Test Bank for Precalculus, 8th edition Robert F. Blitzer Chapter 1-11

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Match the story with the correct figure.

1) The height of an animal as a function of time.

A)

B)

C)

1)

D)

Match the correct viewing rectangle dimensions with the figure.

2)

A) [-25, 25, 5] by [-25, 25, 5] C) [-25, 25, 10] by [-25, 25, 10]

B) [-50, 25, 5] by [-50, 25, 5] D) [-5, 5, 5] by [-5, 5, 5]

1

2)

3)

3)

A) [-4, 4, 2] by [-4, 4, 2] C) [-4, 4, 2] by [-80, 80, 8]

B) [-20, 20, 2] by [-20, 20, 2] D) [-16, 16, 4] by [-4, 4, 2]

Match the story with the correct figure.

4) Mark started out by walking up a hill for 5 minutes. For the next 5 minutes he walked down a steep hill to an elevation lower than his starting point. For the next 10 minutes he walked on level ground. For the next 10 minutes he walked uphill. Determine which graph of elevation above sea level versus time illustrates the story.

A) B)

C)

D)

2

4)

The table of values was generated by a graphing utility with a TABLE feature. Use the following table to solve.

5) Which equation corresponds to Y2 in the table?

A) y2 = 2x - 3

5)

B) y2 = x + 2

C) y2 = 2 - 3x

D) y2 = 2x + 3

Match the story with the correct figure.

6) The amount of rainfall as a function of time, if the rain fell more and more softly.

A) B)

C)

D)

3

6)

Match the correct viewing rectangle dimensions with the figure.

7)

A) [-5, 25, 5] by [-20, 40, 5] C) [-5, 40, 5] by [-5, 40, 5]

7)

B) [-50, 150, 50] by [-2000, 2500, 500] D) [-50, 25, 5] by [-50, 25, 5]

8)

8)

A) [-8, 10, 2] by [-2, 16, 2] C) [-2, 16, 2] by [-2, 16, 2]

B) [-2, 16, 2] by [-8, 10, 2] D) [-20, 10, 2] by [-20, 10, 2]

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

9) One Internet service provider has the following rate schedule for high-speed Internet service: Monthly service charge 1st 50 hours of use Next 50 hours of use Over 100 hours of use

9)

$18.00 free $0.25/hour $1.00/hour

What is the charge for 50 hours of high-speed Internet use in one month?

What is the charge for 75 hours of high-speed Internet use in one month?

What is the charge for 135 hours of high-speed Internet use in one month?

10) A cellular phone plan had the following schedule of charges: Basic service, including 100 minutes of calls 2nd 100 minutes of calls Additional minutes of calls

$20.00 per month $0.075 per minute $0.10 per minute

What is the charge for 200 minutes of calls in one month?

What is the charge for 250 minutes of calls in one month?

Construct a function that relates the monthly charge C for x minutes of calls.

4

10)

11) An electric company has the following rate schedule for electricity usage in single-family residences: Monthly service charge

$4.93

Per kilowatt service charge 1st 300 kilowatts Over 300 kilowatts

$0.11589/kW $0.13321/kW

11)

What is the charge for using 300 kilowatts in one month?

What is the charge for using 375 kilowatts in one month?

Construct a function that gives the monthly charge C for x kilowatts of electricity.

12) The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is if 0 v < 1.79

t W(t) = 33 -

12)

(10.45 + 10 v - v)(33 - t ) 22.04

33 - 1.5958(33 - t)

if 1.79 v < 20 if v 20

where v represents the wind speed (in meters per second) and t represents the air temperature (°C). Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.)

13) A gas company has the following rate schedule for natural gas usage in single-family residences: Monthly service charge

$8.80

Per therm service charge 1st 25 therms Over 25 therms

$0.6686/therm $0.85870/therm

13)

What is the charge for using 25 therms in one month?

What is the charge for using 45 therms in one month?

Construct a function that gives the monthly charge C for x therms of gas.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Give the domain and range of the relation.

14) {(-3, 7), (-2, 2), (0, -2), (2, 2), (4, 14)} A) domain: {-3, -2, 0, 2, 4}; range: {7, 2, -2, 14} B) domain: {-3, -2, 2, 4}; range: {7, 2, -2, 14} C) domain: {7, 2, -2, 14}; range: {-3, -2, 2, 4} D) domain: {7, 2, -2, 14}; range: {-3, -2, 0, 2, 4}

14)

5

Find the inverse of the one-to-one function.

3 15) f(x) = x + 5 1 A) f-1 (x) = B) f-1 (x) = x3 - 5 3 x -5

15) C) f-1 (x) = x - 5

D) f-1 (x) = x3 + 25

Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.

16) f(x) = x , g(x) = x + 2 16)

A) g shifts the graph of f vertically down 2 units

B) g shifts the graph of f vertically up 2 units

6

C) g shifts the graph of f vertically up 2 units

D) g shifts the graph of f vertically down 2 units

17) f(x) =

x, g(x) =

x-1

17)

7

A) g shifts the graph of f vertically down 1 unit

B) g shifts the graph of f 1 unit to the right

C) g shifts the graph of f vertically up 1 unit

8

D) g shifts the graph of f 1 unit to the left

Find the inverse of the one-to-one function.

18) f(x) = (x - 4)3 A) f-1 (x) = C) f-1 (x) =

3

18)

x+4

B) f-1 (x) =

x + 64

D) f-1 (x) =

9

3 3

x-4 x+4

Begin by graphing the standard quadratic function f(x) = x 2 . Then use transformations of this graph to graph the given function.

19) g(x) = x2 + 2 19)

A)

B)

C)

D)

Determine whether the relation is a function.

20) {(-3, -7), (-2, -8), (-1, -2), (-1, 2)} A) Function

20)

B) Not a function

Evaluate the function at the given value of the independent variable and simplify.

21) g(x) = 4x - 2; g(x + 1) 1 A) x - 2 B) 4x - 2 C) 4x - 1 4 Graph the equation.

10

21) D) 4x + 2

22) -4y + 3x + 6 = 0

22)

A)

B)

C)

D)

Graph the linear function by plotting the x- and y-intercepts.

11

23)

1 x+y-4=0 2

23)

A) intercepts: (0, -8), (8, 0)

B) intercepts: (0, 4), (8, 0)

C) intercepts: (0, 4), (-8, 0)

D) intercepts: (0, 4), (-4, 0)

Determine whether the equation defines y as a function of x.

24) x2 + y = 4 A) y is a function of x

24)

B) y is not a function of x

For the given functions f and g , find the indicated composition. g(x) = x2 + 2x + 3 25) f(x) = x2 + 2x - 5, (f g)(4) A)

B) 762

C)

12

25) D) 778

Use the graph to determine the x- and y-intercepts.

26)

26)

A) y-intercepts: -8, 8 C) x-intercepts: -5, 5

B) x-intercepts: -5, 5; y-intercepts: -8, 8 D) x-intercepts: -8, 8; y-intercepts: -5, 5

Use the graph of f to draw the graph of its inverse function.

27)

27)

A)

B)

13

Identify the intervals where the function is changing as requested.

28) Increasing

A) (-3, )

B) (-3, 3)

C) (-2, 2)

Find the inverse of the one-to-one function.

29) f(x) = -8x + 8 -8x - 8 A) f-1 (x) = -8 C) f-1 (x) =

28)

D) (-2, )

29)

y- 8 B) f-1 (x) = -8

x+8 -8

D) f-1 (x) =

x-8 -8

Evaluate the function at the given value of the independent variable and simplify. x2 + 5 ; f(2) 30) f(x) = x3 + 3x A)

B)

C)

Determine whether the given function is even, odd, or neither.

31) f(x) = 4x2 + x4 A) Even

B) Neither

14

9 14

30) D)

C) Odd

9 8

31)

Use the graph to determine the function's domain and range.

32)

A) domain: [0, ) range: (- , )

B) domain: [0, ) range: [0, )

32)

C) domain: (- , ) range: [1, )

D) domain: [0, ) range: [1, )

Find the domain of the function.

5x 33) f(x) = x+5

33)

A) (- , 0) (0, ) C) (- , -5) Solve.

B) (- , ) D) (- , -5)

(-5, )

34) An investment is worth $2458 in 1994. By 1999 it has grown to $3843. Let y be the value of the investment in the year x, where x = 0 represents 1994. Write a linear equation that models the value of the investment in the year x.

1 x + 2458 A) y = B) y = -277x + 5228 277 C) y = -277x + 2458

D) y = 277x + 2458

15

34)

Use the shape of the graph to name the function.

35)

35)

A) Absolute value function C) Standard cubic function

B) Constant function D) Identity function

Determine whether the given function is even, odd, or neither.

36) f(x) = x5 - x4 A) Neither

B) Odd

Given functions f and g, determine the domain of f + g.

37) f(x) = 3x2 - 5, g(x) = 2x3 - 1 A) (0, ) C) (- , -3) or (-3, -2) or (-2, )

C) Even

B) (- , ) D) (- , 0) or (0, )

Solve the problem.

38) A salesperson gets a commission of $1000 for the first $10,000 of sales, and then $500 for each additional $10,000 or partial of sales. Let S(x) represent the commission on x dollars of sales. Find the value of S(45,000).

A) $2750 B) $3000 C) 3250 D) $2250

16

36)

37)

38)

Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g.

39) g(x) = f(x) - 2 y = f(x)

A)

B)

C)

D)

17

39)

Begin by graphing the standard function f(x) = x 3 Then use transformations of this graph to graph the given function.

40) g(x) = x3 - 2 40)

A)

B)

C)

D)

Write the standard form of the equation of the circle with the given center and radius.

41) (-5, -10); 14 A) (x - 10)2 + (y - 5)2 = 196 B) (x + 10)2 + (y + 5)2 = 196 C) (x - 5)2 + (y - 10)2 = 14

D) (x + 5)2 + (y + 10)2 = 14

18

41)

Graph the equation.

42) x2 + y2 - 4x - 2y + 1 = 0

42)

A)

B)

For the given functions f and g , find the indicated composition.

43) f(x) = -4x + 7, g(x) = 3x + 9 (g f)(x) A) -12x + 30 B) -12x - 12 C) -12x + 43 Evaluate the piecewise function at the given value of the independent variable. if x > -3 ; f(-3) 44) f(x) = x + 1 -(x + 1) if x -3 A) -3 B) C) -2

19

43) D) 12x + 30

44) D) 15

The graph of a function f is given. Use the graph to answer the question.

45) Find the numbers, if any, at which f has a relative maximum. What are the relative maxima?

45)

A) f has a relative maximum at x = 3; the relative maximum is 2 B) f has a relative maximum at x = 0; the relative maximum is 2 C) f has a relative maximum at x = -3 and 3; the relative maximum is 0 D) f has no relative maximum Find the domain of the composite function f g.

4 5 , g(x) = 46) f(x) = x+5 x

46)

A) (- , -5) or (-5, 0) or (0, ) C) (- , -1) or (-1, 0) or (0, )

B) (- , -5) or (-5, -1) or (-1, 0) or (0, ) D) (- , )

20

Graph the line whose equation is given.

47) y = 3x + 1

47)

A)

B)

C)

D)

Use the graph of f to draw the graph of its inverse function.

48)

48)

21

A)

B)

C)

D)

Find functions f and g so that h(x) = (f g)(x).

49) h(x) = |2x + 10| A) f(x) = |x|, g(x) = 2x + 10 C) f(x) = -|x|, g(x) = 2x + 10

B) f(x) = |-x|, g(x) = 2x - 10 D) f(x) = x, g(x) = 2x + 10

Use the given conditions to write an equation for the line in point-slope form.

50) Slope = -2, passing through (-6, 6) A) y = -2x - 6 B) x - 6 = -2(y + 6) C) y - 6 = -2(x + 6)

22

D) y + 6 = -2(x - 6)

49)

50)

The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.

51) If f represents the function, find f(1995).

A) approximately 24.5% C) approximately 23%

51)

B) approximately 28% D) approximately 30%

Determine whether the equation defines y as a function of x.

52) x + y3 = 64 A) y is a function of x

52)

B) y is not a function of x

Identify the intervals where the function is changing as requested.

53) Decreasing

A) (- , -2)

B) (- , -3)

C) (-3, -2)

23

53)

D) (0, -2)

Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.

54)

A) not a function

54)

B) function

Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.

55) f(x) = x, g(x) = x - 4 55)

A) g shifts the graph of f vertically up 4 units

24

B) g shifts the graph of f vertically down 4 units

C) g shifts the graph of f vertically down 4 units

D) g shifts the graph of f vertically up 4 units

Write the standard form of the equation of the circle with the given center and radius.

56) (5, 0); 7 A) (x - 5)2 + y2 = 49 B) x2 + (y - 5)2 = 7 C) x2 + (y + 5)2 = 7

D) (x + 5)2 + y2 = 49

25

56)

Use the graph to find the indicated function value.

57) y = f(x). Find f(3).

A) 3

57)

B) 1.5

C) 9

D) -3

Identify the intervals where the function is changing as requested.

58) Decreasing

A) (6, 1)

B) (5, 1)

C) (6, 12)

26

58)

D) (5, 12)

Based on the graph, find the range of y = f(x).

4 if -5 x < -2 |x| if -2 x < 7 59) f(x) = x

59)

if 7 x 14

A) [0, )

C) [0, 14]

B) [0, 7]

Use the given conditions to write an equation for the line in the indicated form.

60) Passing through (2, -4) and parallel to the line whose equation is y = -8x + 2; slope-intercept form 1 3 A) y = - 8x + 12 B) y = - 8x - 12 C) y = - x 8 2

D) [0, 7)

60) D) y = 8x - 12

Solve the problem.

61) The annual yield per apple tree is fairly constant at 330 pounds per tree when the number of trees per acre is 50 or fewer. For each additional tree over 50, the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. a. Express the yield per tree, Y, in pounds, as a function of the number of apple trees per acre, x. b. Express the total yield for an acre, T, in pounds, as a function of the number of apple trees per acre, x.

A) a. Y(x) = -4x + 130 B) a. Y(x) = -4x + 530 2 b. T(x) = -4x + 130x b. T(x) = -4x2 + 530x C) a. Y(x) = 4x + 530 b. T(x) = 4x2 + 530x

D) a. Y(x) = 4x + 130 b. T(x) = 4x2 + 130x

27

61)

Graph the equation.

62) (x - 3)2 + (y - 1)2 = 9

62)

A)

B)

Domain = (-6, 0), Range = (-4, 2)

Domain = (0, 6), Range = (-2, 4)

28

Graph the equation in the rectangular coordinate system.

63) y =

63)

A)

B)

C)

D)

29

Answer Key Testname: C1

1) C 2) A 3) C 4) C 5) D 6) D 7) B 8) B 9) $18.00 $24.25 $65.50 10) $27.50 $32.50;

20 if 0 x 100 C(x) = 20 + 0.075(x - 100) if 100 < x 200 27.50 + 0.1(x - 200) if x > 200 11) $39.70 $49.69 if 0 x 300 C(x) = 4.93 + 0.11589x 39.697 + 0.13321(x - 300) if x > 300 12) 6.0°C 13) $25.52 $42.69 if 0 x 25 C(x) = 8.8 + 0.6686x 25.515 + 0.133219(x - 25) if x > 25 14) A 15) B 16) C 17) B 18) D 19) A 20) B 21) D 22) C 23) B 24) A 25) D 26) B 27) A 28) C 29) D 30) C 31) A 32) D 33) D 34) D 35) A 36) A 37) B 38) B

30

Answer Key Testname: C1

39) B 40) A 41) D 42) B 43) A 44) B 45) B 46) C 47) A 48) B 49) A 50) C 51) B 52) A 53) C 54) A 55) C 56) A 57) B 58) D 59) D 60) A 61) B 62) B 63) B

31

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior to match the function with its graph.

1) f(x) = 2x2 + 2x - 1 1)

A) falls to the left and falls to the right

B) falls to the left and rises to the right

C) rises to the left and falls to the right

D) rises to the left and rises to the right

1

2) f(x) = 4x3 + 2x2 - 2x + 3 A) rises to the left and falls to the right

B) falls to the left and falls to the right

C) rises to the left and rises to the right

D) falls to the left and rises to the right

2

2)

3) f(x) = -8x3 + 3x2 + 4x - 1 A) falls to the left and rises to the right

B) rises to the left and rises to the right

C) falls to the left and falls to the right

D) rises to the left and falls to the right

3

3)

Solve.

4) f(x) = -2x2 - 2x + 1 A) rises to the left and falls to the right

B) falls to the left and rises to the right

C) falls to the left and falls to the right

D) rises to the left and rises to the right

5) The profits (in millions) for a company for 8 years were as follows: Year, x Profits, P 1993, 1 1.1 1994, 2 1.7 1995, 3 2.0 1996, 4 1.4 1997, 5 1.3 1998, 6 1.5 1999, 7 1.8 2000, 8 2.1 Which of the following polynomials is the best model for this data?

A) P(x) = -0.03x4 - 0.3x2 + 1.3x + 0.17 B) P(x) = -0.08x3 + 7x2 + 1.3x - 0.18

C) P(x) = 0.03x3 - 0.3x2 + 1.3x + 0.17

D) P(x) = 0.05x2 - 0.8x + 6

4

4)

5)

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behavior to match the function with its graph.

6) f(x) = 3x4 - 2x2 6)

A) rises to the left and falls to the right

B) falls to the left and rises to the right

C) falls to the left and falls to the right

D) rises to the left and rises to the right

5

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Complete the following: (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. (c) Find the y-intercept. (d) Graph the function.

7) f(x) = -2(x - 1)(x + 3)3 7)

8) f(x) = x2 (x + 3)

8)

6

9) f(x) = (x + 2)(x - 3)2

9)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the range of the quadratic function.

10) f(x) = 2x2 + 3x - 9 A) (- , -

81 ] 8

B) [-

3 , ) 4

C) [-

81 , ) 8

D) (- , -

Find a rational zero of the polynomial function and use it to find all the zeros of the function.

11) f(x) = x3 + 7x2 + 8x + 2 A) {-1, -6 + 2, -6 C) {1, -1, -2}

2}

3 ] 4

10)

11)

B) {1, -6 + 7, -6 - 7} D) {-1, -3 + 7, -3 - 7}

Solve the problem.

12) Is there origin symmetry for the rational function f(x) = A) Yes

-2x2 + 6 ?

4x

12)

B) No

Use the vertex and intercepts to sketch the graph of the quadratic function.

13) f(x) = 3 - x2 - 2x

7

13)

A)

B)

C)

D)

Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.

2x <x 14) 14) x+7

A) (- , 5)

(7, )

B) (- , -7)

(-5, 0)

C) (-7, -5) (0, )

D) (- , -7) (0, )

8

Write an equation that expresses the relationship. Use k as the constant of variation.

15) a varies directly as the square of b. k k A) a = kb2 B) a = C) a = b b2 Determine whether the function is a polynomial function.

16) f(x) = 4x3 + 5x2 - 4x-4 + 80 A) No

15) D) a = k b

16)

B) Yes

Solve the problem.

17) While traveling in a car, the centrifugal force a passenger experiences as the car drives in a circle varies jointly as the mass of the passenger and the square of the speed of the car. If a passenger experiences a force of 225 newtons when the car is moving at a speed of 50 kilometers per hour and the passenger has a mass of 100 kilograms, find the force a passenger experiences when the car is moving at 60 kilometers per hour and the passenger has a mass of 60 kilograms.

A) 194.4 newtons B) 216 newtons C) 252 newtons D) 172.8 newtons Write an equation that expresses the relationship. Use k for the constant of proportionality.

18) r varies directly as a and inversely as the difference between s and t. a ka k A) r = B) r = C) r = ka(s - t) D) r = k(s - t) s- t a(s - t)

17)

18)

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.

19) f(x) = x3 + 10x2 + 33x + 36 19)

A) 3, crosses the x-axis; -3, crosses the x-axis; -4, crosses the x-axis.

C) -3, crosses the x-axis; -4, touches the x-axis and turns around

B) 3, crosses the x-axis; -3, touches the x-axis and turns around; -4, crosses the x-axis.

D) -3, touches the x-axis and turns around; -4, crosses the x-axis.

Find the y-intercept of the polynomial function.

20) f(x) = -x2 + 2x + 3 A) -3

B) 3

C) 0

20)

D) -1

If y varies inversely as x, find the inverse variation equation for the situation.

1 21) y = when x = 35 5 A) y =

x 7

B) y =

7 x

C) y =

Graph the function.

9

1 7x

21) D) y =

1 x 175

22) f(x) =

x2 + 5x - 3 x-8

22)

A)

B)

C)

D)

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.

23) f(x) = 2x3 - 5x2 - 2x + 4 A) rises to the left and rises to the right C) falls to the left and rises to the right

B) rises to the left and falls to the right D) falls to the left and falls to the right

10

23)

Write an equation that expresses the relationship. Use k as the constant of variation.

24) The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 72 ft from a light is 3.2 foot-candles, find the intensity on a screen 90 ft from the light.

A) 5 foot-candles B) 2.048 foot-candles C) 4 foot-candles D) 2.56 foot-candles Solve the problem.

25) You have 120 feet of fencing to enclose a rectangular region. What is the maximum area?

A) 896 square feet B) 3600 square feet C) 900 square feet D) 14,400 square feet Find the x-intercepts (if any) for the graph of the quadratic function.

26) 3x2 + 6x + 2 = 0 Give your answers in exact form.

-3 ± 3 -6 ± 3 ,0 ,0 A) B) 3 3

C)

-3 ± 3 ,0 6

24)

25)

26) D)

-3 ± 15 ,0 3

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.

27) f(x) = -x2 (x + 8)(x2 - 1) 27)

A) 0, touches the x-axis and turns around; 8, crosses the x-axis; -1, touches the x-axis and turns around; 1, touches the x-axis and turns around C) 0, crosses the x-axis; -8, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis

B) 0, touches the x-axis and turns around; -8, crosses the x-axis; -1, crosses the x-axis; 1, crosses the x-axis D) 0, touches the x-axis and turns around; -8, crosses the x-axis; 1, touches the x-axis and turns around

11

Graph the polynomial function.

28) f(x) = -x2 (x + 1)(x + 3)

28)

A)

B)

C)

D)

Find the coordinates of the vertex for the parabola defined by the given quadratic function.

29) f(x) = -7(x - 2)2 - 8

29)

Find a rational zero of the polynomial function and use it to find all the zeros of the function.

30) f(x) = 2x4 - 17x3 + 58x2 - 77x + 26

30)

A) (-7, -2)

B) (2, -8)

C) (-8, 2)

1 A) {-2, , 3 + 2i, 3 - 2i} 2

C) {2, -

1 B) {-2, - , 2 + 3i, 2 - 3i} 2

1 , 2 + 3i, 2 - 3i} 2

D) {2,

12

1 , 3 + 2i, 3 -2i} 2

D) (-2, -8)

Solve the quadratic equation using the quadratic formula. Express the solution in standard form.

31) 8x2 + 3x + 3 = 0 3 87 ± A) 16 16

3 87 ± i B) 16 16

3 87 ± i C) 16 16

Find the range of the quadratic function.

32) f(x) = -7(x - 5)2 - 7 A) (- , 5]

B) [-5, )

C) (- , -7]

D) [-7, )

Divide using long division.

3r3 - 13r2 - 54r - 14 33) r- 7 A) 3r2 + 8r +

Solve.

2 r- 7

31)

3 87 ± D) 16 16

32)

33) B) r2 + 2r + 8

C) 3r2 - 8r - 2

D) 3r2 + 8r + 2

34) Suppose that a polynomial function is used to model the data shown in the graph below.

34)

Determine the degree of the polynomial function of best fit and the sign of the leading coefficient.

A) Degree 3; negative leading coefficient.

B) Degree 3; positive leading coefficient.

C) Degree 4; positive leading coefficient.

D) Degree 4; negative leading coefficient.

Find the indicated intercept(s) of the graph of the function. x2 - 3x + 6 35) y-intercept of f(x) = 8x A) 0,

3 4

B) 0, -

35)

4 3

C) (0, 6)

13

D) none

Solve.

36) Suppose that a polynomial function is used to model the data shown in the graph below.

For what intervals is the function increasing?

A) 0 through 50 C) 0 through 10 and 30 through 50

B) 0 through 10 and 40 through 50 D) 0 through 20 and 30 through 50

Determine the maximum possible number of turning points for the graph of the function.

37) f(x) = (2x + 5)4 ( x4 - 5)(x - 3) A) 4

B) 18

C) 9

D) 8

Divide using long division.

4m 3 + 21m 2 - 47m + 14 38) m+7 A) m 2 + 7m + 4

36)

37)

38) B) 4m 2 + 7m + 2

C) 4m 2 - 7m + 2

D) m 2 + 8m + 9

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.

39) f(x) = -4x4 + 2x3 + 2x2 + 5x - 3

39)

The graph of a quadratic function is given. Determine the function's equation.

40)

40)

A) falls to the left and rises to the right C) rises to the left and falls to the right

B) rises to the left and rises to the right D) falls to the left and falls to the right

A) g(x) = -x2 + 4x + 4 C) h(x) = -x2 - 2

B) f(x) = -x2 - 4x - 4 D) j(x) = -x2 + 2

14

Determine the maximum possible number of turning points for the graph of the function.

41) f(x) = (x - 5)(x + 3)(x - 3)(x + 1) A) 3 B) 4 C) 1 D) 0 Use the vertex and intercepts to sketch the graph of the quadratic function.

42) f(x) = -3(x - 6)2 - 3

A)

B)

C)

D)

Find an nth degree polynomial function with real coefficients satisfying the given conditions.

43) n = 3; - 4 and i are zeros; f(-3) = 60 A) f(x) = 6x3 + 24x2 + 6x + 24 B) f(x) = -6x3 - 24x2 + 6x + 24 C) f(x) = 6x3 + 24x2 - 6x - 24

D) f(x) = -6x3 - 24x2 - 6x - 24

15

41)

42)

43)

Find the vertical asymptotes, if any, of the graph of the rational function. x 44) f(x) = 2 x +9 A) x = 9 C) x = -9

44)

B) x = -9, x = 9 D) no vertical asymptote

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point.

45) f(x) = 4x2 + 8x 45)

A) maximum; 1, - 4 C) minimum; 1, - 4

B) minimum; - 1, - 4 D) maximum; - 1, - 4

Solve the problem.

46) Two people are 41 years old and 21 years old, respectively. In x years from now, their ages can be represented by x + 41 and x + 21. Use long division to find the ratio of the older person's age to the younger person's age in x years.

62 20 62 A) 1.9524 B) 1 + C) 1 + D) 1 + x + 21 x + 21 x + 41 Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither.

47) f(x) = -x2 (x + 5)(x2 - 1) A) y-axis symmetry

Use transformations of f(x) =

48) f(x) =

B) origin symmetry

C) neither

46)

47)

1 1 or f(x) = to graph the rational function. x x2

1 +4 x+2

48)

16

A)

B)

C)

D)

Find the x-intercepts (if any) for the graph of the quadratic function.

49) f(x) = -x2 + 11x - 30 A) No x-intercepts C) (-5, 0) and (-6, 0)

B) (5, 0) and (6, 0) D) (5, 0) and (-6, 0)

Solve the problem.

50) You have 300 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area.

A) 150 ft by 37.5 ft B) 150 ft by 150 ft C) 77 ft by 73 ft D) 75 ft by 75 ft

49)

50)

Find an nth degree polynomial function with real coefficients satisfying the given conditions.

51) n = 3; -2 and -2 + 3i are zeros; leading coefficient is 1 A) f(x) = x3 + 5x2 + 21x - 14 B) f(x) = x3 + 6x2 + 21x + 26

51)

Solve the problem.

52) The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(x) = 4x2 - 24x + 81. Find the number of automobiles that must be produced to

52)

C) f(x) = x3 - 4x2 + 21x + 26

D) f(x) = x3 + 6x2 + 15x + 26

minimize the cost.

A) 12 thousand automobiles C) 3 thousand automobiles

B) 6 thousand automobiles D) 45 thousand automobiles

17

53) If y varies directly as the square of x, and y = 500 when x = 15, find y when x = 12.

3125 A) 625 B) C) 320 D) 400 4 Use the vertex and intercepts to sketch the graph of the quadratic function.

54) f(x) = 3x2 + 24x + 47

A)

B)

C)

D)

18

53)

54)

Divide and express the result in standard form.

2i 55) 1 + 7i A)

1 7 + i 25 25

B) -

55)

7 1 + i 24 24

C) -

Divide using long division.

56) (-20x3 + 26x2 - 13x + 4) ÷ (5x - 4) A) -4x2 + 2x - 1

B) -4x2 - 1

1 7 i 24 24

C) x2 - 2x + 1

D)

+

1 i 25

D) x2 + 2x - 1

56)

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero.

1 2 2 5 (x + 1) 57) f(x) = x + 57) 2

A)

1 , multiplicity 2, touches the x-axis and turns around; 2

1, multiplicity 5, crosses the x-axis 1 B) - , multiplicity 2, crosses the x-axis.

2

C) -

1 , multiplicity 2, touches the x-axis and turns around; 2

-1, multiplicity 5, crosses the x-axis 1 D) - , multiplicity 2, touches the x-axis and turns around.

2

19

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation.

58) (2x + 9)(5x - 4) > 0 58)

A) -

9 4 , 2 5

B) -

9 4 , 2 5

C) - , -

D)

9 2

4 , 5

4 , 5

Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation.

