Study Guide for Math 3 Final Semester 2

(1)

Simplify the expression. 1.

2.

3.

____ 4. Which exponential function is shown in the graph below?

2 4 6

–2 –4

–6 x

2 4 6 8 10

y

a. c.

b. d.

Rewrite the equation in logarithmic form. 5.

Evaluate the logarithm. 6.

7.

8.

(2)

10. Simplify .

Find the inverse of the function. 11.

12.

Write a rule for g that represents the indicated transformations of the graph of f.

13. ; reflection in the y-axis followed by a translation 4 units right

14. ; translation 8 units right followed by a vertical stretch by a factor of 5

15. The height H (in centimeters) of a container being printed by a 3-D printer is related to the radial position r

(in centimeters) by the equation . Describe the transformation of

represented by H. Then use the function to determine the height of the container when the radial position is 0.5 centimeter.

Expand the logarithmic expression.

16.

17.

Condense the logarithmic expression.

18.

19.

20. Use the change-of-base formula to evaluate .

____ 21. The amount of time T (in years) it will take for a population of a certain species to grow from 1550 to a can

be modeled by the function . Which function is equivalent?

(3)

b. d.

22. The amount of time (in years) it takes for an initial investment of p0 to grow to a value of p can be modeled by

. How long will it take the initial investment to triple in value?

Solve the equation.

23.

24.

25.

26.

27.

Newton’s Law of Cooling states that for a cooling substance with initial temperature , the

temperature T after t minutes can be modeled by , where is the surrounding temperature and r is the cooling rate of the substance.

28. You remove a baked potato from a 400°F oven and place it on the counter in your 67°F kitchen. After 20 minutes, you measure the temperature of the potato to be 128°F. What will the temperature of the potato be after 26 minutes?

Solve the inequality. 29.

30.

31.

32.

33. Write an exponential function whose graph passes through (1, 3) and (2, 15).

____ 34. Which statements are true?

(4)

b.

c. The equations and are equivalent. d. The equations and are equivalent. e.

____ 35. Which equations are correct? a.

b.

c. d.

e.

36. The variables x and y vary inversely, and y  –2 when x  5. Write an equation that relates x and y. Then find

y when x  –7.

37. The variables x and y vary inversely, and y  9 when x  2. Write an equation that relates x and y. Then find y

when x  8.

38. The variables x and y vary inversely, and y  –11 when x  16. Write an equation that relates x and y. Then find y when x  –9.

39. Graph . Compare the graph with the graph of .

40. Graph . Compare the graph with the graph of .

41. Graph . State the domain and range.

(5)

Simplify.

45.

46.

Find the product.

47.

48.

49.

50.

51.

Find the quotient.

52.

53.

54.

Find the sum or difference.

(6)

56.

57.

Solve the equation.

58.

59.

60.

61.

62.

63. A company spends $4000 to set up the manufacturing line for a fastener. The cost to produce each fastener is $4.75. Estimate how many fasteners must be produced for the average cost of the fastener to fall to $19.25.

64. A heavy duty cleaning solution is composed of 12.2% water and 87.8% vinegar by weight. You have 32 ounces of a basic cleaning solution, which is 24.4% water and 75.6% vinegar. How much vinegar should you mix with the basic cleaning solution to make a heavy duty solution?

Write the first six terms of the sequence. 65. an = 2n – 4

66.

67.

68.

(7)

70. –15, –24, –33, –42,...

71. 4.8, 5, 5.2, 5.4,...

72. , , , ,...

73. 1, –6, 36, –216, ...

74. 3, 6, 9, 12, ...

Write the series using summation notation. 75. 36 + 69 + 102 +  + 267

76. 9 + 29 + 129 + 

Find the sum.

77.

78.

Match the following with its description below.

a. e.

b. 280 f.

c. 72 g.

d. 43 h. 38

____ 79. 12th term of the sequence –1, 3, 7, ...

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____ 81. sum of the series 2 + 8 + 18 +  + 98

____ 82. summation notation for the series 2 + 8 + 18 +  + 98

____ 83. summation notation for the series –1 + 3 + 7 +  + 67

84. Write a rule for the nth term of the sequence. Then find . –6, –4, –2, 0, ...

Write a rule for the nth term of the arithmetic sequence.

85. ,

Find the sum of the infinite geometric series, if it exists.

86.

