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2.7 Mathematical Models: Constructing Functions October 28, 2008 2­7 Mathematical Models: Constructing Functions Objective: Construct & Analyze Functions Oct 27­11:33 AM 1 2.7 Mathematical Models: Constructing Functions October 28, 2008 Using real­world situations we can construct mathematical models to answer the question being asked. We will take the verbal description and translate that into mathematical concepts that we are familiar with to create equations that can be solved. Oct 27­11:35 AM 2 2.7 Mathematical Models: Constructing Functions October 28, 2008 Examples: Demand Equation The price p, in dollars, and the quantity x sold of a certain product obey the demand equation: R = xp (Revenue equals the number of items sold times the price per item.) The Law of Demand states that x and p are related such that as one increases, the other decreases. (Inverse Function) Oct 27­11:35 AM 3 2.7 Mathematical Models: Constructing Functions October 28, 2008 1. a. Express the revenue R as a function of x. Oct 27­12:17 PM 4 2.7 Mathematical Models: Constructing Functions b. c. October 28, 2008 What is the revenue if 100 units are sold? Graph the function Oct 27­11:36 AM 5 2.7 Mathematical Models: Constructing Functions October 28, 2008 d. What quantity x maximizes the revenue? e. What is the maximum revenue? Oct 27­1:16 PM 6 2.7 Mathematical Models: Constructing Functions October 28, 2008 2. A farmer has 400 yards of fencing to enclose a rectangular garden. Express the area A as a function of the width x of the rectangle. Oct 27­1:19 PM 7 2.7 Mathematical Models: Constructing Functions 3. Let P(x, y) be a point on the graph of October 28, 2008 . a. Express the distance d from P to the origin as a function of x. b. What is d if x = 1? Remember that . Oct 27­1:19 PM 8 2.7 Mathematical Models: Constructing Functions c. October 28, 2008 Graph d = d(x) d) For what value of x is d the smallest? (From the graph we see it is at x = ± 1) Oct 27­1:19 PM 9 2.7 Mathematical Models: Constructing Functions October 28, 2008 4. A rectangle is inscribed in a semicircle of radius 2. Let P be the point in quadrant I that is the vertex of the rectangle and is on the circle. (Refer to figure in problem #16 on page 136.) a) Express the area A of the rectangle as a function of x. b) Express the perimeter P of the rectangle as a function of x. Oct 27­1:19 PM 10 2.7 Mathematical Models: Constructing Functions c) October 28, 2008 Graph A = A(x). For what value x of is A largest? Oct 27­1:19 PM 11 2.7 Mathematical Models: Constructing Functions October 28, 2008 Homework: p. 135/ 5, 8, 9, 17 Oct 27­1:23 PM 12