59) x3 + 6x2 + 11x + 6 = 0 59)

A) -1; The remainder is zero; -1, -2, and -3, or {-3, -2, -1} B) -1; The remainder is zero; -1, 2, and -3, or {-3, -1, 2} C) -1; The remainder is zero; -1, -2, and 3, or {-2, -1, 3} D) -1; The remainder is zero; 1, -2, and -3, or {-3, -2, 1}

20

Answer Key Testname: C2

1) D 2) D 3) D 4) C 5) C 6) D 7) (a) falls to the left and to the right (b) x-intercepts: (-3, 0), crosses x-axis; (1, 0), crosses x-axis (c) y-intercept: (0, 54) (d)

8) (a) falls to the left and rises to the right (b) x-intercepts: (0, 0), touches x-axis and turns; (-3, 0), crosses x-axis (c) y-intercept: (0, 0) (d)

21

Answer Key Testname: C2

9) (a) falls to the left and rises to the right (b) x-intercepts: (3, 0), touches x-axis and turns; (-2, 0), crosses x-axis (c) y-intercept: (0, 18) (d)

10) C 11) D 12) A 13) D 14) C 15) A 16) A 17) A 18) B 19) D 20) B 21) B 22) B 23) C 24) B 25) C 26) A 27) B 28) B 29) B 30) D 31) C 32) C 33) D 34) B 35) D 36) C 37) D 38) C 39) D 40) D 22

Answer Key Testname: C2

41) A 42) B 43) A 44) D 45) B 46) C 47) C 48) C 49) B 50) D 51) B 52) C 53) C 54) B 55) D 56) A 57) D 58) C 59) A

23

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

The graph of a logarithmic function is given. Select the function for the graph from the options.

1)

A) f(x) = log 2 (-x)

B) f(x) = 1 - log 2 x

C) f(x) = - log 2 x

D) f(x) = log 2 x

The graph of an exponential function is given. Select the function for the graph from the functions listed.

2)

A) f(x) = - 2 x

B) f(x) = 2 x

C) f(x) = - 2 -x

1

D) f(x) = 2 -x

1)

2)

The graph of a logarithmic function is given. Select the function for the graph from the options.

3)

A) f(x) = log 3 x - 2

B) f(x) = log 3 (x - 2)

C) f(x) = log 3 x

D) f(x) = log 3 (x + 2)

4)

3)

4)

A) f(x) = log 2 x

B) f(x) = log 2 (x - 1)

C) f(x) = log 2 (x + 1)

D) f(x) = log 2 x + 1

The graph of an exponential function is given. Select the function for the graph from the functions listed.

5)

A) f(x) = 4 x

B) f(x) = - 4 x

C) f(x) = 4 -x

2

D) f(x) = - 4 -x

5)

The graph of a logarithmic function is given. Select the function for the graph from the options.

6)

A) f(x) = log 3 x

B) f(x) = log 3 (x + 2)

C) f(x) = log 3 x - 2

D) f(x) = log 3 (x - 2)

The graph of an exponential function is given. Select the function for the graph from the functions listed.

7)

A) f(x) = - 4 -x

C) f(x) = - 4 x

B) f(x) = 4 -x

B) f(x) = log 2 x

C) f(x) = 1 - log 2 x

D) f(x) = - log 2 x

3

7)

D) f(x) = 4 x

The graph of a logarithmic function is given. Select the function for the graph from the options.

8)

A) f(x) = log 2 (-x)

6)

8)

The graph of an exponential function is given. Select the function for the graph from the functions listed.

9)

A) f(x) = 3 x

B) f(x) = 3 x + 1

C) f(x) = 3 x - 1

D) f(x) = 3 x - 1

The graph of a logarithmic function is given. Select the function for the graph from the options.

10)

A) f(x) = log 2 x

B) f(x) = - log 2 x

C) f(x) = 1 - log 2 x

D) f(x) = log 2 (-x)

The graph of an exponential function is given. Select the function for the graph from the functions listed.

11)

A) f(x) = 5 x

B) f(x) = 5 x - 2

C) f(x) = 5 x - 2

4

9)

D) f(x) = 5 x + 2

10)

11)

12)

12)

A) f(x) = 5 x + 2

B) f(x) = 5 x - 2

C) f(x) = 5 x + 2

D) f(x) = 5 x

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Present data in the form of tables. For the data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.

13) Number of Homes Built in a Town by Year Year Number of Homes 1985 11 1991 91 1994 145 1997 192 2002 223

13)

5

14) Percentage of Population Living in the South Suburbs of a Large City Year Percent 1950 55 1960 69 1970 73 1980 76 2000 77

14)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. x 15) ln 15) y

A)

1 x ln 2 y

B)

1 ln x - ln y 2

Use the compound interest formulas A = P 1 +

C)

1 1 ln x - ln y 2 2

D) ln

x - ln

y

r nt and A = Pe rt to solve. n

16) Find the accumulated value of an investment of $8000 at 9% compounded continuously for 4 years.

A) $11,466.64 B) $11,292.65 C) $10,880.00 D) $11,566.64 Solve the problem.

17) If Emery has $1800 to invest at 7% per year compounded monthly, how long will it be before he has $3300? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.)

A) 0.11 yrs, 0.866 yrs B) 91.159 yrs, 8.945 yrs C) 0.747 yrs, 0.722 yrs D) 8.684 yrs, 8.659 yrs

6

16)

17)

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

7x 18) log w 18) 5

A) log w 2x

B) log w 7x - log w 5

C) log w 7 + log w x + log w 5

D) log w 7 + log w x - log w 5

19) log 3 11

-5

19)

A) -5 log 3 11

B) 11 log 3 5

C) 3 log 5 11

D) -15 log 11

Evaluate the expression without using a calculator.

20) log7 343 A) 21 Solve.

20)

B) 7

C) 3

D) 343

21) The half-life of silicon-32 is 710 years. If 60 grams is present now, how much will be present in 200 years? (Round your answer to three decimal places.)

A) 0 B) 8.515 C) 49.358 D) 58.84

Solve the problem.

22) Use the formula R = log

a + B to find the intensity R on the Richter scale, given that amplitude a is T

21)

22)

368 micrometers, time T between waves is 3.1 seconds, and B is 3.1. Round answer to one decimal place.

A) 8.3 B) 2.1 C) 5.2 D) 7.9

23) The rabbit population in a forest area grows at the rate of 9% monthly. If there are 250 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. Use y = 250(2.7)0.09t.

A) 744

B) 731

C) 718

23)

D) 729

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. x-2 24) log 4 24) x5

A) log 4 (x - 2) + 5 log 4 x

B) 5 log 4 x - log 4 (x - 2)

C) log 4 (x - 2) - 5 log 4 x

D) log 4 (x - 2) - log 4 x

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. x+3 =7 25) e 25) A) 2.30 B) -0.70 C) -0.99 D) -1.05

7

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

26) log 3 x = 5 26)

A) {243}

C) {15}

D) {125}

Evaluate or simplify the expression without using a calculator.

6 27) 10log x A) 6 B) x-1/6 C) x6

D) x1/6

28) ln 1 A) 1

B) {1.46}

B) -1

C) 0

27)

28)

D) e

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

8 29) ln x 29) 1 A) x ln 8 B) 8ln x C) ln x D) 8ln x 8

Find the domain of the logarithmic function.

30) f(x) = log (x2 - 11x + 24) A) (-3, 8)

B) (8, )

C) (- , -3)

D) (- , 3)

(8, )

30)

Graph the function.

31) Use the graph of f(x) = 4 x to obtain the graph of g(x) = 4 -x.

A)

B)

8

31)

C)

D)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

32) log 6 (x + 4) = -2 32)

A) -

143 36

B)

145 64

C)

33) ln (x - 6) + ln (x + 1) = ln (x - 15) A) {3, -3} B) {3} 34) log x + log (x - 63) = 2 8 8 A) {1, -64}

145 36

D) -

C) {-3}

143 64

33)

D)

34) B) {8}

C) {-1, 64}

D) {64}

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. x 35) log 4 35) 16

A) 8 -

1 log 4 x 2

B)

1 log 4 x - 2 2

C) log 4 x - 2

D) - 2 log 4 x

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.

1 36) (log6 x + log6 y) - 5 log6 (x + 2) 36) 5

A) log6

5

xy 5(x + 2)

B) log6

Solve the problem.

37) The logistic growth function f(t) =

5

x+y (x + 2)5

C) log6

5

xy

(x + 2)5

D) log6

5

x+

5

y (x + 2)5

520 describes the population of a species of 1 + 16.3e-0.24t

butterflies t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 19 months?

A) 9880 butterflies B) 520 butterflies C) 444 butterflies D) 570 butterflies

9

37)

Write the equation in its equivalent logarithmic form.

38) a 2 = 81 A) loga 2 = 81

B) log2 81 = a

C) log81 a = 2

D) loga 81 = 2

38)

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

39) log 5 (7 · 11) 39)

A) ( log 5 7)( log 5 11)

B) log 5 77

C) log 5 7 + log 5 11

D) log 5 7 - log 5 11

Graph the function by making a table of coordinates.

40) f(x) = 0.7x

40)

A)

B)

10

C)

D)

Evaluate or simplify the expression without using a calculator.

7 41) ln e 1 e A) B) C) 7e 7 7

11

41) D) 7

Graph the function by making a table of coordinates.

42) f(x) = 3 x

42)

A)

B)

C)

D)

Solve the problem.

43) The population in a particular country is growing at the rate of 1.9% per year. If 7,638,000 people lived there in 1999, how many will there be in the year 2003? Use f(x) = y0 e0.019t and round to the nearest ten-thousand.

A) 8,080,000 B) 8,240,000 C) 9,070,000 D) 9,890,000

43)

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

44) 7ex = 25 44)

A) -0.55

B) -1.27

C) 1.27

12

D) 0.55

Use the compound interest formulas A = P 1 +

r nt and A = Pe rt to solve. n

45) Suppose that you have $4000 to invest. Which investment yields the greater return over 6 years: 7.5% compounded continuously or 7.6% compounded semiannually?

A) Both investment plans yield the same return.

B) $4000 invested at 7.6% compounded semiannually over 6 years yields the greater return.

C) $4000 invested at 7.5% compounded continuously over 6 years yields the greater return.

Graph the function.

46) Use the graph of log 5 x to obtain the graph of f(x) = 2 + log 5 x.

A)

B)

C)

D)

13

45)

46)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

47) log 3 + log x = 1 47) 8 8 3 8 1 3 A) { 8} B) { } C) { } D) { } 3 3 8 Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.

48) log 3 (x - 1) - log 3 (x - 4) 48)

A) log 3 (x2 - 5x + 4)

B) log 3

x-1 x-4

C) log 3 3

D) log 3

x-1 x +4

Graph the function.

49) Use the graph of f(x) = ex to obtain the graph of g(x) = ex/4 + 3.

A)

B)

14

49)

C)

D)

Find the domain of the logarithmic function.

1 50) f(x) = ln x-2 A) (-2, )

50)

B) (2, )

C) (1, )

D) (0, )

Solve the problem.

51) A city is growing at the rate of 0.6% annually. If there were 4,497,000 residents in the city in 1995, find how many (to the nearest ten-thousand) are living in that city in 2000. Use y = 4,497,000(2.7)0.006t.

A) 4,630,000

B) 12,140,000

C) 4,660,000

51)

D) 360,000

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

3 x4 x + 5 52) loga 52) (x -2)2

A) loga x4 + loga (x + 5)1/3 - loga (x - 2)2 C) 4 loga x +

B) 4 loga x - 3 loga (x + 5) - 2 loga ( x - 2)

1 loga (x + 5) - 2 loga (x - 2) 3

D) loga x4 + loga (x + 5)-3 - loga (x - 2)2

Find the domain of the logarithmic function.

53) f(x) = ln (6 - x) A) (- , 6) B) (- , 6) or (6, )

C) (-6, )

D) (- , 0)

53)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

54) log 6 x2 = log 6 (3x + 28) 54) 7 A) {7} B) C) D) {7, -4} 6

15

Solve the problem.

55) The function f(x) = 1 + 1.3 ln (x + 1) models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where x is the number of consecutive days the basketball player has practiced for two hours. After 100 days of practice, what is the average number of consecutive free throws the basketball player makes?

A) 8 consecutive free throws B) 7 consecutive free throws C) 10 consecutive free throws D) 11 consecutive free throws

55)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

56) log 3 (2x + 2) = log 3 (2x + 5) 56)

A) {0}

B) {3}

C)

7 3

D)

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

7 57) log 3 57) 13

A)

log 3 7

B) log 3 13 - log 3 7

log 3 13

C) log 3 7 - log 3 13

D) log 3 7 + log 3 13

Graph the function.

58) Use the graph of f(x) = 3 x to obtain the graph of g(x) = 3 x - 3 + 2.

16

58)

A)

B)

C)

D)

Solve the equation by expressing each side as a power of the same base and then equating exponents.

59) 3 (1 + 2x) = 243 A) {81}

B) {-2}

Solve the problem.

60) The logistic growth function f(t) =

C) {6}

D) {2}

51,000 models the number of people who have become 1 + 2549e-1.4t

59)

60)

ill with a particular infection t weeks after its initial outbreak in a particular community. How many people were ill after 7 weeks?

A) 51,000 people B) 53,550 people C) 140 people D) 44,684 people

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

61) ln x = 4 61) 4 A) B) 4 e C) {ln 4} D) e4 ln 1

Graph the function.

17

62) Use the graph of log 2 x to obtain the graph of f(x) = -

Solve.

1 log 2 x.

2

A)

B)

C)

D)

63) The population of a particular country was 23 million in 1982; in 1996, it was 33 million. The exponential growth function A =23ekt describes the population of this country t years after 1982.

Use the fact that 14 years after 1982 the population increased by 10 million to find k to three decimal places.

A) 0.026 B) 0.036 C) 0.474 D) 0.164

18

62)

63)

Answer Key Testname: C3

1) A 2) C 3) C 4) D 5) C 6) D 7) C 8) D 9) C 10) C 11) C 12) D 13) a.

b. Exponential function 14) a.

b. Logarithmic Function 15) C 16) A 17) D 18) D 19) A 20) C 21) C

19

Answer Key Testname: C3

22) C 23) B 24) C 25) D 26) A 27) D 28) C 29) C 30) D 31) B 32) A 33) D 34) D 35) B 36) C 37) C 38) D 39) C 40) D 41) A 42) A 43) B 44) C 45) C 46) C 47) B 48) B 49) A 50) B 51) A 52) C 53) A 54) D 55) B 56) D 57) C 58) A 59) D 60) D 61) D 62) B 63) A

20

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Match the function to its graph.

1) y = -cot x +

1)

2

A)

B)

C)

D)

1

2) y = -cot (x + ) A)

B)

C)

D)

2

2)

3) y = cot x A)

B)

C)

D)

3

3)

4) y = -cot x -

4)

2

A)

B)

C)

D)

4

5) y = -tan x A)

B)

C)

D)

5

5)

6) y = tan x -

6)

2

A)

B)

C)

D)

6

7) y = tan (x + ) A)

B)

C)

D)

7

7)

8) y = tan x +

8)

2

A)

B)

C)

D)

8

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

9) The following data represents the normal monthly precipitation for a certain city.

Normal Monthly Month, x Precipitation, inches January, 1 6.06 February, 2 4.45 March, 3 4.38 April, 4 2.08 May, 5 1.27 June, 6 0.56 July, 7 0.17 August, 8 0.46 September, 9 0.91 October, 10 2.24 November, 11 5.21 December, 12 5.51 Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form y = A sin( x - ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

9

9)

10) The following data represents the normal monthly precipitation for a certain city.

10)

Normal Monthly Month, x Precipitation, inches January, 1 3.91 February, 2 4.36 March, 3 5.31 April, 4 6.21 May, 5 7.02 June, 6 7.84 July, 7 8.19 August, 8 8.06 September, 9 7.41 October, 10 6.30 November, 11 5.21 December, 12 4.28 Draw a scatter diagram of the data for one period. Find the sinusoidal function of the form y = A sin( x - ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the trigonometric function. Do not use a calculator.

13 11) sec 4 A) -2

B) -

2

C)

2 2

11) D) -

2 3 3

Use the unit circle to find the value of the trigonometric function.

12) csc

12)

3

A)

3 2

B)

2 3 3

C) 2

10

D)

2

Find an equation for the graph.

13)

A) y = 2 sin

1 x 4

13)

B) y = 2 sin 4x

C) y = 4 sin 2x

D) y = 4 sin

Find the exact value of the indicated trigonometric function of .

2 Find sin .

14) cos = , tan < 0 3 A) -

5 2

B) -

5 3

Classify the angle as acute, right, obtuse, or straight.

15) 116.244° A) right B) straight

1 x 2

14) 3 2

C) - 5

D) -

C) obtuse

D) acute

Use a calculator to find the value of the trigonometric function to four decimal places.

16) cos 0.3 A) 0.2955 B) 0.9553 C) 1.0000

D) 0.0052

15)

16)

Find a cofunction with the same value as the given expression.

17) cos

17)

10

A) sin

2 5

B) sin

C) cos

10

Graph the function.

11

10

D) cos

2 5

18) y = 2 cos

1 x 3

18)

A)

B)

C)

D)

12

Find an equation for the graph.

19)

A) y = 3 cos

2

x

19)

B) y = 3 cos 2 x

C) y = 2 cos 3 x

D) y = 2 cos

3

Use reference angles to find the exact value of the expression. Do not use a calculator.

-20 20) cot 3 A) -

Let

3 3

B) - 3

C)

3 3

be an angle in standard position. Name the quadrant in which the angle lies.

21) cos < 0, csc < 0 A) quadrant IV B) quadrant I C) quadrant III

Graph the function.

1 22) y = sin (x + ) 3

x

20) D)

3

D) quadrant II

21)

22)

13

A)

B)

C)

D)

23) y = csc

x 2

23)

14

A)

B)

C)

D)

Find a positive angle less than 360° or 2 that is coterminal with the given angle.

24) -266° A) 266° B) -86° C) 274°

D) 94°

Find the exact value of the expression.

3 25) cos-1 2 A)

4

24)

25) B)

C)

6

11 6

D)

4

Solve the problem.

26) A building 240 feet tall casts a 40 foot long shadow. If a person stands at the end of the shadow and looks up to the top of the building, what is the angle of the person's eyes to the top of the building (to the nearest hundredth of a degree)? (Assume the person's eyes are 4 feet above ground level.)

A) 80.38° B) 80.24° C) 80.54° D) 9.76°

26)

Sin t and cos t are given. Use identities to find the indicated value. Where necessary, rationalize denominators.

-3 5 2 . Find csc t.

27) sin t = - , cos t = 27) 7 7 A)

-7 5 15

B) -

7 2

C)

15

3 5 2

D)

-2 5 15

Find the exact value of the indicated trigonometric function of .

9 in quadrant IV Find tan .

28) sec = , 4 A) -

9 4

B) -

65

C) -

28) 65 9

D) -

65 4

Find a cofunction with the same value as the given expression.

29) csc

29)

16

A) sec

7 16

B) csc

C) csc

16

7 16

D) cot

16

Find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round your answer to the nearest whole number.

30) 30)

a 33° b = 12

A) a = 1 cm

B) a = 18 cm

C) a = 7 cm

D) a = 8 cm

Use a vertical shift to graph the function.

31) y = -3 cos 3x -

3

+2

31)

16

A)

B)

C)

D)

Use a calculator to find the value of the expression rounded to two decimal places.

32) tan-1 (0.6) A)

B)

C)

D)

Determine the amplitude or period as requested.

1 33) Amplitude of y = cos 5x 2 A)

5

32)

33)

B) 4

C)

1 2

D) 5

Solve the right triangle shown in the figure. Round lengths to one decimal place and express angles to the nearest tenth of a degree.

34) B = 28.0°, c = 3.6 mm A) a = 2.4 mm, A = 62.0°, b = 2.7 mm C) a = 1.7 mm, A = 62.0°, b = 3.2 mm

B) a = 3.2 mm, A = 62.0°, b = 1.7 mm D) a = 3.2 mm, A = 62.0°, b = 2.4 mm

Graph the function.

17

34)

35) y = 3 sin x +

35)

2

A)

B)

C)

D)

18

36) y = -

1 sin x + 4 2

36)

A)

B)

C)

D)

Find a cofunction with the same value as the given expression.

37) csc 13° A) sec 103° B) sec 13° C) sin 13°

19

D) sec 77°

37)

Use the Pythagorean Theorem to find the length of the missing side.Then find the indicated trigonometric function of the given angle. Give an exact answer with a rational denominator.

38) Find sin .

38)

5

6 6 61 A) 61

B)

61 6

C)

61 5

Determine the phase shift of the function.

39) y = 3 sin (3 x - 4)

5 61 61

39)

A) 4 units to the left C)

D)

B) 4 units to the right

4 units to the right 3

D)

Draw the angle in standard position.

7 40) 6

4 units to the left 3

40)

A)

B)

C)

D)

20

Use an identity to find the value of the expression. Do not use a calculator.

41) sec2

4

- tan2

41)

4

A) 1

B) -

C)

4

4

D) -1

Draw the angle in standard position.

42) 405° A)

42)

B)

C)

D)

Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.

43) sin(tan-1 x) 43)

A)

x2 + 1 x2 + 1

B)

x x2 + 1 x2 + 1

C) x x2 + 1

21

D)

x x2 - 1 x2 - 1

Solve the problem.

44) From a boat on the lake, the angle of elevation to the top of a cliff is 25°35'. If the base of the cliff is 1325 feet from the boat, how high is the cliff (to the nearest foot)?

A) 634 feet B) 637 feet C) 647 feet D) 644 feet

44)

Find the exact value of the expression, if possible. Do not use a calculator.

45) cos-1 cos A)

6

45)

6

B)

C)

3

D)

6

3

Classify the angle as acute, right, obtuse, or straight.

46)

46)

3

A) acute

B) straight

C) right

D) obtuse

Use even and odd properties of the trigonometric functions to find the exact value of the expression.

47) tan -

47)

6

A) - 3

B)

3 3

C) -

22

3 3

D)

3

Graph the function.

48) y = -3 sin (2 x + 2 )

48)

A)

B)

C)

D)

23

Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g in the same rectangular coordinate system.

49) f(x) = 3 cos x, g(x) = cos 3x; h(x) = (f + g)(x) 49)

A)

B)

C)

D)

24

An object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in meters . Find the maximum displacement, the frequency, and the time required for one cycle.

50) d = 3 cos 5 t meters 50) 5 2 A) displacement = -3 meters; period = seconds; f = oscillations/second 2 5

B) displacement = 3 meters; period =

5 2 seconds; f = oscillations/second 2 5

C) displacement = 3 meters; period =

2 5

D) displacement = 3 meters; period =

2 5 seconds; f = oscillations/second 5 2

seconds; f =

5 oscillations/second 2

Find the exact value of the trigonometric function. Do not use a calculator.

51) cos A)

4

- 1000 2 2

51) B)

2

C) -

2 2

Use a calculator to find the value of the trigonometric function to four decimal places.

52) csc 1 A) 57.2996 B) 1.8508 C) 1.0002

D) -

2

52)

D)

Find the length of the arc on a circle of radius r intercepted by a central angle . Round answer to two decimal places.

53) r = 4 centimeters, = 50° 53) A) 3.49 centimeters B) 3.14 centimeters C) 2.79 centimeters D) 3.84 centimeters Use the given triangles to evaluate the expression. Rationalize all denominators.

54) cos

54)

6

A)

2 2

B)

3

C)

25

3 2

D)

2 3 3

Use even and odd properties of the trigonometric functions to find the exact value of the expression.

55) cos (-150°) A) -

1 2

B)

3 2

3 2

C) -

D)

1 2

Solve the problem.

56) A car wheel has a 16-inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward 3 ft?

A) 128.9° B) 143.9° C) 138.9° D) 133.9° Use reference angles to find the exact value of the expression. Do not use a calculator.

-5 57) sec 4 A) -

2 3 3

B) -

2

C) -2

55)

56)

57) D)

2 2

Find the exact value of the trigonometric function. Do not use a calculator.

58) sec

58)

4

A) -

2 2

B) - 2

C)

2

D)

2 2

Graph the function.

59) y =

cot x +

59)

4

A)

B)

26

C)

D)

Use a sketch to find the exact value of the expression.

3 60) cos tan-1 8 A)

73 8

B)

60)

8 73

C)

3 8

D)

8 73 73

Use a vertical shift to graph the function.

61) y = -4 sin 3x +

3

-2

61)

A)

B)

27

C)

D)

Find the exact value of the expression. Do not use a calculator.

62) 1 - tan2 80° + csc2 10° A) -1

B) 1

C) 0

D) 2

Find the exact value of the trigonometric function. Do not use a calculator.

21 63) cot 4 3

A)

B)

3 3

C) -1

63) D) 1

Use reference angles to find the exact value of the expression. Do not use a calculator.

64) sec 2 A) 0

B) 1

C) -1

62)

64) D) undefined

Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function. x ) 65) sin(tan-1 65) 2

A)

x x2 - 2 x2 - 2

B) x x2 + 2

C)

Graph the function.

28

x2 + 2 x2 + 2

D)

x x2 + 2 x2 + 2

66) y =

1 cot 3x 4

66)

A)

B)

C)

D)

Find the reference angle for the given angle.

5 67) 4 A)

4

B)

67) C)

8

Graph the function.

29

3 4

D)

5 4

68) y = -3 sin

1 x 3

68)

A)

B)

C)

D)

30

Use the given triangles to evaluate the expression. Rationalize all denominators.

69) csc

3

A)

sin

3

3- 3 3

- tan

69)

3

B)

2- 3 2

C) 1 -

3

D) 2 -

3

An object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in meters . Find the maximum displacement, the frequency, and the time required for one cycle.

70) d = 7 + 5 cos 3 t meters 70) 2 3 A) displacement = 5 meters; period = seconds; f = oscillations/second 3 2

B) displacement = 5 meters; period =

3 2 seconds; f = oscillations/second 2 3

C) displacement = 12 meters; period =

2 3 seconds; f = oscillations/second 3 2

D) displacement = 12 meters; period =

2 3

seconds; f =

31

3 oscillations/second 2

Answer Key Testname: C4

1) D 2) D 3) B 4) B 5) D 6) C 7) C 8) C 9) y = 3.14 sin(0.46x + 1.53) + 3.16;

10) y = 2.17 sin(0.49x - 1.88) + 6.02;

11) B 12) B 13) C 14) B 15) C 16) B 17) A 18) B 19) D 20) C 21) C 22) D 23) D 24) D 25) B 26) A 27) B 28) D 29) A 30) D 32

Answer Key Testname: C4

31) A 32) A 33) C 34) B 35) B 36) C 37) D 38) D 39) C 40) C 41) A 42) B 43) B 44) A 45) A 46) A 47) C 48) A 49) A 50) D 51) A 52) D 53) A 54) C 55) C 56) A 57) B 58) C 59) A 60) D 61) B 62) D 63) D 64) D 65) D 66) D 67) A 68) C 69) C 70) A

33

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Verify the identity.

1) cos x +

2

= -sin x

1)

2)

cos x + cos y x-y = cot sin x - sin y 2

2)

3)

cos( + ) = cot cos sin

3)

- tan

4) sin4t = 2 sin2t cos2t

4)

5) 1 + sec2 x sin2 x = sec2 x

5)

6) cot 2 x + csc 2 x = 2 csc 2 x - 1

6)

7) sin

7)

2

-

= -cos

8) cos( - ) - cos( + ) = 2 sin

sin

8)

9) sin( + ) - sin( - ) = 2 cos

sin

9)

10) tan 2 x (1 + cos 2x) = 1 - cos 2x

10)

tan x - 1 1 + tan x

11)

· csc

= sec

12)

-

= -sin

13)

sin sin

- sin + sin

= tan

15) cot

· sec

11) tan x -

12) tan 13) cos

14)

2

4

=

2

cot

+ 2

14)

= csc

15)

16) csc2 u - cos u sec u= cot2 u

16)

1

17) (1 + tan2 u)(1 - sin2 u) = 1

17)

18) csc u - sin u = cos u cot u

18)

19) cos 4 = 2 cos2 (2 ) - 1

19)

20) tan

20)

2

+ x = -cot x

21) sin( - ) cos( + ) = sin

cos

- sin

cos

21)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Complete the identity. cos ( + ) =?

22) cos ( - ) A)

22)

1 - tan ( + ) 1 + tan tan

B)

1 + tan 1 - tan

tan tan

C)

1 + tan tan 1 - tan ( + )

D)

1 - tan 1 + tan

tan tan

Use the given information to find the exact value of the trigonometric function.

3 lies in quadrant IV Find sin .

23) sin = - , 5 2 A) -

5 5

B) -

30 10

23)

10 10

C)

D)

5 5

Use the given information to find the exact value of the expression.

7 , lies in quadrant IV Find sin 2 .

24) cos = 25 A) -

527 625

B)

336 625

C) -

24) 336 625

D)

527 625

Find all solutions of the equation.

25) sin x csc x = -2 sin x

11 + n or x = A) x = 6 6

C) x =

6

+ 2n or x =

25) + n or x = n

B) x =

+ 2n or x = n

D) x =

11 6

Complete the identity.

26) tan2 2x + cos2 2x + sin2 2x = ?

A) sin2 2x

27)

6 6

+ 2n or x = + n or x =

C) cos2 2x

B) 2

6 6

+ 2n or x = n + n or x = n

D) sec2 2x

sin 4x + sin 6x =? cos 3x + cos 5x

A)

cos 10x sin 8x

26)

27) B) tan

4 6 x + tan x 3 5

C)

2

sin 10x cos 8x

D)

sin 5x cos 4x

28)

(cot x + 1)(cot x + 1) - csc2 x =? cot x

A) cot x

28)

B) 1

C) 2

D) 0

Use the given information to find the exact value of the expression.

24 2 , lies in quadrant II, and cos = , lies in quadrant I 29) sin = 25 5 A)

-14 + 24 21 125

B)

48 + 7 21 125

C)

Find cos ( - ).

14 - 24 21 125

D)

29)

48 - 7 21 125

Use the given information to find the exact value of the trigonometric function.

1 Find sin .

30) cos = , csc > 0 4 2 8 + 2 15 4

A)

B)

6 4

C)

30)

8 - 2 15 4

D)

Fill in the blank using the word product, sum, quotient, or difference.

