SECTION 4.2 The Graph of a Polynomial Function; Models 213 45. Temperature The following data represent the temperature T Fahrenheit ( ) ° in Kansas City, Missouri, x hours after midnight on a March day. 3 6 9 12 15 18 21 24 50.2 48.0 49.4 53.0 56.6 57.2 52.1 50.0 Temperature (°F), T Hours after Midnight, x Source: The Weather Underground (a) Draw a scatter plot of the data. Comment on the type of relation that may exist between the two variables. (b) Find the average rate of change in temperature from 9 am to 12 noon. (c) What is the average rate of change in temperature from 3 pm to 9 pm? (d) Decide on a function of best fit to these data (linear, quadratic, or cubic) and use this function to predict the temperature at 5 pm. (e) With a graphing utility, draw a scatter plot of the data and then graph the function of best fit on the scatter plot. (f) Interpret the y-intercept. 46. Future Value of Money Suppose that you make deposits of $500 at the beginning of every year into an Individual Retirement Account (IRA), earning interest r (expressed as a decimal).At the beginning of the first year, the value of the account will be $500; at the beginning of the second year, the value of the account, will be + + = + r r $500 $500 $500 500 1000 Value of 1st deposit Value of 2nd deposit (a) Verify that the value of the account at the beginning of the third year is T r r r 500 1500 1500. 2 ( ) = + + (b) The account value at the beginning of the fourth year is F r r r r 500 2000 3000 2000. 3 2 ( ) = + + + If the annual rate of interest is 5% 0.05, = what will be the value of the account at the beginning of the fourth year? 47. Challenge Problem A Geometric Series In calculus, you will learn that certain functions can be approximated by polynomial functions.We will explore one such function now. (a) Using a graphing utility, create a table of values with Y f x x 1 1 1 ( ) = = − and Y g x x x x 1 2 2 2 3 ( ) = = + + + for x 1 1 − < < and Tbl 0.1. Δ = (b) Using a graphing utility, create a table of values with Y f x x 1 1 1 ( ) = = − and Y g x x x x x 1 2 3 2 3 4 ( ) = = + + + + for x 1 1 − < < and Tbl 0.1. Δ = (c) Using a graphing utility, create a table of values with Y f x x 1 1 1 ( ) = = − and Y g x x x x x x 1 2 4 2 3 4 5 ( ) = = + + + + + for x 1 1 − < < and Tbl 0.1. Δ = (d) What do you notice about the values of the function as more terms are added to the polynomial? Are there some values of x for which the approximations are better? 48. Challenge Problem Tennis Anyone? Assume that the probability of winning a point on serve or return is treated as constant throughout the match. Further suppose that x is the probability that the better player in a match wins a set. (a) The probability P3 that the better player wins a best-ofthree match is P x x x 121 . 3 2 ( ) ( ) [ ] = + − Suppose the probability that the better player wins a set is 0.6. What is the probability that this player wins a best-ofthree match? (b) The probability P5 that the better player wins a best-offive match is P x x x x 1 3 1 6 1 5 3 2 [ ] ( ) ( ) ( ) = + − + − Suppose the probability that the better player wins a set is 0.6. What is the probability that this player wins a bestof-five match? (c) The difference between the probability of winning and losing is known as the win advantage. For a best-of-n match, the win advantage is P P P 1 2 1 n n n ( ) − − = − The edge, E, is defined as the difference in win advantage between a best-of-five and best-of-three match. That is, E P P P P 2 1 2 1 2 5 3 5 3 ( ) ( ) ( ) = − − − = − Edge as a function of win probability of a set x is ( ) ( ) ( ) ( ) = + − + − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − + − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⎧ ⎨ ⎪ ⎩⎪⎪ ⎫ ⎬ ⎪ ⎭⎪⎪ E x x x x x x 2 1 3 1 6 1 1 2 1 2 2 Graph E E x( ) = for x 0.5 1. ≤ ≤ (d) Find the probability of winning a set x that maximizes the edge. What is the maximum edge? (e) Explain the meaning of E 0.5 . ( ) (f) Explain the meaning of E 1 . ( ) Source: Stephanie Kovalchik, “Grand Slams Are ShortChanging Women’s Tennis,” Significance, October 2015. 49. Challenge Problem Suppose ( ) ( )( ) =− − + f x ax x b x c , 2 2 where a b c 0 . < < < (a) Graph f. (b) In what interval(s) is there a local maximum value? (c) Which numbers yield a local minimum value? (d) Where is f x 0? ( ) < (e) Where is f x 4 0? ( ) − − < (f) Is f increasing, decreasing, or neither on c , ? ( ] −∞ −
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