Average Value of a Function

1
Lesson 6-5

• Average Value of a Function

2
Icebreaker

• Evaluate
• Ponder If the slope of the secant represents
  the average rate of change over an interval, what
  corresponds, graphically, to the Average Value
  of the function over the interval?

3
Objectives

• Find the average value of a function

4
Vocabulary

• Average value average height (in the rectangle)
  associated with the area under the curve between
  two points

5
Average Value

• Average value of a function, f(x), on a,b is
• The integral represents the area under the curve
  and f(c) represents the height of rectangle that
  is equal to the area under the rectangle.
• Formerly called the Mean Value Theorem for
  Integrals

f(x)
b
c
a
6
6-5 Example 1
f(x) 3x² 2x on 1, 4
1 1 ------ ------ ? b
a 4 - 1
average value of f(x) ? 48 16
7
6-5 Example 2
f(x) sin x on 0, p
1 1 ------ ------ 1/p b
a p - 0
average value of f(x) 1/p (2) 2/p
8
6-5 Example 3
f(x) x² on 1, 4
1 1 ------ ------ ? b
a 4 - 1
average value of f(x) ? (21) 7 f(c) c²
7 c v7
9
6-5 Example 4
10
Summary Homework

• Summary
• Average value is like the average slope (secant
  line)
• Remember the FTC Part II
• Homework
• pg 467 1, 3, 4, 7, 9, 10, 17, 18