98 CHAPTER 2 Functions and Their Graphs Applications and Extensions 77. Interactive Figure Exercise Exploring the Slope of the Secant Line Open the “Secant” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) The polynomial function shown in blue has a local maximum of 3 at x 1. = − Move Point B to 1, 3 . ( ) − [Note: You may grab the coordinates of Point B and move them around if you are having a difficult time reading the coordinates.] Move Point A so that the x-coordinate of Point A is less than x 1. = − That is, move Point A to the left of Point B. What is the sign of the slope of the secant line for any value of x 1? <− I. Positive II. Negative III. Zero IV. Cannot be determined (b) Is the polynomial function shown in blue increasing or decreasing on the interval , 1? ( ] −∞ − (c) If a function is increasing on the interval , 1, ( ] −∞ − then for any x 1, <− the slope of the secant line, m f x f x 1 1 , sec ( ) ( ) ( ) = − − − − will be (positive, negative, zero). (d) Leave Point B at 1, 3 . ( ) − Move Point A so that the x-coordinate is near x 1 = (but less than x 1 = ). That is, move Point A to the right of Point B. What is the sign of the slope of the secant line? I. Positive II. Negative III. Zero IV. Cannot be determined (e) Is the polynomial function shown in blue increasing or decreasing on the interval 1, 1 ? [ ] − (f) If a function is decreasing on the interval 1, 1 , [ ] − then for any x 1 1, − < ≤ the slope of the secant line, m f x f x 1 1 , sec ( ) ( ) ( ) = − − − − will be (positive, negative, zero). (g) Leave Point B at 1, 3 . ( ) − Now move Point A toward 1, 3 . ( ) − What value does the slope of the secant line approach? 78. Interactive Figure Exercise Exploring the Slope of the Secant Line Open the “Secant” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Move Point A to 2, 1 ( ) − − and Point B to 1, 3 . ( ) − What is the slope of the secant line? (b) Grab Point B and move it toward Point A. As Point B approaches Point A (but remains to the right of Point A), to the nearest integer, what does the value of the slope of the secant line approach? 79. F x x x8 9 4 2 ( ) = − + + (a) Determine whether F is even, odd, or neither. (b) There is a local maximum value of 25 at x 2. = Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of F and the x-axis between x 0 = and x 3 = is 50.4 square units. Using the result from (a), determine the area of the region enclosed by the graph of F and the x-axis between x 3 = − and x 0. = 80. G x x x 32 144 4 2 ( ) = − + + (a) Determine whether G is even, odd, or neither. (b) There is a local maximum value of 400 at x 4. = Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of G and the x-axis between x 0 = and x 6 = is 1612.8 square units. Using the result from (a), determine the area of the region enclosed by the graph of G and the x-axis between x 6 = − and x 0. = 81. Minimum Average Cost The average cost per hour in dollars, C, of producing x riding lawn mowers can be modeled by the function C x x x x 0.3 21 251 2500 2 ( ) = + − + (a) Use a graphing utility to graph C C x . ( ) = (b) Determine the number of riding lawn mowers to produce in order to minimize average cost. (c) What is the minimum average cost? 82. Medicine Concentration The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function C t t t t t 0.002 0.039 0.285 0.766 0.085 4 3 2 ( ) = − + − + + (a) After how many hours will the concentration be highest? (b) A mother nursing a child must wait until the concentration is below 0.5 before the child can be fed. After taking the medication, how long must the mother wait before feeding the child? 83. E. coli Growth A strain of E. coli Beu 397-recA441 is placed into a nutrient broth at 30° Celsius and allowed to grow. The data shown in the table are collected. The population is measured in grams and the time in hours. Since population P depends on time t, and each input corresponds to exactly one output, we can say that population is a function of time, so P t( ) represents the population at time t. Time (hours), t Population (grams), P 0 2.5 3.5 4.5 6 0.09 0.18 0.26 0.35 0.50 (a) Find the average rate of change of the population from 0 to 2.5 hours. (b) Find the average rate of change of the population from 4.5 to 6 hours. (c) What is happening to the average rate of change as time passes? 84. Credit Card Debt In the year 2022, the credit card debt per household (adjusted for inflation) in the United States was $9990. The data in the table show the credit card debt per household in the United States for the years 2012–2022. Since the debt D depends on the year y, and each input
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