Analysis AB



Math Analysis AB

Midterm Review Problems

1. Find the slope-intercept-form equation of the line that passes through [pic] and [pic].

2. Find the slope-intercept-form equation of the line that is parallel to [pic] and passes through [pic].

3. Given [pic], [pic], and [pic]find

a) [pic] b) the domain of f

c) the range of f d) [pic]

e) the domain of [pic]

4. Convert [pic] to radians.

5. Find two standard-position angles (one with positive measure and one with negative measure) that are coterminal with the standard-position angle having measure [pic].

6. What is the amplitude of the function [pic]?

7. What is the period of the function [pic]?

8. What is the phase shift of the graph of the function [pic]?

9. Find the inverse of [pic]. If the domain of the inverse must be restricted, clearly communicate the domain of the inverse.

10. Describe (in some order, if necessary) the transformations of the parent function [pic] that produce the function [pic].

11. Use the graphs of f and g to complete the tasks that follow.

y = f(x) y = g(x)

Evaluate, if possible (if not possible, write "not possible"):

a) [pic] b) [pic] c) [pic]

12. Graph one cycle of the following functions in the space provided. Clearly show the key x-values in the cycle. Note: I have not given you a y-axis; be sure to add an appropriately-placed y-axis.

a.) [pic]

b.) [pic]

c.) [pic]

13. The terminal side of a standard-position angle [pic] passes through the point [pic]. Find [pic].

14. Without using a calculator, find the exact value of [pic].

15. Without using a calculator, find the exact value of [pic].

16. Without using a calculator, find the exact value of [pic].

17. If [pic] and [pic], find [pic].

18. If [pic] and [pic], find [pic].

19. Given the function [pic], find the intervals over which the graph of f is

a) increasing

b) decreasing

c) constant

Simplify the following.

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

Verify the following

25. [pic]

26. [pic]

27. [pic]

28. [pic]

29. [pic]

Solve the following for x in the interval [pic]

30. [pic]

31. [pic]

32. [pic]

33. [pic]

Solve the following. Find all of the solutions. Express the answers in radians.

34. [pic]

35. sin2 x = 3cos2 x

36. [pic]

Write the following as the sine or cosine of an angle.

37. [pic]

38. Find the exact value for [pic][pic]

39. Find [pic] given [pic]

40. [pic]

41. Solve [pic] given [pic]

42. Solve [pic] given [pic]

Find the area of the triangle given the following information

43. a = 4in, b = 5in, c = 8in

44. [pic]

45. Tony must find the distance from A to B on opposite sides of a lake. He locates point C that is 860 ft from A and 175 feet from B. He measures the angle at C to be 78[pic]. Find distance AB.

46. The Goodyear Blimp is located between two observers who are standing 1.75 miles apart. The angle of elevation from observer A is 33[pic] and the angle of elevation from observer B is 37[pic]. How high is the Blimp?

47. A runner decides to go out for a long run. They start off at their house and run [pic] at a rate of 8 mph for 0.75 of an hour. She then changes her course and decides to run [pic] at a rate of 8 mph for 1 hour. How far is she from where she started? What is the total distance that she ran that day?

48. Write [pic] in standard form.

49. Write [pic] in standard form.

50. Write 12 – 12i in trigonometric form. Express your answer in radians.

51. Write [pic] in trigonometric form. Express your answer in radians.

52. Evaluate [pic]

53. Evaluate [pic]. Express your answer in trig form (approximate radians) and in standard form.

54. Solve [pic] Express your answers in trig form and in standard form. Represent the answers graphically.

55. Multiply [pic]. Express your answer in trig form (degrees) and in standard form.

56. Evaluate [pic]. Express your answer in trig form (use exact radians)

57. Evaluate the following. Express your answers in standard form.

a.) [pic] b.) [pic]

c.) [pic] d.) [pic]

e.) [pic]

58. List the possible rational zeros of the function: [pic]

59. Find all of the zeros of the following polynomials.

a.) [pic]

b.) [pic]

c.) [pic]

60. Find all solutions of the equation: [pic]

61. Write [pic] as a product of factors

a.) irreducible over the rational numbers.

b.) irreducible over the real numbers.

c.) irreducible over the complex numbers.