1 31) The formula sin sin = [cos ( - ) - cos ( + )] can be used to change a 2 into the

1 (sin 12x + sin 2x) 2

D)

1 (sin 12x + cos 2x) 2

Use the given information to find the exact value of the expression.

3 21 , lies in quadrant II 33) cos = - , lies in quadrant III, and sin = 5 5 6 + 4 21 25

B)

32)

1 B) (cos 12x - cos 2x) 2

A) sin (cos 35x2 )

-8 + 3 21 25

C)

Express the product as a sum or difference.

34) sin 4x sin 2x 1 A) (cos 6x - cos 2x) 2 C)

31)

B) quotient, difference D) product, difference

Express the product as a sum or difference.

32) sin 7x cos 5x

A)

of two sines

of two cosine expressions.

A) product, sum C) quotient, sum

C)

10 4

B)

1 (sin 6x + cos 2x) 2

-6 - 4 21 25

1 (cos 2x - cos 6x) 2

D) sin2 8x2

3

Find cos ( + ).

D)

33)

8 - 3 21 25

34)

35) cos

5x x cos 2 2

35)

A)

1 (cos 3x - sin 2x) 2

B)

1 (cos 2x + cos 3x) 2

C)

1 cos2 5x 4

D)

1 (cos 6x - sin4x) 4

Use substitution to determine whether the given x-value is a solution of the equation.

-2 3 , x= 36) sin x = 2 3 A) Yes

36)

B) No

Complete the identity.

5 =?

37) cos x 6 3 (cos x - sin x) 2

B) -

1 (- 3 cos x + sin x) 2

D)

A) C)

37)

Express the sum or difference as a product.

38) cos 2x - cos 4x A) cos (-2x) B) 2 sin 3x sin x

3 (cos x + sin x) 2 3 (cos x - sin x) 2

C) -2 cos 3x sin x

D) -2 sin 3x sin x

38)

Complete the identity.

39)

(sin x + cos x)2 =?

1 + 2 sin x cos x

A) 1 40)

39) B) - sec2 x

C) 0

D) 1 - sin x

cos 5x - cos 9x =? cos 5x + cos 9x

A) -tan 7x

40) B) tan 2x tan 7x

C) cot 7x

4

D) 0

Use the graph to complete the identity.

1 + cos x sin x + =?

41) sin x 1 + cos x

A) 2cos x

41)

B) 2csc x

C) 2sin x

D) 2sec x

Use the given information to find the exact value of the trigonometric function.

5 Find sin .

42) sec = - , lies in quadrant II 3 2 A)

5 5

B) -

2 5

C) -

5 5

42) D)

2 5 5

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.

43) cos 2x + cos 6x = 2 cos 4x cos x 43) A) 2

B)

C) 0

4

D) -2

Solve the problem.

44) If a projectile is fired at an angle and initial velocity v, then the horizontal distance traveled by the 1 2 projectile is given by D = v sin cos . Express D as a function of 2 .

16 A) D =

1 2 v cos 2 16

B) D =

1 2 v sin 2 16

C) D =

1 2 v sin 2 32

D) D =

1 2 v sin 2 8

Complete the identity.

45) sin2 x + sin2 x cot2 x = ?

A) cot2 x + 1

B) cot2 x - 1

C) 1

5

D) sin2 x + 1

44)

45)

Solve the problem.

46) Graph in a [- , , y=2

2

] by [-3, 3, 1] viewing rectangle

46)

sin x sin 2x sin 3x sin 4x sin 5x sin 6x sin 7x + + + 1 2 3 4 5 6 7

A)

B)

C)

D)

Use a half-angle formula to find the exact value of the expression.

5 47) cos 12 A) -

1 2

2-

3

B)

1 2

2-

3

C)

47) 1 2

2+

3

D) -

1 2

2+

3

Complete the identity.

1 - cos x =?

48) sin x A) csc x + cot x

48) B) -csc x - cot x

C) csc x - cot x + 1

D) csc x - cot x

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.

49) 2 sin 120° cos 120° 49) 3 1 1 A) B) C) D) 2 2 2

6

Use a half-angle formula to find the exact value of the expression.

3 50) cos 8 A) -

1 2

2+

2

B) -

1 2

2-

2

C)

Find the exact value by using a sum or difference identity.

51) sin 165° 2( 3 - 1) A) - 2( 3 + 1) B) 4 Complete the identity.

52) csc x(sin x + cos x) = ?

A) -2 tan2 x

50) 1 2

2+

2

D)

1 2

2-

2

51) C) - 2( 3 - 1)

2( 3 - 1) 4

D) -

52) B) sec x csc x

C) 1 + cot x

D) sin x tan x

Solve the equation on the interval [0, 2 ).

3 53) sin 4x = 2 A) 0,

4

53)

,

B)

C) 0

D)

Complete the identity.

54) tan ( - ) = ?

A) -tan

B) tan

4

5 4

,

2 7 7 13 5 19 , , , , , , , 12 6 3 12 6 12 3 12

C) cot

D) -cot

Solve the equation on the interval [0, 2 ).

3 55) cos 2x = 2 A) C)

55)

3 2

B) D)

2

6

11 6

,

12

,

11 13 23 , , 12 12 12

Solve the equation on the interval [0, 2 ).

56) tan2 x sin x = tan2 x A)

2

,2

54)

56) C) 0,

B) 0,

2

D)

2

,

Use a calculator to solve the equation on the interval [0, 2 ). Round the answer to two decimal places.

57) sin x = -0.33 A) 0.34, 5.94 B) 0.34, 1.91 C) 0.34, 3.48 D) 3.48, 5.95

7

57)

Use trigonometric identities to find the exact value. tan 160° - tan 40° 58) 1 + tan 160° tan 40° 3 3

A) -

58)

B) -2

1 2

C) -

D) - 3

Complete the identity. sin2 x - cos2 x =?

59) 1 - cot2 x A) sin2 x

59) C) cos2 x

B) 1

Find the exact value of the expression.

60) cos (165°) cos (45°) + sin (165°) sin (45°) 1 A) B) -2 2

D) -1

60) 3 C) 2

D) -

3

Use the graph to complete the identity.

61) cos 3x cos 6x + sin 3x sin 6x = ?

A) y = cos 3x

61)

B) y = cos 9x

C) y = sin 3x

D) y = sin 9x

Solve the equation on the interval [0, 2 ).

2 62) cos 2x = 2 A) C)

4 8

62)

,

3 5 7 , , 4 4 4

B) 0,

,

7 9 15 , , 8 8 8

D) no solution

Find the exact value by using a sum or difference identity.

63) sin (200° - 80°) 3 1 A) B) 2 2

2 4 , , 3 3

63) C)

8

3 2

D) -

10 3

Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

64) sin3 x 64)

A)

3 1 sin x - sin 3x 4 4

B)

3 sin x 4

C)

3 1 sin x + sin 2x 4 4

D)

3 1 1 sin x + sin 2x - sin 3x 4 2 4

Solve the equation on the interval [0, 2 ).

65) cos x +

A)

+ cos x -

3

2

,

3

=1

65)

B)

6

2

,

4

Complete the identity.

66) 8 sin x cos3 x + 8 sin3 x cos x = ?

A) 4 cos x

B) 4 cos 2x

D)

C) 4 sin x

D) 4 sin 2x

Use the given information to find the exact value of the expression.

21 5 , lies in quadrant III, and cos = , lies in quadrant II 67) tan = 20 13 A) -

135 377

B) -

152 377

,

C) 0

C)

6

4

Find sin ( + ).

345 377

D)

Use the graph to complete the identity.

3 3 68) cos 3x cos x + sin 3x sin x = ?

2 2

66)

67)

352 377

68)

A) y = cos y = cos

9 x 2

B) y = sin y = cos

C) y = sin y = cos

3 x 2

D) y = cos

9

3 x 2

9 x 2

Find the exact value under the given conditions.

4 8 , 0< < 69) sin = , 0 < < ; cos = 5 2 17 2 A)

84 85

B) -

Find tan ( + ).

36 85

C)

Solve the equation on the interval [0, 2 ).

70) 2 cos2 x + sin x - 2 = 0 3 2 , , A) , 2 2 3 3 11 6

71) sin x +

A)

2

,

- sin x -

5 B) , 6 6

11 6

77 85

D) -

2 C) , 3 3

77 36

5 D) 0, , , 6 6

=1

B)

4

69)

71)

2

,

C)

3

4

,

D)

3

Use a half-angle formula to find the exact value of the expression.

3 72) sin 8 A)

1 2

2+

2

B) -

1 2

2-

2

72)

C) -

Find the exact value of the expression.

73) sin 260° cos 20° - cos 260° sin 20° 3 13 A) B) 2 3

1 2

2+

2

D)

1 2

2-

75) cos x + A)

2

x

73)

74) C) sec2 x

B) 1

468 493

D) csc2 x

=?

75) B) -cos x

C) -sin x

Use the given information to find the exact value of the expression.

21 15 , lies in quadrant II, and cos = , lies in quadrant I 76) sin = 29 17 A)

2

3 D) 2

1 C) 2

Complete the identity. csc x cot x =?

74) sec x A) cot2 x

70)

B)

C)

Express the sum or difference as a product.

77) sin 9x - sin 3x A) 2 sin 3x cos 6x B) 2 sin 6x cos 3x

D) cos x

Find sin ( - ).

132 493

C) 2 cos 3x cos 6x

10

D)

76)

475 493

D) 2 sin 3x

77)

Use the graph to complete the identity. (sec x + tan x) (sec x - tan x) =?

78) sec x

A) sin x

78)

B) cos x

C) sec x

D) csc x

Use a calculator to solve the equation on the interval [0, 2 ). Round to the nearest hundredth of a radian.

79) sin 2x - sin x = 0 A) 0, 2.09, 4.19 B) 0, 1.05, 3.14, 5.24 C) 1.05, 3.14, 5.24 D) 0, 2.09, 3.14, 4.19 Solve the equation on the interval [0, 2 ).

80) cos2 x + 2 cos x + 1 = 0 A)

4

,

7 4

B)

,

3 2

C) 2

D)

2+ 3 C) 4

2- 3 D) 4

2

Find the exact value by using a difference identity.

81) tan 105° A) 2 +

3

B) -2 -

79)

80)

81)

3

Complete the identity.

82) (sin x + cos x)2 = ?

A) 1 + cos 2x

B) 1 - cos 2x

C) sin 2x

D) 1 + sin 2x

Solve the equation on the interval [0, 2 ). x x 83) sec = cos 2 2

82)

83)

A)

2 7 7 13 5 , , , , , , 12 6 3 12 6 12 3

B) 0

C)

5 4 4

D) 0,

,

11

4

, ,

5 3

Use trigonometric identities to find the exact value. tan 55° + tan 95° 84) 1 - tan 55° tan 95° A) -

1 2

84)

B) - 3

C) -2

Express the product as a sum or difference.

85) cos 4x cos 8x 1 A) (cos 4x + cos 12x) 2 C)

D) -

3 3

85)

1 B) (cos 12x - cos 4x) 2

1 (cos 12x - sin 4x) 2

D) cos2 20x2

Rewrite the expression as a simplified expression containing one term.

-11 -11 -11 -11 - cos + + cos - sin + 86) sin 6 6 6 6 3 2

A)

3 2

B) -

C)

Express the product as a sum or difference.

87) sin 4x cos 5x 1 A) (cos 9x + sin x) 2 C)

1 2

86) D) -

1 2

87)

1 B) (sin 9x - sin x) 2

1 (cos 9x - cos x) 2

D) sin (cos 20x2 )

Complete the identity. cos ( - ) =?

88) cos sin A) tan C) tan 89) tan

2

A)

88)

+ cot + cot

B) tan + tan D) cot + tan tan

=? tan sec + 1

89) B)

tan csc + 1

C)

tan csc - 1

D)

tan sec - 1

Use the given information to find the exact value of the trigonometric function.

4 Find cos .

90) cos = , lies in quadrant IV 5 2 A) -

3 10 10

B) -

2 5

C)

Complete the identity.

91) sec4 x + sec2 x tan2 x - 2 tan4 x = ?

A) 3 sec4 x - 2

B) 4 sec4 x

2 5

C) tan2 x - 1

12

90) D)

3 10 10

D) sec4 x + 2

91)

Answer Key Testname: C5

1) cos x +

2

= cos x cos

cos x + cos y = sin x - sin y

3)

cos( + ) cos = cos sin

- sin x sin

cos =

x-y x+y 2 sin cos 2 2

cos - sin cos sin

= (cos x)(0) - (sin x)(1) = - sin x.

2

x+y x-y cos 2 2

2 cos

2)

2

sin

x-y 2

x-y sin 2

= cot

x-y 2

-

sin cos

cos cos cos sin

=

sin sin

=

cos sin

-

sin cos

= cot

- tan

4) sin 4t = sin [2(2t)] = 2 sin2t cos2t. sin2 x = 1 + tan2 x = sec2 x.

5) 1 + sec2 x sin2 x = 1 + cos2 x 6) cot 2 x + csc 2 x = csc 2 x - 1 + csc 2 x = 2 csc 2 x - 1.

7) sin

2

= sin

-

cos

2

- cos

sin

2

= (-1) · cos

- 0 · sin

= - cos

8) cos ( - ) - cos ( + ) = cos cos + sin sin - ( cos cos - sin sin ) = 2 sin sin .

9) sin ( + ) - sin ( - ) = sin cos + cos sin - sin cos + cos sin = 2 cos sin .

1 - cos 2x (1 + cos 2x) = 1 - cos 2x 10) tan 2 x (1 + cos 2x) = 1 + cos 2x 11) tan x 12) tan 13) cos

4

· csc 2

-

sin sin

- sin + sin

15) cot

· sec

14)

tan x - tan /4 tan x - 1 = .

1 + (tan x)(tan /4) 1 + tan x

=

=

sin cos

1 sin

=

cos

+ sin

2 sin

2

cos

+ 2

2 sin

+ 2

cos

2

1 cos

=

= cos

=

=

cos sin

· 2

·

1 cos

1 sin

= sec 2

sin

=

sin

2

cos

2

- 1 · sin

·

cos

+ 2

sin

+ 2

= - sin

= tan

2

cot

+ 2

= csc

16) csc2 u - cos u sec u = csc2 u - cos u ·

1 = csc2 u - 1 = cot2 u cos u

17) (1 + tan2 u)(1 - sin2 u) = sec2 u · cos2 u = 18) csc u - sin u =

= 0 · cos

1 · cos2 u = 1 cos2 u

1 1 - sin2 u cos2u cos u - sin u = = = cos u · = cos u cot u sin u sin u sin u sin u

19) cos 4 = cos[2(2 )] = 2 cos2 (2 ) - 1 sin (( /2) + x) sin ( /2) cos x + sin x cos ( /2) 1 · cos x + sin x · 0 +x = = = = -cot x.

20) tan 2 cos (( /2) + x) cos ( /2) cos x - sin ( /2) sin x 0 · cos x - 1 · sin x 21) sin( - ) cos ( + ) = (sin cos - cos sin )(cos cos - sin sin ) = sin cos cos2 - sin2 cos sin - cos2 sin cos + cos sin sin2 = sin cos (cos2 + sin2 ) - sin cos (sin2 + cos2 ) = sin cos - sin 22) D

13

cos

Answer Key Testname: C5

23) C 24) C 25) C 26) D 27) D 28) C 29) A 30) B 31) D 32) C 33) A 34) B 35) B 36) A 37) C 38) B 39) A 40) B 41) B 42) D 43) B 44) C 45) C 46) D 47) B 48) D 49) A 50) D 51) B 52) C 53) D 54) A 55) D 56) B 57) D 58) D 59) A 60) A 61) A 62) C 63) C 64) A 65) C 66) D 67) A 68) D 69) D 70) D 71) D 72) A 14

Answer Key Testname: C5

73) D 74) A 75) A 76) D 77) A 78) B 79) B 80) B 81) B 82) D 83) B 84) D 85) A 86) A 87) B 88) A 89) A 90) A 91) A

15

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Select the representation that does not change the location of the given point.

1) (-4, 8 ) A) (-4, 7 ) B) (4, 6 ) C) (4, 7 )

D) (-4, 9 )

The graph of a polar equation is given. Select the polar equation for the graph.

2)

A) r = 4 sin

B) r = 4 cos

C) r = 2

1)

2)

D) r sin

=2

3)

3)

A) r = 2 + cos

B) r = 2 + sin

C) r = 4 cos

1

D) r = 4 sin

4)

4)

A) r = 2 sin(2 )

B) r = 2 cos(2 )

C) r = 2

D) r = 2 + cos(2 )

5)

5)

A) r = 6 cos

B) r = 6 sin

C) r sin

=3

Select the representation that does not change the location of the given point.

6) (9, 50°) A) (9, 410)° B) (-9, 140)° C) (-9, 410)°

D) r = 3

D) (9, 230)°

The graph of a polar equation is given. Select the polar equation for the graph.

7)

A) r = -4 cos

B) r = -4 sin

C) r sin

2

= -2

6)

7)

D) r = -2

8)

8)

A) r = 8 sin

B) r = 4 + sin

C) r = 8 cos

D) r = 4 + cos

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

9) Plot the point 4, (a) r > 0, (b) r < 0, (c) r > 0

-2 0 2

6

and find other polar coordinates (r, ) of the point for which:

9)

<0 <2 <4

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the specified vector or scalar.

10) v = -7i + 3j; Find 10v .

A) 20i 10

10) B) 10 58

C) -10 58

3

D) 20 10

Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle measures to the nearest degree.

11) B = 41°, a = 4, b = 3 11) A) A1 = 61°, C1 = 78°, c1 = 4.5; B) A1 = 61°, C1 = 78°, c1 = 0.1; A2 = 119°, C2 = 20°, c2 = 1.6 C) no triangle

A2 = 119°, C2 = 20°, c2 = 0.1 D) A = 29°, C = 110°, c = 5.7

Let v be the vector from initial point P1 to terminal point P2 . Write v in terms of i and j.

12) P1 = (-4, -3); P2 = (5, 3) A) v = 9i + 6j

12) B) v = 7i + 8j

C) v = 8i + 7j

Use a polar coordinate system to plot the point with the given polar coordinates.

-5 13) 2, 4

A)

B)

4

D) v = 6i + 9j

13)

C)

D)

Use a graphing utility to graph the polar equation.

1 14) r = 2 - 4 sin

14)

A)

B)

5

C)

D)

Solve the problem.

15) Two airplanes leave an airport at the same time, one going northwest (bearing 135°) at 418 mph and the other going east at 341 mph. How far apart are the planes after 4 hours (to the nearest mile)?

A) 2204 miles B) 2807 miles C) 2337 miles D) 702 miles Find a. If necessary, round your answer to the nearest hundredth.

16)

15)

16)

45°

61

A) 7.28

36°

B) 6.22

C) 9.66

D) 16.68

Find the product of the complex numbers. Leave answer in polar form.

7 7 + i sin 17) z 1 = 3 cos 4 4 z2 =

6 cos

17)

9 9 + i sin 4 4

B) 3 2 sin

A) 3 2(sin 0 + i cos 0) C) 3 2 cos

5 5 + i sin 4 4

5 5 + i cos 4 4

D) 3 2(cos 0 + i sin 0)

Test the equation for symmetry with respect to the given axis, line, or pole.

18) r = -2 cos ; the polar axis A) has symmetry with respect to polar axis B) may or may not have symmetry with respect to polar axis

6

18)

Rectangular coordinates of a point are given. Use a graphing utility in radian mode to find polar coordinates of the point.

19) (-16, -12) Express to three decimal places.

19) A) (20, -2.214) B) (20, -5.356) C) (28, -2.214) D) (20, -2.498) Test the equation for symmetry with respect to the given axis, line, or pole.

20) r cos = ; the polar axis A) may or may not have symmetry with respect to polar axis B) has symmetry with respect to polar axis Find the unit vector that has the same direction as the vector v.

21) v = 8i 1 A) u = i B) u = 8i C) u = 64i 8

20)

21) D) u = i

Find the area of the triangle having the given measurements. Round to the nearest square unit.

22) A = 37°, b = 18 meters, c = 11 meters A) 60 square meters B) 122 square meters C) 30 square meters D) 120 square meters Match the point in polar coordinates with either A, B, C, or D on the graph.

23) (4, 0)

A) C

B) A

C) B

Find the unit vector that has the same direction as the vector v.

24) v = -7j A) u = 49j

B) u = -7j

1 C) u = - j 7

23)

D) D

24) D) u = -j

Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.

25) v = 2j, w = 4i A) parallel B) orthogonal C) neither Graph the polar equation.

7

22)

25)

26) r = 3 sin2

26)

cos

A)

B)

C)

D)

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

27) (1 - i)10 27) A) 32

B) -32i

C) 32 - 32i

8

D) -32 + 32i

Graph the polar equation.

28) r = 3 cos + 4 sin

28)

A)

B)

C)

D)

9

Use the vectors v, u, w, and z to draw the indicated vector.

29) 3w

29)

A)

B)

C)

D)

Graph the polar equation.

10

30) r = 5 cos2

30)

sin

A)

B)

C)

D)

Solve the problem.

31) Find the work done by a force of 2 pounds acting in the direction of 35° to the horizontal in moving an object 10 feet from (0, 0) to (10, 0).

A) 17.3 ft-lb B) 32.8 ft-lb C) 11.5 ft-lb D) 16.4 ft-lb Polar coordinates of a point are given. Find the rectangular coordinates of the point.

32) ( , -27°) A) (-2.3, 4.5) B) ( , -2.3) C) (-4.5, 2.3)

11

D) (

, -4.5)

31)

32)

Solve the problem.

33) Two sailboats leave a harbor in the Bahamas at the same time. The first sails at 23 mph in a direction 350°. The second sails at 33 mph in a direction 200°. Assuming that both boats maintain speed and heading, after 3 hours, how far apart are the boats?

A) 129.2 miles B) 116 miles C) 113.8 miles D) 162.5 miles

33)

Find another representation, (r, ), for the point under the given conditions.

34) 1,

3

, r > 0 and 2 <

A) 1, -

<4

2 3

34) B) 1,

4 3

C) 1,

7 3

D) 1, -

5 3

Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.

35) v = 3i + 4j, w = 3i - 2j A) parallel B) orthogonal C) neither

35)

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

3 3 3 + i sin ) 36) 10 (cos 36) 4 4 A) -500 2 + 500 2i C) 5 2 + 5 2i

B) 50 2 + 50 2i D) 15 2 + 15 2i

Use Heron's formula to find the area of the triangle. Round to the nearest square unit.

37) a = 10 inches, b = 14 inches, c = 6 inches A) 26 square inches B) 35 square inches C) 24 square inches D) 33 square inches Test the equation for symmetry with respect to the given axis, line, or pole.

38) r = 4 + 4 cos ; polar axis A) may or may not have symmetry with respect to the polar axis B) has symmetry with respect to the polar axis Solve the problem.

39) A guy wire to a tower makes a 65° angle with level ground. At a point 38 ft farther from the tower than the wire but on the same side as the base of the wire, the angle of elevation to the top of the tower is 30°. Find the length of the wire (to the nearest foot).

A) 66 feet B) 71 feet C) 33 feet D) 38 feet

37)

38)

39)

Test the equation for symmetry with respect to the given axis, line, or pole.

40) r = 4 + 4 cos ; the line

=

40)

2

A) may or may not have symmetry with respect to the line B) has symmetry with respect to the line

=

2

12

=

2

Solve the problem.

41) A power boat in still water maintains a speed of 45 miles per hour. The boat heads directly across a river perpendicular to the current which has a speed of 8 miles per hour. Find the actual speed and direction of the boat.

A) 46 miles per hour; 10° off course B) 22 miles per hour; 21° off course C) 17 miles per hour; 28° off course D) 45 miles per hour; 10° off course Solve the equation in the complex number system.

42) x7 - 1 = 0

A) 1, cos(51.4°) + i sin(51.4°), cos(102.9°) + i sin(102.9°), cos(154.3°) + i sin(154.3°), cos(205.7°) + i sin(205.7°), cos(257.1°) + i sin(257.1°), cos(308.6°) + i sin(308.6°) B) 1, cos(51.4°) + i sin(51.4°), cos(102.9°) + i sin(102.9°), cos(154.3°) + i sin(154.3°), cos(205.7°) + i sin(205.7°), cos(257.1°) + i sin(257.1°), cos(308.6°) + i sin(308.6°), -1 C) 1, cos(25.7°) + i sin(25.7°), cos(51.4°) + i sin(51.4°), cos(77.1°) + i sin(77.1°), cos(102.9°) + i sin(102.9°), cos(128.6°) + i sin(128.6°), cos(154.3°) + i sin(154.3°) D) -1, cos(25.7°) + i sin(25.7°), cos(51.4°) + i sin(51.4°), cos(77.1°) + i sin(77.1°), cos(102.9°) + i sin(102.9°), cos(128.6°) + i sin(128.6°), cos(154.3°) + i sin(154.3°)

41)

42)

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

7 7 5 + i sin ) 43) 2 2 (cos 43) 4 4 A) -64 + 64i

B) -128 + 128i

C) -

2+

2i

D) -64 2 + 64 2i

Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.

44) a = 8, b = 13, c = 15 A) A = 32°, B = 59°, C = 89° B) A = 30°, B = 59°, C = 91° C) A = 34°, B = 57°, C = 89° D) no triangle Find all the complex roots. Write the answer in the indicated form.

45) The complex fourth roots of -16 (rectangular form)

44)

45)

A) 2 + i, 2 - i, -2 + i, -2 - i B) 1 + i, 1 - i, -1 + i, -1 - i C) 2 + 2i, 2 - 2i, - 2 + 2i, - 2 - 2i D) 8 2 + 8 2i, 8 2 - 8 2i, -8 2 + 8 2i, -8 2 - 8 2i 46) The complex cube roots of -8i (rectangular form) A) -2i, 3 + i, 3 + i C) 2i, - 3 - i, 3 - i

46) B) -2i, 3 - i, 3 - i D) 2i, - 3 + i, - 3 + i

Find projwv.

47) v = i - 3j; w = 5i + 12j 31 93 i+ j A) 10 10

155 372 ij B) 13 13

155 372 ij C) 169 169

13

31 186 ij D) 2 5

47)

Plot the complex number.

48) -2 + i

48)

A)

B)

C)

D)

14

Graph the polar equation.

49) r = 3 + sin

49)

A)

B)

C)

D)

Polar coordinates of a point are given. Find the rectangular coordinates of the point.

50) (-4, -90°) A) ( , ) B) (0, -4) C) (0, 4)

D) (-4, 0)

50)

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

51) (1 + i)20 51) A) -1024

B) -1024i

C) 1024i

15

D) 1024

Write the vector v in terms of i and j whose magnitude v and direction angle are given.

52) v = 10, = 120° A) v = 5i - 5 3j B) v = -5i + 5 3j C) v = 5 3i - 5j D) v = -5 2i + 5 2j Find the unit vector that has the same direction as the vector v.

53) v = -3i + j 10 -3 i+ i + 10j A) u = B) u = -3 11 -3 i+ 10

C) u =

1 j 10

53) 1 j 11

D) u = -3 10i +

10j

Write the complex number in polar form. Express the argument in radians.

54) 2 - 2i 7 7 7 7 + i sin + i sin A) 2 2 cos B) 2 cos 4 4 4 4 C) 2 cos

5 5 + i sin 4 4

D) 2 2 cos

52)

54)

5 5 + i sin 4 4

Solve the problem.

55) A painter needs to cover a triangular region 62 meters by 66 meters by 74 meters. A can of paint covers 70 square meters. How many cans will be needed?

A) 3 cans B) 14 cans C) 317 cans D) 28 cans

55)

Polar coordinates of a point are given. Use a graphing utility to find the rectangular coordinates of the point to two decimal places.

56) 2.1, A) (

56)

9

, -1.35)

B) (

, 1.35)

C) (-1.35,

Convert the polar equation to a rectangular equation.

57) r = -2 cos A) x2 + y2 = 2 B) x - 1 2 + y2 = 4

)

C) x + 1 2 + y2 = 1

Write the complex number in polar form. Express the argument in degrees.

58) i A) 6(cos 90° + i sin 90°) B) 6(cos 0° + i sin 0°) C) 6(cos 270° + i sin 270°) D) 6(cos 180° + i sin 180°) Find the product of the complex numbers. Leave answer in polar form.

59) z 1 = 4i z 2 = -6 + 6i

57) D) x = -2

58)

59)

A) 24 2 sin

3 3 + i cos 8 8

B) 24 2 sin

5 5 + i cos 4 4

C) 24 2 cos

5 5 + i sin 4 4

D) 24 2 cos

3 3 + i sin 8 8

16

D) (1.35, 1.61)

Convert the rectangular equation to a polar equation that expresses r in terms of .

60) y2 = 3x A) r2 (cos

B) r = 3 cot2 x

+ sin ) = 3

C) r = 3 cot x cscx

60)

D) r = 9 cot x cscx

Test the equation for symmetry with respect to the given axis, line, or pole.

61) r2 = sin 2 ; the pole

61)

Solve the equation in the complex number system.

62) x5 - 1 = 0

62)

A) has symmetry with respect to the pole B) may or may not have symmetry with respect to the pole

A) 1, cos(72°)+ i sin(72°), cos(144°) + i sin(144°), cos(216°) + i sin(216°), cos(288°) + i sin(288°) B) -1, cos(72°)+ i sin(72°), cos(144°) + i sin(144°), cos(216°) + i sin(216°), cos(288°) + i sin(288°) C) 1, cos(36°) + i sin(36°), cos(108°) + i sin(108°), cos(180°)+ i sin(180°), cos(252°) + i sin(252°) D) 1, cos(72°) + i sin(72°), cos(144°) + i sin(144°), cos(216°) + i sin(216°),-1

Test the equation for symmetry with respect to the given axis, line, or pole.

