MAC1147 Exam#1 Name L

MAC1147 Exam#1

Name___________________________________

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

Graph the function.

1) f(x) = -x + 3 if x < 22x - 3 if x 2

Objective: (1.3) Understand and Use Piecewise Functions

2) f(x) =x + 5 if -7 x < 2-6 if x = 2-x + 4 if x > 2


1

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

3) f(x) =4 if -5 x < -3|x| if -3 x < 8

x if 8 x 14


Find and simplify the difference quotient f(x + h) - f(x)h

, h 0 for the given function.

4) f(x) = 16x

Objective: (1.3) Find and Simplify a Function's Difference Quotient

5) f(x) = x2 + 4x - 8Objective: (1.3) Find and Simplify a Function's Difference Quotient

Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the givenfunction.

6) h(x) = (x + 6)2 - 6

Objective: (1.6) Graph Functions Involving a Sequence of Transformations

2

7) g(x) = -12

(x + 7)2 + 3


Begin by graphing the standard square root function f(x) = x . Then use transformations of this graph to graph the givenfunction.

8) g(x) = - x + 1 - 2


9) h(x) = -x + 1 - 2


3

Begin by graphing the standard absolute value function f(x) = x . Then use transformations of this graph to graph thegiven function.

10) g(x) = 13

x + 5 - 5


Add Note HereBegin by graphing the standard cubic function f(x) = x3. Then use transformations of this graph to graph the givenfunction.

11) g(x) = -(x - 5)3 - 2


Find the domain of the composite function f g.

12) f(x) = x + 2, g(x) = 8x + 3

Objective: (1.7) Determine Domains for Composite Functions

13) f(x) = 3x + 8

, g(x) = 48x

Objective: (1.7) Determine Domains for Composite Functions

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

14) f(x) = 7x + 43

Objective: (1.8) Find the Inverse of a Function

4

15) f(x) = 54x + 7

Objective: (1.8) Find the Inverse of a Function

16) f(x) =3

x - 5Objective: (1.8) Find the Inverse of a Function

Complete the square and write the equation in standard form. Then give the center and radius of the circle.17) x2 + y2 - 8x - 10y = -5

Objective: (1.9) Convert the General Form of a Circle's Equation to Standard Form

18) x2 + y2 - 10x - 6y + 20 = 0Objective: (1.9) Convert the General Form of a Circle's Equation to Standard Form

Solve the problem.19) An open box is made from a square piece of sheet metal 21 inches on a side by cutting identical squares from the

corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of thesquare cut from each corner, x.Objective: (1.10) Construct Functions from Formulas

20) A kennel owner has 1600 feet of fencing to enclose a rectangular dog exercise pen. Express the area of theexercise pen, A, as a function of one of its dimensions, x.Objective: (1.10) Construct Functions from Formulas

21) Two apartment tenants have a total of 400 feet of fencing to enclose a rectangular garden and subdivide intotwo smaller gardens, one for each of them, by placing the fencing parallel to one of the sides. Express the area ofthe entire garden, A, as a function of x.

Objective: (1.10) Construct Functions from Formulas

5

22) The figure shows a rectangle with two vertices on a semicircle of radius 8 and two vertices on the x-axis. LetP(x, y) be the vertex that lies in the first quadrant. Express the perimeter of the rectangle, P, as a function of x.

y = 64 - x2

-8 8

Objective: (1.10) Construct Functions from Formulas

Find the product and write the result in standard form.23) (9 + 2i)2

Objective: (2.1) Multiply Complex Numbers

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in acircuit, I, in amperes, the voltage of the circuit, E, in volts, and the resistance of the circuit, R, in ohms, by the formulaE = IR. Solve the problem using this formula.

24) Find E, the voltage of a circuit, if I = (9 + 7i) amperes and R = (3 + 4i) ohms.Objective: (2.1) Multiply Complex Numbers

Divide and express the result in standard form.

25) 1 + 3i5 + 4i

Objective: (2.1) Divide Complex Numbers

26) 5 + 2i2 + 4i

Objective: (2.1) Divide Complex Numbers

Perform the indicated operations and write the result in standard form.27) ( 10 - - 49)( 10 + - 49)

Objective: (2.1) Perform Operations with Square Roots of Negative Numbers

28) -36 - -1806

Objective: (2.1) Perform Operations with Square Roots of Negative Numbers

Solve the quadratic equation using the quadratic formula. Express the solution in standard form.29) 7x2 = 5x - 2

Objective: (2.1) Solve Quadratic Equations with Complex Imaginary Solutions

6

Use the vertex and intercepts to sketch the graph of the quadratic function.30) f(x) = -2x2 - 24x - 73

