AB Calculus (Vahsen) Power Packet DUE SEPTEMBER

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Garey McBride
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1 1. Definition of Average Rate of Change Name: The table below shows the times, t, in seconds achieved by Usain Bolt in the 100 meter final at the 2008 Olympics. Distance, d, is in meters. (a) For each successive time interval, calculate the average rate of change of distance. Create a new table showing the time interval and this average rate of change. What is a common name for the average rate of change of distance? (b) Where did Usain Bolt attain his maximum speed during this race? Some runners are running their fastest as they cross the finish line. Does this seem to be true in this case? t d Finding a Formula for a Linear Function from a Graph The function is 2 f x 1 x. Find the equation of the line PQ. y = 1 x 2 Q x = 2 P (0, -8) 1

2 3. Formulas for Linear Functions An accountant works 9 hours a day, 5 days a week. He divides his time between data entry and meeting with co-workers. A session of data entry takes 3 hours while a co-worker meeting requires 90 minutes. Let x be the number of co-worker meetings the accountant holds during a given week. If y is the number of data entry sessions for which he has time remaining, then y is a function of x. Assume that this relationship is linear and that sessions can be split up and continued on different days. (a) Graph the relationships between x and y. [Hint: Consider the maximum number of sessions that can be held. What if only co-worker meetings were held? Only data entry?] (b) Find a formula for y as a function of x. (c) Explain what the x- and y- intercepts represent in the context of the accountant s schedule. (d) A new program is implemented so that the data entry takes 90 minutes instead of 3 hours. Graph this situation. Describe those features of this graph that have changed from the one sketched in part (a) and those that have remained the same. 2

3 4. Input and Output Virginia state income tax is based on taxable income, which is part of a person s total income. The tax owed to the state is calculated using the taxable income (not total income). For a single person with a taxable income above $17,000, the tax owed is $720 plus 5.75% of the taxable income over $17,000. (a) Compute the tax owed by a Ms. V whose taxable income is approximately $42,400. (b) Consider Ms. V whose taxable income is 80% of her total income, $x where x is greater than $21,250. Write a formula for T(x), the taxable income (don t think too hard). (c) Write a formula for L(x) to evaluate the tax liability for x = $53,000 and compare your results to part (a). 3

4 5. Concavity and Rates of Change Match each of the following descriptions with an appropriate graph and table of values. (a) The weight of your jumbo box of Cinnamon Toast Crunch decreases by an equal amount each week. (b) The value of your new car depreciated rapidly at first, but its value declined more slowly as time went on. (c) While sky-diving, your distance from the ground decreases at an increasing rate. (d) For a while, it looked like the decline in profits from newspaper sales was going down, but then they began declining ever more rapidly as the Washington Post started posting articles online. (e) x (f) y x y Record your three letter answer combinations here: a (g) (h) x y x y b c d (i) (k) (j) (l) 4

5 6. The Short-Run Behavior of Polynomial/Rational Functions Give a possible formula for the following polynomials. You may leave your answers in factored form Points: (1, 0), (-2, 0), and ( ) Points: (-2, 0), (-1, -3), and (2, 0) Point: ( ) Points: (0, 1) and (4, 0) x 2x is a line with a hole in it. What is the equation of the line? 5 x What are the coordinates of the hole? c. The graph of f x 5

6 7. Graphs of Exponential Functions The table below shows the population of people infected with Ebola is growing exponentially, and that those infected with the common cold is growing linearly. Date August 1 September 1 People infected with Ebola 2 38 People infected with common cold (a) Find a formula for the population infected with each disease as a function of the number of days since August 1. (b) Use this function to predict the population infected with each disease on November 1. (c) The local government will declare a state of emergency if more than 1000 people have died from a specific disease. Estimate the day when the government will make this declaration for Ebola. Ebola has a mortality rate of 90%. (d) Estimate the day in which number of Ebola patients overtakes the number of people who have a cold. You may use your calculator. 6

7 8. The Leesburg Premium Outlets are holding a Black Friday shopping event. Security wants to estimate the number of customers to expect. They collect data from three previous years, and determine that the crowds follow the same general pattern. When the store opens at midnight, 1000 people enter, and the total number in the store triples every hour and 15 minutes. When the number of people in the outlets reaches 16,000, security guards need to be stationed at the entrances to control the crowds. At what time should the guards be commissioned? 7

8 9. Without a calculator, match the formulas (a) (e) to one of the graphs (i) (v). (a) y x a, a 0 x (b) y, a 1 a (d) y x a, a 0 (e) y a x, a 0 (c) y 1 x (i) (iv) (ii) (v) Record your two letter answer combinations here: a b c (iii) d e 8

9 10. A local government official is proposing that fines should be determined based on the number of miles per hour the person was going over the speed limit. They are proposing the following rules: People must pay at least $50 for their fine. For regular speeding, people will pay $10 times the miles per hour over the speed limit. People who are convicted of reckless driving (at least 20 mph over the speed limit) will pay $50 on top of their fine for regular speeding. (a) Suppose x is the difference between the miles per hour a driving was going and the posted speed limit, and y is the fine the driver must pay. Write a piecewise-defined formula for y in terms of x. (b) What fine would a person get for going 33 mph on Catoctin Circle (25 mph zone)? (c) How fast would a driver have to go in a 65 mph zone to get a fine of $280? (d) Graph y as a function of x. 9

b) According to the statistics above the graph, the slope is What are the units and meaning of this value?

b) According to the statistics above the graph, the slope is What are the units and meaning of this value? ! Name: Date: Hr: LINEAR MODELS Writing Motion Equations 1) Answer the following questions using the position vs. time graph of a runner in a race shown below. Be sure to show all work (formula, substitution,