1. Sullivan PreCalculus Section 2.6 Mathematical Models: Constructing Functions Objectives Construct and analyze functions

2. Example: The price p and the quantity x sold of a certain product obey the demand equation a.) Express the revenue R as a function of the quantity of items sold (x). Revenue = (Price)(Items sold) = (x)(p)

3. b.) What is the revenue if 150 items are sold? c.) Graph the function R(x) on a graphing utility. d.) Using the graph, find the number of items x that will maximize revenue. What is the maximum revenue? Quantity that maximizes revenue: 200 items Maximum Revenue: $10,000

4. e. What price should be charged for each item to achieve maximum revenue? Maximum Revenue occurs when x = 200 items Price = $50 should be charged to achieve maximum revenue.

5. Example: An open box with a square base is to made from a square piece of cardboard 30 inches on a side by cutting out a square from each corner and turning up the sides. a.) Express the volume V of the box as a function of the length x of the side of the square cut from each corner. The volume of a box is given by: V = (length)(width)(height)

6. 30in. x x x x x x x x Length = 30 - 2x Width = 30 - 2x Height = x So, Volume = (30 - 2x)(30 - 2x)(x) c.) Graph V(x) using a graphing utility and estimate what value of x will maximize V. At x = 5 inches, the volume is maximum (2000 cubic inches)

7. A rectangle has one corner on the graph of, another at the origin, and third on the positive y axis and fourth on the positive x axis. (0,16) (x,y) Example

8. a. Express the area of the rectangle as a function of x. A = lw A = xy A = since b. Find the domain of the function. Since Area must be positive, then x and y are positive.