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
This can be seen in the following diagram
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.
| Rate | Time | Distance | |
|---|---|---|---|
| One-Way Trip | 30 mph | 5 hours | 150 miles |
| Return Trip | 50 mph | 3 hours | 150 miles |
| Total Trip | ? mph | 8 hours | 300 miles |
Calculus students can solve the problem using the formula for Average Value of a Function:
