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Understanding Rolle's Theorem: Conditions and Applications in Calculus

Rolle's Theorem states that if a function f is continuous on [a,b] and differentiable on (a,b), there exists at least one number c in (a,b) such that f’(c) = 0. This theorem ensures the existence of at least one horizontal tangent line between the endpoints of the interval. It's important to note that while the theorem guarantees the existence of such a c, it does not identify its exact location. Through several examples, we can determine when the theorem applies and find values of c that satisfy f’(c) = 0.

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Understanding Rolle's Theorem: Conditions and Applications in Calculus

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  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

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