Growth 23%

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1 y x Decay 23% Math 109C - Fall 2012 page Write the quantity 12,600,000,000 miles in scientific notation. The result is: (A) x 10 9 miles (B) x 10 9 miles (C) x miles (D) x billion miles (E) x billion miles 40. New York City s population grew at a rate of approximately 0.43% each year since In 2006, approximately 8.2 million people lived within the city limits. a. Construct a function to model the population of New York City in millions, P t, as a function of the number of years since 2000, t. b. What is the practical interpretation of the y-intercept of your equation above? c. What does your model predict the population to be for 2010? 41. Fill in the table. f x Ca x Growth or decay? Growth/Decay factor Growth/Decay rate y x Growth 23%

2 Math 109C - Fall 2012 page For each of the following, identify whether the function is linear or exponential and explain why. Algebraically determine a formula each. a. x f x b. x f x One of the functions given in the table below is linear (of the form y ax b) and the other is exponential (of the form y ab x ). Determine which is which and find the values of a and b in each case. Explain your reasoning. x h x z x

3 Math 109C - Fall 2012 page Novocain is a common local anesthesia drug used during dental procedures. Below is a graph of the amount of Novocain in effect as a function of the number of hours from the beginning of the procedure: a. What is the practical implication of the decay rate in the exponential model above? b. Rewrite the equation so that the model gives the amount in effect as a function of time in minutes since the beginning of the dental procedure, m. c. What is the practical implication of the decay rate in the exponential model you found in (b) above?

4 Math 109C - Fall 2012 page Rewrite as indicated: a. Write in logarithmic form: b. Write in exponential form: ln e For the natural log function y ln x, which of the following is/are correct? (1) The graph is increasing (2) The x-intercept is 1, 0 (3) The graph has a minimum (4) The y-intercept is 0, e (A) All of them (B) 1, 2 and 3 only (C) 2, 3 and 4 only (D) 2 and 3 only (E) 1 and 2 only 47. For the function R x logx, which of the following is/are correct? (1) The domain is 0, (2) The range is, (3) R x has a x-intercept (A) 2 and 3 only (B) 1 and 2 only (C) 3 only (D) 1 and 3 only (E) All of them 48. Find the x-intercept of the graph of y ln x a 2 (A) ln a 2, 0 (B) 2 ln a, 0 (C) e 2 ln a, 0 (D) e 2 a, 0

5 Math 109C - Fall 2012 page 20 It has been rumored that college costs have been growing exponentially. Suppose the cost of four years of college (in thousands of dollars) can be expressed as y Ce kt where t 0 corresponds to 1992, t 4 corresponds to 1996, and so on. The graph of this function is shown below. Use this graph to answer the next two questions. 49. Use the graph (above) to find the value of C. (A) 1992 (B) 75 (C) 0 (D) What is the value of k? (A) (B) (C) (D) Rewrite f t 150e 0.025t in the form f t Ca t 52. Express as a sum or difference of the natural logarithms of x, y, and z: ln x 3 y z 4 (A) 3 ln x ln y 4 ln z (C) 3 2 ln x 1 2 ln y 2 ln z (B) 3 2 ln x ln y 4 ln z (D) 6 ln x 2 ln y 4 ln z 53. Express as a single logarithm and simplify if possible:

6 Math 109C - Fall 2012 page logx 4 logy 2 logz (A) log x 1/3 y 4 z 2 (B) log 1 3 x 4y 2z (C) 7 3 (D) log x1/3 y 4 z 2 log xy z 54. Find the exact value of ln 4 e 5. (A). 8 (B) (C) e (D). 8e 55. If M and N are positive, which of the following is/are correct? (1) ln MN ln M ln N (2) ln M N ln M ln N (3) ln M ln N ln M ln N (A) All of them (B) None of them (C) 1 only (D) 1 and 3 only (E) 2 and 3 only 56. Express as a sum, difference and/or multiple of logarithms: log xy2 100 (A) log x 2y 2 (B) 2 log x 2 log y 100 (C) 2 log xy 100 (D)log x 2 log y 2

