MATH Final Eam Study Questions Spring 08 Note: Certain eam questions have been more challenging for students. Questions marked (***) are similar to those challenging eam questions.. A company produces a pair of skates for $.5 and sells each pair for $89.95. If the fied costs are $7.7, how many pairs must the company produce and sell in order to break even? Between 9 and Between and 7 Between 7 and Between and 5 (E) More than 5. Determine which one the following represents y as a function of. 0 y 0 y 5 0 y 5. (***) The concentration of a certain drug, in mg/l, can be modeled by the following 0.8t function: C t 5t e, where t is the number of hours after the drug was administered. The drug is effective if its concentration is greater than. mg/l. Use the graph of the function to determine the interval of time when the drug is effective. 0, 0..6, 0, 0.,.6 (E) 0., Page
. Determine which one of the following represents y as a function of. 5 8 6-7 -6-5 - - - - - 5 6 7 - - - -5-6 -7-6 - - 6 - - -6 5. Solve each of the following equations for y and graph the formula(s) on your graphing calculator. Which one of these equations represents y as a function of? y 9 y 0 y 5 5 y (E) y y 7 6. (***) Use the function f 5 to evaluate f f h h. h 9 56 h h h 9 56 h h h 9 h 56 (E) None of these R. 7. Determine the domain of the function 5 0.,, 0. 0.,, 0. (E) 0, Page
8. (***) Which one of the following functions has domain all real numbers ecept and? f f f f f (E) 9. Find the domain and range of the function shown in the graph below: Domain: Domain: Domain: Domain: (E) Domain:, Range: 5, 5, Range:,, Range: 5, 5, 0-5 - - - - - 0 5 - - - -5-6 Range:,, Range: 5, 5 Page
0. Identify the interval(s) on which the function below is increasing. 6 5 0-5 - - - - - 0 5 6 - -,,,.,, (E) 0,. Use the graph of y f given below to determine the interval on which f negative., 0, (E), 5 0-5 - - - - 0 5 - - - - -5,, 0 is Page
. (***) Determine the zeros of p 6. (E) only and, 0 0 only, and only 0 only and The following table shows the number of Starbucks locations worldwide for certain years: Year Number of Starbucks locations 0 8,066 0,66 05,06 Use this table to answer the net two questions.. What is the slope of the line that passes through these points, and what does it mean in practical terms? m 00. The number of Starbucks locations worldwide increases by 00 per year. m 00. The number of Starbucks locations worldwide increases by 00 for years. m 950. The number of Starbucks locations worldwide increases by 950 per year. m 650. The number of Starbucks locations worldwide increases by 650 for years. (E) m 650. The number of Starbucks locations worldwide increases by 650 per year.. Which linear function would best represent the number of Starbucks locations as a function of time in years since 0? S t 650t 8,066 S t 00t 8,066 S t 950t 8,066 S t 650t,66 (E) S t 00t,66 Page 5
5. Determine an equation in slope-intercept form for the line graphed below. y y y 5 (E) None of these y 9 Between the years 990 and 000, the child mortality rate (the number of children per,000 live births who die before reaching the age of 5) in India was decreasing linearly. In 990, the child mortality rate was 6, and in 000, it was 9. Use this information to answer the net two questions. 6. What is the slope of this function, and what does it tell you in practical terms? m 5. The child mortality rate in India increased by 5 children per year between the years 990 and 000. m 5. The child mortality rate in India decreased by 5 children per year between the years 990 and 000. m.5. The child mortality rate in India increased by.5 children per year between the years 990 and 000. m.5. The child mortality rate in India decreased by.5 children per year between the years 990 and 000. (E) None of these 7. Epress the child mortality rate as a function of the time in years since 990. C t C t 5t 9 5t 6.5t 9 C t.5t 6 (E) C t.5t 9 C t Page 6
8. (***) What are the intercepts of the function f? (E), 0 0,, 0, 0 0, 0, and and, and and 0,,0,0,0 only only only 0,, only 0, and only 9. Consider the piecewise function Evaluate f. (E) undefined 5 f. 0. What is the range of the function 5, 5,,, (E), 6 0 f 0? 5 Page 7
. Give a rule for the function graphed below. f f f f 5 5 (E) None of these Page 8
