PDMPDM Midterm Exam Review Chapters 2, 3, 4-1 to 4-4, 6-1 to 6-6

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Chapter 2

1. Consider the function h(x) = x5 –5x + 4.a) Describe the intervals on which h is increasing and on which h is decreasing. Use correct notation.

b) Find any relative maximum or minimum values – present answers as ordered pairs.

2. Solve for x: log1264 + log123x = 2.

3. Solve for x: log12(x+ 5) + log12(x – 5) = 2.

4. Multiple Choice – Which of the described functions is increasing on its entire domain?a) f(x) = x3 + 9 b) c(x) = 10 –

c) t(x) = x2 d) h(x) =

5. Let h be the real function defined by .

a) Use interval notation to describe the domain of h.b) Use interval notation to describe the range of h.

6. Consider the function f, where .

a) What is the ?

b) Write an equation for the horizontal asymptote.

7. Let h be the real function defined by .a) Use interval notation to describe the domain of h.b) Use interval notation to describe the range of h.

8. State the restrictions of each function.

9. Let f be the real function defined by f(x) = x4.a) Use interval notation to describe the domain of f.

b) Use interval notation to describe the range of f.

10. Consider the function .

a) Use limit notation to describe the end behavior of the function p. b) Give equations for any horizontal asymptotes to the graph of y = p(c).

11. If h is an even function and , what is ?

12. Multiple Choice – Which of the following describes the domain of the real function f with rule

?

a) [0, ∞) b) (1, ∞) c) (-∞, -1), (-1, 1), (1, ∞) d) [0, 1], (1, ∞)

13. A vial contains a 300 mg sample of C10, a substance with a half-life of 20 seconds.a) Write a formula for a function that expresses the number of grams of C10 left in the vial after t seconds.

b) Approximately how many seconds will it take for the sample to decay to 200 mg.

14. ) Multiple Choice – Suppose $4500 is deposited in an interest bearing account with an annual rate of 4.5%, compounded continuously. How much will be in the account after 6 years?

a) $4662.00 b) $5860.17 c) $4533.88 d) $5894.84

15. Consider the function g graphed below.

Identify the interval(s) on which g is decreasing.Find any relative maximum or relative minimum values as ordered pairs.Might the function be even, odd, or neither?

16. Consider the continuous function k(x) which has exactly three zeros. Determine if each statement is true, false, or unable to determine.

x k(x)4 -12.55 26 197 -208 -65

It is impossible for a zero to exist over the interval (5, 6). True False Unable To Determine

There exists an x, where 5 < x < 7 where k(x) = 0. True False Unable To Determine

There exists an x, where x > 10 such that k(x) < 0. True False Unable To Determine

17. Graph

Domain: Range: Horizontal Asymptotes:Vertical Asymptotes:Relative Maximums:Relative Minimums:Interval of Increase:Interval of Decrease:

x

y

Chapter 3

18. Solve

19. Solve for all real values of r:

20. Let r and v be the functions defined by and , respectively. Find a rule and state the domain of the given function.

a)

b)

21. Find all zeros of the function f given by the rule f(x) = (log3x)2 – 3log3x – 4.

22. Solve for all real values of k: k8 – 2k4 = -1.

23. Solve m2 – 15 > -2m for all real number solutions.

24. Solve for all real values of z: |2z – 5| > 7.

25. Solve for all real values of z:

26. Solve for all real values of x: 2x3 + 3x2 < 2x.

27. Multiple Choice – Let h(x) = -2x + 1 and k(x) = x2 + 2 for all real numbers x. Compute k h(2).

a) -18 b) -11 c) 11 d) -13

28. Multiple Choice – Given f(x) = x2 + 1 and g(x) = , what is f(g(x))?

a) f(g(x)) = + 1 b) f(g(x)) = c) f(g(x)) = 3x + d) f(g(x)) = + 1

29. Is the inverse of the function a function? Justify your answer.

30. If , then f(3) = 6 and f(5) = -10. Does f(x) have zero between 3 and 5? Explain.

31. On the grid below are graphed the functions f and g. On the blank grid, sketch the graph of the function f + g.

32. For functions f(x) = x2 – 7x and g(x) = 5x – 4.

a) Find the formula for .

b) Find the formula for .

