Lesson Plans

Lesson 2

Average Rates – The Difference Equation

 

Introduction:

 

In this lesson the concept of average rates will be developed.  The lesson will start with the concept of average velocity and conclude by introducing the difference equation, which will be used in the next lesson to develop the concept of instantaneous rates.  Use the material here to demonstrate the models and to generate discussion about rates in general.  Everything here can be extended to many other topics besides physics.

 

Average Velocity:

 

Suppose you were traveling from Atlanta, Georgia to West Palm Beach, Florida, at distance of about 625 miles, in your car.  During the trip you would be traveling at various rates of speed because of differing speed limits, stops for food and gasoline and other things.  Your average speed for the trip would be 625 miles divided by the time it took you to make the trip including stops. So if you made the trip in 9 ½ hours your average speed would be 625 ¸9 ½ = 65.8 miles per hour.  That is, your average speed equals the distance traveled divided by the elapsed time it took to travel that distance.  In calculus we will call that the change in distance divided by the change in time and write it as shown below:

 

 

Suppose you could write a function that would describe your distance as a function of time.  Let  represent this function.  The distance traveled from time t to time t + 2 would be d(t + 2) – d(t).  If we let 2 be represented by Dt then we can write an equation to calculate average speed between any two times for any function d(t).  See below

 

 

Example:

 

Suppose d(t) = 5t +3 meters.  Find the average speed from times t = 3 to t = 7.

 

Solution:

 

Example:

 

If d(t) = 6t + 1 meters, what is the average speed from t = 1 to t = 3 seconds?

 

Solution:

 

 

Average Power:

 

Power is the analog to speed in work problems.  Power is the rate at which work is done.  Average power can be calculated in the same manner as average speed.  That is:

 

 

Example:

 

Let W(t) = 3t² + 12t +4 watt-seconds.  Find the average power between t = 3 and t = 8 seconds.

 

Solution:

 

 

Difference Quotient:

 

If c is in the domain of a function y = f(x), then the average rate of change between c and x is defined as

 

 

This expression is called the difference quotient (or difference equation) of f at c.  We can use this quotient to find a function that will give us the average rate of change from any point c in the domain of f to any arbitrary x in f.  For example, using the work function given above and c = 2, we have:

 

 

What is the significance of this function that is calculated from the difference quotient?  More on that in the next lesson.  However, using this result we can calculate the average power from t = 2 to t = 6 by simply substituting 6 in 3t + 18.  So the number is 36 watts.

 

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