e. 8.5 f. 12 PAGE 4

5 8. Which of the following integrals are improper? Select the correct answers. a. b. c. 9. Find the volume of the solid obtained by rotating the region bounded by and about the y axis. Select the correct answer. a. b. c. 10. Find the average value of the function on the interval.. Select the correct answer. a. b. c. 11. Which equation does the function satisfy? a. b. c. d. PAGE 5

6 12. Which of the following functions is a solution of the differential equation? Check all that apply. a. b. c. d. 13. Which of the given series is(are) convergent? a. b. 14. Which of the given series are absolutely convergent? a. b. 15. For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer). a. b. c. 16. Find the interval of convergence of the series. a. ( 1, 1] b. diverges everywhere c. ( 1, 1) d. [ 1, 1] e. [ 1, 1) PAGE 6

7 17. Find the Maclaurin series for f(x). a. b. c. d. e. none of those f. 18. Find the Maclaurin series for f (x) by any method.. a. none of those b. c. d. 19. Evaluate the indefinite integral as an infinite series.. a. b. c. d. e. f. none of those 20. Find the cosine of the angle between the vectors. (approximate to the nearest hundredth.) a b c a = 7j + 5k b = 5i + 4j + 6k PAGE 7

8 21. For what values of a are the vectors orthogonal? a. 4, 0 b. 0 c. 20 d. 4, 4, The figure shows a vector a in the xy plane and a vector b in the direction of k. Their lengths are and. Find. a b c. 54 d e Find a vector equation for the line through the points and. a. b. c. d. e. 24. Find the distance between the skew lines with parametric equations and. a. 24 b c For what values of const is the following integral improper? Enter your answer as an inequality. PAGE 8

9 26. Evaluate the definite integral: If entering a decimal, round to three decimal places. 27. Evaluate the integral. 28. Use the Trapezoidal Rule to approximate for n = 4. Round the result to four decimal places. 29. Evaluate the following integral if it is convergent. 30. Evaluate the integral. 31. Evaluate the indefinite integral. 32. Set up, but do not evaluate, an integral that represents the length of the parametric curve PAGE 9

10 33. Set up, but do not evaluate, an integral for the length of the curve 34. Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. 35. Find the average value of the function on the interval [7, 10]. Round your answer to the nearest tenth. 36. A spring has a natural length of 12 cm. If a force of 29 N is required to keep it stretched to a length of 22 cm, how much work (in J) is required to stretch it from 12 cm to 20 cm? 37. A heavy rope, 80 ft long, weighs 0.9 lb/ft and hangs over the edge of a building 150 ft high. How much work (in ft lb) is done in pulling the rope to the top of the building? 38. Calculate the center of mass of the lamina with density shown in the figure below. PAGE 10

11 39. Calculate the center of mass of a lamina with density in the shape of a quarter circle with radius 9 as shown below. 40. For what values of r does the function satisfy the differential equation? 41. Use Euler s method with step size h = 0.3 to compute the approximate y value of the solution of the initial value problem 42. Use Euler s method with step size h = 0.6 to compute the approximate value of y(1.8) where y(x) is the solution of the initial value problem 43. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? 44. Find the interval of convergence of the series. 45. Find the interval of convergence of the series. PAGE 11

12 46. Find the Maclaurin series for f (x) Use series to approximate the definite integral to within the indicated accuracy. error < Find the Taylor series for f (x) centered at a = Evaluate the indefinite integral as a power series. 50. Find a unit vector that is orthogonal to both 6i + 6j and 6i + 6k. 51. Find the scalar projection of b onto a. a = <5, 7> b = <10, 8> Round your answer to the nearest hundredth. 52. Use a scalar projection to find the distance from the point P (6, 9) to the line 2x + 17y + 10 = 0. Express your answer in terms of square roots, if necessary. 53. Find the cross product.. PAGE 12

13 54. Let P be a point not on the line L that passes through the points Q and R. Use the formula where to find the distance from the point P ( 0, 1, 3 ) to the line through Q ( 5, 4, 4 ) and R ( 5, 1, 3 ). 55. Find the point of intersection. 56. Find an equation of the plane through the origin and parallel to the plane. 57. Find the point at which the line given by the parametric equations below intersects the plane. PAGE 13

14 ANSWER KEY Name: Class: Math 7B Date: 1., 2., 3.,,, ( 1, 1) PAGE 1

15 ANSWER KEY Name: Class: Math 7B Date: , 4, PAGE 2

16 ANSWER KEY Name: Class: Math 7B Date: PAGE 3