Chapter 4

33. Consider the polynomial p(x) = 4x5 – 7x3 + 2x2 + 1 and the factor x + 5.Is the polynomial evenly divisible by the factor?

34. Use the Quotient-Remainder Theorem for Integers to express 734 divided by 4.

35. A thumb tack manufacturer packages 30 thumb tacks in every box. Suppose 500,000 thumb tacks are manufactured.

a) How many boxes can be filled, each with 30 thumb tacks?

b) How many thumb tacks will be left unboxed?

c) Write an equation in the form of the Quotient-Remainder Theorem to describe the situation.

36. Multiple Choice Which quadratic expression is not factorable over the set of rational numbers?

a) 2x2 + 5x – 18

b) 2x3 + 11x2 + 15x

c) 16x2 – 38x + 12

d) x2 – 8x + 8

37. When n is divided by d, the quotient is 72 and the remainder is 23. Give one possible pair of values for n and d.

38. Find the quotient and the remainder when x3 + 2x2 + 5x – 24 is divided by x – 2.

39. Find the quotient and the remainder when 3x4 + 7x3 – x2 + 14x – 3 is divided by x + 3.

40. Find all zeros and their corresponding multiplicities for the polynomial p(x) = x4 – 6x3 + 17x2 – 28x + 20 given that 2 is a zero of k(x) with multiplicity 2.

41. Find all zeros and their corresponding multiplicities for the polynomial p(x) = 3x3 + 6x2 + 10x.

42. Find a polynomial with integer coefficients of lowest degree that has real zeros at 4, 0.75, and -0.2.

43. One solution to the equation 0 = x4 – 2x3 + 27x2 – 2x + 26 is 1 + 5i. Find all other solutions in the set of complex numbers.

44. Find a polynomial of smallest degree with real coefficients that has zeros 3 and 2 + i.

45. Use long division to divide. 2x5 + x4 – 6x3 + 16x2 – 21x + 10 by x2 – 3x + 5

Chapter 6

46. Convert 126o to an exact radian measure.

47. If and , then _______________.G

48. If and , then _______________.

49. If , what is the value of ?

50. For all , = ______________________.

51. Write an equation for the image of y = sin x under the scale change S:(x, y) = ( x, 2y) followed by the

translation T:(x, y) = (x + 3, y + 7).

52. Prove the identity: sec2x – tan2x = 1 and give its domain.

53. Prove the identity:

54. Given the triangle shown, describe the function values in terms of the triangle’s two legs.

a) sec

b) csc

55. Prove the identity cos(A + B) + cos(A – B) = 2cosAcosB and give it domain.

56. Consider the cosine function.

a) Give an equation for the graph of the image of y = cos x under the transformation

b) Give the amplitude, period, phase shift and vertical shift for your answer to part a).

57. Give an exact value for sin 165.

58. Give an exact value for tan 15.

59. If cos = - and , find cos(2).

60. Write an equation whose graph is a transformation image of the graph y = sinx with amplitude = 4,

period = , vertical shift = -2, and phase shift = 5.

61. Find the exact, simplified value of sin(285o).

62. Find the exact value of .

63. If , and , find .

64. If and , and and are both in Quadrant I, find the exact value of cos( + ).

65. If and , find the exact value of .

66. Find the simplified, exact value of tan(75o) using an angle sum/difference formula.

67. Graph the given function. y = 2sin(2θ + ) – 1

Amp Per Phase Shift Vert Shift

68. Graph the given function. y = 3sec(θ - ) + 1

Amp Per Phase Shift Vert Shift

Now – redo your quizzes and tests!!

Mathematics exams will be administered1ST BLOCK Wednesday, January 30th, 2013.

90 minutes = 1 midterm exam MC + OE Pencils, erasers and BATTERIED calculators ready?

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