7 Math 109C - Fall 2012 page Solve for x : 3 2x 27 2x 1 The answer is a number: (A) between 0. 5 and 1 (B) between 0 and 0.5 (C) between 0. 5 and 0 (D) between 1 and Use natural logarithms to solve for x : 3 6e 2x 5 (A) x 1 2 ln 3 (B) x 2 ln 3 (C) x 1 2 ln 1 3 (D) x ln Solve for x : 3 x 5 x 1 The solution is a number: (A) between 2 and 4 (B) between 5 and 3 (C) between 1 and 0 (D) between 3 and Solve for x: log 2 4 x log 2 3 x 3 The solution is a number: (A) between 9 and 6 (B) between 6 and 4 (C) between 4 and 1 (D) between 1 and 2 (E) between 2 and 5

8 Math 109C - Fall 2012 page Solve for x : ln 2x 1 2 (A) x e2 2 (B) x e2 1 2 (C) x e (D) x e A math teacher compiles some data on the number of hours, t, his students spent studying for the exam and the score they received, S t. He found that it fit the logarithmic model: S t 40 log 2t 6 40 If a student wishes to earn a score of 90 on the exam, how long should she study? 63. Assume that $12000 is invested at 5.5% compounded continuously. a. Determine the nominal interest rate. b. Determine the effective interest rate. 64. Johnny invested money at 3.8% compounded monthly. a. Determine the nominal interest rate. b. Determine the effective interest rate. 65. The effective rate on an investment compounded continuously is 6.4%. a. Determine the nominal rate. Solve algebraically. b. Using the rate found in part (a), how long would it take for your money to double? Solve algebraically.

9 Math 109C - Fall 2012 page The graph below shows the exponential growth of an investment of P dollars over time (in years). Estimate the time it will take for this investment to double. (A) 10 years (B) 20 years (C) 30 years (D) 40 years (E) Not enough information to know 67. A bank quotes a 4.75% effective annual interest rate on a savings account where interest is compounded continuously. Find the nominal interest rate for this account and round to the nearest 0.01%. 68. The number of California gray whales is growing according to the formula A Pe 0.015t, where t is measured in years. How long will it take the number of whales to double? (A) less than 40 years (C) between 50 and 60 years (E) more than 70 years (B) between 40 and 50 years (D) between 60 and 70 years

10 Math 109C - Fall 2012 page When a murder is committed, the temperature of the body cools according to the equation: H 22 15e rt where H is the temperature of the body and t is the time in hours since the murder was committed. a. Suppose after two hours, the body cools to 35 degrees C. Find r. b. Suppose the police find a dead body with a temperature of 30 degrees C at 4 pm. Use this information to determine when the murder was committed. 70. The proportion of carbon-14, an isotope of carbon, in living plant matter is constant. Once a plant dies, the carbon-14 in it begins to decay with a half-life of 5570 years. An archaeologist measures the remains of carbon-14 in a prehistoric hut and finds it to be one-tenth the amount of carbon-14 in the living wood. How old is the hut? 71. In 1980, the population of the United States was approximately million people. In 1990, the population had grown to approximately million. Assuming an exponential growth model y Ca x, what is the projected population of the U.S. in the year 2000? (A) million (B) 310 million (C) million (D) million 72. Mr. Smart decided to invest $20,000 in a savings account. At what annual percentage rate, compounded monthly, did he invest his money in order to have $36,500 at the end of 10 years? The interest rate is: (A) less than 6.1% (B) between 6.1% and 6.3% (C) between 6.3% and 6.5% (D) between 6.5% and 6.7% (E) greater than 6.7%