An online wholesaler sells flag t-shirts in bulk. When customers order more shirts, they receive a discounted price on all of the shirts ordered. The wholesaler s pricing structure is as follows: Number of shirts ordered Price per shirt - $5.99-7 $.9 7 or more $.5 Use this information to answer the net two questions.. Which of the following functions represents the total cost of an order of flag t-shirts as a function of the number of shirts purchased? 5.99 C.9 7.5 7 5.99 C.9 65.89 7.58.79 7 5.99 C.9 65.89 7.55.9 7 5.99 C.9 65.89 7.569.0 7 (E) None of these. How many flag t-shirts can be purchased for a total cost of $00? The number of flag t-shirts is: Less than 0 Between 0 and 0 Between 0 and 50 Between 50 and 60 (E) More than 60 Page 9
, 5. (***) If the point the graph of y g. 5 is on the graph of y g, determine a point that MUST be on (E), 5, 0, 0,, 5. Given the graph of y f below, what is the domain of f?, (E), 6 5, 0, 0, 6 5 0-6 -5 - - - - - 0 5 6 - - - -5-6 Page 0
6. (***) The graph of y f is given below. Sketch a graph of the -intercepts of y f? 6 5 0-6 -5 - - - - - 0 5 6 - - - -5-6 y f. What are,0 and, 0, 0 and, 0,0 and 5,0 (E), 0 and, 0, 0 and, 0 7. Describe the function y g 5 as a transformation of the function y g Stretch horizontally, reflect across -ais, shift left 5 Compress vertically, reflect across -ais, shift up 5 Stretch vertically, reflect across -ais, shift up 5 Compress vertically, reflect across y-ais, shift up 5 (E) Stretch horizontally, reflect across y-ais, shift up 5. Page
The following graph shows the average daily TV and internet consumption per person worldwide (in minutes) for several years. Let represent the average daily TV consumption per person (in minutes), and gt f t represent the average daily internet consumption per person (in minutes), where t is measured in years since 009. Use this graph to answer the net two questions. 8. Evaluate f g and give a practical interpretation. f g 09. In 0, people spent 09 more minutes per day watching TV than on the internet. f g 09. In 0, people spent 09 fewer minutes per day watching TV than on the internet. f g 90. In 0, people spent 90 more minutes per day watching TV than on the internet. f g 90. In 0, people spent 90 fewer minutes per day watching TV than on the internet. (E) None of these 9. Is the function f g t always increasing, always decreasing, or neither, over the years shown? Always increasing Always decreasing Neither always increasing nor always decreasing Page
0. (***) Use the functions g and h given below to determine h g. h 5 0 g -5 - - - - - 5 - - - -5 0 (E) 5 f g.. Consider the functions f and g. Find f g 7 5 f g 5 f g f g 6 (E) f g 5 7 Page
. Use the functions f 5 and h to determine domain of?, h f h f. What is the,,, (E),. Suppose h f g. If h g and, find f. f f f f (E) f. (***) Kohl s allows customers to use both a dollar off coupon and a percentage off coupon on the same transaction, but the store stipulates that the dollar off coupon must be used first. If represents the original price of an item, then f 0 represents the price after a $0 off coupon, and g 0.70 represents the price after a 0% off coupon. Which of the following functions could be used to represent the price of an item after both coupons are applied? f g g f f g (E) g f f g Page
5. The function T f 5 0 represents the amount of time an employee at a grocery store will take to unload boes from a delivery truck. Determine a formula for the inverse function. (E) 0 f T f T T 5 0 f T f T T 5 f T 5T 0 T 0 5 T 0 5 6. Determine which one of the functions described below represents a one-to-one function. g is the function that assigns to each Math student his/her Math section number S is the function that assigns to each UA student the last four digits of his/her social security number R R 0 5 0 0 5 0 7. (***) Find the inverse of the function f. f f f 0.5 f (E) None of these Page 5
8. Determine the value of f for f. 7 7 7 7 (E) None of these 9. (***) Determine the value of f 5 for the function f whose graph is shown below. -5 - - - - 5-0 Undefined (E) None of these - - - -5 Page 6
0. Find a formula for the parabola graphed below. y y y y (E) y. Find the verte of the quadratic function, 6, 8,6, 8 (E) None of these f.. Find a formula for a parabola that goes through the points 5, 0,, 0, and, f 5 f 5 f 5 f 5 (E) f 5. Page 7
