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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
Objective: (1.3) Understand and Use Piecewise Functions
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
Objective: (1.3) Understand and Use Piecewise Functions
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
Objective: (1.6) Graph Functions Involving a Sequence of Transformations
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
Objective: (1.6) Graph Functions Involving a Sequence of Transformations
9) h(x) = -x + 1 - 2
Objective: (1.6) Graph Functions Involving a Sequence of Transformations
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
Objective: (1.6) Graph Functions Involving a Sequence of Transformations
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
Objective: (1.6) Graph Functions Involving a Sequence of Transformations
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
Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions
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
Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions
46) f(x) = x4 - 64x2
Objective: (2.3) Recognize Characteristics of Graphs of Polynomial Functions
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
Objective: (2.3) Graph Polynomial Functions
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)
Objective: (2.3) Graph Polynomial Functions
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
Objective: (2.7) Solve Rational Inequalities
83) 4xx + 6
< x
Objective: (2.7) Solve Rational Inequalities
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
Objective: (3.2) Find the Domain of a Logarithmic Function
16
94) f(x) = ln 1x + 10
Objective: (3.2) Find the Domain of a Logarithmic Function
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
Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations
17
106) log8
(x2 - 7x) = 1
Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations
107) log6
x + log6
(x - 35) = 2
Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations
108) log3 (x + 3) - log3 (x - 5) = 3
Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations
109) log 3 (x + 6) + log 3 (x - 6) - log 3 x = 2
Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations
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
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
Answer KeyTestname: 18-19 EXAM #1 REVIEW
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
Answer KeyTestname: 18-19 EXAM #1 REVIEW
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
Answer KeyTestname: 18-19 EXAM #1 REVIEW
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
Answer KeyTestname: 18-19 EXAM #1 REVIEW
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
Answer KeyTestname: 18-19 EXAM #1 REVIEW
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