63) r = -4 cos ; the line

=

63)

2

A) has symmetry with respect to the line

=

2

B) may or may not have symmetry with respect to the line

17

=

2

Use the result e i = cos 64) 2e( i)/2

+ i sin

to plot the complex number.

64)

A)

B)

C)

D)

Use a graphing utility to graph the polar equation.

18

65) r = 2 sin

-

65)

4

A)

B)

C)

D)

19

Graph the polar equation.

66) r = 3 sin 2

66)

A)

B)

C)

D)

20

67) r = 4 cos

67)

A)

B)

C)

D)

Test the equation for symmetry with respect to the given axis, line, or pole.

68) r = 3 - 6 sin ; the polar axis A) has symmetry with respect to the polar axis B) may or may not have symmetry with respect to the polar axis Polar coordinates of a point are given. Find the rectangular coordinates of the point.

69) (-3, -135°) -3 2 -3 2 3 2 -3 2 3 2 3 2 , , , A) B) C) 2 2 2 2 2 2

21

68)

69) D)

-3 2 3 2 , 2 2

Find all the complex roots. Write the answer in the indicated form.

70) The complex cube roots of 8 (rectangular form) A) 2, -1 - 3i, 1 - 3i B) 2, 1 + C) 2, -1 + 3i, -1 - 3i D) 2, 1 +

70) 3i, -1 - 3i 3i, 1 - 3i

Find the angle between the given vectors. Round to the nearest tenth of a degree.

71) u = -i + 3j, v = 4i - 2j A) 45° B) 0.7° C) 83.6°

D) 135°

Solve the problem.

72) The magnitude and direction of two forces acting on an object are 35 pounds, N45°E, and 55 pounds, S30°E, respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

A) F = 49.17; = -11.3° B) F = 43.30; = 2.7° C) F = 57.04; = -23.6° D) F = 65.19; = -7.5° The rectangular coordinates of a point are given. Find polar coordinates of the point. Express in radians.

73) (0, - 3) A) (- 3, 90°) B) (- 3, 180°) C) (- 3, 270°) D) ( 3, 90°)

22

71)

72)

73)

Plot the complex number.

74) 3 + 6i

74)

A)

B)

C)

D)

The rectangular coordinates of a point are given. Find polar coordinates of the point. Express in radians.

75) (5, -5) A) (-5 2, 135°) B) (-5 2, 45°) C) (-5 2, 225°) D) (5 2, 135°) Sketch the vector as a position vector and find its magnitude.

23

75)

76) v = -5i + 3j

76)

A) v = -2

B) v = 8

C) v = 2 2

D) v =

Write the complex number in rectangular form.

77) -5(cos 120° + i sin 120°) 5 5 3 5 3 i i A) + B) - + 2 2 2 2

34

77) 5 -5 3 i C) - + 2 2

24

5 -5 3 i D) + 2 2

Test the equation for symmetry with respect to the given axis, line, or pole.

78) r = 6 + 2 cos ; the pole A) may or may not have symmetry about the pole B) has symmetry about the pole

78)

Find the area of the triangle having the given measurements. Round to the nearest square unit.

79) B = 20°, a = 4 feet, c = 8 feet A) 22 square feet B) 11 square feet C) 15 square feet D) 5 square feet Solve the problem.

80) To find the distance AB across a river, a distance BC of 1176 m is laid off on one side of the river. It is found that B = 105.0° and C = 14.9°. Find AB. Round to the nearest meter.

A) 302 meters B) 352 meters C) 349 meters D) 299 meters Test the equation for symmetry with respect to the given axis, line, or pole.

81) r = -4 sin ; the pole A) may or may not have symmetry about the pole B) has symmetry about the pole Perform the indicated operation.

82) u = 2i + j, v = -7i - 3j, w = i - 8j; Find v - (u - w).

A) -8i - 12j B) -10i + 4j

C) -4i - 10j

79)

80)

81)

D) -8i + 12j

The rectangular coordinates of a point are given. Find polar coordinates of the point. Express in radians.

83) (2, -2 3) 11 5 5 11 A) 4, B) 4, C) 2, D) 2, 6 3 3 6

82)

83)

Decompose v into two vectors v1 and v2 , where v1 is parallel to w and v2 is orthogonal to w.

84) v = i + 2j, w = i + j 11 23 13 (3i + j), v2 = i+ j A) v1 = 10 10 10 C) v1 =

11 9 13 (3i + j), v2 = i+ j B) v1 = 10 10 10

11 2 11 (3i + j), v2 = i + j 9 3 9

D) v1 =

25

11 3 9 (3i + j), v2 = i+ j 10 10 10

84)

Use the vectors v, u, w, and z to draw the indicated vector.

85) z - v

85)

A)

B)

C)

D)

26

Answer Key Testname: C6

1) C 2) B 3) B 4) B 5) D 6) A 7) B 8) D 9)

(a) (4, -

11 ) 6

(b) (-4,

7 ) 6

(c) (4,

13 ) 6

10) B 11) A 12) A 13) C 14) C 15) B 16) D 17) D 18) A 19) D 20) B 21) D 22) A 23) A 24) D 25) B 26) B 27) B 28) C 29) D 27

Answer Key Testname: C6

30) D 31) D 32) B 33) D 34) C 35) C 36) A 37) A 38) B 39) C 40) A 41) A 42) A 43) B 44) A 45) C 46) C 47) C 48) D 49) A 50) C 51) A 52) B 53) C 54) A 55) D 56) A 57) C 58) A 59) C 60) C 61) A 62) A 63) B 64) A 65) C 66) B 67) B 68) B 69) C 70) C 71) D 72) C 73) A 74) C 75) A 76) D 77) D 78) A 79) D 28

Answer Key Testname: C6

80) C 81) A 82) A 83) B 84) D 85) A

29

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Write the partial fraction decomposition of the rational expression. x2 - 111 1) x4 - x2 - 72 A)

1 1 7 + x + 3 x - 3 x2 + 8

B)

1 1 7 + x + 3 x - 3 x2 + 8

C)

1 1 7 x + 3 x - 3 x2 + 8

D)

1 1 7 + + x + 3 x - 3 x2 + 8

1)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

2) Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

3) Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?

1

2)

3)

4) A coffee store has available 75 pounds of A grade coffee and 120 pounds of B grade coffee.

These will be blended into 1 pound packages as follows: an economy blend that contains 4 ounces of A grade coffee and 12 ounces of B grade coffee and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of B grade coffee. Using x to denote the number of packages of the economy blend and y to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible number of packages of each blend. Graph the system of inequalities.

4)

5) The Jillson's have up to $75,000 to invest. They decide that they want to have at least $25,000 invested in stable bonds yielding 6% and that no more than $45,000 should be invested in more volatile bonds yielding 12%. How much should they invest in each type of bond to maximize income if the amount in the more volatile bond should not exceed the amount in the more stable bond? What is the maximum income?

5)

6) Eric's Carpentry manufactures two types of bookshelves that are 4 feet tall and 3 feet wide, a basic model and a deluxe model. Each basic bookshelf requires 1.5 hours for assembly and 1 hour for finishing; each deluxe model requires 2.5 hours for assembly and 1 hour for finishing. Two assemblers and one finisher are employed by the company, and each works 40 hours per week. Eric can sell more basic models than deluxe models, so he wants the number of basic models produced to be 50% more than the number of deluxe models produced. If he makes $50 profit on the basic models and $65 profit on the deluxe models, how many should he make to maximize the profit? What is the maximum profit?

6)

7) The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? What is the maximum income?

7)

2

8) The Jillson's have up to $75,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 6% and that no more than $20,000 should be invested in more volatile bonds yielding 12%. (a) Using x to denote the amount of money invested in the stable bonds and y the amount invested in the more volatile bonds, write a system of linear inequalities that describe the possible amounts of each investment. (b) Graph the system of inequalities.

8)

9) An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have words written on them and the artist wants the words to all be horizontal in the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost $3.50 each and the larger tiles cost $4.50 each, how many of each should be used to minimize the cost? What is the minimum cost?

9)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the system of equations.

10) x + y + z = -4 x - y + 5z = 5x + y + z = -24 A) {( , -4, -5)}

10)

B) {(-5, -4, 5)}

C) {(-4, -5, )}

D) {( , -5, -4)}

Solve the problem.

11) A man is planting a section of garden with tomatoes and cucumbers. The available area of the section is 130 square feet. He wants the area planted with tomatoes to be more than 20% of the area planted with cucumbers. Write a system of inequalities to describe the situation. Let x = amount to be planted in tomatoes and y = amount to be planted in cucumbers.

A) x + y 130 B) x + y = 130 C) x + y 130 D) x + y 130 x < 0.20y x 0.20y 0.20x > y x > 0.20y x 0 x 0 x 0 x 0 y 0 y 0 y 0 y 0

3

11)

Solve the system of equations.

12) x + 2y + 5z = 13 5y + 3z = 21 z=2 A) {(-3, 2, 3)}

12)

B) {(2, 3, -3)}

C) {(3, -3, 2)}

D) {(-3, 3, 2)}

Solve the problem.

13) A cruise boat traveled 24 miles downstream, with the current, in 4 hours. The return trip upstream, against the current, covered the same distance but took 12 hours. Find the average velocity of the current.

A) 10 mph B) 6 mph C) 2 mph D) 4 mph Graph the inequality.

14) y x - 2

13)

14)

A)

B)

C)

D)

4

Graph the solution set of the system of inequalities or indicate that the system has no solution.

15) y > x2 3x + 2y 6

A)

B)

C)

D)

Solve the problem.

16) A vineyard produces two special wines, a white and a red. A bottle of the white wine requires 14 pounds of grapes and 1 hour of processing time. A bottle of red wine requires 25 pounds of grapes and 2 hours of processing time. The vineyard has on hand 2,198 pounds of grapes and can allot 160 hours of processing time to the production of these wines. A bottle of the white wine sells for $11.00, while a bottle of the red wine sells for $20.00. How many bottles of each type should the vineyard produce in order to maximize gross sales?

A) 14 bottles of white and 132 bottles of red B) 42 bottles of white and 59 bottles of red C) 76 bottles of white and 42 bottles of red D) 132 bottles of white and 14 bottles of red

5

15)

16)

Solve the system by the addition method.

17) x2 + y2 = 4 4x2 + 9y2 = 36

A) {(0, 2), (0, -2)}

17)

B) {(2, 0), (-2, 0)}

C) {(3, 0), (-3, 0)}

D) {(0, 3), (0, -3)}

Solve by the method of your choice.

18) 2x2 + xy - y2 = 3

18)

x2 + 2xy + y2 = 3

A)

2 3 3 2 3 3 , , ,3 3 3 3

B)

2 3 3 2 3 3 ,, , 3 3 3 3

C)

-2 1 2 3 1 ,- , , 3 3 3 3

D)

-2 1 2 3 1 , , ,3 3 3 3

Solve the system by the addition method.

19) -6x + 6y = -3x - 3y = A) {(-7, )} B) {(-8, )}

19) C) {(-7, )}

D)

Solve the problem.

20) A right triangle has an area of 10 square inches. The square of the hypotenuse is 104. Find the lengths of the legs of the triangle. Round your answer to the nearest inch.

A) 1 inches and 20 inches B) 4 inches and 100 inches C) 4 inches and 5 inches D) 2 inches and 10 inches

20)

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers.

21) The sum of the squares of two numbers is 58. The sum of the two numbers is 4. Find the two 21) numbers.

A) -7 and 3 B) -3 and C) -3 and ; -7 and 3 D) and ; -7 and -3

Determine if the given ordered triple is a solution of the system.

22) (2, -3, 5) x - y + z = 10 x+y+z=4 x + y - z = -6 A) not a solution B) solution

6

22)

Graph the inequality.

23) (x + 2)2 + (y + 5)2 > 4

23)

A)

B)

C)

D)

Solve the problem.

24) A doctor has told a sick patient to take vitamin pills. The patient needs at least 34 units of vitamin A, at least 7 units of vitamin B, and at least 22 units of vitamin C. The red vitamin pills cost 10¢ each and contain 8 units of A, 1 unit of B, and 2 units of C. The blue vitamin pills cost 20¢ each and contain 3 units of A, 1 unit of B, and 6 units of C. How many pills should the patient take each day to minimize costs?

A) 5 red and 2 blue B) 6 red and 1 blue C) 7 red and 0 blue D) 2 red and 5 blue

7

24)

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

25) 9x - 3y = 7 25) -36x + 12y = -21 27 9 ,A) {(4, 3)} B) 14 14

C) {(x, y)| 9x - 3y = 7}

D)

Graph the inequality.

26) x - y > -2

26)

A)

B)

C)

D)

8

Write the partial fraction decomposition of the rational expression. x-6 27) (x - 2)(x - 4) A)

2 1 + x-2 x-4

B)

1 -2 + x-2 x-4

C)

Graph the inequality.

28) 2x + 5y 10

2 -1 + x-2 x-4

27) D)

2 -1 + x-2 x-4

28)

A)

B)

C)

D)

9

An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.

Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs. z = 6x + 6y 29) Objective Function 29) Constraints x 0 0 y 5 2x + 3y 12 2x + 3y 20 A) Maximum: 60; at (10, 0) B) Maximum: 42; at (2, 5) C) Maximum: -24; at (4, 0) D) Maximum: 36; at (6, 0)

30) Objective Function Constraints

30)

z = 16x + 10y 0 x 10 0 y 5 3x + 2y 6 A) Maximum: 30; at (0, 3) C) Maximum: 160; at (10, 0)

B) Maximum: 50; at (0, 5) D) Maximum: 210; at (10, 5)

Solve the problem.

31) A deli sells three sizes of turkey sandwiches: the small turkey sandwich contains 6 ounces of meat and sells for $2.50; the regular turkey sandwich contains 9 ounces of meat and sells for $3.00; and the large turkey sandwich contains 13 ounces of meat and sells for $3.50. A customer requests a selection of each size for a reception. She and the manager agree on a combination of 40 sandwiches made from 22 pounds 14 ounces of turkey for a total cost of $117. How many of each size sandwich will be in this combination? (Note: 1 pound = 16 ounces) A) 12 small sandwiches, 24 medium sandwiches, 4 large sandwiches.

B) 10 small sandwiches, 14 medium sandwiches, 16 large sandwiches.

C) 16 small sandwiches, 12 medium sandwiches, 12 large sandwiches.

D) 14 small sandwiches, 18 medium sandwiches, 8 large sandwiches.

Write the partial fraction decomposition of the rational expression.

10x + 2 32) (x - 1)(x2 + x + 1) A)

4 -4x + 2 + x - 1 x2 + x + 1

B)

4 2x - 4 + x - 1 x2 + x + 1

C)

4 2 -4 + + x-1 x+1 x-1

D)

4x + 2 -4 + x - 1 x2 + x + 1

10

31)

32)

Solve the problem.

33) You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, y, after x seconds. Consider the following data:

33)

x, seconds after y, ball's height, in feet, ball is thrown above the ground 1 84 3 116 5 20 Find the quadratic function y = ax2 +bx + c whose graph passes through the given points.

A) y = -16x2 + 60x + 40 B) y = -12x2 + 80x + 16

C) y = -16x2 + 80x + 20

D) y = -10x2 + 100x - 6

Solve the system by the substitution method.

34) x2 + y2 = 85

34)

x + y = 13 A) {(7, 6), (6, 7)} C) {(7, -6), (6, -7)}

B) {(-7, -6), (-6, -7)} D) {(-7, ), (-6, )}

Solve the problem.

35) You invested $34,200 and started a business selling vases. Supplies cost $17 per vase and you are selling each vase for $36. Let x represent the number of vases produced and sold and write the cost function, C, and revenue function, R. A) C(x) = 17x + 36 B) C(x) = 17x + 34,200x R(x) = 34,200x R(x) = 36x C) C(x) = 17x + 34,200 D) C(x) = 17x + 34,200 R(x) = 36 R(x) = 36x 36) A basketball player scored 39 points in a game. The number of three-point field goals the player made was 25 less than three times the number of free throws (each worth 1 point). Twice the number of two-point field goals the player made was 9 more than the number of three-point field goals made. Find the number of free-throws, two-point field goals, and three-point field goals that the player made in the game.

A) 10 free throws; 5 two-point field goals; 7 three-point field goals B) 11 free throws; 7 two-point field goals; 8 three-point field goals C) 10 free throws; 7 two-point field goals; 5 three-point field goals D) 10 free throws; 8 two-point field goals; 7 three-point field goals

35)

36)

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers.

37) The difference between the squares of two numbers is 39. Twice the square of the second number 37) subtracted from the square of the first number is 14. Find the numbers.

A) 8 and 5 B) 8 and 5; -8 and 5; 8 and -5; -8 and -5 C) 8 and 5; -8 and 5; 8 and -5 D) 8 and 5; -8 and -5

11

Solve the problem.

38) A person with no more than $7000 to invest plans to place the money in two investments, telecommunications and pharmaceuticals. The telecommunications investment is to be no more than 5 times the pharmaceuticals investment. Write a system of inequalities to describe the situation. Let x = amount to be invested in telecommunications and y = amount to be invested in pharmaceuticals.

A) x + y = 7000 B) x + y 7000 C) x + y = 7000 D) x + y 7000 x 5y x 5y y 5x 5x y x 0 x 0 x 0 x 0 y 0 y 0 y 0 y 0 Graph the inequality. x y 1 39) + 4 3

38)

39)

A)

B)

C)

D)

12

Solve the problem.

40) An office manager is buying used filing cabinets. Each small file cabinet costs $7, takes up 7 square feet of floor space, and holds 10 cubic feet of files. Each large file cabinet costs $8, takes up 10 square feet of floor space, and holds 13 cubic feet of files. The total cost cannot exceed $97, and the office has no more than 109 square feet of floor space available for the cabinets. How many of each file cabinet should the manager buy to maximize storage capacity? What is the maximum storage capacity?

A) 7 small file cabinets and 6 large file cabinets; 148 cubic feet B) 6 small file cabinets and 7 large file cabinets; 161 cubic feet C) 13 small file cabinets and 0 large file cabinets; 130 cubic feet D) 0 small file cabinets and 13 large file cabinets; 169 cubic feet Determine if the given ordered triple is a solution of the system.

41) (-2, -4, 4) x + 5y + 3z = -10 4y + 4z = 0 z=4 A) solution B) not a solution

40)

41)

Find the maximum or minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of feasible points.

42) Objective Function: z = -5x - y Find maximum.

A) maximum: -15 B) maximum: -22

42) C) maximum: -18

D) no maximum

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.

2x - 5 43) 43) (x + 1)(x - 4)

A)

A B + x+1 x-4

B)

A B Cx + D + + x + 1 x - 4 (x + 1)(x - 4)

C)

A B C + + x + 1 x - 4 (x + 1)(x - 4)

D)

A B C + + x + 1 x - 4 (x + 1)2 (x - 4)2

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

44) x + y = -8 44) x - y = 14 A) {(3, 11)} B) {(3, -11)} C) {(x, y) | x + y = -8} D)

13

Determine if the given ordered triple is a solution of the system.

45) (-3, -5, -1) x + 2y + 5z = -26 5y + 3z = -34 z = -3 A) not a solution B) solution

45)

Solve the system by the addition method.

46) x2 + y2 - 4x - 2y - 31 = 0 x2 - y2 - 4x + 2y - 33 = 0

A) {( , 7), ( , -5)}

46)

B) {(8, 1), (-4, 1)}

C) {( , -7), ( , 5)}

D) {(8, 1), (4, 1)}

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars. Use the information in the figure to answer the question.

47) More than how many binoculars must be produced and sold for the company to have a profit gain?

A) 1500 binoculars B) 2700 binoculars C) 2250 binoculars D) 750 binoculars

47)

48) Use the revenue and cost functions to write the profit function from producing and selling x binoculars.

A) P(x) = 4x + 1500 B) P(x) = 2x + 1500 C) P(x) = 4x - 1500 D) P(x) = 2x - 1500

48)

Solve the problem.

49) Ellen wishes to mix candy worth $2.23 per pound with candy worth $3.78 per pound to form 25 pounds of a mixture worth $3.47 per pound. How many pounds of the more expensive candy should she use?

A) 25 pounds B) 5 pounds C) 20 pounds D) 7 pounds 50) A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.35 and an ounce of potatoes for $0.02. Let x = the number of ounces of chicken and y = the number of ounces of potatoes purchased per patient. Write the objective function that describes the total cost per patient per meal.

A) z = 0.35x + 0.02y B) z = 35x + 2y C) z = 0.02x + 0.35y D) z = 2y + 35y

14

49)

50)

51) A bank teller has 47 $10 and $5 bills in her cash drawer. The value of the bills is $290. How many $ 10 bills are there?

A) 13 $10 bills B) 34 $10 bills C) 11 $10 bills D) 36 $10 bills

51)

52) The sum of two numbers is -3. If one number is subtracted from the other, their difference is 9.

Find the numbers.

A) 3, -6 B) -2, -1 C) 3, 6 D) -3, -6

52)

Graph the solution set of the system of inequalities or indicate that the system has no solution.

53) x + 2y 2 x-y 0

A)

B)

15

53)

C)

D)

Solve the problem.

54) A person at the top of a 600 foot tall building drops a yellow ball. The height of the yellow ball is given by the equation h = -16t2 + 600 where h is measured in feet and t is the number of seconds

54)

since the yellow ball was dropped. A second person, in the same building but on a lower floor that is 468 feet from the ground, drops a white ball 1.5 seconds after the yellow ball was dropped. The height of the white ball is given by the equation h = -16(t - 1.5)2 + 468 where h is measured in feet and t is the number of seconds since the yellow ball was dropped. Find the time that the balls are the same distance above the ground and find this distance.

A) 4 seconds; 344 feet B) 3.5 seconds; 404 feet C) 2.5 seconds; 500 feet D) 3 seconds; 456 feet

55) Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 5 hours to make a hat and 4 hours to make an afghan, and she has no more than 52 hours available. She has material for no more than 12 items and no more than 9 afghans. The bazaar will sell the hats for $21 each and the afghans for $12 each. How many of each should she make to maximize the income for the bazaar? What is the maximum income?

A) 6 hats and 6 afghans; $198 B) 8 hats and 4 afghans; $216 C) 4 hats and 8 afghans; $180 D) 9 hats and 3 afghans; $225

55)

56) Two cars leave a city and head in the same direction. After 2 hours, the faster car is 22 miles ahead of the slower car. The slower car has traveled 84 miles. Find the speeds of the two cars.

A) 110 mph and 121 mph B) 42 mph and 53 mph C) 44 mph and 55 mph D) 31 mph and 42 mph

56)

16

Graph the solution set of the system of inequalities or indicate that the system has no solution.

57) x2 + y2 64 x2 + y2 49

A)

B)

C)

D) no solution

Solve the problem.

58) A real estate investor is examining a triangular plot of land. She measures each angle of the field.

The sum of the first and second angles is 88° more than the measure of the third angle. If the measure of the third angle is subtracted from the measure of the second angle, the result is thrice the measure of the first angle. Find the measure of each angle. (Note: The sum of the angles of a triangle is 180°.)

A) First angle is 28°, second angle is 108°, third angle is 44° B) First angle is 20°, second angle is 112°, third angle is 44° C) First angle is 26°, second angle is 114°, third angle is 40° D) First angle is 22°, second angle is 112°, third angle is 46°

17

57)

58)

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars. Use the information in the figure to answer the question.

59) What is the profit when 998 binoculars are produced?

A) $3496 B) $496 C) $5492 Solve the system of equations by the substitution method.

60) y = 2x - 2 4y + 16x = 112 A) {( , )} B) {(-5, 8)}

D) $2492

59)

60)

C) {( , )}

D)

Solve the problem.

61) In 1985, in the town of Appleby, 21.2% of Hispanics were overweight, increasing by an average of 0.48% per year. In 1985, in the town of Appleby, 0.16% of whites were overweight, increasing by an average of 32.6% per year. Write a function that models the percentage, y, of Hispanics who are overweight x years after 1985. Write a function that models the percentage, y, of whites who are overweight x years after 1985.

A) Hispanics: y + 0.48x = 21.2 B) Hispanics: y = 21.2x + 0.48 Whites: y + 32.6x = 0.16 Whites: y = 0.16x + 32.6 C) Hispanics: y = 0.48x + 21.2 D) Hispanics: y = 21.68x Whites: y = 32.6x + 0.16 Whites: y = 32.76x

61)

Find the maximum or minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of feasible points.

62) Objective Function: z = 5x + 6y Find minimum.

A) minimum: 33 B) minimum: 40

62) C) minimum: 32

18

D) no minimum

Determine if the given ordered triple is a solution of the system.

63) (2, -2, -3) x-y+z=3 x + y + z = -3 x + y - z = -7 A) not a solution B) solution Determine whether the given ordered pair is a solution of the system.

64) (3, 3) 3x = -6 - y 4x = -3 - 3y A) not a solution B) solution Solve the problem.

65) The following is known about three numbers: If the second number is subtracted from the sum of the first number and 3 times the third number, the result is -14. The third number plus 2 times the first number is -3. The first number plus 3 times the second number plus the third number is 12.

Find the three numbers.

[Hint: let x represent the first number, y the second number, and z the third number. Use the given conditions to write and solve a system of equations.]

A) x = 1, y = 6, z = -3 B) x = 0, y = 5, z = -3 C) x = 0, y = 4, z = 0 D) x = -1, y = , z = -1 66) A flat rectangular piece of aluminum has a perimeter of 50 inches. The length is 9 inches longer than the width. Find the width.

A) 8 inches B) 26 inches C) 17 inches D) 25 inches Graph the inequality.

67) x + 5y -5

63)

64)

65)

66)

67)

19

A)

B)

C)

D)

20

68) y > x2 + 7

68)

A)

B)

C)

D)

21

69) y > 2 x

69)

A)

B)

C)

D)

Solve the system by the substitution method.

70) x - y = 13 (x - 2)2 + (y + 5)2 = 20 A) {(6, -7), (0, -9)}

70)

B) {(-7, 6), (-9, )}

C) {(-7, 6), (-9, 0)}

Determine if the given ordered triple is a solution of the system.

71) (0, 3, -3) x - y + 3z = -12 3x + z = -3 x + 2y + z = 3 A) not a solution B) solution

22

D) {(6, -7), (4, -9)}

71)

Determine whether the given ordered pair is a solution of the system.

72) (3, -4) 4x + y = 16 3x + 4y = 25 A) solution B) not a solution

72)

Solve the system by the addition method.

73) x2 + y2 = 181

73)

x2 - y2 = 19

A) {(10, 9), (-10, 9), (10, -9), (-10, -9)} C) {(10, 9), (9, 10), (-10, -9), (-9, -10)}

B) {(10, -9), (10, 9)} D) {(-10, -9), (-9, -10)}

Solve the system by the substitution method.

74) 3y - x = 10 x2 + y2 - 100 = 0

74)

A)

0,

10 16 , 6, 3 3

B) {(10, 0), (-10, 0), (8, 6)}

C)

0,

10 10 16 , 0, , 6, 3 3 3

D) {(-10, 0), (8, 6)}

Solve the problem.

75) The sum of three numbers is -2. If the second number is subtracted from the sum of the first and third numbers, the result is 4. If the third number is subtracted from the sum of the first and second numbers, the result is 6. Find the three numbers.

[Hint: let x represent the first number, y the second number, and z the third number. Use the given conditions to write and solve a system of equations.]

A) x = 5, y = -3, z = -4 B) x = 3, y = -1, z = -4 C) x = 6, y = -2, z = -6 D) x = 6, y = -3, z = -5

75)

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

76) 5y = 35 - 8x 76) 2x = 65 - 5y A) {(-5, 15)} B) {(5, -15)} C) {(x, y) 8x + 5y = 35 } D)

Solve the system of equations.

77) x - y + 2z = 3x + z= x + 4y + z = -8 A) {(4, 0, -3)}

77)

B) {(-3, 0, )}

C) {(0, -3, 4)}

Solve the system by the addition method.

78) 2x + 63y = 63 5x - 9y = -9 A) {(1, 0)} B) {(1, 1)}

D) {(4, -3, 0)}

78) C) {(0, 1)}

23

D) {(0, 0)}

Solve the problem.

79) A system for tracking ships indicated that a ship lies on a hyperbolic path described by 5x2 - y2 = 25. The process is repeated and the ship is found to lie on a hyperbolic path described by

79)

y2 - 3x2 = 25. If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.

A) (5, 10) B) (-5, -10) C) (-10, -5) D) (10, 5)

Solve the system by the addition method.

80) x2 - y2 = 21 x2 + y2 = 47 A) {( 34, B) {( 34, C) {( 34, D) {(0, 0)}

80)

13), (- 34, 13)} 13), (- 34, 13), ( 34, - 13), (- 34, - 13)} 13), (- 34, - 13)}

Write the partial fraction decomposition of the rational expression.

8x + 1 81) x3 - 1

81)

A)

3x + 2 -3 + x - 1 x2 + x + 1

B)

3 -3x + 2 + x - 1 x2 + x + 1

C)

3 2x - 3 + x - 1 x2 + x + 1

D)

3 2 -3 + + x-1 x+1 x-1

Solve the problem.

82) When a plane flies with the wind, it can fly 1088 miles in 4 hours. When the plane flies in the opposite direction, against the wind, it takes 8 hours to fly the same distance. What is the average velocity of the wind?

A) 85 mph B) 102 mph C) 68 mph D) 51 mph

82)

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x y =1 83) + 83) 5 15 x y - =0 2 6

A)

15 5 , 2 2

C) (x, y)

B) x y + =1 5 15

5 15 , 2 2

D)

Solve the system by the addition method.