,. Find a formula for a parabola whose verte is at 0,. The coefficient of is: and goes through the point 6 5 (E). A stone is thrown upward; its height in meters t seconds after release is given by h t.9t 9t 77.. How long will it take the stone to hit the ground? Less than 9 seconds Between 9 and seconds Between and seconds Between and 5 seconds (E) More than 5 seconds 5. (***) A coffee company currently charges $.0 per bag and sells,000 bags per week. The company plans to decrease the price per bag. Past eperience indicates that each $0.0 price decrease raises weekly sales by 00 bags. How much should be charged per bag in order to attain the maimum revenue? The amount that should be charged per bag to attain the maimum revenue is: More than $.5 Between $.75 and $.5 Between $.5 and $.75 Between $.95 and $.5 (E) Less than $.95 Page 8
6. (***) Suppose a sunglass manufacturer determines the demand function for a certain line of sunglasses is given by p50, where p is the price per pair and is the number 000 of pairs sold. The fied cost of producing a line of sunglasses is $5,000 and each pair of sunglasses costs $ to make. How many sunglasses should be produced to maimize profit? 00,000 sunglasses 87,000 sunglasses 9,000 sunglasses 00,000 sunglasses (E) 88,000 sunglasses 7. (***) A concert venue holds a maimum of 000 people. With ticket prices at $0, the average attendance is 650 people. It is predicted that for each dollar the ticket price is lowered, approimately 5 more people attend. What is the maimum possible revenue from this concert? Less than $0,000 Between $0,000 and $,000 Between $,000 and $8,000 Between $8,000 and $,000 (E) More than $,000 8. Determine which one of the following represents a polynomial function. P Q 5 6 R 7 S / T 7 (E) Page 9
The number of monthly active Facebook users worldwide (in millions) between the year 009 and the year 07 can be modeled approimately by the function f t. 7. 0.8 80., where t is time in years since 009. Use this model to answer the net two questions. 9. Evaluate f and give a practical interpretation. (E) f 56.57. In the year 00, there were an average of 56.57 Facebook users added each month. f 56.57. In the year 00, there were 56.57 monthly Facebook users. f 56.57. In the year 00, there were 56.57 million monthly Facebook users. f 56.57. In the year 00, there were an average of 56.57 million Facebook users added each month. f 56.57. To reach million monthly Facebook users, it took 56.57 months. 50. (***) According to this model, in what year did the number of monthly active Facebook users reach,000,000,000? 0 0 0 0 (E) 05 Page 0
5. Consider the polynomial function graphed below. Determine whether the leading coefficient is positive or negative and whether the degree is even or odd. 0 6 8 - - - - - -8 - The leading coefficient is positive, the degree is odd The leading coefficient is positive, the degree is even The leading coefficient is negative, the degree is odd The leading coefficient is negative, the degree is even 5. Which one of the following is a polynomial function of degree 5 with zeros at and 7 only? f 7 f 7 f 7 f 7 (E) f 7 Page
5. Determine a possible equation for the polynomial function graphed below. 8 6 0, 6 - - - - - - -6-8 y y y y (E) y Page
5. Determine a possible equation for the polynomial function graphed below. 6, -5 - - - - 5 - - -6 y y y 6 y (E) y R 5 6 on your calculator. Given that 55. Graph what are the other factors? is a factor of R, (E) and and and and and Page
56. Use your graphing calculator to graph of P? P 6 5. What are the zeros 5,, and only 5,, and only 5,, and only (E) 5, 5,, and, and only only 6t The height, in feet, of a dwarf evergreen tree can be modeled by the function h f t, t. where t is time in years since sprouting. Use this information to answer the net two questions. 57. Determine what happens to the function f t as t gets very large. f t f t. f t.8 f t 6 (E) None of these 58. (***) Evaluate f 5, and give a practical interpretation. (E) f 5.8. After 5 years, the height of the tree is.8 feet. f 5.. After 5 years, the height of the tree is. feet. f 5 6. After 5 years, the height of the tree is 6 feet 5.8 f 5 6. When a tree is 5 feet tall, 6 years have passed. f. When a tree is 5 feet tall,.8 years have passed. Page
