Mean-Value Theorem for Integrals

1. Mean-Value Theorem for Integrals Lesson 5.7

2. This is Really Mean

3. Mean Value for Derivatives • Given f(x) on interval [a,b] • Slope of secant through end points is • Theorem says there is at least one value, c for which

4. b a Mean Value for Derivatives • Theorem says there is at least one value, c for which c

5. c b a Mean Value for Integrals • We claim there is a c such that

6. Mean Value for Integrals • Since • We can solve for • The theorem does not give us the value of c • How can we find it?

7. Finding the Mean Value Point • We know the value • Set that value equal to f(x) … solve for x • Try f(x) = x2 + 4x + 1 on [0,2]Find the value of c so that

8. Example • f(x) = x2 + 4x + 1on [0,2]

9. Example • Consider on the interval [0.5, 2] • What is the area? • Draw the rectangle where

10. Example • Given • What is the average value of the function?

11. Mean Value in Modeling • If s(t) is the speed of the traffic at time t • If T(x) = temperature at time x

12. Mean Value in Modeling • Suppose the population of a city is given byfrom year 1900 (year 0 = 1900). • What is the average population for the years 1950 to 1960?

13. Assignment • Lesson 5.7 • Page 381 • Exercises: 3, 7, 9, 13, 15, 19, 21, 23, 27, 31, 33, 35