Objective: (2.2) Graph Parabolas

Solve the problem.31) You have 208 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize

the enclosed area.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

32) A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has360 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

33) You have 328 feet of fencing to enclose a rectangular region. What is the maximum area?Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

34) You have 112 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the sidealong the river, find the length and width of the plot that will maximize the area.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

35) A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form rightangles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatestamount of water to flow.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

36) A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of theplayground. 576 feet of fencing is used. Find the dimensions of the playground that maximize the totalenclosed area.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

37) A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of theplayground. 624 feet of fencing is used. Find the maximum area of the playground.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

38) The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the functionC(x) = 4x2 - 16x + 36. Find the number of automobiles that must be produced to minimize the cost.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

7

39) The owner of a video store has determined that the profits P of the store are approximately given byP(x) = -x2 + 80x + 60, where x is the number of videos rented daily. Find the maximum profit to the nearestdollar.Objective: (2.2) Solve Problems Involving a Quadratic Function's Minimum or Maximum Value

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

40) f(x) = 7x2 - x3

Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions

41) x5 - 6x3 + 5x = 0Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions

42) f(x) = (x - 3)2(x2 - 16)Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions

Find the y-intercept of the polynomial function.43) f(x) = (x + 1)(x - 4)(x - 1)2


44) f(x) = (x - 2)2(x2 - 9)Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither.45) f(x) = 3x2 - x3


46) f(x) = x4 - 64x2


47) f(x) = x3 - 3xObjective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions

Solve the problem.48) A herd of bison is introduced to a wildlife refuge. The number of bison, N(t), after t years is described by the

polynomial function N(t) = -t4 + 19t + 80. Use the Leading Coefficient Test to determine the graph's endbehavior. What does this mean about what will eventually happen to the bison population?Objective: (2.3) Determine End Behavior

Find the zeros of the polynomial function.49) f(x) = x3 - 5x2 - 9x + 45

Objective: (2.3) Use Factoring to Find Zeros of Polynomial Functions

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

Objective: (2.3) Use Factoring to Find Zeros of Polynomial Functions

8

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

51) f(x) = 2(x + 5)(x + 3)4

Objective: (2.3) Identify Zeros and Their Multiplicities

52) f(x) = x3 + 5x2 - x - 5Objective: (2.3) Identify Zeros and Their Multiplicities

Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and makethe degree of the function as small as possible.

53) Crosses the x-axis at -4, 0, and 2; lies above the x-axis between -4 and 0; lies below the x-axis between 0 and 2.Objective: (2.3) Identify Zeros and Their Multiplicities

54) Touches the x-axis at 0 and crosses the x-axis at 2; lies below the x-axis between 0 and 2.Objective: (2.3) Identify Zeros and Their Multiplicities

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the givenintegers.

55) f(x) = 9x3 + 9x2 + 5x + 1; between -1 and 0Objective: (2.3) Use the Intermediate Value Theorem

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

56) g(x) = 43

x + 3

Objective: (2.3) Understand the Relationship Between Degree and Turning Points

57) f(x) = (7x + 4)5( x5 - 7)(x - 5)Objective: (2.3) Understand the Relationship Between Degree and Turning Points

Graph the polynomial function.58) f(x) = x3 + 9x2 - x - 9

Objective: (2.3) Graph Polynomial Functions

9

59) f(x) = 6x3 - 6x - x5


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 eachintercept.(c) Find the y-intercept.(d) Graph the function.

60) f(x) = x2(x + 2)


Divide using long division.61) (-5x5 - x3 - 2x2 + 260x + 14) ÷ (x2 - 7)

Objective: (2.4) Use Long Division to Divide Polynomials

Divide using synthetic division.

62) 2x2 - 19x + 35x - 7

Objective: (2.4) Use Synthetic Division to Divide Polynomials

Use synthetic division and the Remainder Theorem to find the indicated function value.63) f(x) = x4 + 7x3 - 5x2 + 7x + 9; f(-2)

Objective: (2.4) Evaluate a Polynomial Using the Remainder Theorem

10

Use synthetic division to show that the number given to the right of the equation is a solution of the equation, then solvethe polynomial equation.

64) 2x3 + 5x2 - 13x - 30 = 0; -3Objective: (2.4) Use the Factor Theorem to Solve a Polynomial Equation

Use the Rational Zero Theorem to list all possible rational zeros for the given function.65) f(x) = 3x4 + 3x3 - 5x2 + 2x - 12

Objective: (2.5) Use the Rational Zero Theorem to Find Possible Rational Zeros

Find a rational zero of the polynomial function and use it to find all the zeros of the function.66) f(x) = 2x4 + 11x3 + 39x2 + 43x + 13

Objective: (2.5) Find Zeros of a Polynomial Function

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.67) x3 + 4x2 - x - 4 = 0

Objective: (2.5) Solve Polynomial Equations

Find the domain of the rational function.

