968 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous rate of change of revenue at = x 25 bicycles. 0 Number of Bicycles, x Total Revenue, R (in dollars) 0 25 28,000 60 45,000 102 53,400 150 59,160 190 62,360 223 64,835 249 66,525 (a) Find the average velocity from = t 1 to = t 4 seconds. (b) Find the average velocity from = t 1 to = t 3 seconds. (c) Find the average velocity from = t 1 to = t 2 seconds. (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous velocity at = t 1 second. 50. Instantaneous Rate of Change The data to the right represent the total revenue R (in dollars) received from selling x bicycles at Tunney’s Bicycle Shop. (a) Find the average rate of change in revenue from = x 25 to = x 150 bicycles. (b) Find the average rate of change in revenue from = x 25 to = x 102 bicycles. (c) Find the average rate of change in revenue from = x 25 to = x 60 bicycles. Retain Your Knowledge Problems 51–54 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for a final exam, or subsequent courses such as calculus. 51. Find the vertex and focus of the parabola − − + = x x y 2 2 7 0. 2 52. Solve the system by substitution: = + − = − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x x y 4 2 2 53. Evaluate the combination: ( ) C 5, 3 54. Find the area of the given triangle, rounded to two decimal places. 13 6 9 A B C ‘Are You Prepared?’ Answers 1. = − y x5 14 2. False 14.5 The Area Problem; The Integral Now Work the ‘Are You Prepared?’ problems on page 973. • Geometry Formulas (Section A.2, pp. A15–A16) • Summation Notation (Section 12.1, pp. 859–862) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Approximate the Area under the Graph of a Function (p. 969) 2 Approximate Integrals Using a Graphing Utility (p. 972) The Area Problem The development of the integral, like that of the derivative, was originally motivated to a large extent by a problem in geometry: the area problem . Area Problem Suppose a function f is nonnegative and continuous on a closed interval [ ] a b , . Find the area enclosed by the graph of f, the x -axis, and the vertical lines = x a and = x b. Figure 21 illustrates the area problem. We refer to the area A shown as the area under the graph of f from a to b . Figure 21 Area problem Area A y 5 f (x) Area A 5 area under the graph of f from a to b x a b y
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