"Calculus and its applications: " - Calculus and its applications rates of change have objective: compute an average rate of change, find a simplified difference quotient. | 2012 Pearson Education, Inc. All rights reserved Slide Average Rates of Change OBJECTIVE Compute an average rate of change. Find a simplified difference quotient. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: As x approaches a, the limit of f (x) is L, written if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: The average rate of change of y with respect to x, as x changes from x1 to x2, is the ratio of the change in output to the change in input: where x2 ≠ x1. Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION (concluded): If we look at a graph of the function, we see that which is both the average rate of change and the slope of the line from P(x1, y1) to Q(x2, y2). The line through P and Q, is called a secant line. Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide Example 1: For find the average rate of change as: a) x changes from 1 to 3. b) x changes from 1 to 2. c) x changes from 2 to 3. a) When x1 = 1, When x2 = 3, Thus, the average rate of change is Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (concluded): b) When x1 = 1, When x2 = 2, Thus, the average rate of change is c) When x1 = 2, When x2 = 3, Thus, the average rate of change is Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide Average Rates of Change Quick Check 1 State the average rate of change for each situation in a short sentence. Be sure to include units. a.) It rained 4 inches over a period of 8 hours. The average rate of change is The average rate of rain fall was inches of rain every hour. b.) Your car travels 250 miles on 20 gallons of gas. The average rate of change is The average . | 2012 Pearson Education, Inc. All rights reserved Slide Average Rates of Change OBJECTIVE Compute an average rate of change. Find a simplified difference quotient. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: As x approaches a, the limit of f (x) is L, written if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: The average rate of change of y with respect to x, as x changes from x1 to x2, is the ratio of the change in output to the change in input: where x2 ≠ x1. Average Rates of Change 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION (concluded): If we look at a graph of the function, we see that which is both the average rate of change and the slope of the line from P(x1, y1) to Q(x2, y2). The line through P and Q, is called a secant line. Average Rates
