Calculus for Business and Social Sciences Day 3 Section 3.3 Rates of Change

Example If it takes you 30 minutes to drive the 15 miles from CCU to the beach, what is the average speed during your trip? So your average speed is? What does average speed tell you about the trip? Were you actually travelling this speed for part of the trip?

Average Rate of Change The average rate of change of f(x) with respect to x for a function f as x changes from a to b is How does this relate to the example of driving from CCU to Myrtle Beach?

Example The percentage of men aged 65 and older in the workforce has been declining over the last century. The percent can be approximated by the function where x is the number of years since Find the average rate of change of this percent from 1960 to 2000.

Instantaneous rate of change Think back to the driving example. We computed the average rate of change by dividing the total distance traveled by the total time of the trip. Think about the speedometer in your car. Is it working the same way? What happens if we break the distance and time into small intervals?

Instantaneous rate of change As mentioned on the previous slide, we can take time in small intervals. Suppose we measure from a to a+h where h is a small number. Then the average rate of change becomes If we take the limit of this as h goes to zero, we get the instantaneous rate of change at x=a.

Example A ball is tossed in the air and its distance from the ground t seconds after being thrown is Find the average velocity of the ball from 2 seconds to 4 seconds. Find the instantaneous velocity of the ball at 4 seconds.

Practice Find the average rates of change for the following functions over the given intervals. between x=2 and x=6 between x=1 and x=8

More Practice Find the instantaneous rate of change for the following functions at t=2 at x=0

Homework On pages , do 2-20 odd 24, 28, 32, 36, 38