87.

Write a recursive rule for the sequence. 88. 2, 6, 10, 14, 18, . . .

(9)

Math 3 Final Semester 2

Answer Section

1. ANS:

PTS: 1 DIF: Level 1 REF: Math3 Sec. 5.1

KEY: simplifying natural base exponential expressions | natural base exponential expression | natural base e |

properties of exponents NOT: Example 1

2. ANS:

3. ANS:

4. ANS: B PTS: 1 DIF: Level 1 REF: Math3 Sec. 5.1

KEY: natural base exponential function | graphing natural base exponential functions | graph of an

exponential function NOT: Example 2

5. ANS:

KEY: rewriting exponential equations in logarithmic form NOT: Example 2 6. ANS:

–2

KEY: evaluating logarithms | logarithmic expression NOT: Example 3 7. ANS:

1 2

KEY: evaluating logarithms | logarithmic expression NOT: Example 3 8. ANS: 3

KEY: evaluating logarithms | logarithmic expression NOT: Example 3 9. ANS:

(10)

KEY: using inverse properties of logarithmic and exponential functions | simplifying logarithmic expressions

NOT: Example 5

10. ANS: 2x

KEY: using inverse properties of logarithmic and exponential functions | simplifying logarithmic expressions

NOT: Example 5

11. ANS:

KEY: inverse functions | finding inverse functions NOT: Example 6 12. ANS:

KEY: inverse functions | finding inverse functions NOT: Example 6 13. ANS:

KEY: writing transformations of exponential functions | exponential function | writing exponential functions | transformation NOT: Example 5

14. ANS:

KEY: writing transformations of logarithmic functions | logarithmic function | transformation NOT: Example 6

15. ANS:

The graph of H is a vertical stretch by a factor of 1.8 followed by a translation 1.6 centimeters down of the graph of f ; about 0.7 cm

KEY: describing transformations of exponential functions | transformation | translation NOT: Application-1

16. ANS:

KEY: properties of logarithms | expanding logarithmic expressions NOT: Example 2

17. ANS:

(11)

KEY: properties of logarithms | expanding logarithmic expressions NOT: Example 2

18. ANS:

KEY: properties of logarithms | condensing logarithmic expressions NOT: Example 3

19. ANS:

KEY: properties of logarithms | condensing logarithmic expressions NOT: Example 3

20. ANS: 2.738

KEY: change-of-base formula | evaluating logarithms NOT: Examples 4 and 5

21. ANS: B PTS: 1 DIF: Level 1 REF: Math3 Sec. 5.4

KEY: application | logarithmic function NOT: Example 6-1 22. ANS:

yrs

KEY: application | logarithmic function NOT: Example 6-2 23. ANS:

x = –1

KEY: exponential equations | solving exponential equations NOT: Example 1 24. ANS:

KEY: exponential equations | solving exponential equations NOT: Example 1 25. ANS:

x = 4

KEY: logarithmic equations | solving logarithmic equations NOT: Example 3 26. ANS:

x = 631

(12)

KEY: logarithmic equations | solving logarithmic equations NOT: Example 4 28. ANS:

about 104°F

KEY: exponential equations | solving exponential equations NOT: Example 2-2 29. ANS:

KEY: solving exponential inequalities | exponential inequality NOT: Example 5 30. ANS:

KEY: solving exponential inequalities | exponential inequality NOT: Example 5 31. ANS:

KEY: solving logarithmic inequalities | logarithmic inequality NOT: Example 6 32. ANS:

KEY: solving logarithmic inequalities | logarithmic inequality NOT: Example 6 33. ANS:

KEY: writing exponential functions using two points | writing exponential functions | exponential function NOT: Example 2

34. ANS: A, B, C PTS: 1 DIF: Level 2 REF: Math3 Sec. 5.2

logarithmic function NOT: Combined Concept

35. ANS: C, D, E PTS: 1 DIF: Level 2 REF: Math3 Sec. 5.4

NOT: Combined Concept 36. ANS:

; y 

10 7

(13)

37. ANS:

; y 

9 4

KEY: writing inverse variation equations | inverse variation NOT: Example 3 38. ANS:

; y 

176 9

KEY: writing inverse variation equations | inverse variation NOT: Example 3 39. ANS:

4 8

–4

–8 x

4 8

–4

–8

y

The graph of g lies farther from the axes than the graph of f, lies in the same quadrants, and has the same asymptotes, domain, and range. It is a vertical stretch by a factor of 21 of the graph of f.