84) 2x2 + y2 = 17

84)

3x2 - 2y2 = -6 A) {(2, -3), (-2, 3)} C) {(1, 3), (-1, -3)}

B) {(2, 3), (2, -3), (-2, 3), (-2, -3)} D) {(1, 3), (1, -3), (-1, 3), (-1, -3)}

24

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

85) x - 7y = -2 85) -3x + 21y = A) {(-2, 0)} B) {(0, 0)} C) {(x, y) | x - 7y = -2} D)

Write the partial fraction decomposition of the rational expression. x 86) (x - 6)(x - 7) A)

-6 -7 + x-6 x-7

B)

6 -7 + x-6 x-7

C)

6 -7 + x-6 x-7

86) D)

7 -6 + x-6 x-7

An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.

Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs. z = 7x + 6y 87) Objective Function 87) Constraints x 0 y 0 3x + y 21 x + y 10 x + 2y 12 A) Maximum 60; at (6, 3) B) Maximum 65.5; at (5.5, 4.5) C) Maximum 66; at (6, 4) D) Maximum 68; at (8, 2)

Solve the problem.

88) A volunteer wants to crochet beach hats and baby afghans for a church fund-raising bazaar. She needs 8 hours to make a hat and 2 hours to make an afghan and she has 16 hours available. Write an inequality that describes the situation and use the inequality to decide whether she can make 9 hats and 8 afghans in the time allowed. Let x represent the number of hats and y the number of afghans that she makes.

A) 8x + 2y 16; no B) 8x + 2y 16; yes C) 8x + 2y 16; no D) 8x + 2y 16; yes

25

88)

Graph the inequality.

89) x + y < -5

89)

A)

B)

C)

D)

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers.

90) The sum of two numbers is 1 and their product is -342. Find the numbers.

90) A) -18 and 19; 18 and -19 B) -18 and 19 C) and -19 D) -18 and -19; 18 and 19

26

Answer Key Testname: C7

1) A 2) 75 packages of the breakfast blend, 40 packages of the afternoon blend; maximum profit $192.50 3) order 100 type A printers 4) x 0 y 0 3x + 2y 480 x + 2y 300

5) $37,500 in the stable bonds and $37,500 in the volatile bonds; maximum income $6750 6) 24 basic models, 16 deluxe models; maximum profit $2240 7) $60,000 in the stable bonds and $60,000 in the volatile bonds; maximum income $9900 8) (a) x + y 75,000 x 40,000 y 20,000 x 0 y 0 (b)

9) 0 small tile, 6 large tiles; minimum cost $27 10) B 11) D 12) D 27

Answer Key Testname: C7

13) C 14) D 15) C 16) D 17) A 18) A 19) C 20) D 21) B 22) B 23) C 24) A 25) D 26) C 27) C 28) C 29) A 30) D 31) D 32) A 33) C 34) A 35) D 36) C 37) B 38) B 39) A 40) A 41) A 42) A 43) A 44) B 45) A 46) B 47) D 48) D 49) C 50) A 51) C 52) A 53) D 54) B 55) C 56) B 57) B 58) D 59) B 60) A 61) C 62) C 28

Answer Key Testname: C7

63) A 64) A 65) B 66) A 67) D 68) B 69) A 70) D 71) B 72) B 73) A 74) D 75) A 76) A 77) C 78) C 79) A 80) B 81) B 82) C 83) B 84) B 85) C 86) D 87) B 88) C 89) C 90) B

29

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the product AB, if possible.

1) 9 A = -7 2 5 , B = 0 -3 A) -63 0 -15 Evaluate the determinant.

2) -3 -4 1 3 05 -4 0 5 A) -140

1)

B) 309

C) -78

D) -63 0 -15

2)

B)

C)

D) -20

Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.

3) x + y + z - w = 6 2x - y + 3z + 4w = -4 4x + 2y - z - w = -13 -x - 2y + 4z + 3w = 12 1 1 1 1 A) {(- , , , - )} B) {(4, -3, -5, 2)} 4 3 5 2 1 1 1 1 D) {( , - , - , )} 4 3 5 2

C) {(-4, 3, 5, -2)}

1

3)

The

shape in the figure below is shown using 9 pixels in a 3 × 3 grid. The color levels are given to the right of the 131 figure. Use the matrix 1 3 1 that represents a digital photograph of the shape to solve the problem.

333

4) Adjust the contrast by changing the black to white and the light grey to dark grey. Use matrix addition to accomplish this.

A) B) 131 202 131 0 -3 0 101 -1 3 -1 + = + = 131 202 131 0 -3 0 101 -1 3 -1 -3 -3 -3 333 33 3 000 333 000 C) D) 131 020 131 1 -3 1 202 -1 -1 -1 1 3 1 + -1 -1 -1 = 0 2 0 1 3 1 + 1 -3 1 = 2 0 2 -1 -1 -1 -3 -3 -3 333 222 333 000 Perform the matrix row operation (or operations) and write the new matrix.

5) -40 55 45 -30 1 R 1 13 -3 0 5 1 2 -7 4 21 A) B) -40 55 45 -30 -8 11 9 -30 1 13 3 1 13 -3 0 0 5 5 5 2 -7 4 21 2 -7

C)

D)

9 -8 11 1 13 3 5 5 5

-6

2 7 5 5

21 5

4 5

0

2

4 21

-8 11 9 -6 1 13 -3 0 2 -7 4 21

4)

5)

Find the inverse of the matrix, if possible.

6) A = -5 -2 -4 -3 A) B) 3 2 3 7 7 7 -

4 5 7 7

6)

C)

2 7

4 5 7 7

5 7

2 7

D)

4 3 7 7

4 5 7 7 -

3 7

2 7

Use Cramer's rule to solve the system.

7) -5x + 3y - z = x + 7y + 4z = -5x + y + z = -3 A) {(5, 2, 5)} B) {(2, -5, -2)}

C) {(3, 3, 2)}

D) {(2, 5, 2)}

Solve the problem.

8) Let B = [-1 1 4 -3]. Find -2B. A) -3 -1 2 -5 B) 2 -2 -8 6

C) 2 1 4 -3

D) -2 2 8 -6

7)

Write the augmented matrix for the system of equations.

9x + 5z = 8 9) 6y + 3z = 42 3x + 8y + 2z = 66 A) B) 905 8 903 8 0 6 3 42 0 6 8 42 3 8 2 66 5 3 2 66

8)

9)

C)

905 063 382

D)

950 8 6 3 0 42 3 8 2 66

Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations.

10) 3x + y = 10 3x + y = 28 A) system contains dependent equations B) system is inconsistent

3

10)

The

shape in the figure below is shown using 9 pixels in a 3 × 3 grid. The color levels are given to the right of the 131 figure. Use the matrix 1 3 1 that represents a digital photograph of the shape to solve the problem.

333

11) Using the same color levels from the instructions, write a 3 × 3 matrix A that represents the letter L in dark grey on a white background. Then find a 3 × 3 matrix B so that A + B lightens only the letter L from dark grey to light grey.

A) B) 211 100 100 -1 0 0 A= 2 1 1 ;B= 1 0 0 A = 1 0 0 ; B = -1 0 0 -1 -1 -1 222 111 111 C) D) 311 200 -2 -1 -1 -1 0 0 A = 3 1 1 ; B = -2 -1 -1 A = 2 0 0 ; B = -1 0 0 -2 -2 -2 -1 -1 -1 333 222 Evaluate the determinant.

12) 22 -1 3 A) 8

11)

12)

B) 4

C)

Write the augmented matrix for the system of equations.

3x - 2y + 8z = 7 13) 5x + 7y - 2z = 32 3x - 2y + 7z = 9 A) B) 3 5 3 7 3 -2 8 5 7 -2 -2 7 -2 32 8 -2 7 9 3 -2 7

D) -8

13)

C)

4

3 -2 8 7 5 7 -2 32 3 -2 7 9

D)

7 8 -2 3 32 -2 7 5 9 7 -2 3

Find the inverse of the matrix, if possible.

14) A= 3 5 -3 2 A) B) 3 -5 -3 5 -3 2 -3 2

14)

C)

1 3 -

1 5

D) No inverse

1 1 3 2

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

15) 2x + y + 2z - 4w = 10 x + 3y + 2z - 11w = 17 3x + y + 7z - 21w = 0 A) {(w + 5, 8w + 4, -3w - 2, w)} B) {(3w + 5, 6w + 6, -4w - 3, w)} C) {(w - 5, 8w - 4, -3w + 2, w)} D) {(-3w + 5, 2w + 6, 4w - 3, w)} Use Cramer's rule to solve the system.

16) -4x - 7y + 2z = -65 -2x + 7y + 7z = 80 6x - 7y - 2z = -57 A) {(2, -9, -3)} B) {(3, 7, 3)}

16)

C) {(2, 9, 3)}

D) {(9, 3, 9)}

Solve the problem.

17)

Let A = -3 1 . Find 2A. 02 A) -6 2 02

15)

17)

B)

C)

-1 3 24

-6 1 02

D)

-6 2 04

Write a system of linear equations in three variables, and then use matrices to solve the system.

18) Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 122 g protein, 118 g fat, and 159 g carbohydrate. According to the health conscious hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal?

A) 9 mushrooms; 4 meatballs; 3 eggs B) 10 mushrooms; 5 meatballs; 4 eggs C) 4 mushrooms; 3 meatballs; 9 eggs D) 3 mushrooms; 9 meatballs; 4 eggs Write the matrix equation as a system of linear equations without matrices.

19) 4 8 x = 5 13 2 y 15 A) 4x + 8y = 5 B) 8x + 4y = 5 C) 4x + 8y = - 5 2x + 13y = 15 13x + 2y = 15 13x + 2y = -15

5

18)

19)

D) 4x + 8y = 5 13x + 2y = 15

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

20) x + y + z + w = 7 3x - 2z + 5w = 11 -4x + 3y + w = 4 -x - y - z - w = 6 3 1 A) B) {( , 1, , -2)} 2 3 C) {(-11,

7 6 , , -4)} 19 19

7 1 1 D) {( , - , 5, - )} 4 2 6

Find the products AB and BA to determine whether B is the multiplicative inverse of A. 21) A = 10 1 , B= 0 1 -1 0 -1 10 -1 A) B A B) B = A-1 Find the product AB, if possible.

22) A = -1 3 , B = 0 -2 5 14 1 -3 2 A) 3 4 -7 -14 1 13 C) AB is not defined.

21)

22)

B)

D)

3 -7 1 4 -14 13

0 -6 15 1 -12 8

Solve the problem.

23)

6 4 3 8 -4 3 Let A = 12 2 -3 and B = -2 0 2 . Find A + B. -2 6 9 1 9 -6 A) B) C) 10 -8 6 10 -8 6 14 0 6 10 2 -5 14 2 -1 14 2 -5 -1 15 3 1 15 3 1 15 3

Evaluate the determinant.

24) 67 72 A) 28

20)

23)

D)

14 0 6 10 2 -1 -1 15 3

24)

B) -37

C)

6

D) 61

Find the product AB, if possible.

25) A = -1 3 , B = -2 0 -1 2 22 A) -1 6 -6 4 26)

25)

B)

C)

2 -6 -1 1

B)

15 -10 -6 12

5 -4

D) AB is not defined.

15 -10 5

-6 12 -4

Find the inverse of the matrix, if possible.

27) A = -5 2 -3 0 A) B) 1 5 2 6 0-

20 -2 4

26)

A = 3 -2 1 , B = 5 0 -2 2 0 4 -1 A) 15 0 0 8

C)

D)

6 -1 4 -6

1 3

27)

0 -

C)

1 3

1 5 2 6

1 03 1 5 2 6

Find the product AB, if possible.

28) 4 -4 -1 A = [8 4 -4], B = 1 -8 5 -6 -8 -2 A) 8 4 -4 4 -4 -1 1 -8 5 -6 -8 -2 C) 60 -32 20

D) -

5 1 6 3 1 2

0

28)

B)

32 -16 4 8 -32 -20 -48 -32 8

D) 60 -32 20

Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.

29) Use the coding matrix A = 2 1 and its inverse A-1 = 3 -1 to decode the cryptogram 9 6 .

29) 53 25 17 -5 2 A) DARE B) CURB C) BEAD D) CARE

7

Solve the problem.

30) -8 -2 -3 Let A = 3 7 -3 -6 -1 -7 A) 8 20 13 -3 -29 25 8 5 13

-8 7 2 and B = 3 -4 8 . Find -3A + 2B. -5 1 -4 B) C) 16 13 11 -16 5 -1 63 5 -6 -25 17 -11 0 -11 13 4 17

30)

D)

-16 6 -11 53 0 -1 5 -11

Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.

31) Use the coding matrix A = 1 -4 and its inverse A-1 = 9 4 to decode the cryptogram -7 -8 .

31) 21 16 21 -2 9 A) ALAS B) ACTS C) ARMS D) ABLE Write a system of linear equations in three variables, and then use matrices to solve the system.

32) There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36- to 59-year-old buyers and the percentage of buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 34%. If the percentage of buyers in the oldest group is doubled, it is 25% less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups.

A) 27% 35 and younger; 55% 36-59 year olds; 18% 60 and older B) 33% 35 and younger; 53% 36-59 year olds; 14% 60 and older C) 14% 35 and younger; 53% 36-59 year olds; 33% 60 and older D) 35% 35 and younger; 50% 36-59 year olds; 15% 60 and older Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

33) 5x - y + z = 8 7x + y + z = 6 1 7 1 A) {( z + , z, z)} B) 6 6 6 C) {(-

1 7 1 13 z+ , z, z)} 6 6 6 6

32)

33)

D) {(-z + 3, 4z + 7, z)}

Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.

34) 5x - y + 7z = 77 2x + 8z = 64 7y + z = 41 A) {(-8, 16, 5)} B) {(-8, 5, 16)} C) {(8, 6, 5)} D) {(8, 5, 6)}

8

34)

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.

35) 35) 3 1 0 6 -1 6 -1 8 1 0 7 0 0 5 -4 0 2 0 -2 -9 A) B) 3x + y + z + 6w = -1 3x + y + 6z = -1 -x + 8y + z + w = 6 -x + 8y + z = 6 7x + y + z + 5w = -4 7x + 5y = -4 x + 2y + z - 2w = -9 2x - 2y = -9 C) D) 3x + y + 6w = -1 3x + y + 6w = -1 x + 8y + z = 6 -x + 8y + z = 6 7x + 5w = -4 7x + 5w = -4 2y + 2w = -9 2y - 2w = -9

Solve the problem.

36)

Let A = -1 0 61 A) -3

36)

and B = -1 6 . Find A - B. 31 B) -2 6 92

C)

Find the product AB, if possible.

37) 30 A = 1 3 -3 , B = -3 1 20 5 05 A) -6 -12 6 25 C) 3 -9 0 0 0 25

06 -3 0

D)

0 -6 3 0

37)

B) -12 -6 25 6 D) AB is not defined.

Give the order of the matrix, and identify the given element of the matrix.

38) 7 -11 -6 9 -11 -5 -e -2 -14 15 10 9 4 5 ;a 34 1 -12 14 -1 -8 3 A) 20; 5

B) 5 × 4; 14

C) 4 × 5;

9

38)

D) 4 × 4;

Solve the system using the inverse that is given for the coefficient matrix.

39) x + 2y + 3z = -6 12 3 -2 4 -1 The inverse of 1 1 1 is 3 -5 2 . x + y + z = -10 - 2z = 4 -1 2 -1 x 1 0 -2 A) {(-32, 40, -18)} B) {(-48, -60, -22)} C) {(-22, 34, -18)} Write the augmented matrix for the system of equations.

5x + 4y - 7z + w = 9 40) 8y + z = 3 x - y - 6z = -9 12x - 12y + 9z = 10 A) 5 4 -7 1 9 0 8 1 0 3 -1 -6 1 0 -9 12 -12 9 0 10 C)

5 4 0 8 -1 1 12 -12

-7 1 -6 9

39)

D) {(-6, 0, 0)}

40)

B)

D)

9 6 -9 10

5 0 1 12

4 8 1 12

7 1 6 9

1 9 0 3 0 -9 0 10

5 4 -7 1 9

0 8 1 0 3

1 9 -1 -12 9 -6 0 0 -9 10

Find the products AB and BA to determine whether B is the multiplicative inverse of A. 41) 2 0 1 1 2 0 0 -2 1 1 1 0 3 -1 -1 -2 A= , B= -1 -2 1 0 -2 2 1 5 0 0 1 1 2 -2 -1 -3 -1 A) B A B) B = A-1

41)

Solve the problem.

42)

1 Let A = -3 2 A) 3 -9 6

42) -1 and B = 3 . Find A - 2B. -2 B) -3 9 -6

C)

3 -6 4

Find the products AB and BA to determine whether B is the multiplicative inverse of A. 43) 100 1 00 A= 1 1 0 , B = -1 1 0 111 0 -1 1 -1 A) B A B) B = A-1

10

D) -1 3 -2

43)

Use Cramer's rule to solve the system.

44) 3x + 2y = 38 3x - 2y = 10 A) {(8, 7)} B) {(-8, -7)}

44) C) {(7, 8)}

D) {(-7, 8)}

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

3x + y + z - 2w = 10 45) 2x + 3y + 3z + w = -5 2x + y + 4z + 11w = 11 A) {(6, -4, -2, 1)} B) {(7, -1, -6, 2)} C) {(w + 5, 3w - 7, -4w + 2, w)} D) {(2w + 3, 6w - 7, -10w + 8, w)} Use Cramer's rule to solve the system.

46) 11x + 3y= 1 5x + 2y = 4 7 7 10 39 , )} , )} A) {(B) {(10 39 7 7

45)

46) C) {(

,-

10 )} 7

D) {(

14 49 , )} 37 37

Find the products AB and BA to determine whether B is the multiplicative inverse of A. 47) 1 0 0 -1 1 00 1 2 1 0 0 1 0 -2 -2 A= , B= 1 1 1 -2 1 -1 1 3 000 1 0 00 1 A) B = A-1 B) B A-1

47)

Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.

1 0 -2 48) Use the coding matrix A = 1 2 3 to encode the message COME_HERE. 48) 11 1 19 27 -21 35 5 -7 -21 3 -23 -11 -5 A) -14 -30 39 B) 28 69 44 C) 15 0 18 D) 72 29 56 8 11 -13 13 33 26 13 8 5 31 13 28 Solve the problem.

49)

-1 5 Let A = 0 4 9 -4 A) -8 3 -17 0 6 -6

49)

72 and B = 17 4 . Find A - B. 32 B) 3 -4 7 0 -6 6

C)

1 6 7 8 12 -1

D)

1 3 7 0 6 -2

Solve the system of equations using matrices. Use Gaussian elimination with back-substitution. x + y + z = -1 50) x - y + 2z = -7 2x + y + z = 1 A) {(2, 1, -4)} B) {(-4, , )} C) {(2, -4, 1)} D) {(-4, , )}

11

50)

Evaluate the determinant.

51) 1 1 2 10 -

5 3

51)

2 5

A)

B)

11 30

C)

Use Cramer's rule to solve the system.

52) 3x + 4y - z = 25 x - 4y + 2z = 11 3x + y + z = 32 A) {(7, -3, -8)} B) {(7, 3, 8)}

7 300

C) {(3, 8, 3)}

Solve the matrix equation for X. 54) Let A = 1 -2 and B = 1 -2 ; X+A=B 1 -3 1 -1 A) B) X= 0 0 X= 0 0 2 0 0 2

Let A = -4 1 25 A) 13

131 150

52)

Write the matrix equation as a system of linear equations without matrices.

53) -9 -4 x = -5 -7 -6 y 6 A) -9x - 4y = -5 B) -9x - 4y = 5 C) -9x - 4y = -5 6y 6 6y -7x = -7x = -6 -6x - 7y = 6

Solve the problem.

55)

D) -

D) {(8, 1, 8)}

53)

D) -4x - 9y = -5 -7x - 6y = 6

54)

C)

X= 0 2 0 0

D)

X= 2 0 0 0

55)

and B = 6 2 . Find A + B. 3 -2 B) 2 -2 -1 -6

C)

23 53

Solve the system of equations using matrices. Use Gauss-Jordan elimination.

56) x = - y - z x - y + 2z = 4x + y =12 - z A) {(-5, , 4)} B) {(4, -5, 1)} C) {( , -5, 4)}

12

D)

34 -1 3

56)

D) {( , 4, -5)}

Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.

57) x + 6y = 57) 9x - 2y = 65 A) B) 3 9 x = 75 3 6 x = 75 6 -2 y 65 9 -2 y 65 C) D) 3 6 x = 65 75 6 x = 3 -2 9 y -2 75 65 9 y

Solve the problem.

58) Determinants are used to show that three points lie on the same line (are collinear). If x1 y1 1

58)

x2 y2 1 = 0, x3 y3 1 then the points (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ) are collinear. If the determinant does not equal 0, then the points are not collinear. Are the points (-1, -8), (0, 6), and (-3, -34) collinear?

A) No B) Yes

Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.

59) Use the coding matrix A = 3 7 to encode the message LIFE. 59) 25 A) 78 62 B) 54 28 C) -3 -5 D) 99 53 54 43 129 67 3 3 69 37 Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.

60) 60) 9 8 8 -2 2 0 9 4 9 5 0 2 A) 9x + 8y + 8z = -2 B) 9x - 8y + 8z = -2 2x + y + 9z = 4 2x + 9z = -4 9x + 5y + z = 2 9x + 5y = -2 C) 9x + 8y + 8z = -2 D) 9x + 8y + 8z = -2 2x + 9z = 4 2x + 9z = 4 9x + 5z = 2 9x + 5y = 2

Evaluate the determinant.

61) -9 2 -8 -3 A) -43

61)

B)

C) -42

D) 11

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

4x - y + 3z = 12 62) x + 4y + 6z = -32 5x + 3y + 9z = 20 A) {(-8, -7, 9)} B) C) {(2, -7, -1)} D) {(8, -7, -2)}

13

62)

Find the product AB, if possible.

63) 3 7 -2 -6 8 2 A = -5 7 2 , B = 4 8 -2 -5 9 2 9 -2 -5 A) 3 7 -2 -5 7 2 9 -2 -5 -6 8 2 4 8 -2 -5 9 2 C) 20 48 -37 62 34 11 -12 -20 12

63)

B)

D)

-18 56 -4 -20 56 -4 -45 -18 -10

20 48 -37

62 -12 34 -20 11 12

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. x + 8y + 8z = 8 64) 7x + 7y + z = 1 8x + 15y + 9z = -9 A) {(0, 0, 1)} B) {(1, -1, 1)} C) D) {(-1, 0, 1)} Write a system of linear equations in three variables, and then use matrices to solve the system.

65) A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 16 hours to prepare, 3 hours to paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 14 hours to paint, and 7 hours to fire. If the workshop has 117 hours for prep time, 85 hours for painting, and 118 hours for firing, how many of each can be made?

A) 4 wreaths; 7 trees; 5 sleighs B) 5 wreaths; 4 trees; 7 sleighs C) 8 wreaths; 6 trees; 5 sleighs D) 7 wreaths; 5 trees; 4 sleighs Find the inverse of the matrix, if possible.

66) 100 A = -1 1 0 111 A) B) 111 -1 0 0 011 -1 -1 0 -1 -1 -1 001

64)

65)

66)

C)

14

1 -1 1 0 1 -1 0 0 1

D)

1 0 0 1 1 0 -2 -1 1

Evaluate the determinant.

67) 0002 2471 7284 9899 A) -420

67)

B)

C) -18

D) -16

Perform the matrix row operation (or operations) and write the new matrix.

68) 2 -5 1 3 -4 0 4 -2 -4R1 + R 2 -1 5 -3 -1 A) B) 2 -5 1 3 18 -5 -15 11 -12 20 0 -14 -4 0 4 -2 -1 5 -3 -1 -1 5 -3 -1 C) D) 2 -5 1 3 -12 20 0 -14 4 -20 8 10 -4 0 4 -2 -1 -1 5 -3 -1 5 -3 -1

68)

Solve the system of equations using matrices. Use Gauss-Jordan elimination.

69) x + y - z + w = -5 3x - y + 3z - 2w = 7 -2x + 2y + z - w = 16 -x - 2y - 3z + 3w = -22 A) {(2, -3, -4, -2)}

B) {(-2, 3, 4, -2)}

1 1 1 1 C) {( , - , - , - )} 2 3 4 2

D) {(-2, -3, 5,

Find the inverse of the matrix, if possible.

70) A= 0 3 -6 3 A) B) 1 1 6 6 1 3

0

69)

1 )} 2

70)

C)

1 1 6 6 -

1 3

1 0 6 1 3

0

1 6

D)

1 3

0

1 1 6 6

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

71) 5x + 2y + z = -11 2x - 3y - z = 17 7x - y = 12 A) {(1, -5, 0)} B) {(0, -6, 1)} C) D) {(-2, 0, -1)}

15

71)

Solve the problem using matrices.

72) The final grade for an algebra course is determined by grades on the midterm and final exam. The grades for four students and two possible grading systems are modeled by the following matrices.

Student 1 Student 2 Student 3 Student 4

72)

Midterm Final 73 79 44 62 83 89 98 96

System System 1 2 Midterm 0.2 0.5 Final 0.8 0.5 Find the final course score for Student 3 for both grading System 1 and System 2.

A) System 1: 39.8; System 2: 53 B) System 1: 77.8; System 2: 76 C) System 1: 61.1; System 2: 110.9 D) System 1: 87.8; System 2: 86

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

73) x + y + z = 7 x - y + 2z = 7 2x + 3z = 14 3z z 3z - 7, , z)} + 7, 2z, z)} A) {(B) {(2 2 2 C) {(-

3z z + 7, , z)} 2 2

D) {(-

Find values for the variables so that the matrices are equal.

74) x y+6 = 5 9 4z 1 8 1 A) x = 5; y = 9; z = 8 C) x = 5; y = 3; z = 2

3z - 7, 2z, z)} 2

74)

B) x = 9; y = 1; z = 5 D) x = 1; y = 15; z = 32

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. x+ y+z = 9 75) 2x - 3y + 4z = 7 x - 4y + 3z = -2 7z 34 2z 11 z 34 2z 11 + + , , z)} , , z)} A) {( B) {( + 5 5 5 5 5 5 5 5 C) {(-

73)

7z 34 2z 11 + , , z)} 5 5 5 5

D) {(-

16

7z 34 2z 11 + + , , z)} 5 5 5 5

75)

Find the product AB, if possible.

76) -7 A = -6 6 5 , B = 1 -4 9 2 9 A) 93 55 C)

76)

B) AB is not defined.

D) 93 55

-6 6 5 -4 9 2 -7 1 9

Evaluate the determinant.

77) 1 1 8 3 3

77)

8

A) -2

B) 2

C) 0

D)

73 24

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. x - y + z - w = 10 78) -2x + 3y + 5w = -28 x + 2y + 8z + 3w = -10 x - 4y - 6z - 5w = 30 A) {(24, 10, -6, -2)} B) {(3w - 2, -8w + 3, 4w + 9, w)} C) {(-17w - 10, -13w - 16, 5w + 4, w)} D) Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations.

79) -6x + 6y = 90 -12x + 12y = 88 A) system contains dependent equations B) system is inconsistent Use Cramer's rule to solve the system.

80) 4x + 5y = 67 -2x + 2y= -2 A) {(-7, 8)} B) {(8, 7)}

78)

79)

80) C) {(7, 8)}

Solve the matrix equation for X. 81) 8 -6 -2 1 Let A = 0 -1 and B = 2 1 ; B - X = 3A 6 -8 0 5 A) B) 2 -3 14 -9 X = -2 -2 X = -2 4 -18 29 18 -19

D) {(8, -7)}

81)

C)

17

14 -9 X= 2 4 -18 29

D)

2 -3 X = -2 -2 6 -19

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. x+y+z+w=8 82) 3x + 2y + z + 4w = 21 4x + 4y + 5z + 8w = 30 2x + 3y + 6z + 9w = 15 A) B) {(-3, 16, -6, 1)} C) {(-6w + 3, 9w + 7, -4w - 2, w)} D) {(5w + 11, -3w - 7, -3w + 4, w)}

82)

Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.

83) x + 9y + 8z = 24 83) -2x + 9y + 4z = -12 -2x + 2y - 2z = -22 A) B) 2 -2 -2 x 24 2 9 x 8 9 9 2 y = -12 -2 9 y = 4 -22 -2 2 z -2 8 4 -2 z C) D) 24 8 9 x 2 2 9 8 x 24 -12 4 9 y = -2 -2 9 4 y = -12 -22 -2 2 z -2 -2 2 -2 z -22

Solve the problem.

84) The area of a triangle with vertices (x1 , y1 ), (x2 , y2 ), and (x3, y3 ) is x1 y1 1 1 x y 1 Area = ± , 2 2 2 x3 y3 1 where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area.

Use this formula to find the area of a triangle whose vertices are (9, 4), (8, -7), and (-6, -4).

9 157 A) B) 9 C) 157 D) 2 2

18

84)

The

shape in the figure below is shown using 9 pixels in a 3 × 3 grid. The color levels are given to the right of the 131 figure. Use the matrix 1 3 1 that represents a digital photograph of the shape to solve the problem.

333

85) Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix addition to accomplish this.

A) B) 131 0 -1 0 020 131 0 -1 0 121 + = + = 131 0 -1 0 020 131 0 -1 0 121 -1 -1 -1 -1 -1 -1 333 222 333 222 C) D) 131 111 242 131 020 -1 -1 -1 131 + 111 = 242 1 3 1 + -1 -1 -1 = 0 2 0 -1 -1 -1 333 111 444 333 222 Solve the problem.

86) Determinants are used to show that three points lie on the same line (are collinear). If x1 y1 1

85)

86)

x2 y2 1 = 0, x3 y3 1 then the points (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ) are collinear. If the determinant does not equal 0, then the points are not collinear. Are the points (-7, 10), (0, -5), and (-21, 40) collinear?

A) Yes B) No

Find the products AB and BA to determine whether B is the multiplicative inverse of A. 87) 2 -1 0 1 -1 2 A = -1 1 -2 , B = -3 -2 4 -1 1 1 1 0 -1 -1 A) B A B) B = A-1

19

87)

Solve the system of equations using matrices. Use Gauss-Jordan elimination.