11 Math 109C - Fall 2012 page The release of fluorocarbons used in household sprays destroys the ozone layer in the upper atmosphere. Suppose the amount of ozone is given by P Ce t where t is measured in years. How long will it take for 70% of the ozone to disappear? (Round to the nearest yr.) (A) About 143 yrs. (B) About 1699 yrs. (C) About 1360 yrs. (D) About 482 yrs. 74. A computer virus has been introduced into a system of computers. The data below was collected over the next 5 months: Months since virus was introduced Number of computers infected Which of the following equations best describes the number of computers infected, y, as a function of the time, t (measured in months), since the virus was introduced? (A) y 6 4 t (B) y 24 4 t (C) y 24 72t (D) y 24 t 75. Based on the pattern of the data from the previous question, find the y-intercept of this function, and describe its practical significance. (A) (0,6); there were no computers infected at time t 6 (B) (0,6); 6 computers were initally infected (C) (0,4); there were no computers infected at time t 6 (D) (0,4); 4 computers were initially infected

12 Math 109C - Fall 2012 page The rate of inflation is often used to describe the value of currencies over time. For example, if you buy something now for $100 and the rate of inflation is 5% a year, this means that one year from now, you ll need $105 to buy the same item you bought a year ago. With this in mind, we consider the impact of inflation on the price of buying a home. In 1967, the price of a new home was about $24,600. In 2006, the price of a new home was $290,600. Assuming that home prices are increasing exponentially, find: a. Find a formula for the Price of a home, P, using n thirty-nine year periods since In other words, when n 0 this corresponds to the year When n 1 this corresponds to the year When n 2 this corresponds to the year 2045, and so on. b. Use your formula in part (c) to complete the table below: Year Price c. Find a formula for the price of a home, P, using t one-year periods since In other words, when t 0 this corresponds to the year 1967, when t 1 this corresponds to the year 1968, and so on. d. What is the one-year rate of inflation?

13 Math 109C - Fall 2012 page One measure of inflation is the cost of a consumer bundle. A consumer bundle is the mean cost for a group of goods and services that an average consumer unit would use throughout the given year. We want to model the cost of a consumer bundle from the years 1980 through 2008 and are debating whether to use a linear, exponential, power or quadratic function to do so. We construct the following plots and regression equations: a. Based on these plots, which type of model will best fit the consumer data? Explain. b. Use the model from part (a) to predict what the cost of a consumer bundle is in 2010.

14 Math 109C - Fall 2012 page In the following graph, Ray Kurzweil displays the growth of computer power per $1000, P, as a function of time in his book The Singularity is Near: Focusing on the portion of the graph from 1950 to 1990, the data can be summarized by the model where t is measured in years since log P t a. Create a model for P as a function of the years since b. What is the practical interpretation of the initial value of your model? c. What is the practical interpretation of the growth rate of your model?

15 Math 109C - Fall 2012 page The graph below displays an exponential regression equation for the cost of a McDonald s single hamburger over time. a. If the data were to instead be plotted on a semi-log plot, what type of function should we use to model the data? b. Find the regression equation of the type chosen in (a) above. What would the correlation coefficient be for this equation?

16 Math 109C - Fall 2012 page Consider the following set of data: Year Median Age in U.S a. Find a linear regression y ax b equation for the median age in the United States as a function of the year. i. What is the correlation coefficient? What does it tell you about the data (include 2 things)? ii. What is the practical interpretation of the value of a? iii. What is the practical interpretation of the value of b? Does this value make sense? Explain. b. Find an exponential regression equation of the form y a b x. i. What is the correlation coefficent? What does it tell you about the data (include 2 things)? ii. What is the practical interpretation of b? iii. What is the practical interpretation of the value of a? Does this value make sense? Explain. iv. Rewrite your equation in the form y Pe rt. c. Which regression equation appears to best fit the data? Explain. Which equation would you thus use for interpolation? d. Statisticians have made the following predictions for the median age in the United States. With this new information, which regression equation appears to be better at extrapolation? Year Median Age in U.S

17 Math 109C - Fall 2012 page Which of the following graphs best represents: f x 600 x 50 2 (A) (B) (C) (D) 82. For the function f x x 2 6x 62, which of the following statements is/are true? (1) the x-coordinate of the vertex is 3 (2) f x has a maximum value of 71 (3) one of the x-intercepts is 3 71, 0 (A) 1 and 2 only (B) 1 and 3 only (C) 2 and 3 only (D) 2 only (E) 1 only