68) h(x) = x + 8x2 + 1

Objective: (2.6) Find the Domains of Rational Functions

69) h(x) = x + 4x2 - 36x

Objective: (2.6) Find the Domains of Rational Functions

Use the graph of the rational function shown to complete the statement.70)

As x -2-, f(x) ?Objective: (2.6) Use Arrow Notation

11

71)

As x 2+, f(x) ?Objective: (2.6) Use Arrow Notation

72)

As x - , f(x) ?Objective: (2.6) Use Arrow Notation

73)

As x 0+, f(x) ?Objective: (2.6) Use Arrow Notation

12

Use transformations of f(x) = 1x

or f(x) = 1x2

to graph the rational function.

74) f(x) = 1x2

- 4

Objective: (2.6) Use Transformations to Graph Rational Functions

Graph the rational function.

75) f(x) = 3xx2 - 1

Objective: (2.6) Graph Rational Functions

13

76) f(x) = x2 - 2x - 8x2 + 4

Objective: (2.6) Graph Rational Functions

Find the slant asymptote, if any, of the graph of the rational function.

77) f(x) = x2 - 5x + 2x + 3

Objective: (2.6) Identify Slant Asymptotes

78) g(x) = x3 - 9x2 + 6x

Objective: (2.6) Identify Slant Asymptotes

Solve the problem.79) A company that produces inflatable rafts has costs given by the function C(x) = 30x + 25,000 , where x is the

number of inflatable rafts manufactured and C(x) is measured in dollars. The average cost to manufacture eachinflatable raft is given by

_C (x) = 30x + 25,000

x.

Find _C (250). (Round to the nearest dollar, if necessary.)

Objective: (2.6) Solve Applied Problems Involving Rational Functions

80) A drug is injected into a patient and the concentration of the drug is monitored. The drug's concentration, C(t),in milligrams after t hours is modeled by

C(t) = 8t3t2 + 1

.

What is the horizontal asymptote for this function? Describe what this means in practical terms.Objective: (2.6) Solve Applied Problems Involving Rational Functions

14

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

81) x + 7x + 4

< 4

Objective: (2.7) Solve Rational Inequalities

82) xx + 2

2


83) 4xx + 6

< x


Solve the problem.

84) The average cost per unit, y, of producing x units of a product is modeled by y = 450,000 + 0.25xx

. Describe the

company's production level so that the average cost of producing each unit does not exceed $1.75.Objective: (2.7) Solve Problems Modeled by Polynomial or Rational Inequalities

Graph the function.85) Use the graph of f(x) = 3x to obtain the graph of g(x) = 3x - 1.

Objective: (3.1) Graph Exponential Functions

15

86) Use the graph of f(x) = 4x to obtain the graph of g(x) = 4x - 2 - 2.

Objective: (3.1) Graph Exponential Functions

Solve the problem.87) The size of the raccoon population at a national park increases at the rate of 4.5% per year. If the size of the

current population is 106, find how many raccoons there should be in 6 years. Use the function f(x) = 106e0.045tand round to the nearest whole number.Objective: (3.1) Evaluate Functions with Base e

88) The population in a particular country is growing at the rate of 1.5% per year. If 2,546,000 people lived there in1999, how many will there be in the year 2005? Use f(x) = y0e0.015t and round to the nearest ten-thousand.

Objective: (3.1) Evaluate Functions with Base e

Write the equation in its equivalent logarithmic form.89) 123 = y

Objective: (3.2) Change From Exponential to Logarithmic Form

90) 5x = 25Objective: (3.2) Change From Exponential to Logarithmic Form

Find the domain of the logarithmic function.91) f(x) = log 8 (x - 7)2

Objective: (3.2) Find the Domain of a Logarithmic Function

92) f(x) = log (x2 - 12x + 32)Objective: (3.2) Find the Domain of a Logarithmic Function

93) f(x) = log x + 4x - 9


16

94) f(x) = ln 1x + 10


Evaluate or simplify the expression without using a calculator.

95) 7 10log 3.9

Objective: (3.2) Use Common Logarithms

96) 10log 5

x

Objective: (3.2) Use Common Logarithms

97) ln 6

eObjective: (3.2) Use Natural Logarithms

Evaluate the expression without using a calculator.98) eln 239

Objective: (3.2) Use Natural Logarithms

99) eln 15x5

Objective: (3.2) Use Natural Logarithms

Solve the problem.100) The long jump record, in feet, at a particular school can be modeled by f(x) = 18.5 + 2.5 ln (x + 1) where x is the

number of years since records began to be kept at the school. What is the record for the long jump 20 years afterrecord started being kept? Round your answer to the nearest tenth.Objective: (3.2) Use Natural Logarithms

Solve the exponential equation. Express the solution set in terms of natural logarithms.