88) -3x - 5y - z = -36 x - 8y - 2z = -43 8x + y + z = 35 A) {(-3, 4, 6)} B) {(3, 7, 4)} C) {(3, 4, 7)} Write the augmented matrix for the system of equations. x - 6y + z = 11 89) y + 3z = 15 z = 17 A) B) 1 -6 1 11 1 -6 1 11 0 1 3 15 1 1 3 15 0 0 1 17 1 1 1 17

88)

D) {(6, 4, -3)}

89)

C)

1 6 1 11 0 1 3 15 0 0 1 17

D)

0 -6 0 11 0 0 3 15 0 0 0 17

Solve the problem using matrices.

90) State University has a College of Arts & Sciences, a College of Business, and a College of Engineering. The percentage of students in each category are given by the following matrix.

Arts & Sciences Business Engineering

Freshman 60% 20% 20%

Sophomore 50% 40% 10%

Junior 40% 30% 30%

90)

Senior 70% 10% 20%

The student population is distributed by class and age as given in the following matrix.

Female Freshman 460 Sophomore 550 Junior 890 Senior 630

Male 790 750 630 480

How many female students are in the College of Business? How many male students are in the College of Arts & Sciences?

A) 1348 students; 642 students B) 642 students; 1437 students C) 695 students; 518 students D) 540 students; 695 students

Solve the problem.

91)

Let A = 3 3 25 A) 6 10 3 16

91)

and B = 0 4 . Find 2A + B. -1 6 B) 6 10 1 11

C)

20

6 7 3 11

D)

6 14 2 22

The

shape in the figure below is shown using 9 pixels in a 3 × 3 grid. The color levels are given to the right of the 131 figure. Use the matrix 1 3 1 that represents a digital photograph of the shape to solve the problem.

333

92) Adjust the contrast by leaving the black alone and changing the light grey to dark grey. Use matrix addition to accomplish this.

A) B) 131 0 -1 0 121 131 101 232 + = + = 131 0 -1 0 121 131 101 232 -1 -1 -1 333 322 333 000 333 C) D) 131 111 232 131 1 -1 1 121 131 + 111 = 232 1 3 1 + 1 -1 1 = 1 2 1 -1 -1 -1 333 111 333 333 322 Solve the system of equations using matrices. Use Gauss-Jordan elimination. + 2w = -12 93) 3x + 5y 2x + 6z - w = -5 -2y + 3z - 3w =-3 -x + 2y + 4z + w = -2 A) {(-1, 3, 0, -3)} B) {(-1, -3, 0, 3)} C) {(1, -3, 0, 3)} Solve the matrix equation for X. 94) 2 -2 5 0 Let A = -8 0 and B = 0 -7 ; 4X + A = B 5 6 2 6 A) B) 3 2 3 1 - X = 8 -7 4 2 -3 0 7 X = -2 4 3 4

92)

93)

D) {(1, 3, 0, -3)}

94)

C)

X=

0

-

21

3 4

1 2

2 -

7 4

3 4

0

D)

-3 2 X = -8 7 3 0

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.

95) 95) 1 7 1 1 2 2 0

1

0

0

9 2

6

1 -4 7 A) {(- , , -4)} 2

B) {(-

27 , 24, -4)} 2

C) {(- 6,

, -5)}

D) {(-

23 , 2

Solve the problem.

96)

96)

3 -5 Let A = -2 and B = 1 . Find A + B. -4 9 A) B) 3 -5 -2 -2 1 -1 -4 9 5

Evaluate the determinant.

97) 512 214 521 A) -19

, -4)}

C)

2 5 6

D)

[-2 -1 5]

97)

B) 85

C) -59

D)

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.

98) 98) 1 1 9 4 0 1 -3 -7 0 0 1 -6 A) {(-39, 11, -6)} B) {(4, -7, -6)} C) {(-7, -5, -7)} D) {(83, -25, -6)}

22

Solve the problem using matrices.

99) The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving?

99)

A) t - 2 cars/min between I2 and I1 ; t + 1 cars/min between I1 and I3 B) t + 8 cars/min between I2 and I1 ; t + 3 cars/min between I1 and I3 C) t + 2 cars/min between I2 and I1 ; t - 3 cars/min between I1 and I3 D) t + 1 cars/min between I2 and I1 ; t + 4 cars/min between I1 and I3 Find values for the variables so that the matrices are equal.

100) x+3 y+4 = 6 1 7 5 7z A) x = 6; y = 1; z = 5 C) x = -3; y = 3; z = -5

100)

B) x = 3; y = -3; z = 5 D) x = 3; y = 5; z = 6

23

Answer Key Testname: C8

1) C 2) C 3) C 4) D 5) D 6) B 7) D 8) B 9) A 10) B 11) D 12) A 13) C 14) D 15) D 16) C 17) D 18) A 19) D 20) A 21) A 22) B 23) D 24) B 25) A 26) D 27) C 28) D 29) C 30) A 31) D 32) B 33) C 34) D 35) D 36) D 37) A 38) C 39) A 40) A 41) A 42) A 43) B 44) A 45) C 46) B 47) A 48) D 49) A 50) A 24

Answer Key Testname: C8

51) B 52) B 53) A 54) B 55) C 56) B 57) B 58) A 59) D 60) D 61) B 62) B 63) D 64) C 65) D 66) D 67) A 68) A 69) B 70) A 71) C 72) D 73) C 74) C 75) D 76) A 77) B 78) C 79) B 80) B 81) C 82) C 83) D 84) D 85) D 86) A 87) A 88) C 89) A 90) B 91) A 92) B 93) B 94) C 95) D 96) B 97) A 98) D 99) B 100) B 25

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Match the equation to the graph.

1) x2 = 7y A)

B)

C)

D)

1

1)

2) y2 = -7x A)

B)

C)

D)

3) y2 = 2x A)

B)

C)

D)

2

2)

3)

4) (y - 2)2 = 8(x - 2) A)

B)

C)

4)

D)

5) (x + 2)2 = 8(y + 1) A)

B)

C)

D)

3

5)

6)

y2 x2 =1 16 9

6)

A)

B)

C)

D)

4

7)

x2 y2 =1 16 9

7)

A)

B)

C)

D)

5

8) x2 = -10y A)

B)

C)

D)

8)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

The parametric equations of four curves are given. Graph each of them, indicating the orientation.

3 t 9) C1 : x = 7 sin t, y = 7 - 7 cos2 t; 2 2

9)

C2 : x = ln t, y = ln t2 ; e-4 t e3 C3 : x = t2 - 8, y = t - 3; -4 t 4 C4 : x = t - 5, y = t + 2; -4 t 7

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the standard form of the equation of the parabola using the information given.

10) Focus: (6, 0); Directrix: x = -6 A) y2 = 24x B) y2 = 6x C) x2 = 24y

6

D) y2 = -24x

10)

Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located at the pole.

3 11) r = 11) 12 + 4 cos

A) ellipse; The directrix is

3 3 unit(s) to the right of the pole at x = .

4 4

B) ellipse; The directrix is

3 3 unit(s) below the pole at y = - .

4 4

C) ellipse; The directrix is

3 3 unit(s) above the pole at y = .

4 4

D) ellipse; The directrix is

3 3 unit(s) to the left of the pole at x = - .

4 4

Graph the ellipse. (x + 2)2 (y - 1)2 + =1 12) 9 4

12)

A)

B)

7

C)

D)

Graph the parabola.

13) x2 = 5y

13)

A)

B)

C)

D)

8

Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.

14) x = t, y = 4t + 2; 0 t 4 14) A) y = -2x + 19; 0 x 2 B) y = 2x2 + 4; -1 x 2 C) y = -2x2 + 19; 0 x 2

D) y = 4x2 + 2; 0 x 2

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.

15) x = -(y + 8)2 - 5

15)

Use a graphing utility to graph the equation.

16) 16x2 - 24xy + 9y2 - 3x - 4y = 0

16)

A) Domain: (- , -5) Range: (- , ) C) Domain: (- , ) Range: (- , )

B) Domain: (- , ) Range: (- , -5] D) Domain: (- , -5] Range: (- , )

A)

B)

C)

D)

9

Graph the ellipse and locate the foci.

17) 4x2 = 36 - 9y2

17)

A) foci at ( 5, 0) and (- 5, 0)

B) foci at (2 3, 0) and (-2 3, 0)

C) foci at (0,

D) foci at ( 13, 0) and (- 13, 0)

5) and (0, - 5)

10

Use the relation's graph to determine its domain and range. x2 y2 + =1 18) 16 4

A) Domain: [-4, 4] Range: [-2, 2]

18)

B) Domain: [-4, 4] Range: (- , )

C) Domain: [-2, 2] Range: [-4, 4]

D) Domain: (-4, 4) Range: (-2, 2)

Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.

19) x = (70 cos 30°)t, y = 2 + (70 sin 30°)t - 16t2 ; t = 7 19)

A) (245 3, -537)

B) (-245 3, 312)

C) (245 3, 135)

D) (490 3, 459)

Rewrite the equation in a rotated x'y'-system without an x'y' term. Express the equation involving x' and y' in the standard form of a conic section.

20) x2 + 2xy + y2 + x - y - 4 = 0 20)

A) y'2 =

2 x' 2

B)

x'2 y'2 =1 2 4

C) x'2 =

2 y' + 2 2

D)

x'2 y'2 + =1 2 4

Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.

21) x = et, y = e-t; - < t < 21) B) y = ex; - < x < D) y = e-ln x; 0 < x <

A) y = e-x; - < x < C) y = eln x; 0 < x <

11

Use point plotting to graph the plane curve described by the given parametric equations.

22) x = t, y = 2t + 3; 0 t 4

A)

B)

C)

D)

Graph the polar equation.

12

22)

23) r =

9 3 - cos

23)

Identify the directrix and vertices.

A) directrix: 9 unit(s) to the right of the pole at x = 9 9 9 vertices: - , , , 4 2

B) directrix: 9 unit(s) above the pole at y = 9 9 9 vertices: , , - , 4 2 2 2

C) directrix: 9 unit(s) to the left of the pole at x = -9 9 9 vertices: , , , 0 4 2

D) directrix: 9 unit(s) below the pole at y = -9 9 9 vertices: - , , , 4 2 2 2

13

Find the vertex, focus, and directrix of the parabola with the given equation.

24) (y - 4)2 = 4(x + 2)

24)

Find the vertices and locate the foci for the hyperbola whose equation is given.

25) 16x2 - 25y2 = 400

25)

A) vertex: (-2, ) focus: (-3, ) directrix: x = -1 C) vertex: (2, -4) focus: (3, -4) directrix: x = 1

B) vertex: (-2, ) focus: (-1, ) directrix: x = -3 D) vertex: ( , -2) focus: ( , -2) directrix: x =

A) vertices: (-5, 0), (5, 0) foci: (-3, 0), (3, 0) C) vertices: (0, -5), (0, 5) foci: (0, - 41), (0, 41)

B) vertices: (-5, 0), (5, 0) foci: (- 41, 0), ( 41, 0) D) vertices: (-4, 0), (4, 0) foci: (- 41, 0), ( 41, 0)

Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located at the pole.

3 26) r = 26) 1 - 3 sin

A) hyperbola; The directrix is 1 unit(s) below the pole at y = -1.

B) hyperbola; The directrix is 1 unit(s) to the right of the pole at x = 1.

C) hyperbola; The directrix is 1 unit(s) above the pole at y = 1.

D) hyperbola; The directrix is 1 unit(s) to the left of the pole at x = -1.

Convert the equation to the standard form for a hyperbola by completing the square on x and y.

27) 9y2 - 4x2 - 36y + 16x - 16 = 0 (x + 2)2 (y + 2)2 =1 A) 9 4

C)

(y + 2)2 (x + 2)2 =1 B) 4 9

(y - 2)2 (x - 2)2 =1 4 9

D)

(y - 2)2 (x - 2)2 =1 9 4

Determine the direction in which the parabola opens, and the vertex.

28) y2 - 12y - x + 41 = 0 A) Opens to the left; (- 5, -6) C) Opens downward; (-6, - 5)

B) Opens downward; ( , - 5) D) Opens to the right; (5, 6)

Use a graphing utility to graph the equation.

14

27)

28)

29) 6x2 + 5 3xy + y2 = 20

29)

A)

B)

C)

D)

Find the standard form of the equation of the ellipse satisfying the given conditions.

30) Endpoints of major axis: (-4, -8) and (-4, 4); endpoints of minor axis: (-9, -2) and (1, -2); (x + 2)2 (y + 4)2 (x - 4)2 (y - 2)2 + =1 + =1 A) B) 25 36 25 36 C)

(x + 4)2 (y + 2)2 + =1 25 36

D)

15

(x - 5)2 (y - 6)2 + =1 25 36

30)

Find the standard form of the equation of the ellipse and give the location of its foci.

31)

A)

x2 y2 + =1 9 49

B)

foci at (0, 7) and (3, 0) x2 y2 + =1 C) 9 49

31)

x2 y2 + =1 9 49

foci at (0, -7) and (0, 7) x2 y2 + =1 D) 49 9

foci at (0, -2 10) and (0, 2 10)

foci at (0, -2 10) and (0, 2 10)

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.

32) y2 - 12y - x + 32 = 0

32)

Use vertices and asymptotes to graph the hyperbola. Find the equations of the asymptotes.

33) 16x2 - 4y2 = 64

33)

A) Domain: (- , ) Range: (- , )

B) Domain: (-4, ] Range: (- , )

C) Domain: (- , 4) Range: (- , )

16

D) Domain: (- , ) Range: (- , 4]

A) Asymptotes: y = ±

1 x 2

B) Asymptotes: y = ± 2x

C) Asymptotes: y = ±

1 x 2

D) Asymptotes: y = ± 2x

Solve the problem.

34) A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 30 feet. If the distance across the top of the mirror is 50 inches, how deep is the mirror in the center?

125 125 125 in. in. in.

A) B) 9 in.

C) D) 3456 288 24 Use the center, vertices, and asymptotes to graph the hyperbola. (y - 1)2 (x + 1)2 =1 35) 4 16

17

34)

35)

A)

B)

C)

D)

Find the standard form of the equation of the parabola using the information given.

36) Vertex: (1, -7); Focus: (9, -7) A) (x + 1)2 = 8(y - 7) B) (y + 7)2 = -32(x - 1) C) (x + 1)2 = -8(y - 7)

D) (y + 7)2 = 32(x - 1)

Identify the equation without completing the square.

37) 2x2 - 4x + y - 3 = 0 A) circle

36)

B) hyperbola

C) ellipse

D) parabola

Find the standard form of the equation of the ellipse satisfying the given conditions.

38) Endpoints of major axis: (8, -4) and (-2, -4); endpoints of minor axis: (3, -1) and (3, -7) (x - 3)2 (y + 4)2 (x + 3)2 (y + 3)2 + =1 + =0 A) B) 25 9 25 9 C)

(x + 4)2 (y - 3)2 + =1 9 25

D)

18

(x + 3)2 (y + 3)2 + =1 25 9

37)

38)

Solve the problem.

39) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 34 feet and the height of the arch over the center of the roadway is 10 feet. Two trucks plan to use this road. They are both 10 feet wide. Truck 1 has an overall height of 9 feet and Truck 2 has an overall height of 10 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge.

A) Neither Truck 1 nor Truck 2 can pass under the bridge.

B) Truck 2 can pass under the bridge, but Truck 1 cannot.

C) Truck 1 can pass under the bridge, but Truck 2 cannot.

D) Both Truck 1 and Truck 2 can pass under the bridge.

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term.

40) 15x2 - 24xy + 8y2 + 30x - 40y - 45 = 0 89 9 9 89 x' y'; y = x' + y' A) x = 100 20 20 100

C) x =

39)

40)

7 24 24 7 x' y'; y = x' + y' B) x = 25 25 25 25

3 4 4 3 x' - y'; y = x' + y' 5 5 5 5

D) x =

1 3 3 1 x' y'; y = x' + y' 2 2 2 2

Solve the problem.

41) An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower.

The towers stand 60 inches apart. At a point between the towers and 18 inches along the road from the base of one tower, the cable is 1.44 inches above the roadway. Find the height of the towers.

A) 8.5 in.

B) 11 in.

C) 9 in.

D) 9.5 in.

Graph the polar equation.

12 Identify the directrix and vertices.

42) r = 4 + 7 sin

19

41)

42)

A) directrix:

12 unit(s) to the left of 7

the pole at x = vertices:

C) directrix:

12 7

12 unit(s) above 7

the pole at y =

12 , 0 , 4, 11

vertices:

12 unit(s) below 7

the pole at y = vertices:

B) directrix:

D) directrix:

12 7

12 3 , , 4, 11 2 2

12 unit(s) to the right of 7

the pole at x =

12 3 , , 4, 11 2 2

vertices:

12 7

12 7

12 , 0 , 4, 11

Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.

43) x = 7 sin t, y = 7 cos t; 0 t 2 43) A) y = a 2 - x2 = 49; - < x < B) y = x2 - 9; -2 x 2 C) y2 - x2 = 49; - < x <

D) x2 + y2 = 49; -7 x 7

Find the foci of the ellipse whose equation is given. (x + 3)2 (y + 1)2 + =1 44) 9 16 A) foci at (-3, -1 C) foci at (-1, -3 -

7) and (-3, -1 + 7) and (-1, -3 +

44)

7) 7)

B) foci at (-2, -1 - 7) and (-2, -1 + 7) D) foci at ( , -1 - 7) and ( , -1 + 7)

Graph the ellipse and locate the foci.

20

45)

x2 y2 + =1 11 36

45)

A) foci at (0, 6) and (0, -6)

B) foci at (0,

C) foci at (5, 0) and (-5, 0)

D) foci at (0, 5) and (0, -5)

11) and (0, - 11)

Eliminate the parameter. Write the resulting equation in standard form.

46) A hyperbola: x = 1 + 6 sec t, y = 5 + 2 tan t (x - 5)2 (y - 1)2 (x - 1)2 (y - 5)2 + =1 =1 A) B) 4 36 36 4 C)

(x - 1)2 (y - 5)2 + =1 36 4

D)

Use a graphing utility to graph the equation.

21

(x + 5)2 (y + 1)2 =1 36 4

46)

47) 2x2 + 4 3xy + 6y2 +

3x - y = 0

47)

A)

B)

C)

D)

Find the standard form of the equation of the ellipse satisfying the given conditions.

48) Foci: (-3, 0), (3, 0); x-intercepts: -6 and 6 x2 y2 x2 y2 x2 y2 + =1 + =1 + =1 A) B) C) 27 36 9 27 9 36

22

x2

y2 + =1 D) 36 27

48)

Convert the equation to the standard form for a hyperbola by completing the square on x and y.

49) x2 - y2 - 2x - 6y - 9 = 0 A) (x - 1) 2 + (y + 3) 2 = 1

B) (x - 1) 2 - (y + 3) 2 = 1

(y - 1) 2 (x + 3) 2 =1 36 4

D) (y - 1) 2 - (x + 3) 2 = 1

C)

Use the rotated system to graph the equation.

50) 3x2 - 2 3xy + y2 - 8x - 8 3y = 0

49)

50)

A)

B)

C)

D)

23

Find a set of parametric equations for the rectangular equation.

51) y = x4 - 1 A) x = t2 ; y = t2 - 1 C) x = t2 ; y = t4 - 1

B) x = t; y = t4 - 1 D) x = t; y = t2 - 1

Graph the parabola with the given equation.

52) (y + 2)2 = -6(x + 1)

51)

52)

A)

B)

C)

D)

Graph the ellipse and locate the foci.

24

53)

x2 y2 + =1 64 36

53)

A) foci at (2 7, 0) and (-2 7, 0)

B) foci at (3 5, 0) and (-3 5, 0)

C) foci at (0, 3 5) and (0, -3 5)

D) foci at (0, 2 7) and (0, -2 7)

Solve the problem.

54) An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower.

The towers are both 16 inches tall and stand 80 inches apart. Find the vertical distance from the roadway to the cable at a point on the road 12 inches from the lowest point of the cable.

A) 1.64 in.

B) 1.44 in.

C) 5.76 in.

D) 1.24 in.

25

54)

Find the standard form of the equation of the hyperbola satisfying the given conditions.

55) Foci: (-10, 0), (10, 0); vertices: (-4, 0), (4, 0) y2 x2 y2 x2 x2 y2 x2 y2 =1 =1 =1 =1 A) B) C) D) 16 84 16 100 16 84 16 100

55)

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.

56) x = y2 - 7 56) x = y2 - 7y

A) {(-6, 1)}

B) {(6, 1)}

Identify the equation without applying a rotation of axes.

57) 4x2 + 11xy + 4y2 - 2x - 4y + 9 = 0 A) hyperbola

B) parabola

C) {(1, -6)}

D) {(-6, -1)}

C) ellipse

D) circle

Find two sets of parametric equations for the given rectangular equation.

58) y = x2 - 5 A) x = -t, y = t2 + 5; x = t, y = t - 5 C) x = t, y = t2 - 5; x = t, y = t - 5, t 0

B) x = t, y = t2 - 5; x = t, y = t + 5, t 0 D) x = -t, y = t2 - 5; x = t, y = -t + 5, t 0

57)

58)

Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located at the pole.

3 59) r = 59) 1 - 3 cos

A) ellipse; The directrix is 1 unit(s) to the left of the pole at x = -1.

B) hyperbola; The directrix is 1 unit(s) to the right of the pole at x = 1.

C) hyperbola; The directrix is 1 unit(s) to the left of the pole at x = -1.

D) ellipse; The directrix is 1 unit(s) to the right of the pole at x = 1.

26

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.

60) x = (y + 9)2 - 1 60) (x - 9)2 + (y + 9)2 = 1

A) {(-1, -9)} C) {(-1, -9), (9, -9)}

B) {(9, -9)} D)

Is the relation a function?

61) y2 + 2y - x + 6 = 0 A) Yes

61)

B) No

Find the standard form of the equation of the hyperbola satisfying the given conditions.

62) Foci: (0, -8), (0, 8); vertices: (0, -5), (0, 5) y2 x2 x2 y2 y2 x2 x2 y2 =1 =1 =1 =1 A) B) C) D) 25 39 25 39 25 64 25 64 Solve the problem. Express answers relative to an x'y'-system in which the given equation has no x'y'-term.

63) x2 + 2xy + y2 - 8x + 8y = 0; Find the coordinates of the vertex.

A) (0, 0)

B) (2, 0)

C) (-2, 0)

62)

63)

D) - 2, 0

Write the equation in terms of a rotated x'y'-system using , the angle of rotation. Write the equation involving x' and y' in standard form.

64) 14x2 + 12 3xy + 2y2 - 17 = 0; = 30° 64)

A)

x'2 y'2 = -1 1 17 20 4

B)

x'2 y'2 =1 1 1 20 4

C)

x'2 y'2 + =1 17 17 20 4

Find the standard form of the equation of the hyperbola satisfying the given conditions.

65) Center: (9, 7); Focus: (5, 7); Vertex: (8, 7) (y - 7)2 (x - 9)2 =1 - (y - 7)2 = 1 A) (x - 9)2 B) 15 15 C)

(x - 7)2 - (y - 9)2 = 1 15

(y - 9)2 =1 D) (x - 7)2 15

27

D)

x'2 y'2 =1 17 17 20 4

65)

Solve the problem.

66) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 32 feet and the height of the arch over the center of the roadway is 9 feet. Two trucks plan to use this road. They are both 12 feet wide. Truck 1 has an overall height of 9 feet and Truck 2 has an overall height of 8 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge.

A) Truck 1 can pass under the bridge, but Truck 2 cannot.

B) Both Truck 1 and Truck 2 can pass under the bridge.

C) Neither Truck 1 nor Truck 2 can pass under the bridge.

D) Truck 2 can pass under the bridge, but Truck 1 cannot.

66)

Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.

67) x = 4 + 5 cos t, y = 6 + 4 sin t; t =

67)

2 5 2 ,6 +2 2 2

A) (4, 8)

B) 4 +

C) (9, 6)

D) (4, 10)

Find the foci of the ellipse whose equation is given.

68) 9(x - 1)2 + 36(y + 2)2 = 324

68)

A) foci at (-2 + 3 3, ) and (-2 - 3 3, ) C) foci at (- 3 3, -2) and (3 3, -2)

B) foci at (1 + 3 3, -2) and (1 - 3 3, -2) D) foci at (1 + 3 3, 1) and (1 - 3 3, 1)

Find the vertices and locate the foci for the hyperbola whose equation is given.

69) y = ± x2 - 8 A) vertices: (-2 2, 0), (2 2, 0) foci: (-4, 0), (4, 0)

B) vertices: (0, -2 2), (0, 2 2) foci: (0, -4), (0, 4)

C) vertices: (-8, 0), (8, 0) foci: (-2 2, 0), (2 2, 0)

D) vertices: (-8, 0), (8, 0) foci: (-4, 0), (4, 0)

Convert the equation to the standard form for an ellipse by completing the square on x and y.

70) 25x2 + 16y2 + 150x + 64y - 111 = 0 (x + 2)2 (y + 3)2 + =1 A) 16 25

C)

(x - 3)2 (y - 2)2 + =1 B) 16 25

(x + 3)2 (y + 2)2 + =1 16 25

D)

28

(x + 3)2 (y + 2)2 + =1 25 16

69)

70)

Use a graphing utility to obtain the plane curve represented by the given parametric equations.

71) Cycloid: x = 2(t - sin t), y= 2(1 - cos t), 0 t 6

A)

B)

C)

D)

Use the rotated system to graph the equation.

29

71)

72) 7x2 + 2xy + 7y2 = 24

72)

A)

B)

C)

D)

Determine the direction in which the parabola opens, and the vertex.

73) y = x2 - 4x + 3 A) Opens upward; (2, -1) C) Opens to the right; (-1, -2)

B) Opens upward; (-2, -1) D) Opens to the right; (-1, 2)

Find the focus and directrix of the parabola with the given equation.

74) y2 = 12x A) focus: (0, 3) directrix: y = -3

B) focus: (3, 0) directrix: x = 3

C) focus: (3, 0) directrix: x = -3

30

D) focus: (0, -3) directrix: y = -3

73)

74)

Use the center, vertices, and asymptotes to graph the hyperbola.

75) (y + 4)2 - 9(x - 4)2 = 9

A)

B)

C)

D)

75)

Write the equation in terms of a rotated x'y'-system using , the angle of rotation. Write the equation involving x' and y' in standard form.

76) 5x2 - 6xy + 5y2 - 8 = 0; = 45° 76)

A)

x'2 + y'2 = 1 4

B)

x'2 y'2 + =1 4 4

C) x'2 + 4y'2 = 1

31

D)

x'2 + 2y'2 = 1 4

Answer Key Testname: C9

1) D 2) B 3) B 4) A 5) C 6) B 7) A 8) C 9)

10) A 11) A 12) D 13) B 14) D 15) D 16) C 17) A 18) A 19) A 20) C 21) D 22) C 32

Answer Key Testname: C9

23) C 24) B 25) B 26) A 27) C 28) D 29) A 30) C 31) C 32) B 33) B 34) C 35) B 36) D 37) D 38) A 39) C 40) C 41) C 42) B 43) D 44) A 45) D 46) B 47) B 48) D 49) B 50) B 51) B 52) D 53) A 54) B 55) C 56) A 57) A 58) C 59) C 60) D 61) B 62) A 63) A 64) D 65) A 66) D 67) D 68) B 69) A 70) C 71) B 72) B 33

Answer Key Testname: C9

73) A 74) C 75) D 76) A

34

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

A statement Sn about the positive integers is given. Write statements Sk and Sk+1 , simplifying Sk+1 completely.

1) Sn : 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) =

n(n + 1)(n + 2) 3

Use mathematical induction to prove that the statement is true for every positive integer n.

2) 10 + 20 + 30 + . . . + 10n = 5n(n + 1)

1)

2)

A statement Sn about the positive integers is given. Write statements Sk and Sk+1 , simplifying Sk+1 completely.

3) Sn : 12 + 4 2 + 7 2 + . . . + (3n - 2)2 =

n(6n 2 - 3n - 1) 2

3)

A statement Sn about the positive integers is given. Write statements S1 , S2 , and S3 , and show that each of these statements is true.

4) Sn : 2 is a factor of n 2 + 11n

4)

Use mathematical induction to prove that the statement is true for every positive integer n.

4 4 4 1 +...+ =5 15) 4 + + 5 25 n 1 5 5n 6) 2 is a factor of n 2 - n + 2

5)

6)

A statement Sn about the positive integers is given. Write statements S1 , S2 , and S3 , and show that each of these statements is true.

n(6n 2 - 3n - 1) 7) Sn : 12 + 4 2 + 7 2 + . . . + (3n - 2)2 = 2

8) Sn : 2 + 7 + 12 + . . . + (5n - 3) =

7)

n(5n - 1) 2

8)

Use mathematical induction to prove that the statement is true for every positive integer n. n(5n + 1) 9) 3 + 8 + 13 + . . . + (5n - 2) = 2 10) 1 · 4 + 2 · 4 + 3 · 4 + . . . + 4n =

4n(n + 1) 2

9)

10)

A statement Sn about the positive integers is given. Write statements Sk and Sk+1 , simplifying Sk+1 completely.

11) Sn : 3 + 8 + 13 + . . . + (5n - 2) =

n(5n + 1) 2

11)

1

A statement Sn about the positive integers is given. Write statements S1 , S2 , and S3 , and show that each of these statements is true.

12) Sn : 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) =

n(n + 1)(n + 2) 3

12)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the indicated sum.

4 1 13) 3i i=1 5 A) 12

13) B)

25 36

C)

11 18

Find the term indicated in the expansion.

14) (5x + 2)5 ; 5th term A) 200x

B) 400x

C) 160

D)

1 12

D) 1000x2

Solve the problem.

