SECTION 2.3 Properties of Functions 99 (e) Find the average rate of change of the debt from 2016 to 2018. (f) What appears to be happening to the average rate of change in the years from 2012 to 2018? (g) Find the average rate of change of the debt from 2019 to 2021. What does the negative sign represent? (h) What factor(s) do you think caused this average rate of change from 2019 to 2021? 85. For the function f x x ,2 ( ) = compute the average rate of change: (a) From 0 to 1 (b) From 0 to 0.5 (c) From 0 to 0.1 (d) From 0 to 0.01 (e) From 0 to 0.001 (f) Use a graphing utility to graph each of the secant lines along with f. (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number? 86. For the function f x x ,2 ( ) = compute the average rate of change: (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with f. (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number? corresponds to exactly one output, the debt is a function of the year. D y( ) represents the debt for each year y. Year, y Debt, D (in dollars) 2012 9273 2013 9259 2014 9512 2015 9523 2016 9938 2017 10,271 2018 10,409 2019 10,507 2020 9218 2021 9177 2022 9990 Source : U.S Federal Reserve. (a) Plot all the points: (2012, 9273), (2013, 9259), and so on. (b) Draw a line segment from the point (2012, 9273) to the point (2019, 10 507). What does the slope of this line segment represent? (c) Find the average rate of change of the debt from 2012 to 2014. (d) Find the average rate of change of the debt from 2014 to 2016. m f x h f x x h x f x h f x h h 0 sec ( ) ( ) ( ) ( ) ( ) = + − + − = + − ≠ Problems 87–94 require the following discussion of a secant line. The slope of the secant line containing the two points x f x , ( ) ( ) and x h f x h , ( ) ( ) + + on the graph of a function y f x( ) = may be given as (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find msec for h 0.5, = 0.1, and 0.01 at x 1. = What value does msec approach as h approaches 0? (c) Find an equation for the secant line at x 1 = with h 0.01. = (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 87. f x x2 5 ( ) = + 88. f x x3 2 ( ) = − + 89. f x x x2 2 ( ) = + 90. f x x x 2 2 ( ) = + 91. f x x x 2 3 1 2 ( ) = − + 92. f x x x3 2 2 ( ) = − + − 93. f x x 1 ( ) = 94. f x x 1 2 ( ) = 95. Challenge Problem Mean Value Theorem Suppose f x x x x 2 6. 3 2 ( ) = + − + From calculus, the Mean Value Theorem guarantees that there is at least one number in the open interval 1, 2 ( ) − at which the value of the derivative of f, given by f x x x 3 4 1, 2 ( ) ′ = + − is equal to the average rate of change of f on the interval. Find all such numbers x in the interval. 96. Challenge Problem If f is an odd function, determine whether g x f x 2 3 ( ) ( ) = − − is even, odd, or neither. Explaining Concepts 97. Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: 0, 3 ( ) − and 3,0 ; ( ) a local maximum value of 2− at 1; − a local minimum value of 6− at 2. Compare your graph with those of others. Comment on any differences. 98. Redo Problem 97 with the following additional information: increasing on , 1, 2, ; ( ] [ ) −∞ − ∞ decreasing on 1, 2 . [ ] − Again compare your graph with others and comment on any differences. 99. How many x -intercepts can a function defined on an interval have if it is increasing on that interval? Explain. 100. Suppose that a friend of yours does not understand the idea of increasing and decreasing functions. Provide an explanation, complete with graphs, that clarifies the idea. 101. Can a function be both even and odd? Explain. 102. Using a graphing utility, graph y 5 = on the interval 3, 3 . [ ] − Use MAXIMUM to find the local maximum values on 3, 3 . [ ] − Comment on the result provided by the graphing utility.
RkJQdWJsaXNoZXIy NjM5ODQ=