Calculus Sample Exam
Calculus Sample Exam Chapter 1: Limits, Continuity, 1 sided Limits, and Limits to Directions: Use the graph to determine the limit, if it exists. 1. lim x 2 3 x 2 3 x, x 2 2. lim x 2 0, x2 3. lim x 4 Directions: Determine the limit algebraically, if it exists 4. lim x 5 5x 5. limtan x 3 x 139 x 2 x 10 x 10 x 2 100 6. lim 7. lim 6 1 cos x x2 x 0 8. Let f(x) = 4x – 5 and g(x) = x2. Find lim f g x lim f g x x 2 x 2 1 x 4 Directions: Find the limit. 9. lim x x 2 3 x x 4 x 2 3x 4 x x 0 10. lim f x x f x x x 0 , where f(x) = 5x – 5 Directions: Use the Graph to determine limits and discuss continuity 11. a) lim x 4 b) lim x 4 c) lim x 4 12. a) lim x 4 b) lim x 4 c) lim x 4 Directions: Find the limit if it exists 13. lim x 49 x 3 1, x 0 14. lim f(x) x 0 x 1, x 0 x 7 x 49 Directions: Determine the values where the function is not continuous. Evaluate whether the points of discontinuity are removable or non-removable. 15. f(x) = -12x2 + 8x + 9 16. f x x x 5x 2 17. f x x 1 x 1 Directions: Find the limit x 6 18. lim x 8 x 8 x 2 x 1 19. lim 3 x 1 x 1 1 20. lim x 2 x 0 x Chapter 2: Derivatives & Related Rates Directions: Match the graph to its derivative. 21. _____ A. 22. _____ B. 23. _____ C. 24. _____ D. 25. _____ E. Directions: Find the derivative. 26. f(x) = -7x2 + 2cos x 27. f x x4 8 28. f x x3 29. f x 30. f x x 5 cos x 31. f x 32. f x sinx 8x cos x 2 x3 2 7 x 3cos x 9 x x2 3 33. f x x 6 4 34. f x x 4 7 9x 4 Directions: Solve 35. A projectile is shot upwards from the surface of the earth with an initial velocity of 126 meters per second. The position function is s(t) = -4.9t2 + vot + so, where vo refers to the initial velocity, and so refers to the starting height. What is the velocity after 6 seconds? 36. A ball is thrown straight down from the top of a 300 foot building with an initial velocity of -22 feet per second. The position function is s(t) = -16t2 + vot + so, where vo refers to the initial velocity, and so refers to the starting height. What is the velocity of the ball after 1 second? Directions: Find dy by implicit differentiation dx 37. x8 + 9x + 7xy – y8 = 4 38. x = tan(x + y) Directions: Solve 39. The radius of a sphere is increasing at a rate of 9 inches per minute. Find the rate of change of the volume when r = 15 inches. 40. A point moves along the curve y = 2x2 + 1 in such a way that the “y” value is decreasing at the rate of 2 units per second. At what rate is “x” changing when 3 x= ? 2 Chapter 3: Applications of Derivatives Directions: Find the critical numbers for the function. 42. f x x 2 x 1 41. f x x 2x 1 4 3 Directions: State why Rolle’s Theorem does NOT apply for the given interval 43. f x 1 x 3 2 , [2, 4] 44. f x x 2 3x , [0, 2] Directions: Determine whether the Mean Value Theorem can be applied. If it can, find f b f a the values of “c” in which f ' c over the given interval. ba 45. f x x 2 2 2x 1 2x 1 , [-1, 3] 46. f x 8 7 , [1,7] x Directions: Find the horizontal asymptote. 47. f x 5x 2 x 3 48. f x 2x 2 6x 1 1 x 2 49. f x 2 x x 3 x 2 Directions: Analyze the function. Identify intercepts, vertical and/or horizontal asymptotes, use the 1st and 2nd derivative tests to and identify and test critical values for possible relative maximum and minimum points. Then sketch a graph. 50. f x 1 x 2 2 Directions: Solve 51. The management of a large store has 1600 feet of fencing to enclose a rectangular storage yard using the building as one side of the yard. If the fencing is used for the remaining 3 sides, find the area of the largest possible yard. 52. A page is to contain 45 square inches of print. The margins at the top and bottom of the page are each 11/2 inches wide. The margins on each side are 1 inch. What should be the dimensions of print so that a minimum amount of paper is used? Directions: Use Newton’s Method to approximate the zero in the given interval to 0.001 53. x3 + 4x + 2, [-1, 0] Directions: Find dy and y for each. 54. y = x3 – 2x when x = 2 and x = 0.1 55. f x 1 when x = 2 and x = -0.01 x Directions: Find dy 56. y x2 2x 2 1 57. y 1 4x 2
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