Quote of the Day:
"When I was four years old they tried to test my IQ. They 
   showed me a picture of three oranges and a pear. They 
   asked me, "which one is different and does not belong?" 
   They taught me different was wrong."   -- Ani DiFranco   

Objectives:
The student will determine the average value of a function 
   over a given interval.

1. What is meant by the term 'Average?"
   (A) What is the average of 98 and 92?  
       (DISCRETE DATA) Answer is 95.

   (B) If a car travels at a rate of 30 mph for 150 miles,
       then travels at 50 mph for 150 miles,
       what is its average speed in mph?
       (CONTINUOUS DATA) Answer is 37.5 mph 
       (see earlier lesson for an explanation)     

   (C) What is the average value of the function y = x from
       x = 0 to x = 4?
       (CONTINUOUS DATA)  Area is intuitive – avg = 2.

   (D) What is the average value of the function y = sin(x) 
       from x = 0 to x = 2 pi?
       (CONTINUOUS DATA)  Area is intuitive – avg = 0.
  
   (E) What is the average temperature in Winchester in 
       January?
       (CONTINUOUS DATA) 

2. Definition for Average Value of a Function

    We are looking for a y-value which is the average of 
    all the y-values from x = a to x = b.

    In other words, we are looking for a horizontal line 
    where the area above the line inside the curve equals 
    the area below the line outside the curve.  In the 
    diagram bow, the green area equals the yellow area.

    The left side represents the area of the rectangle 
    with base (b – a) and height equal to the Average 
    Value.  The right side represents the area under 
    the curve from x = a to x = b.    

3. Example

4. The Average Value of a Function is called "The Mean 
   Value Theorem for Integrals"

   Look at the diagram below showing the velocity of your 
   car from time t = a to time t = b.
   Let's say that your average velocity is 60 mph (OK, so 
   you're still in the Handley Parking Lot).
   (1) Does it mean that you always went 60 mph?      (NO)
   (2) Did you ever go 60 mph?        (YES, at least once) 

5. Examples
       

6. Now back to the classic problem:
   If one travels 30 mph over a trip of 150 miles and 
   then returns over the same 150 miles at a rate of 
   50 mph, what is the average rate of speed over the 
   whole trip?

   Most students will immediately answer 40 mph instead of 
   correctly responding 37.5 mph.  Let's solve this problem 
   at two levels – algebra and calculus.

   Students know that average speed is defined as the total 
   distance traveled divided by the total time elapsed.  
   The table below can be set up to solve the problem.  
   Since the total distance traveled is 300 miles, and the 
   total time is 8 hours, the average speed is 37.5 mph.

    Calculus students can solve the problem using the 
    formula for Average Value of a Function:

7. Assignment
   p. 440 (49, 50, 51, 53, 54)
   Mini Exam #3 Review Sheets  

Click here to go to the next page