18 Math 109C - Fall 2012 page Write f x 3x 2 60x 1 in vertex form. The SUM of the x and y coordinates of the vertex is: (A) 473 (B) 365 (C) 365 (D) Find the vertex of the quadratic function The vertex is located: f x 3x 2 7x 4. (A) Above the x-axis (C) At the origin (B) Below the x-axis (D) On the x-axis but not at the origin 85. Find the vertex of the quadratic function f t 4 7 t The y-coordinate of the vertex is: t 3. (A) 1 2 (B) 5 7 (C) 6 7 (D) Find the equation of the parabola that has a vertex of 3, 1 and passes through the point 4, 1. The coefficient of x 2 is a number: (A) between 2 and 0. 5 (B) between 0. 5 and 1. 5 (C) between 1. 5 and 3 (D) between 3 and 4.5

19 Math 109C - Fall 2012 page A slim line fluorescent bulb with ½ inch diameter needs 1 inch clearance top and bottom in a parabolic reflecting shade. This means the depth of the shade is 2.5 inches. a. What is the equation for this reflector? b. What are the coordinates of the focus? c. What is the diameter of the opening of the shade? 88. Which of the following has two x-intercepts? (1) f x x 2 40 (2) g x x 2 60x 903 (3) h x x (A) 1 only (B) 2 and 3 only (C) 2 only (D) 1 and 3 only 89. A horticulturist has determined that the number of inches a young oak tree grows in one year is a function of the annual rainfall, r, given by g r 0. 01r r 2. What is the maximum number of inches a young oak can grow in a year? The maximum number of inches is: (A) less than 1 (B) between 1 and 2 (C) between 2 and 3 (D) between 3 and 4 (E) between 4 and 5

20 Math 109C - Fall 2012 page A ball is dropped from the top of the Empire State Building (1250 feet tall) in New York City. Its height t seconds after being dropped is given by h t t². a. Find a reasonable domain for this function. Explain your choice. b. Find the time it takes for the ball to hit the ground. Round your answer to the nearest 0.01 seconds. Consider the following problem: In 1990 Joe bought a BMW for $40,000; it depreciates $4,000 each year. Tom bought a classic 57 Chevy for $10,000 and it appreciates $1,000 each year. Use this information to answer the next two questions. 91. For the years 1975 through 1990, the average price, p (in dollars per million British thermal units), of fuel used to generate electricity in the U.S. can be modeled by the function p t t t where t is time in years since Estimate the maximum average price, p, of fuel used according to this model. (A) $11.9 per million Btu (B) $1.94 per million Btu (C) $2.1 per billion Btu (D) $10.2 per million Btu 92. A parabolic mirror has diameter 50 cm and depth 10 cm. Recalling that the focal length of the mirror is 1, find this distance. Answers are rounded to three 4a decimal places where appropriate. (A) 0.2 cm (B) cm (C) 10 cm (D) cm (E) 25 cm 93. Convert the following quadratic function to vertex form: f x 3x² 9x 7

21 Math 109C - Fall 2012 page Find the equation that vertically compresses the graph of y x 2 and shifts the graph up twenty units. (A) y 5 3 x2 20 (B) y 7 2 x 20 2 (C) y 3 4 x2 20 (D) y 2 x You can get the graph of y 2 f x by transforming the graph of y f x in the following way: (A) Compress vertically and reflect across the x-axis (B) Compress vertically and reflect across the y-axis (C) Expand vertically and reflect across the x-axis (D) Expand vertically and reflect across the y-axis 96. For f x x² find the equation for each of the following transformations: a. Shift f x 2 units to the right. b. Shift f x up vertically by 2 units. c. Reflect f x across the x-axis.

22 Math 109C - Fall 2012 page The table below shows the relationship between a car s gas consumption G (in gallons per hour) and the speed of the car v (in miles per hour): v (mph) G (gph) a. Make a scatter plot showing the relationship between G and v. b. Decide on and find a suitable model for the data contained in the table above.