KEY: graphing simple rational functions | rational function NOT: Example 1 40. ANS:

4 8

–4

–8 x

4 8

–4

–8

y

(14)

KEY: graphing simple rational functions | rational function NOT: Example 1 41. ANS:

4 8 12 16 –4

–8 –12

–16 x

4 8 12 16 20 24 28 32 36

–4

y

The domain is all real numbers except 0, and the range is all real numbers except 17.

KEY: translating simple rational functions | rational function NOT: Example 2 42. ANS:

4 8 12 16 20 24 –4

–8 x

4 8 12 16 20

–4

–8

y

(15)

4 8 12 –4

–8 –12

–16 x

4 8 12 16

–4

–8

–12

y

The domain is all real numbers except –1.25, and the range is all real numbers except 2.25.

KEY: graphing rational functions of the form y = (ax + b)/(cx + d) |domain | range | rational function NOT: Example 3

44. ANS:

4 8 12 16 –4

–8 x

4 8 12 16

–4

–8

y

KEY: graphing rational functions of the form y = (ax + b)/(cx + d) |domain | range | rational function NOT: Example 3

45. ANS:

,

KEY: simplifying rational expressions | rational expression NOT: Example 1 46. ANS:

(16)

KEY: multiplying rational expressions | rational expression NOT: Example 2 48. ANS:

5

4,

KEY: dividing rational expressions | rational expression NOT: Example 5 53. ANS:

KEY: dividing rational expressions | rational expression NOT: Example 5 54. ANS:

(17)

KEY: adding or subtracting rational expressions NOT: Example 4 56. ANS:

KEY: solving rational equations | rational equations NOT: Example 1 59. ANS:

KEY: solving rational equations NOT: Example 4 62. ANS:

x

9 5

KEY: solving rational equations NOT: Example 4 63. ANS:

about 276 fasteners

(18)

64. ANS:

You should mix 32 ounces of vinegar with the 32 ounces of basic cleaning solution.

KEY: rational equations | application NOT: Example 2-2 65. ANS:

–2, 0, 2, 4, 6, 8

KEY: terms of a sequence | sequence | writing terms of sequences NOT: Example 1

66. ANS:

5, 2, –3, –10, –19, –30

67. ANS:

1, –2, 4, –8, 16, –32

68. ANS:

, , , , ,

69. ANS:

,

KEY: terms of a sequence | sequence | writing rules for sequences NOT: Example 2

70. ANS:

arithmetic, ,

71. ANS:

arithmetic, ,

(19)

NOT: Example 2 72. ANS:

, ,

73. ANS:

geometric, ,

74. ANS:

, ,

75. ANS:

KEY: summation notation | series NOT: Example 4 76. ANS:

KEY: summation notation | series NOT: Example 4 77. ANS:

55

KEY: finding sums of series | series NOT: Example 5 78. ANS:

441

KEY: finding sums of series | series NOT: Example 5

79. ANS: D PTS: 1 DIF: Level 2 REF: Math3 Sec. 7.1

KEY: arithmetic sequence | arithmetic series NOT: Combined Concept

80. ANS: C PTS: 1 DIF: Level 2 REF: Math3 Sec. 7.1

(20)

81. ANS: B PTS: 1 DIF: Level 2 REF: Math3 Sec. 7.1 KEY: summation notation | series NOT: Combined Concept

82. ANS: E PTS: 1 DIF: Level 2 REF: Math3 Sec. 7.1

KEY: summation notation | series NOT: Combined Concept

83. ANS: A PTS: 1 DIF: Level 2 REF: Math3 Sec. 7.1

KEY: summation notation | series NOT: Combined Concept 84. ANS:

,

KEY: arithmetic sequence | writing rules for arithmetic sequences NOT: Example 2

85. ANS:

KEY: arithmetic sequence | writing rules for arithmetic sequences NOT: Example 4

86. ANS:

KEY: infinite geometric series | finding sums of infinite geometric series NOT: Example 2

87. ANS:

KEY: infinite geometric series | finding sums of infinite geometric series NOT: Example 2

88. ANS: ,

KEY: recursive rule | writing recursive rules for sequences NOT: Example 2 89. ANS: