Unformatted text preview: 5.6_Rates_of_Change_in_Rational_Functions.notebook October 30, 2009 5.6 Rates of Change in Rational Functions
(1) Average Rate of Change The average rate of change represents the slope of the secant over the interval x1 ≤ x ≤ x2. Average
Rate of Change (2) Instantaneous Rate of Change
The instantaneous rate of change represents the slope of the tangent to the curve at the point x = a.
To determine the instantaneous rate of change at a point, we need to choose really small intervals on either side of x = a. Instantaneous Rate of Change
(use ) 1 5.6_Rates_of_Change_in_Rational_Functions.notebook October 30, 2009 Example #1:
5x Using the function f(x) = ,
2x + 4
(a) Determine the average rate of change (AROC) of f(x) over the interval [1, 2]. (b) Estimate the instantaneous rate of change (IROC) of f(x) at x = 3. (c) Explain why you cannot calculate the IROC at x = 2. Definitions
Demand Function •
the price per unit, p(x), that the marketplace is willing to pay for a given product or service. (ie. the price)
• Revenue Function the total revenue when x units are sold at price p(x). Revenue = Number of Items Sold x Price
= x p(x) • Marginal Revenue Function the instantaneous rate of change at a point. It is a measure of the estimated additional revenue from selling one more item (or the amount that the revenue is changing at that particular point. 2 5.6_Rates_of_Change_in_Rational_Functions.notebook October 30, 2009 Example #2:
A fastfood restaurant has conducted a survey that shows that the demand function, p(x), based on the monthly sales of x chicken 30 000 x sandwiches is p(x) = .
(a) Find the revenue function. (b) Estimate the marginal revenue for monthly sales of 5000 sandwiches. Explain its significance. Assigned Work:
#4, 5b, 6bc, 7, 8, Chapter 5 Review:
#1 3, 5, 6, 7bd, 8, 9, 10ac, 11 13, 15 3 ...
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