1. Rolle’s and The Mean Value Theorem BC Calculus

2. Mean Value and Rolle’s Theorems The Mean-Value Theorem ( and its special case ) Rolle’s Theorem are Existence Theorems - - - The basis of many other concepts Existence Theorems insure the existence of one of more numbers having a specific property. They DO NOT identify the point...... [instead - - - It leads to attempts to find the value guaranteed by the theorems]

4. Existence Theorems: Completeness Postulate and Exponents Zero Locator Theorem - Intermediate Value Theorem If f has the following values:

5. Mean Value (Very Important) IF f (x) is: 1. Continuous on [a,b], and 2. Differentiable on (a,b) THEN There exists a point c in (a,b) such that *LAYMAN: The slope of the tangent at c equals the slope of the secant through f (a), and f (b) *[The instantaneous rate of change equals the average rate of change]

6. Example 1: Mean Value Theorem Determine whether satisfies the conditions of the Mean Value Theorem on [ 0, 2] Determine whether satisfies the conditions of the Mean Value Theorem on 1) Is continuous 2) Not differential -- we have a cusp

7. Example 2: M V T Find the “ c ” guaranteed by the Mean Value Theorem. 1) cont.? 2) diff? Part 2: (find it)

8. Example 3: M V T Find the “ c ” guaranteed by the Mean Value Theorem. > Step 1: cont.? diff.?

9. Example 4: MVT Two police patrol a highway with a 70 mph speed limit. The cars have radar and are in radio contact. They are stationed 5 miles apart. As a truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car its speed is clocked at 50 mph. The second patrolman pulls the truck over and issues a citation for excessive speed. WHY? Cont? Diff? Step 1: d=rt

10. Rolle’s ROLLE’S THEOREM: IF f (x) is 1. Continuous on closed interval [a,b], 2. Differentiable on (a,b), and 3. f (a) = f (b) THEN: There exists at least one pt. “c”in (a,b) Such that f / (c) = 0

11. Example 1: Rolle’s Theorem Find the “ c” guaranteed by Rolle’s Therorem. Cont? Diff? Step 1: MVT

12. Example 2: Rolle’s Theorem Show that satisfies the conditions of Rolle’s Theorem on [ 1, 2] Cont? Diff? Step 1:

13. Example 3: Rolle’s Theorem Find the “ c” guaranteed by Rolle’s Theorem. Cont? Diff? Step 1: Rolle’s MVT b/c 0 Rolle’s

14. Example 3: Rolle’s Theorem Find the “ c” guaranteed by Rolle’s Theorem. Cont? Diff? Step 1:

15. EXTRA: Connection to APPROXIMATION

16. EXTRA: Connection to L’Hopital’s Rule

17. Last Update: 12/10/07