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3.2 Rolles & Mean Value Theorem. 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. What does Rolle’s Thrm do?.
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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
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
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)
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!
Example 1 Continued • Therefore by Rolle’s Thrm,
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!
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!
Example 3 Continued • Therefore
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