59. Determine the domain and zeros of the function f 6. Domain: Zeros: Domain: Zeros: Domain: Zeros: Domain: Zeros: 6, 6, 6, 6,, (E) Domain: 6, Zeros: 60. Find the vertical asymptote(s) of L 56. only only and 6 only 6 only (E) There are no vertical asymptotes. Page 5
6. (***) Find the horizontal asymptote of p. (E) y y y y y R 6. (***) Find the horizontal asymptote of y0.5 0.5 y y y 0 (E) y. 6 Page 6
6. (***) Find an equation of the rational function graphed below. 0 8 6 6, 5-6 - - 6 8 0 - - -6 f f 5 f f 5 (E) None of these 6. (***) Determine an equation of a rational function if the zero of the function is, the y-intercept is 0,, and the equations of the asymptotes are and y. y y y y (E) y Page 7
65. Determine the range of J (E),,,,,. 66. Determine the asymptote for L (E) y y 0 y 5 y 0 5. 67. (***) Determine a formula of the form below. What is the value of b?, 0 6 8 y C b for the eponential function graphed - - - - - b b b b 6 (E) b 6 Page 8
68. (***) The table below gives values for a function of the form values of C and b. y C b. Determine the y 8 8 7 (E) C 6 C C 8,,, 6 C, C 8, b 8 b b b b 69. Suppose Melissa invests $900 into a high-yield savings account that pays 5.7% interest compounded quarterly. Her brother, Billy, invests $0,00 into a different account that pays.8% compounded monthly. If no other investments are made, who will have more money in their account at the end of 0 years? How much more money will that person have? Melissa will have $ to $50 more than Billy. Billy will have $ to $50 more than Melissa. Melissa will have $50 to $00 more than Billy. Billy will have $50 to $00 more than Melissa. 70. Phillip wants to have $0,000 in 6 years, so he will place money into a savings account that pays.% interest compounded weekly. How much should Phillip invest now to have $0,000 in 6 years? $,66.07 $,7.7 $8,5.07 $8,5.56 (E) $,5.99 Page 9
7. Dmitry invests $00 in a savings account that earns.6% interest compounded continuously. How much money would Dmitry have in the account after.5 years? $69.6 $758.99 $75.5 $7. (E) $6,008.99 7. Anna has a choice between two investment options for a $000 gift she received. The first option earns 7.8% interest compounded continuously. The second option earns 7.9% interest compounded semi-annually. Which option would yield the greatest amount of money after 5 years and by how much? Option, between $0 and $5 more. Option, between $0 and $5 more. Option, between $5 and $0 more. Option, between $5 and $0 more. 7. Arturo wants to have $5,000 in 6 years, so he will place money into a savings account that pays.7 % interest compounded continuously. How much should Arturo invest now to have $5,000 in 6 years? $,06.98 $69. $8,09.96 $8,78.57 (E) $,0.7,500e 7. The population of a city can be measured by 0.0 P t t, where t represents time in years after 985. What does the model predict the population to be in the year 00? Less than 0,000 Between 0,000 and 0,000 Between 0,000 and 0,000 Between 0,000 and 0,000 (E) More than 0,000 Page 0
75. An invasive beetle was discovered in a small Pacific island 5 years ago. It is estimated that there are,00 beetles on the island now, with a relative (continuous) growth rate of 6%. How many beetles will there be after another 5 years? 6,687 beetles,89 beetles,06,58 beetles,506,79 beetles (E) None of these 76. Change the eponential equation log 9 log 9 log /9 log /9 (E) log 9 9 into logarithmic form. 77. Evaluate the epression log 8. (E) 0 Page
78. (***) Find the domain of the logarithmic function f log.,,,, (E) None of these 79. Write the equation in eponential form: log 8 T 8 T T 8 8 T T 8 (E) 8 T 80. Write the equation in logarithmic form: 6 N log6 N log N 6 log N 6 log 6 N (E) log N 6 Page
8. Use properties of logarithms to epand the epression as much as possible: ln ln y ln ln y ln ln y ln ln ln (E) ln ln ln y y y ln e 8. Rewrite as a single logarithm: log log log log log y P Q z log P yz log zp y Q Q yz log PQ y log zpq (E) None of these 8. Solve for : log log 5 log 8 Page
8 8. Solve for : e ln ln ln ln (E) ln log 5 log 85. (***) Solve for : 6, only only only 6, only (E) 6 only 9 9 Solve each equation eactly. 86. t/ 0 50 t log/ 5 5 t log t log t log/ 5/ 5 Page
