Calculus for Business and Social Sciences

1
Calculus for Business and Social Sciences

• Day 3
• Section 3.3 Rates of Change

2
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?

3
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?

4
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 1900. Find
  the average rate of change of this percent from
  1960 to 2000.

5
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?

6
Instantaneous rate of change

• As mentioned on the previous slide, we can take
  time in small intervals. Suppose we measure from
  a to ah 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 xa.

7
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.

8
Practice

• Find the average rates of change for the
  following functions over the given intervals.
• between x2 and x6
• between x1 and x8

9
More Practice

• Find the instantaneous rate of change for the
  following functions
• at t2
• at x0

10
Homework

• On pages 173-178, do
• 2-20 odd
• 24, 28, 32, 36, 38