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
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:
7. Assignment
p. 480 (3, 5, 6, 7, 8)
Mini Exam #3 Review Sheets
Click here to go to the next page