15) A company models its yearly expenses in millions of dollars using the equation f(t) = 0.07t3 - 0.7t2 + 1.34t + 2.8 where t = 0 represents 1990. The company's account manager

14)

15)

decides to adjust the model so that t = 0 corresponds to 1995 rather than 1990. To do this, she obtains g(t) = f(t + 5). Use the Binomial Theorem to express g(t) in descending powers of t.

A) 0.07t3 + 0.35t2 - 0.41t + 0.75 B) 0.07t3 + 0.35t2 + 13.59t + 6.35

C) 0.07t3 - 1.75t2 - 2.16t + 13.35

D) 0.07t3 - 1.75t2 + 13.59t + 2.39

Use the Binomial Theorem to expand the binomial and express the result in simplified form.

16) (x + 5)4 A) x4 + 20x3 + 250x2 + 500x + 625 C) x4 + 20x3 + 150x2 + 20x + 625

B) x4 + 20x3 + 150x2 + 500x + 625 D) x4 + 5x3 + 150x2 + 250x + 625

17) (2x + 4)4

16)

17)

A) 16x4 + 256x4

B) 16x4 + 128x3 + 384x2 + 512x + 256

C) 16x3 + 128x2 + 384x + 512

D) 64x4 + 512 x3 + 384x2 + 2048x + 256

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation.

18) 7 + 8 + 9 + 10 + . . . + 28 18) 21 26 30 28 (k - 2) (k - 2) (k - 2) (k - 2) A) B) C) D) k=1 k=5 k=9 k=7

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

1 4 16 -...

19) - + 5 5 5 A) -

16,384 5

B)

19)

65,536 5

C)

2

1 25

D) does not exist

Find the probability.

20) What is the probability that a card drawn from a deck of 52 cards is not black?

25 3 A) B) 2 C) 52 4

1 D) 2

20)

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

21) a n = 4n 2 21) 4 A) geometric, r = B) geometric, r = 4 2

C) arithmetic, d = 4

D) neither

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

48 (6i + 1) 22) i=1 A) 7320 B) 7104 C) 7488 D) 6960 Solve the problem.

23) Marina takes a job with a starting salary of $40,000 for the first year with an annual increase of 6% beginning in the second year. What is Marina's salary, to the nearest dollar, in the sixth year?

A) $56,023 B) $54,213 C) $53,529 D) $52,298 Find the indicated sum.

24) Find the sum of the even integers between 35 and 59.

A) 658 B) 517 C) 611

D) 564

Use the Binomial Theorem to expand the binomial and express the result in simplified form.

25) (5x - 5y)3 A) 25x3 y - 25x2 y2 + 25xy3 C) 125x3 - 125x2 y + 125xy2 - 125y3

B) 125x3 - 375x2 y + 375xy2 - 125y3 D) 25x3 y - 50x2 y2 + 25xy3

Find the probability.

26) Give the probability that the roll of a die will show 4 or 5.

5 1 2 A) B) C) 6 3 3

22)

23)

24)

25)

26) D) 2

Evaluate the expression.

27)

5 C3 · 6 C1

27)

16C13

A)

3 28

B)

3 112

C)

3

45 28

D)

3 14

Write the first four terms of the sequence whose general term is given. n4 28) a n = (n - 1)!

A)

1 16 81 128 , , , 0 0 2 3

B)

4 8 8 , , 6, 0 0 3

C) 1, 16,

28) 81 128 , 2 3

D) 4, 8, 6,

8 3

Write a formula for the general term (the nth term) of the geometric sequence.

1 1 1 1 , ,,...

29) , 2 18 162 1458 A) a n =

1 n-1 5 2 9

B) a n =

1 1 - (n - 1) 2 9

C) a n =

1 1 n-1 2 9

D) a n =

1 1 n-1 9 2

29)

Solve the problem.

30) A church has 9 bells in its bell tower. Before each church service 3 bells are rung in sequence. No bell is rung more than once. How many sequences are there?

A) 120,960 B) 504 C) 60,480 D) 84

30)

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a 1 , and common ratio, r.

31) Find a 8 when a 1 = 6, r = 2.

A) 20

31) B) 1536

C) 768

D) 772

Find the probability.

32) A bag contains 6 red marbles, 4 blue marbles, and 1 green marble. What is the probability of choosing a marble that is not blue when one marble is drawn from the bag?

11 7 4 A) B) 7 C) D) 7 11 11 Solve the problem.

33) As part of her retirement savings plan, Patricia deposited $250 in a bank account during her first year in the workforce. During each subsequent year, she deposited $30 more than the previous year. Find how much she deposited during her twentieth year in the workforce. Find the total amount deposited in the twenty years.

A) $820; $10,700 B) $850; $22,000 C) $850; $11,000 D) $820; $21,400 Write the first four terms of the sequence whose general term is given.

2 n 34) a n = 5 2 4 8 , , 5 25 125

B)

2 4 8 16 , , , 5 25 125 625

D) 1,

A) 1, C)

4

2 2 2 2 , , , 5 10 15 20 4 8 16 , , 25 125 625

32)

33)

34)

35) a n = (-1)n (n + 6)

35)

A) 7, 8, 9, 10 C) -7, -8, -9, -10

B) -7, 8, -9, 10 D) -7, -16, -27, -40

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

36)

70

5i i=1 A) 12,425

36) B) 12,603

C) 12,420

D) 12,248

Write a formula for the general term (the nth term) of the geometric sequence.

1 1 1 1 , ,...

37) - , , 3 6 12 24 A) a n = -

1 1 n-1 3 2

B) a n = -

1 1 n-1 2 3

C) a n = -

1 n-1 1 + 3 2

D) a n = -

1 1 - (n - 1) 3 2

37)

Write the first five terms of the geometric sequence.

38) a 1 = -5; r = -3

38)

A) -5, -15, -45, 135, -405 C) -3, 15, -45, 135, -405

B) -5, 15, -45, 135, -405 D) -5, -8, -11, -14, -17

Use the formula for the sum of the first n terms of a geometric sequence to solve.

39) Find the sum of the first six terms of the geometric sequence: -2, -8, -32, . . . .

A) -42 B) 2730 C) - 1365

D) - 2730

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.

5 4 i 40) 5 i=1 369 8404 1476 2101 A) B) C) D) 625 3125 3125 625 Evaluate the factorial expression. n(n + 2 )!

41) (n + 3 )!

A)

n (n + 3)!

39)

40)

41) B)

1 n+3

C)

n 3

D)

n n+3

Use the Binomial Theorem to expand the binomial and express the result in simplified form.

42) (x + 3)3 A) x3 + 3x2 + 9x + 27

B) 3x + 9

C) x3 + 9x2 + 27x + 27

D) x3 + 27

5

42)

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

38 (-5i - 4) 43) i=1 A) -3610 B) -3762 C) -3743 D) -3857 Write the first four terms of the sequence whose general term is given.

(-1)n + 1 44) a n = n+2 A)

1 1 1 1 ,- , ,3 8 15 24

B) -

1 1 1 1 , ,- , 4 5 6 7

D)

C) -

44)

1 1 1 1 , ,- , 3 4 5 6

1 1 1 1 ,- , ,3 4 5 6

Write the first four terms of the sequence defined by the recursion formula.

45) a 1 = -4 and a n = -2a n-1 for n 2 A) -4, -8, -16, -32

B) -4, 8, -16, 32

C) 4, -8, 16, -32

45) D) -4, 10, -18, 34

Solve the problem.

46) A stack of 7 different cards are shuffled and spread out face down. If 5 cards are turned face up, how many different 5-card combinations are possible?

A) 21 B) 42 C) 2520 D) 1260 Write the first five terms of the geometric sequence.

47) a 1 = 6; r = 3 B) 6, 9, 12, 15, 18 D) 18, 54, 162, 486, 1458

Find the probability.

48) What is the probability that a card drawn from a deck of 52 cards is not a heart?

2 4 3 1 A) B) C) D) 5 13 4 4

48)

Solve the problem.

49) How many 3-letter codes can be formed using the letters A, B, C, D, and E? No letter can be used more than once.

A) 20 B) 10 C) 60 D) 40 Use the formula for the sum of the first n terms of a geometric sequence to solve.

1 2 4 8 16 ,....

50) Find the sum of the first 8 terms of the geometric sequence: , - , , - , 3 3 3 3 3 91 3

B) - 26

C) - 29

D) -

49)

50) 85 3

Evaluate the factorial expression.

8!

51) 4!4!

A) 35

46)

47)

A) 3, 18, 108, 648, 3888 C) 6, 18, 54, 162, 486

A) -

43)

51) B) 70

C) 140

6

D) 1680

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.

5 52) 2 · 2i i=1 A) 20 B) 170 C) 12 D) 124 Find the term indicated in the expansion.

53) (3x + 4)5 ; 5th term A) 2880x2

52)

53)

B) 3840x

C) 960x

D) 5120

Write the first four terms of the sequence whose general term is given.

54) a n = (-1)n + 1(n + 5) A) -6, 7, -8, 9 B) 6, -14, 24, -36 C) -7, 8, -9, 10

54) D) 6, -7, 8, -9

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a 1 , and common ratio, r.

55) Find a 8 when a 1 = 40,000, r = -0.1.

A) -0.04

55)

B) -0.004

C) 0.004

D) 0.0004

Write the first five terms of the geometric sequence.

56) a n = 4a n-1 ; a 1 = -4

56)

A) -16, -64, -256, -1024, -4096 C) -4, -16, -64, -256, -1024

B) 4, 16, -64, -256, -1024 D) -4, 0, 4, 8, 12

Does the problem involve permutations or combinations? Do not solve.

57) A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 14 members and any member can be elected to each position? No person can hold more than one office.

A) permutations B) combinations Write the first four terms of the sequence whose general term is given.

58) a n = 4(n + 1)!

A) 8, 48, 288, 1920

B) 4, 16, 72, 384

58)

C) 8, 24, 96, 480

D) 4, 8, 24, 96

59) a n = 3 n A) 9, 27, 81, 243 60) a n = A)

59) B) 3, 9, 27, 81

C) 1, 8, 27, 64

D) 1, 3, 9, 27

(n - 1)! n3

1 1 2 1 , , , 3 6 9 2

57)

60) B) 1,

1 2 3 , , 8 27 32

C) 0, 0,

2 1 , 9 2

D) 0, 0,

2 3 , 27 32

Find the probability.

61) A lottery game has balls numbered 1 through 19. What is the probability of selecting an even numbered ball or a 7?

8 9 10 9 A) B) C) D) 19 19 19 10

7

61)

Solve the problem. Round to the nearest hundredth of a percent if needed.

62) Measurements of the height of a group of men entering a particular college produced the following table. What is the probability that a man entering the college is 68-69 inches tall?

Height (inches) 60-61 62-63 64-65 66-67 68-69 70-71 72-73 74-75 76+ Number 2 11 85 232 257 174 88 37 4 A) 19.55% B) 26.07% C) 29.07%

D) 28.88%

Find the probability.

63) A 6-sided die is rolled. What is the probability of rolling a number that is even and a 3?

1 1 A) 1 B) C) 0 D) 6 2 Find the indicated sum.

6 5 64) i=3 A) 90

63)

64) B) 20

C) 75

D) 15

Find the probability.

65) Urn A has balls numbered 1 through 6. Urn B has balls numbered 1 through 3. What is the probability that a 4 is drawn from A followed by a 2 from B? 1 1 1 1 A) B) C) D) 9 18 6 2 Use the formula for the sum of the first n terms of a geometric sequence to solve.

3 3 3 ,....

66) Find the sum of the first five terms of the geometric sequence: , , 2 8 32 A) 341

B)

1023 512

C)

5 512

D)

1 16

Write the first four terms of the sequence whose general term is given.

3 n 68) a n = 4 A) -

3 9 27 81 ,,,4 16 64 256

B) -

C) -

3 9 27 81 , ,, 4 16 64 256

D)

Find the term indicated in the expansion.

69) (x + 2y)11; 7th term B)

x6 y5

67)

68)

3 3 3 3 , ,,4 8 12 16

3 3 3 3 ,- , ,4 8 12 16

C) 14,784x6 y5

8

65)

66)

Solve the problem.

67) A student must choose 1 of 5 science electives, 1 of 8 social studies electives, and 1 of 7 language electives. How many possible course selections are there?

A) 40 course selections B) 20 course selections C) 280 course selections D) 560 course selections

A) 14,784x5 y7

62)

D)

x5 y6

69)

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

5 70) 180 + 30 + 5 + + . . .

6 A) - 36

70)

B) 215

C) 216

D) does not exist

Solve the problem.

71) In how many ways can 6 players be assigned to 6 positions on a baseball team, assuming that any player can play any position?

A) 30 ways B) 1440 ways C) 720 ways D) 360 ways Find the probability.

72) A game spinner has regions that are numbered 1 through 9. If the spinner is used twice, what is the probability that the first number is a 3 and the second is a 7?

1 10 1 7 A) B) C) D) 18 81 81 10 Write a formula for the general term (the nth term) of the geometric sequence.

73) -2, -6, -18, -54, -162, . . .

A) a n = -2(3)n - 1 B) a n = a 1 + 3 n C) a n = -2(3n)

D) a n = -2(3)n

71)

72)

73)

Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.

1 1 3 13 74) + + + . . . + 74) 3 2 5 15 13

A)

i=2

i i+1

B)

13 i=0

i i+2

C)

n i=1

i i+2

Write a formula for the general term (the nth term) of the geometric sequence.

75) 4, 12, 36, 108, 324, . . .

A) a n = 4(3)n B) a n = a 1 + 3 n C) a n = 4(3n)

D)

13 i=1

i i+2

D) a n = 4(3)n - 1

Evaluate the factorial expression. (n + 6 )!

76) n+6 A) 1

75)

76) B) (n + 5)!

C) 6!

D) n + 6!

Solve the problem.

77) A deposit of $8000 is made in an account that earns 6% interest compounded quarterly. The balance in the account after n quarters is given by the sequence 0.06 n a n = 8000 1 + , n = 1, 2, 3, ...

4 Find the balance in the account after 5 years.

A) $8618.27 B) $10,774.84

C) $6734.28

Write the first five terms of the arithmetic sequence.

78) a 1 = -23; d = 2

77)

D) $5500.28

78)

A) -19, -17, -15, -13, -11 C) -15, -17, -19, -21, -23

B) -23, -21, -19, -17, -15 D) -19, -21, -23, -25, -27 9

Use the Binomial Theorem to expand the binomial and express the result in simplified form.

79) (x + 2y)3 A) 3x + 6y

B) x3 + 6x2 y + 12xy2 + 8y3

C) x3 + 2x2 y + 4xy + 4xy2 + 8y2 + 8y3

D) x3 + 8y3

Evaluate the factorial expression.

7!

80) 6!

A) 7!

Find the indicated sum.

5 (i + 10) 81) i=1 A)

79)

80) B)

7 6

C) 7

D) 1

81) B)

C)

D)

Find the probability.

82) A card is drawn from a deck of 52 cards. What is the probability that it is a picture card (Jack, Queen, King) or a diamond?

7 11 39 25 A) B) C) D) 52 26 52 52 Use the Binomial Theorem to expand the binomial and express the result in simplified form.

83) (x - 9)5 A) x5 - 45x4 + 1620x3 - 14,580x2 + 32,805x - 59,049 B) x5 - 45x4 + 810x3 - 7290x2 + 32,805x - 9

82)

83)

C) x5 - 45x4 + 1620x3 - 14,580x2 + 32,805x - 9 D) x5 - 45x4 + 810x3 - 7290x2 + 32,805x - 59,049

Solve the problem.

84) A pendulum bob swings through an arc 50 inches long on its first swing. Each swing thereafter, it swings only 60% as far as on the previous swing. How far will it swing altogether before coming to a complete stop? Round to the nearest inch when necessary.

A) 63 inches B) 167 inches C) 125 inches D) 83 inches Evaluate the given binomial coefficient.

145 85) 143 A) 1,492,920

B)

84)

85) 145 143

C) 10,440

If the given sequence is a geometric sequence, find the common ratio.

4 8 16 32 64 , 86) , , , 3 3 3 3 3 A) not a geometric sequence C) 6

B) 4 D) 2

10

D)

10440 143

86)

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a 1 , and common ratio, r.

87) Find a 10 when a 1 = 5, r = -3.

A) -98,411

87)

B) -22

C) -98,415

D) 295,245

Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.

88) 4 2 + 8 3 + 124 + . . . + 329 88) A)

C)

8

i=1 8

(4i)i

B)

(4i)i + 1

D)

i=1

8

i=1 8

4i2i - 1 2(i - 1)i + 1

i=1

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a 1 , and common ratio, r.

89) Find a 5 when a 1 = 4, r = -5.

A) -2500

89) B) -12,500

C)

D)

Find the probability.

90) Given the sequence 3, 4, 5, 6 ... 42, what is the probability that a number in the sequence is odd or that it is greater than 12 and less then 25?

4 13 51 17 A) B) C) D) 5 20 80 26 Solve the problem. Round to the nearest dollar if needed.

91) To save for retirement, you decide to deposit $2000 into an IRA at the end of each year for the next 40 years. If the interest rate is 9% per year compounded annually, find the value of the IRA after 40 years.

A) $60,819 B) $618,133 C) $24,330,719 D) $675,765 Write the first five terms of the geometric sequence.

92) a n = -8a n-1 ; a 1 = -4

90)

91)

92)

A) -4, -12, -20, -28, -36 C) -4, 32, -256, 2048, -16,384

B) -8, -32, -256, -2048, -16,384 D) 32, -256, 2048, -16,384, 131,072

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation.

5 7 93) 2 + + 3 + + . . . + 9 93) 2 2

A)

18 k=4

k 2

B)

18 k=1

k 2

C)

18 k=2

11

k 2

D)

8 k=4

k 2

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a 1 , and common ratio, r.

94) Find a 6 when a 1 = 9600, r = A) -150

1 .

2

94)

B) 300

C) -300

D) -30

Find the probability.

95) Give the probability that the roll of a die will show a number less than 6.

5 A) B) 0 C) 1 6

95)

Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.

96) 2 + 4 + 6 + . . . + 16 96) 8 8 8 8 2i i2 2i2 2i A) B) C) D) i=0 i=1 i=1 i=1 Solve the problem.

97) A small business owner made $60,000 the first year he owned his store and made an additional 5% over the previous year in each subsequent year. Find how much he made during his fourth year of business. Find his total earnings during the first four years. (Round to the nearest cent, if necessary.)

A) $7.50; $63,157.50 B) $303,750.00; $791,250.00 C) $202,500.00; $487,500.00 D) $69,457.50; $258,607.50 98) To train for a race, Will begins by jogging 13 minutes one day per week. He increases his jogging time by 5 minutes each week. Write the general term of this arithmetic sequence, and find how many whole weeks it takes for him to reach a jogging time of one hour.

A) a n = 5n + 13; 10 weeks B) a n = 5n + 8; 10 weeks C) a n = 5n + 8; 11 weeks

97)

98)

D) a n = 5n + 13; 11 weeks

99) The following table shows a country's population from 1995 to 1998: Year 1995 1996 1997 1998 Population in millions 12.40 12.77 13.15 13.54 Divide the population for each year by the population in the preceding year. Use this ratio to write the general term of the geometric sequence describing the country's population growth n years after 1994. Then estimate the country's population, in millions, in 2004.

A) a n = 12.40(1.02)n - 1; 18.32 million B) a n = 12.40(1.02)n - 1; 19.24 million C) a n = 12.40(1.03)n - 1; 16.18 million D) a n = 12.40(1.03)n - 1; 16.66 million

12

99)

If the given sequence is a geometric sequence, find the common ratio.

1 1 1 1 , , , 100) -2 -5 -8 -11 A) not a geometric sequence C)

100)

B) -3

1 -3

D) 3

1 3

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a 1 , and common difference, d.

101) Find a8 when a1 = -8 , d = 4 .

A) -40

101)

B) -36

C) 24

D) 20

Write the first three terms in the binomial expansion, expressing the result in simplified form.

102) ( 3 x + 2 y) 7 A) 7 x 7 y + 6 x 6 y2 + 5 x 5 y3

B) 2187 x 7 + 10,206 x 6 y + 20,412 x 5 y2

C) 2187 x 7 y + 10,206 x 6 y2 + 20,412 x 5 y3

Find the indicated sum.

4 i 3 103) i=1 A) 30

102)

D) 7 x 7 + 6 x 6 y + 5 x 5 y2

103) B) 84

C) 39

D) 120

Write the first three terms in the binomial expansion, expressing the result in simplified form.

104) (x2 + 2 )9 A) x18 + 20 x16 + 144 x14 C) x18 + 20 x16 + 288 x14

B) x18 + 18 x16 + 288 x14 D) x18 + 18 x16 + 144 x14

104)

Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for an to find a 20, the 20th term of the sequence.

105) a 1 = -4, d = 0.7

105)

A) a n = 0.7n - 4; a 20 = 10

B) a n = -4n + 0.7; a 20 = -79.3

C) a n = 0.7n - 4.7; a 20 = 9.3

D) a n = -4n + 4.7; a 20 = -75.3

Solve the problem.

106) Ron finds 8 books at a bookstore that he would like to buy, but he can afford only 5 of them. In how many ways can he make his selection? How many ways can he make his selection if he decides that one of the books is a must?

A) 6720; 840 B) 336; 210 C) 3360; 420 D) 56; 35

106)

Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for an to find a 20, the 20th term of the sequence.

107) a n = a n-1 - 4, a 1 = 32

107)

A) a n = 32n + 36, a 20 = 676

B) a n = 28n + 32, a 20 = 592

C) a n = -4n + 36, a 20 = -44

D) a n = -4n + 28, a 20 = -52

13

Find the indicated sum.

4 1 i (- ) 108) 2 i=1 5 A) 16

108) B)

5 16

C)

15 16

D) -

1 16

Solve the problem.

109) A basketball player signs a contract with a starting salary of $830,000 per year and an annual increase of 4% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the eighth year?

A) $1,092,223 B) $1,094,428 C) $1,092,828 D) $1,090,938

109)

Use the formula for nPr to evaluate the expression.

110) 4 P0 A) 40

B) 24

C) 1

D) 0

Solve the problem.

111) The population of a town is increasing by 500 inhabitants each year. If its population at the beginning of 1990 was 27,186, what was its population at the beginning of 2000?

A) 543,270 inhabitants B) 31,686 inhabitants C) 32,186 inhabitants D) 271,635 inhabitants

110)

111)

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

112) a n = 2 n 112) A) arithmetic, d = 2 B) geometric, r = 3 C) geometric, r = 2 D) neither

14

Answer Key Testname: C10

1) Sk: 1 · 2 + 2 · 3 + 3 · 4 + . . . + k(k + 1) =

k(k + 1)(k + 2) 3

Sk+1 : 1 · 2 + 2 · 3 + 3 · 4 + . . . + (k + 1)(k + 2) =

(k + 1)(k + 2)(k + 3) 3

?

2) S1 : 10 = (5 · 1)(1 + 1) ?

10 = 5 · 2 10 = 10 Sk: 10 + 20 + 30 + . . . + 10k = 5k(k + 1) Sk+1 : 10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2) We work with Sk. Because we assume that Sk is true, we add the next multiple of 10, namely 10(k+1), to both sides.

10 + 20 + 30 + . . . + 10k + 10(k + 1) = 5k(k + 1) + 10(k + 1) 10 + 20 + 30 + . . . + 10(k + 1) = (k + 1)(5k + 10) 10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2) We have shown that if we assume that Sk is true, and we add (10(k+1) to both sides of Sk, then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n.

k(6k2 - 3k - 1) 3) Sk: 1 2 + 4 2 + 7 2 + . . . + (3k - 2)2 = 2 Sk+1 : 1 2 + 4 2 + 7 2 + . . . + (3k + 1)2 =

(k + 1)(6k2 + 9k + 2) 2

4) S1 : 2 is a factor of 1 2 + 11 · 1 = 1 + 11 = 12 S2 : 2 is a factor of 2 2 + 11 · 2 = 4 + 22 = 26 S3 : 2 is a factor of 3 2 + 11 · 3 = 9 + 33 = 42

15

Answer Key Testname: C10

?

1 5) S1 : 4 = 5 1 51 ?

4 4 =5 5 4=4 4 4 4 1 +...+ =5 1Sk: 4 + + 5 25 5k - 1 5k Sk+1 : 4 +

4 4 4 1 + +...+ =5 15 25 5k 5k + 1

We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely

4+

4 4 4 4 1 4 + +...+ + =5 1+ 5 25 5 k - 1 5k 5k 5k

4+

4 4 4 5 4 + +...+ =5+ 5 25 k k 5 5 5k

4+

4 4 4 5 · 5k - 5 + 4 + +...+ = 5 25 5k 5k

4+

4 4 4 5k + 1 - 1 + +...+ = 5 25 5k 5k

4+

4 4 4 5k + 1 1 + +...+ = 5 25 k k 5 5 5k

4+

4 4 4 1 + +...+ =55 25 k 5 5k

4+

4 4 4 1 + +...+ =5 15 25 k k 5 5 +1

We have shown that if we assume that Sk is true, and we add

4 to both sides of Sk, then Sk+1 is also true. By the 5k

principle of mathematical induction, the statement Sn is true for every positive integer n.

6) Show S1 is true: 1 2 - 1 + 2 = 2; 2 is a factor of 2 Assume that Sk is true: k2 - k + 2; 2 is a factor of k2 - k + 2 Show that if Sk is true, then Sk+1 is true: Sk+1 = (k + 1)2 - (k + 1) + 2 = k2 + k + 2 = (k2 - k + 2) + 2k (k2 - k + 2) is Sk and 2 is a factor of 2k Since 2 is a factor of Sk+1 is true, the statement is true for all values of n.

16

4 to both sides.

5k

Answer Key Testname: C10

?

1(6 · 1 2 - 3 · 1 - 1) 7) S1 : 1 2 = 2 ?

1 ·2 1 = 2 1=1 ?

2(6 · 2 2 - 3 · 2 - 1) S2 : 1 2 + 4 2 = 2 ?

2 · 17 17 = 2 17 = 17 ?

3(6 · 3 2 - 3 · 3 - 1) S3 : 1 2 + 4 2 + 7 2 = 2 ?

3 · 44 66 = 2 66 = 66 ?

1(5 · 1 + -1) 8) S1 : 2 = 2 ?

1 ·4 2 = 2 2=2 ?

2(5 · 2 + -1) S2 : 2 + 7 = 2 ?

2 ·9 9 = 2 9=9 ?

3(5 · 3 + -1) S3 : 2 + 7 + 12 = 2 ?

3 · 14 21 = 2 21 = 21

17

Answer Key Testname: C10

?

1(5 · 1 + 1) 9) S1 : 3 = 2 ?

1 ·6 3 = 2 3=3 Sk: 3 + 8 + 13 + . . . + (5k - 2) =

k(5k + 1) 2

Sk+1 : 3 + 8 + 13 + . . . + (5k + 3) =

(k + 1)(5k + 6) 2

We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely 5(k+1) - 2, to both sides.

3 + 8 + 13 + . . . + (5k - 2) + (5(k + 1) - 2) =

k(5k + 1) + (5(k + 1) - 2) 2

3 + 8 + 13 + . . . + (5k + 3) =

k(5k + 1) + (5k + 3) 2

3 + 8 + 13 + . . . + (5k + 3) =

k(5k + 1) 2(5k + 3) + 2 2

3 + 8 + 13 + . . . + (5k + 3) =

5k2 + 11k + 6) 2

3 + 8 + 13 + . . . + (5k + 3) =

(k + 1)(5k + 6) 2

We have shown that if we assume that Sk is true, and we add 5(k+1) - 2 to both sides of Sk, then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n.

18

Answer Key Testname: C10

?

(4 · 1)(1 + 1) 10) S1 : 1 · 4 = 2 ?

4 ·2 4 = 2 4=4 Sk: 1 · 4 + 2 · 4 + 3 · 4 + . . . + 4k =

4k(k + 1) 2

Sk+1 : 1 · 4 + 2 · 4 + 3 · 4 + . . . + 4(k + 1) =

4(k + 1)(k + 2) 2

We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely 4(k+1), to both sides."

1 · 4 + 2 · 4 + 3 · 4 + . . . + 4k + 4(k + 1) =

4k(k + 1) + 4(k + 1) 2

1 · 4 + 2 · 4 + 3 · 4 + . . . + 4(k + 1) =

4k(k + 1) 8(k + 1) + 2 2

1 · 4 + 2 · 4 + 3 · 4 + . . . + 4(k + 1) =

(k + 1)(4k + 8) 2

1 · 4 + 2 · 4 + 3 · 4 + . . . + 4(k + 1) =

4(k + 1)(k + 2) 2

We have shown that if we assume that Sk is true, and we add (4(k+1) to both sides of Sk , then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n.

11) Sk: 3 + 8 + 13 + . . . + (5k - 2) =

k(5k + 1) 2

Sk+1 : 3 + 8 + 13 + . . . + (5k + 3) =

(k + 1)(5k + 6) 2

?

1(1 + 1)(1 + 2) 12) S1 : 1 · 2 = 3 ?

1 ·2 ·3 2 = 3 2=2 ?

2(2 + 1)(2 + 2) S2 : 1 · 2 + 2 · 3 = 3 ?

2 ·3 ·4 8 = 3 8=8 ?

3(3 + 1)(3 + 2) S3 : 1 · 2 + 2 · 3 + 3 · 4 = 3 ?

