MAT 3749 Introduction to Analysis

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Presentation transcript:

MAT 3749 Introduction to Analysis Section 2.3 Part III The Mean Value Theorem http://myhome.spu.edu/lauw

Important Result a b

Preview Extreme Value Theorem Fermat’s Theorem Rolle’s Theorem The Mean Value Theorem

References Section 2.3

Maximum Value

Local Maximum

T or F An absolute max is a local max.

The Extreme Value Theorem

Fermat’s Theorem

Lemma (HW)

Fermat’s Theorem

Conceptual Diagrams

Fermat’s Theorem

Fermat’s Theorem

Fermat’s Theorem

Fermat’s Theorem

Proof

Proof

Proof

Proof

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Rolle’s Theorem

Proof

Proof

The Mean Value Theorem

Proof

The Mean Value Theorem

The Mean Value Theorem

The Mean Value Theorem

Theorem (Consequence) If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b). Q: Can we apply the MVT directly?

Corollary (Important) b

Corollary (Important) b