3.2 Rolles & Mean Value Theorem

1. 3.2 Rolles& Mean Value Theorem

2. Rolle’s Theorem Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If Then there is at least one number c in (a,b) such that f’(c)=0

3. What does Rolle’s Thrmdo? • Rolle’s theorem states some x value exists (x=c) so that the tangent line at that specific x value is a horizontal tangent (f’(c)=0) Horizontal Tangent Line ie: f’(c)=0 f(a)=f(b) a c b

4. Notes about Rolle’s Thrm • It is an EXISTENCE Theorem, it simply states that some c has to exist. It does NOT tell us exactly where that value is located. • In order to find the location x=c, we would take f’(x)=0 and find critical numbers like in section 3.1 (Extremaon a closed Interval)

5. Example 1 of Rolle’s Thrm • Determine whether Rolle’s thrm can be applied. If it can be applied, find all values of c such that f’(c)=0 • Ex: • Since f is a polynomial, it is continuous on [1,4] and differentiable (1,4). • Therefore, Rolle’s Theorem can be applied and states there must be some x=c on [1,4] such that f’(c)=0. • Lets find those x values!

6. Example 1 Continued • Therefore by Rolle’s Thrm,

7. Example 2 of Rolle’s Thrm • Determine whether Rolle’s thrm can be applied. If it can be applied, find all values of c such that f’(c)=0 • Ex: • f is continuous on [-2,3] • f is not differentiable on (-2,3) • ROLLE’s cannot be used!

8. Example 3 of Rolle’s Thrm • Determine whether Rolle’s thrm can be applied. If it can be applied, find all values of c such that f’(c)=0 • Ex: • f is continuous on • f is differentiable on • ROLLE’S APPLIES!

9. Example 3 Continued • Therefore

10. Mean Value Theorem • If f is continuous on [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) s.t. f(b) f(a) b a

11. MVT Examplepg 177 #37 parts a,b,c,d