3 ·4 ·5 20 = 3 20 = 20

19

Answer Key Testname: C10

13) B 14) B 15) A 16) B 17) B 18) C 19) D 20) D 21) D 22) B 23) C 24) D 25) B 26) B 27) A 28) C 29) C 30) B 31) C 32) C 33) A 34) C 35) B 36) A 37) A 38) B 39) D 40) B 41) D 42) C 43) D 44) D 45) B 46) A 47) C 48) C 49) C 50) D 51) B 52) D 53) B 54) D 55) B 56) C 57) A 58) C 59) B 60) B 61) C 62) D 20

Answer Key Testname: C10

63) C 64) B 65) B 66) B 67) C 68) C 69) D 70) C 71) C 72) C 73) A 74) D 75) D 76) B 77) B 78) B 79) B 80) C 81) A 82) B 83) D 84) C 85) C 86) D 87) C 88) C 89) C 90) B 91) D 92) C 93) A 94) C 95) A 96) D 97) D 98) C 99) C 100) A 101) D 102) B 103) D 104) D 105) C 106) D 107) C 108) A 109) A 110) C 111) C 112) C 21

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Choose the table which contains the best values of x for finding the requested limit of the given function. sin 2x 1) lim x x 0 -0.01 A) x f(x)

-0.02

-0.03

0.03

B) x f(x)

0.03

0.02

0.01

0.001 0.002 0.003

-0.03 C) x f(x)

-0.02

-0.01

0.01

0.02

0.03

D) x f(x)

-0.2

-0.1

0.1

0.2

0.3

-0.3

0.02

0.01

2) lim (x2 + 8x - 2) x 2

2)

A)

x -1.9 f(x)

-1.99 -1.999

2.001

2.01

2.1

B)

x -0.9 f(x)

-0.99 -0.999

1.001

1.01

1.1

C)

x f(x)

0.9

0.99

0.999

1.001

1.01

1.1

D)

x f(x)

1.9

1.99

1.999

2.001

2.01

2.1

x4 - 1 3) lim x 1 x-1

A)

x f(x)

B)

1)

3) 0.1

0.19

0.119

1.99

1.09

1.9

x -0.9 f(x)

-0.99 -0.999 -1.001

-1.01

-1.1

C)

x f(x)

0.9

0.99

0.999

1.001

1.01

1.1

D)

x f(x)

1.9

1.99

1.999

1.001

1.01

1.1

1

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

4) The figure shows the cost of renting a car, f(x), as a function of distance driven, x, in miles, for distances not exceeding 70 miles.

4)

49 47

Cost (dollars)

45 43 41 39 37

20

40 Distance (miles)

60

(i) Find lim f(x). x 60(ii) Find lim f(x). x 60+ (iii) What can you conclude about lim f(x)? How is this shown by the graph? x 60 (iv) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities?

5) The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.

0.11x if 0 < x 7593 759.30 + 0.18(x - 7593) if 7593 < x 31,687 + 5096.22 0.26(x 31,687) if 31,687 < x 72,784 T(x) = 15,781.44 + 0.29(x - 72,784) if 72,784 < x 149,897 38,144.21 + 0.32(x - 149,897) if 149,897 < x 325,127 94,217.81 + 0.36(x - 325,127) if x > 325,127 (i) Determine whether T is continuous at 7593. (ii) Determine whether T is continuous at 31,687. (iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

2

5)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Determine for what numbers, if any, the given function is discontinuous.

8x if x < 7 if x = 7 6) f(x) = 55 2 x + 7 if x > 7 A) 0, 7

B) 7

C) 8, 7

6) D) None

Graph the function. Then use your graph to find the indicated limit. lim f(x) 7) f(x) = 2x - 3, x -1

A) -2

B) -4

C) -1

7)

D) -5

A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not exist.

4x - 5 if x < 1 8) f(x) = 1 8) if x = 1 -3x - 4 if x > 1 a. lim f(x) b. lim f(x) c. lim f(x) x 1 x 1x 1+

A) a. -1 b. -7 c. -8

B) a. -7 b. -1 c. -8

C) a. -7 b. -1 c. does not exist

D) a. -1 b. -7 c. does not exist

Use the definition of continuity to determine whether f is continuous at a. x2 - 16 9) f(x) = x-4 a=0 A) Continuous

9)

B) Not continuous

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

10) lim (x - 4)( x - 2) 10) x 4

A) does not exist

B) 4

C) 0

3

D) 2

Graph the function. Then use your graph to find the indicated limit. x2 - 36 , lim f(x) 11) f(x) = x-6 x 6

A) does not exist

B) 1

11)

C) 12

D) 6

Use the definition of continuity to determine whether f is continuous at a.

5 12) f(x) = x-8 a=0 A) Continuous

12)

B) Not continuous

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. lim 2x - 7 13) 13) x 1 4x + 5

A) -

1 2

B) does not exist

C) -

5 9

Find the slope of the tangent line to the graph of f at the given point.

14) f(x) = x at (64, 8) 1 1 A) 16 B) C) 16 4

D) -

7 5

14) D) 4

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. lim x2 - 25 15) 15) x 5 x3 - 125

A) -

2 15

B)

1 15

C)

2 15

Find the slope of the tangent line to the graph of f at the given point.

16) f(x) = x2 + 11x - 15 at (1, -3) A) 11

B) -9

C) 26

4

D) does not exist

D) 13

16)

Use the definition of continuity to determine whether f is continuous at a. x+7 17) f(x) = x+2 a = -7 A) Not continuous

17)

B) Continuous

Find the derivative of f at x. That is, find f '(x).

18) f(x) = x; x = 36 1 A) 6 B) 6

C) 12

Complete the table for the function and find the indicated limit.

19) lim (x2 - 5) x 0

1 D) 12

18)

19)

x 0.1 -0.1 -0.01 -0.001 0.001 0.01 f(x) A) -4.9900; -4.9999; -5.0000; -5.0000; -4.9999; -4.9900 limit = -5.0 B) 1.4970; 1.4999; 1.5000; 1.5000; 1.4999; 1.4970 limit = 1.50 C) -1.4970; -1.4999; -1.5000; -1.5000; -1.4999; -1.4970 limit = -1.50 D) -2.9910; -2.9999; -3.0000; -3.0000; -2.9999; -2.9910 limit = -3.0

Use the definition of continuity to determine whether f is continuous at a.

20) f(x) = 3x4 - 9x3 + x - 5 a=5 A) Continuous

B) Not continuous

5

20)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(1) 21) a. lim f(x) 21) x 1

A) a. lim f(x) = 1 x 1

B) a. lim f(x) = -3 x 1

b. f(1) = -3 C) a. lim f(x) = -3 x 1

b. f(1) does not exist D) a. lim f(x) = -3 x 1

b. f(1) = 1

b. (1) = -3

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. x3 + 5x2 + 3x - 9 22) lim 22) x-1 x 1

A) 0

B) -16

C) does not exist

D) 16

Use the graph and the viewing rectangle shown below the graph to find the indicated limit.

23) lim sin x x

- , ,

A) lim sin x = 2 x

2

by [-2, 2, 1]

B) lim sin x = -1 x

C) lim sin x = 1 x

6

D) lim sin x = 0 x

23)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. x3 - 6x + 8 24) lim 24) x-2 x 0

A) does not exist

B) -4

C) 4

D) 0

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(-1/2) 25) a. lim f(x) 25) x -1/2

A) a.

B) a.

lim f(x) = 0 x -1/2

b. f(-1/2) does not exist C) a. lim f(x) does not exist x -1/2

lim f(x) = -2 x -1/2

b. f(-1/2) = -1 D) a. lim f(x) = -1 x -1/2

b. f(-1/2) = -1

b. f(-1/2) = -1

Determine for what numbers, if any, the given function is discontinuous.

26) f(x) = 2x + 5 A)

B) -5

C) None

5 D) 2

26)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

27) lim (x2 + 8x - 2) 27) x 2

A) does not exist

B) 18

C) 0

7

D) -18

Complete the table for the function and find the indicated limit. x2 - 49 28) lim x -7 x + 7 -7.1 -7.01 -7.001 -6.999 -6.99 x f(x) A) 7.1, 7.01, 7.001, 6.999, 6.99, 6.9 limit = 7 B) -14.1, -14.01, -14.001, -13.999, -13.99, -13.9 limit = -14 C) -14.1, -14.01, -14.001, -13.999, -13.99, -13.9 limit = -14 D) -7.1, -7.01, -7.001, -6.999, -6.99, -6.9 limit = -7

28)

-6.9

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(1) 29) a. lim f(x) 29) x 1-

A) a. lim f(x) = 1 x 1-

B) a. lim f(x) = 2 x 1-

b. f(1) = 2 C) a. lim f(x) does not exist x 1-

b. f(1) = 2 D) a. lim f(x) = 1 x 1-

b. f(1) = 2

b. f(1) = 1

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

3 30) lim (x2 - 2) 30) x 1

A) 1

B) -3

C) 3

8

D) -1

Solve the problem.

31) The function f(x) = x2 describes the area of a circle, f(x), in square meters, whose radius measures x meters. If x is changing, (i) Find the average rate of change of the area with respect to the radius as the radius changes from 8 meters to 8.1 meters. (ii) Find the average rate of change of the area with respect to the radius as the radius changes from 8 meters to 8.01 meters. (iii) Find the instantaneous rate of change of the area with respect to the radius at the moment when the radius is 8 meters.

Express all answers in terms of .

A) (i) 1296.1 square meters per meter; B) (i) 1296.1 square meters per meter; (ii) 12,816.01 square meters per meter; (ii) 12,816.01 square meters per meter; (iii) 16 square meters per meter (iii) 32 square meters per meter C) (i) 16.1 square meters per meter; D) (i) 16.1 square meters per meter; (ii) 16.01 square meters per meter; (ii) 16.01 square meters per meter; (iii) 16 square meters per meter (iii) 32 square meters per meter

31)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. x3 + 12x2 - 5x 32) lim 32) 5x x 0

A) does not exist

B) 5

C) 0

9

D) -1

Solve the problem.

33) You rent a moving van from a company that charges $10 per hour plus $0.50 per mile. The van is driven 40 miles in the first hour. The figure below shows the graph of the cost, f(x), in dollars, as a function of the miles, x, that you drive the van.

33)

(i) Find lim f(x). Interpret the limit, referring to miles driven and cost. x 20 (ii) For the first hour hour only, what is the rental cost approaching as the mileage gets closer to 40? (iii) What is the cost to rent the van at the start of the second hour?

A) (i) 40; the cost to rent the van for one hour and drive it 20 miles is $40. (ii) $30 (iii) $40 B) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20. (ii) $30 (iii) $40 C) (i) 30; the cost to rent the van for one hour and drive it 20 miles is $30. (ii) $40 (iii) $40 D) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20. (ii) $30 (iii) $30

Find the slope-intercept equation of the tangent line to the graph of f at the given point.

34) f(x) = 2x2 + x - 3 at x = (4, 33) A) y = 17x - 35

B) y = 4x + 3

C) y = 4x + 1

D) y = 2x - 3

34)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

4+x-2 35) lim 35) x x 0

A)

1 2

B) 4

C) 0

10

D)

1 4

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(-1) 36) a. lim f(x) 36) x -1

A) a. lim f(x) = -2 x -1

B) a. lim f(x) = 0 x -1

b. f(-1) = -1 C) a. lim f(x) does not exist x -1

b. f(-1) = -1 D) a. lim f(x) = -1 x -1

b. f(-1) = -1

b. f(-1) does not exist

Find the slope of the tangent line to the graph of f at the given point.

37) f(x) = 5x2 + x at (-4, 76) A) -14

B) -39

C) -41

Complete the table for the function and find the indicated limit. sin 2x 38) lim x x 0 -0.03 -0.02 -0.01 x 0.01 0.02 0.03 f(x) A) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 1 B) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = -2 C) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 0 D) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 2

11

D) 6

37)

38)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(1) 39) a. lim f(x) 39) x 1

A) a. lim f(x) = 1 x 1

B) a. lim f(x) = 2 x 1

b. f(1) = 2 C) a. lim f(x) = 1 x 1

b. f(1) = 2 D) a. lim f(x) does not exist x 1

b. f(1) = 0

b. f(1) = 2

Find the slope-intercept equation of the tangent line to the graph of f at the given point.

40) f(x) = x at (64, 8) 1 1 1 1 x+4 A) y = 16x + B) y = C) y = 4x + 4 D) y = x + 4 16 4 4

40)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

41) lim ( x - 2) 41) x 0

A) 2

B) does not exist

C) -2

Find the slope-intercept equation of the tangent line to the graph of f at the given point.

42) f(x) = x2 + 5x at (4, 36) A) y = 2x + 5

B) y = 2x - 5

C) y = 13x - 16

12

D) 0

D) y = 13x - 72

42)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(1) 43) a. lim f(x) 43) x 1

A) a. lim f(x) = 1 x 1

B) a. lim f(x) does not exist x 1

b. f(1) does not exist C) a. lim f(x) = 0 x 1

b. f(1) does not exist D) a. lim f(x) = -1 x 1

b. f(1) = 1

44) a.

lim f(x) x 2.5-

A) a.

b. f(1) = 1

44)

b. f(2.5)

B) a.

lim f(x) = 2 x 2.5-

b. f(2.5) = 1 C) a. lim f(x) does not exist x 2.5-

lim f(x) = 1 x 2.5-

b. f(2.5) = 1 D) a. lim f(x) = 0 x 2.5-

b. f(2.5) = 2

b. f(2.5) = 2

Use the definition of continuity to determine whether f is continuous at a.

1 45) f(x) = x2 - 3x a = -1 A) Not continuous

B) Continuous

13

45)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(-1) 46) a. lim f(x) 46) x -1 -

A) a.

B) a.

lim f(x) = -1 x -1 -

b. f(-1) = 1 C) a. lim f(x) does not exist x -1 -

b. f(-1) does not exist D) a. lim f(x) = 0 x -1 -

b. f(-1) does not exist

47) a. lim f(x) x 3-

lim f(x) = -1 x -1 -

b. f(-1) = 1

47)

b. f(3)

A) a. lim f(x) = 3 x 3-

B) a. lim f(x) = 3 x 3-

b. f(3) does not exist C) a. lim f(x) = 3 x 3-

b. f(3) = 3 D) a. lim f(x) does not exist x 3-

b. f(3) = 5

b. f(3) = 5

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

48) lim (3x2 - 19)2 48) x 1

A) 100

B) 256

C) -352

14

D) 144

Find the slope of the tangent line to the graph of f at the given point.

3 49) f(x) = at (1, 3) x A)

3 2

B) 3

C) -3

49) D) 6

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

50) a. lim f(x) 50) b. f(1) x 1

A) a. lim f(x) does not exist x 1

B) a. lim f(x) = 1 x 1

b. f(1) does not exist C) a. lim f(x) = 0 x 1

b. f(1) = 3 D) a. lim f(x) = 3 x 1

b. f(1) = 0

b. f(1) = 3

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. x-2 51) lim 51) x 4 x-4

A) 1/4

B) 0

C) does not exist

D) 4

Use the definition of continuity to determine whether f is continuous at a. x+7 52) f(x) = x+3 a = -3 A) Not continuous

52)

B) Continuous

Solve the problem.

53) An explosion causes debris to rise vertically with an initial velocity of 7 feet per second. The function s(t) = -16t2 + 112t describes the height of the debris above the ground, s(t), in feet, t seconds after the explosion. What is the instantaneous velocity of the debris when it hits the ground?

A) -112 feet per second B) -224 feet per second C) 224 feet per second D) 112 feet per second

15

53)

Graph the function. Then use your graph to find the indicated limit. x2 - 4 , lim f(x) 54) f(x) = x + 2 x -2

A) 1

B) -4

C) -2

Translate the given limit notation into a sentence. sin 2x =2 55) lim x x 0

54)

D) does not exist

55)

A) The limit of

sin 2x as x approaches 2 equals the number 0. x

B) The limit of

sin 2x as x approaches 0 from the right equals the number 2. x

C) The limit of

sin 2x as x approaches 0 equals the number 2. x

D) The limit of

sin 2x as x approaches 0 from the left equals the number 2. x

16

Use the graph and the viewing rectangle shown below the graph to find the indicated limit.

56) lim (x2 - 2) x -2

56)

[-6, 6, 1] by [-6, 6, 1]

A) lim (x2 - 2) = 6 x -2

B) lim (x2 - 2) = 2 x -2

C) lim (x2 - 2) = -6 x -2

D) lim (x2 - 2) = -2 x -2

Graph the function. Then use your graph to find the indicated limit.

57) f(x) = sin x - 2, lim f(x) x /2

A) -1

B)

C) 1

57)

D) -2

A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not exist.

2 58) f(x) = x - 1 if x < 0 58) if x 0 -4 a. lim f(x) b. lim f(x) c. lim f(x) x -5 x -5 x -5 +

A) a. -1 b. -4 c. does not exist

B) a. -4 b. -1 c. does not exist

C) a. -1 b. -4 c. -4

17

D) a. b. c.

Solve the problem.

59) The function f(x) = x3 describes the volume of a cube, f(x), in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 6 inches to 6.1 inches.

A) -109.81 cubic inches per inch B) 4429.81 cubic inches per inch C) 109.81 cubic inches per inch D) 73.83 cubic inches per inch

Determine for what numbers, if any, the given function is discontinuous.

2x - 1 60) f(x) = x2 - 25 A) -5 and 5 and -

1 2

60)

B) -5 and 5

C) None

D) 5

Graph the function. Then use your graph to find the indicated limit.

61) f(x) = 3 - x2 , lim f(x) x 2

A) 4

59)

B) -1

C) 3

61)

D)

Solve the problem.

62) An explosion causes debris to rise vertically with an initial velocity of 4 feet per second. The function s(t) = -16t2 + 64t describes the height of the debris above the ground, s(t), in feet, t seconds

62)

after the explosion. What is the instantaneous velocity of the debris 3.3 second(s) after the explosion?

A) feet per second B) -105.6 feet per second C) 105.6 feet per second D) -41.6 feet per second

Find the slope-intercept equation of the tangent line to the graph of f at the given point.

63) f(x) = x2 + 11x - 15 at (1, -3) A) y = 11x

B) y = 13x - 16

C) y = 11x + 15

18

D) y = 2x + 11

63)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(1) 64) a. lim f(x) 64) x 1

A) a. lim f(x) does not exist x 1

B) a. lim f(x) = 3 x 1

b. f(1) = -3 C) a. lim f(x) = 1 x 1

b. f(1) = 3 D) a. lim f(x) = -3 x 1

b. f(1) = 1

b. f(1) = -3

Complete the table for the function and find the indicated limit.

65) lim (x2 + 8x - 2) x 2

65)

x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) A) 6.810; 7.880; 7.988; 8.012; 8.120; 9.210 limit = 8.0 B) 16.692; 17.592; 17.689; 17.710; 17.808; 18.789 limit = 17.70 C) 5.043; 5.364; 5.396; 5.404; 5.436; 5.763 limit = 5.40 D) 16.810; 17.880; 17.988; 18.012; 18.120; 19.210 limit = 18.0

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. x-7 66) lim 66) - 49 x x 7

A) does not exist

B)

1 7 6 42

C)

19

7 1 42 6

D) 0

A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not exist.

1 if x > 5 x -5 67) f(x) = 67) 2 x - 2x if x 5 a. lim f(x) x 5-

b. lim f(x) x 5+

c. lim f(x) x 5

A) a. does not exist b. 15 c. 15 C) a. 15 b. does not exist c. 15

B) a. does not exist b. 15 c. does not exist D) a. 15 b. does not exist c. does not exist

Determine for what numbers, if any, the given function is discontinuous. x- 3 if x 3 68) f(x) = x2 if x > 3 A) [the set of numbers 1, 2, 3, . . . 8] C) 3

B) [the set of numbers 1, 2, 3, . . . 9] D) None

Find the derivative of f at x. That is, find f '(x).

69) f(x) = x2 + 5x + 20; x = 8 A)

68)

B) 16

C) 7

D) 21

Solve the problem.

70) You are building a screened-in porch attached to your house. Because the house will be used for one side of the enclosure, only three sides will need to be enclosed. You have 24 yards of material to enclose the three walls. The function f(x) = x(24 - 2x) describes the area of the screened-in porch that you can enclose, f(x), in square yards, if the width of the screened-in porch is x yards X 5.7 5.8 5.9 6 6.1 6.2 6.3

Y1 71.82 71.92 71.98 71.98 71.92 71.82

Y 1 = X * (24 - 2X) (i) Use the table shown to find lim f(x). x 6 (ii) Use the graph of f(x) = x(24- 2x) shown to find lim f(x). Do you get the same limit as you did x 6 in part (i)? What information about the limit is shown by the graph that might not be obvious from the table?

20

69)

70)

[-5, 15, 5] by [0, 80, 10] A) (i) 72 (ii) 80; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of material that can be used.

B) (i) 72 (ii) 72; Yes, the same limit is obtained. The graph shows that the limit is the maximum area.

C) (i) 80 (ii) 80; Yes, the same limit is obtained. The graph shows that the limit is the maximum area.

D) (i) 80 (ii) 72; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of material that can be used.

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

71) a. lim f(x) 71) b. f(0) x 0

A) a. lim f(x) = 1 x 0

B) a. lim f(x) = 0 x 0

b. f(0) = 1 C) a. lim f(x) does not exist x 0

b. f(0) = 0 D) a. lim f(x) = -1 x 0

b. f(0) does not exist

b. f(0) = -1

21

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

1 1 lim x 9 72) 72) x 9 x-9

A) does not exist

B)

1 9

Find the derivative of f at x. That is, find f '(x).

73) f(x) = -4x + 20; x = 4 A) -16 B) 20

D) -

C) 80

D) -4

Find the slope of the tangent line to the graph of f at the given point.

74) f(x) = -4x2 + 7x at (5, -65) A) 33

B) 3

1 81

C) 0

C) -33

D) -13

73)

74)

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

75) lim -6 75) x 10

A) -6

B) 0

C) 10

D)

Use the definition of continuity to determine whether f is continuous at a.

7 76) f(x) = 2 x + 8x a=0 A) Continuous

76)

B) Not continuous

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

3x2 + 7x - 2 77) lim 77) x 1 3x2 - 4x + 2

A) does not exist

B) 8

C) -

7 4

D) 0

Solve the problem.

78) A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 32 feet per second. The function s(t) = -16t2 + 32t describes the ball's height above the ground, s(t), in feet, t seconds after it was hit. What is the instantaneous velocity of the ball 1.1 seconds after it was hit?

A) 35.2 feet per second B) feet per second C) -35.2 feet per second D) -3.2 feet per second

22

78)

Graph the function. Then use your graph to find the indicated limit.

79) f(x) = 5x , lim f(x) x -3

A) 15

B) 3

C) -3

79)

D) -15

Translate the given limit notation into a sentence.

80) lim ( x - 2) = 0 x 4 A) The limit of B) The limit of C) The limit of D) The limit of

80)

x - 2 as x approaches 4 equals the number 0. x - 4 as x approaches 0 equals the number 2. x - 2 as x approaches 0 equals the number 4. x - 4 as x approaches 2 equals the number 0.

Find the slope of the tangent line to the graph of f at the given point.

81) f(x) = 2x2 + x - 3 at (4, 33) A) 15

B) 5

C) 19

23

D) 17

81)

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

82) a. lim f(x) 82) b. f(0) x 0

A) a. lim f(x) does not exist x 0

B) a. lim f(x) = 1 x 0

b. f(0) does not exist C) a. lim f(x) = 0 x 0

b. f(0) does not exist D) a. lim f(x) = -1 x 0

b. f(0) = 0

b. f(0) does not exist

Use the definition of continuity to determine whether f is continuous at a. x2 - 64 83) f(x) = x-8 a=8 A) Continuous

84) f(x) =

B) Not continuous

x-6 x+3

84)

a=0 A) Continuous

85) f(x) =

83)

1 , x+4

B) Not continuous if x > -4

85)

x2 - 4x,

if x -4 a = -4 A) Not continuous

B) Continuous

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

86) lim 15x3 86) x -7

A) -5145

B)

C) -343

24

D) -7

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

87) a. lim f(x) 87) b. f(0) x 0

A) a. lim f(x) = 0 x 0

B) a. lim f(x) does not exist x 0

b. f(0) = 6 C) a. lim f(x) = 6 x 0

b. f(0) = 6 D) a. lim f(x) = 6 x 0

b. f(0) = 6

b. f(0) = 0

Find the slope of the tangent line to the graph of f at the given point.

88) f(x) = x2 + 5x at (4, 36) A) 9

B) 13

C) 21

D) 3

Determine for what numbers, if any, the given function is discontinuous.

89) f(x) = -3x2 + 5x A) -

5 6

B) None

C) -3

D)

5 6

Complete the table for the function and find the indicated limit. x4 - 1 90) lim x 1 x-1 x 0.9 0.99 0.999 1.001 1.01 f(x) A) 3.439; 3.940; 3.994; 4.006; 4.060; 4.641 limit = 4.0 C) 1.032; 1.182; 1.198; 1.201; 1.218; 1.392 limit = 1.210

6

if x = 3

89)

90)

1.1

B) 7.439; 7.940; 7.994; 8.006; 8.060; 8.641 limit = 8.0 D) 4.595; 5.046; 5.095; 5.105; 5.154; 5.677 limit = 5.10

Determine for what numbers, if any, the given function is discontinuous. x2 - 9 if x 3 91) f(x) = x - 3 A) 3

88)

B) 3, 9

C) 9

25

91) D) None

Graph the function. Then use your graph to find the indicated limit. x < -7 , lim f(x) 92) f(x) = x + 7 7-x x -7 x -7

A) 14

B) does not exist

C) -7

92)

D)

Complete the table for the function and find the indicated limit. x2 93) lim x 0 sin x

93)

-0.03 -0.02 -0.01 x 0.01 0.02 0.03 f(x) A) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = -1 B) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 1 C) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 0.1 D) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 0

Find the slope-intercept equation of the tangent line to the graph of f at the given point.

6 94) f(x) = at (1, 6) x A) y = 12x - 6

B) y = 6x + 12

C) y = -6x + 12

26

94) D) y = 3x - 6

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. b. f(-2) 95) a. lim f(x) 95) x -2 +

A) a.

B) a.

lim f(x) does not exist x -2 +

b. f(-2) = -1 C) a. lim f(x) = -1 x -2 +

b. f(-2) = -2 D) a. lim f(x) = -2 x -2 +

b. f(-2) does not exist

96) a. lim f(x) x 1+

lim f(x) = -2 x -2 +

b. f(-2) = -1

96)

b. f(1)

A) a. lim f(x) does not exist x 1+

B) a. lim f(x) = 1 x 1+

b. f(1) = 2 C) a. lim f(x) = 2 x 1+

b. f(1) = 0 D) a. lim f(x) = 2 x 1+

b. f(1) = 1

b. f(1) = 2

27

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. (x + h)3 - x3 97) lim 97) h h 0

A) does not exist

B) 3x2

C) 3x2 + 3xh + h 2

D) 0

2x - 7 98) lim x 1 4x + 5

A) -

99)

98)

1 2

B) -

5 9

C) -

7 5

D) does not exist

lim [(x - 3)3 (5x + 1)2 ] x 2

A) -288

99) B) -121

C)

Determine for what numbers, if any, the given function is discontinuous.

100) f(x) = -3 cos x A) -3 B) 0 C) None 101) f(x) =

D) does not exist

D) 1

x- 4 if x 4 x2 -8 if x > 4

A) None

100)

101) B) 0

C) -4, 4

D) 4

Use the definition of continuity to determine whether f is continuous at a.

102) f(x) = 5x4 - 9x3 + x - 6 a=0 A) Not continuous

102)

B) Continuous

A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not exist. x2 - 9 if x 3 103) f(x) = x - 3 103) 5 a. lim f(x) x 3-

A) a. 6 b. 6 c. 5

if x = 3 b. lim f(x) x 3+

c. lim f(x) x 3

B) a. 3 b. 3 c. 5

C) a. 3 b. 3 c. does not exist

28

D) a. 6 b. 6 c. 6

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

104) a. lim f(x) 104) b. f(3) x 3

A) a. lim f(x) does not exist x 3

B) a. lim f(x) = 3 x 3

b. f(3) = 5 C) a. lim f(x) = 4 x 3

b. f(3) = 5 D) a. lim f(x) = 5 x 3

b. f(3) does not exist

105) a. lim f(x) x 2

b. f(3) = 5

105)

b. f(2)

A) a. lim f(x) = 2 x 2

B) a. lim f(x) = 1 x 2

b. f(2) = 1 C) a. lim f(x) does not exist x 2

b. f(2) = 1 D) a. lim f(x) = 1 x 2

b. f(2) = 1

b. f(2) = 2

29

Answer Key Testname: C11

1) C 2) D 3) C 4) (i) 45 (ii) 49 (iii) lim f(x) does not exist; the graph shows a discontinuity at x = 60 x 60 (iv) The graph jumps at its discontinuities due to per-mile charges.

5) (i) continuous (ii) continuous (iii) Answers may vary. For instance if T(x) were partly defined as follows: 0.11x if 0 < x 7593 10,000 if 7593 < x 31,687 T(x) = . . . . . . . . . then there would be a discontinuity at 7593 and a large jump in tax owed for an amount just slightly larger than $7593 in taxable income.

6) B 7) D 8) D 9) A 10) C 11) C 12) A 13) C 14) B 15) C 16) D 17) B 18) D 19) A 20) A 21) C 22) D 23) D 24) B 25) D 26) C 27) B 28) C 29) A 30) D 31) C 32) D 33) B 34) A 35) D 36) C 37) B

30

Answer Key Testname: C11

38) D 39) D 40) B 41) C 42) C 43) A 44) B 45) B 46) B 47) C 48) B 49) C 50) D 51) A 52) A 53) A 54) B 55) C 56) B 57) A 58) D 59) C 60) B 61) B 62) D 63) B 64) D 65) D 66) B 67) D 68) C 69) D 70) B 71) A 72) D 73) D 74) C 75) A 76) B 77) B 78) D 79) A 80) A 81) D 82) B 83) B 84) A 85) A 86) A 87) A 31

Answer Key Testname: C11

88) B 89) B 90) A 91) D 92) B 93) D 94) C 95) B 96) D 97) B 98) B 99) B 100) C 101) D 102) B 103) D 104) A 105) A

32