87. log u 7 6 u 07 u 9 u 007 u 99 / 88. e 0 00 500 ln 75 ln ln 75 ln 5 8 89. 00 t e (E) t ln 00 t ln 0 t ln 0 t ln 00 t ln 00 Page 5
A bakery purchased a commercial grade oven for $0,000 in 0, and was valued at $6,000 in 05. Use this information to answer the net three questions. 90. Assuming the value of the oven depreciates eponentially, epress the value, V, of the oven as a function of time, t, in years since 0. V f t 6,000. t V f t 0,000. t V f t 6,0000.8 t V f t 0,0000.8 t (E) None of these 9. (***) Which of the following statements is true? f f f 6,000 f 6,000 05 05 6,000 6,000 (E) None of these 9. Using the eponential depreciation model, what is the approimate value of the oven in the year 07? $9,06 $,800 $0,0 $89 (E) $8000 Page 6
9. The world population has been growing roughly eponentially for the past 0 years. In 987, the world population was approimately 5 billion. In 998, the world population was approimately 6 billion. Find an eponential equation of the form which models the world population with t representing the number of years since 987. What does this model predict the population was in 00? Less than 6 billion Between 6 and 6.5 billion Between 6.5 and 7 billion Between 7 and 7.5 billion (E) More than 7.5 billion y Ce kt 9. Suppose Matt initially invests $000 in an account bearing % interest compounded monthly. How long will it take for the deposit to double in value, rounded to the nearest 0.0 years? 7. years 7.6 years 9. years 9.7 years (E) None of these 95. (***) Radioactive bismuth has a half-life of 5 days. Estimate how much of a 00 gram sample will be radioactive in days. Less than 0 grams Between 0 and 0 grams Between 0 and 60 grams Between 60 and 80 grams (E) More than 80 grams 96. A certain lake is stocked with 000 fish. The population is growing according to the 0,000 logistic curve: P, where t is measured in months since the lake was initially /5 9e t stocked. After how many months will the fish population be 000? Less than months Between and months Between and 6 months Between 6 and 8 months (E) More than 8 months Page 7
97. (***) Suppose a turkey is removed from an oven when its inner temperature reaches 85 F and placed into a 75 F room to cool. The temperature, in degrees Fahrenheit can be 0.077t modeled by the function H 75 0e, where t is measured in minutes. Use the function to determine when the turkey will cool to 00 F. Less than 60 minutes Between 60 and 90 minutes Between 90 and 0 minutes Between 0 and 50 minutes (E) More than 50 minutes 98. (***) Atmospheric pressure is related to height above sea level according to an eponential model. Suppose the pressure at 8,000 feet is half that at sea level. Find the value of k in the continuous model and eplain what the value of k tells you. Find the equation and use it to estimate the pressure at 000 feet, as a percentage of the pressure at sea level. Less than 0% Between 0% and 0% Between 0% and 60% Between 60% and 80% (E) More than 80% Page 8
99. (***) A cup of tea that is initially 5 F is placed in a room kept at a constant 7 F. The temperature of the tea, T, as a function of time is given by T f where is measured in minutes. Determine the value of interpretation. f 90 f 90 0.5. After 90 minutes, the temperature was 0.5 F. f 90 0.5. After 0.5 minutes, the temperature was 90 F. f 90 7.00. After 90 minutes, the temperature was 7 F. f 90 7.00. After 7 minutes, the temperature was 90 F. 7 5 0.9, and give a practical 00. The number of smartphone users (in millions) in the United States from 00 to 06 is f t 75ln t 50, where t represents the given approimately by the function number of years since 00. According to this model, in what year did the number of smartphone users in the United States reach 50,000,000? 00 0 0 0 (E) 0 Page 9
Spring 08 Final Eam Study Guide Answers Question Answer Question Answer Question Answer B 5 B 69 C A 6 D 70 D D 7 B 7 B A 8 D 7 A 5 E 9 C 7 E 6 C 0 E 7 C 7 B A 75 A 8 D E 76 E 9 E B 77 B 0 A D 78 C B 5 E 79 B D 6 C 80 E E 7 A 8 D A 8 B 8 A 5 C 9 C 8 C 6 D 50 B 8 A 7 D 5 C 85 E 8 E 5 E 86 B 9 B 5 C 87 A 0 D 5 D 88 C A 55 A 89 E A 56 D 90 D E 57 D 9 A C 58 E 9 C 5 E 59 E 9 D 6 D 60 B 9 B 7 B 6 A 95 B 8 C 6 D 96 C 9 B 6 A 97 C 0 B 6 E 98 E E 65 E 99 B D 66 C 00 C C 67 A A 68 D Page 0