101) 2 x + 8= 7

Objective: (3.4) Use Logarithms to Solve Exponential Equations

102) 4x + 4 = 52x + 5

Objective: (3.4) Use Logarithms to Solve Exponential Equations

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

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

Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations

104) 7 + 9 ln x = 9Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations

105) log2

11 + log2

x = 1


17

106) log8

(x2 - 7x) = 1


107) log6

x + log6

(x - 35) = 2


108) log3 (x + 3) - log3 (x - 5) = 3


109) log 3 (x + 6) + log 3 (x - 6) - log 3 x = 2


Solve the problem.110) Larry has $2500 to invest and needs $3100 in 13 years. What annual rate of return will he need to get in order to

accomplish his goal, if interest is compounded continuously? (Round your answer to two decimals.)Objective: (3.4) Solve Applied Problems Involving Exponential and Logarithmic Equations

18

Answer KeyTestname: 18-19 EXAM #1 REVIEW

1)

2)

3) [0, 8)

4) -16x (x + h)

5) 2x + h + 46)

19


7)

8)

9)

10)

20


11)

12) (- , -3) or (-3, )13) (- , -6) or (-6, 0) or (0, )

14) f-1(x) = 3x - 47

15) f-1(x) = 54x

- 74

16) f-1(x) = x3 + 517) (x - 4)2 + (y - 5)2 = 36

(4, 5), r = 618) (x - 5)2 +(y - 3)2 = 14

(5, 3), r = 1419) V(x) = x(21 - 2x)220) A(x) = x(800 - x)

21) A(x) = x 400 - 3x2

22) P(x) = 4x + 2 64 - x223) 77 + 36i24) (-1 + 57i) volts

25) 1741

+ 1141 i

26) 910

- 45 i

27) 5928) -6 - i 5

29) 514

± i 3114

21


30)

31) 52 ft by 52 ft32) 16,200 ft233) 6724 square feet34) length: 56 feet, width: 28 feet35) 4.5 inches36) 96 ft by 144 ft37) 16,224 ft238) 2 thousand automobiles39) $166040) 0, touches the x-axis and turns around;

7, crosses the x-axis41) 0, crosses the x-axis;

1, crosses the x-axis;-1, crosses the x-axis;

5, crosses the x-axis;- 5, crosses the x-axis

42) 3, touches the x-axis and turns around;-4, crosses the x-axis;4, crosses the x-axis

43) -444) -3645) neither46) y-axis symmetry47) origin symmetry48) The bison population in the refuge will die out.49) x = 5, x = -3, x = 350) x = -1, x = -7,51) -5, multiplicity 1, crosses x-axis; -3, multiplicity 4, touches x-axis and turns around52) -1, multiplicity 1, crosses the x-axis;

1, multiplicity 1, crosses the x-axis; - 5, multiplicity 1, crosses the x-axis.

53) f(x) = x3+ 2x2 - 8x54) f(x) = x3 - 2x255) f(-1) = -4 and f(0) = 1; yes56) 0

22


57) 10

58)59)

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

61) -5x3 - 36x - 2 +8x

x2 - 762) 2x - 563) -65

23


64) 52

, -2, -3

65) ±1, ± 2, ± 3, ± 4, ± 6, ± 12, ± 13

, ± 23

, ± 43

66) {-1, - 12 , -2 + 3i, -2 - 3i}

67) {1, -1, -4}68) all real numbers69) {x|x 0, x 36}70) -71) -72) 173) +74)

75)

24


76)

77) y = x - 878) y = x - 679) $13080) y = 0; 0 is the final amount, in milligrams, of the drug that will be left in the patient's bloodstream.81) (- , -4) or (- 3, )

82) [-4, -2)

83) (-6, -2) (0, )

84) At least 300,000 units85)

25


86)

87) 13988) 2,790,00089) log 12 y = 3

90) log 5 25 = x

91) (- , 7) or (7, )92) (- , 4) (8, )93) (- , -4) (9, )94) (-10, )95) 27.396) x1/5

97) 16

98) 23999) 15x5

100) 26.1 feet

101) ln 7ln 2

- 8

102) 5 ln 5 - 4 ln 4ln 4 - 2 ln 5

103) - 14336

104) e 2/9

105) { 211

}

106) {8, -1}107) {36}

108) { 6913

}

109) {12}110) 1.65%

26

Documents

MAC1147 Exam#1 Name L