62. Describe the end behavior for the graphs of the following functions.

a.) [pic]

b.) [pic]

c.) [pic]

d.) [pic]

63. Given [pic], where R is the total revenue in dollars and x is the number of units sold, answer the questions that follow:

a. What is the number of units that produce maximum revenue?

b. What is the maximum revenue produced?

64. The perimeter of a rectangle is 300 feet. What is the width of the rectangle of maximum area?

65. Divide [pic]

66. Find the remainder when [pic] is divided by [pic]. Show 2 different methods for finding the remainder.

67. Write a cubic (degree 3) polynomial function that has zeros -5, [pic], [pic] and a y-intercept of (0,-10).

68. Write a quadratic function with zeros 3, -2 that passes through (4,5).

69. An open box is made from a 16-inch square piece of material by cutting equal squares with sides of length, x, from all corners and turning up the sides. The volume of the box is [pic]. Estimate the value of x for which the volume is maximized.

70. Graph [pic]

Vertex:

Axis of Symmetry:

Max/Min value:

Zeros:

71. Graph [pic]

Vertex:

Axis of Symmetry:

Max/Min value:

Zeros:

Match each graph with the most appropriate equation below.

A B C D

_____ 72.) [pic] _____ 73.) [pic]

_____ 74.) [pic] _____ 75.) [pic]

76. Which of the following are factors of [pic]?

a.) (x – 2) b) (x + 2) c) (x + 12) d) (x + 4) e) (x – 1)

77. According to Descartes’ Rule of signs, [pic] has

a) 1 positive real root b) 2 positive real roots

2 or 0 negative real roots 3 negative real roots

c) 3 or 1 positive real roots d) 2 or 0 positive real roots

3 or 1 negative real roots 3 or 1 negative real roots

78. Which of the following is an upper bound for [pic]?

a) 2 b) -1 c) 4 d) 0 e) 1

79. Which of the following is a lower bound for [pic]?

a) -2 b) -4 c) 0 d) -3 e) 4

80. If [pic] are zeros of a given function, which of the following MUST also be zeros of the function?

a) [pic] b) [pic] c) -1, -3 d) [pic]

Answer Key:

1.) [pic] 2.) [pic] 3.) a. 1103 b. (-∞, ∞) c. [14, ∞) d. [pic]

e. [pic] 4.) [pic] 5.) possible answers: 90º, -270º 6.) 5 7.) [pic] 8.) left [pic]

9.) [pic] 10.) Reflection over the y-axis, Vertical Translation Up 7, Reflection over the x-axis, Horizontal Translation Left 2 11.) a. 4 b. 2 c. 1 12.) a. Key Pts: [pic]

b. [pic] c. [pic] 13.) [pic] 14.) [pic] 15.) [pic] 16.) [pic] 17.) [pic] 18.) [pic] 19.) a. (5,∞) b. (-∞,-2)

c. (-2,5) 20.) [pic] 21.) [pic] 22.) [pic] 23.) [pic] 24.) [pic] 25.) – 29.) solutions will vary

30.) [pic] 31.) [pic] 32.) [pic] 33.) [pic] 34.)[pic]

35.) [pic] 36.)[pic] 37.) [pic] 38.) [pic]

39.) [pic] 40.) 2 Δs:[pic]

41.) No triangle possible 42.) [pic] 43.) 8.182 in2 44.) 222.332 ft2

45.) 841.216 ft 46.) 0.610 mi 47.) 13.748 miles from start; she ran 14 miles 48.) [pic] 49.) [pic]

50.) [pic] 51.) [pic] 52.)

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