Differentiability and the Mean Value Theorem

GRE Math Subject · Axiom Academy

LESSON Differentiability & the Mean Value Theorem Rolle's theorem, MVT, L'H pital's rule, and Taylor's theorem with remainder Step 1: Differentiability — Definition and Implications Definition: A function is differentiable at if the limit exists and is finite. When it exists, this limit is denoted . Theorem: If is differentiable at , then is continuous at . The converse is false. The classic counterexample is , which is continuous everywhere but not differentiable at (the left and right derivatives disagree). GRE Trap: The Weierstrass function is continuous everywhere but differentiable nowhere . The GRE occasionally tests whether you know such pathological functions exist. Differentiability on an interval: We say is differentiable on if exists for every . On a closed interval , we require one-sided derivatives at the endpoints. Step 2: Rolle's Theorem and the Mean Value Theorem Rolle's Theorem: If is continuous on , differentiable on , and , then there exists such that . Rolle's theorem is a special case of the more powerful result: Mean Value Theorem (MVT): If is continuous on and differentiable on , then there exists such that: Geometric interpretation: There is at least one point where the instantaneous rate of change equals the average rate of change over the interval. GRE Application: "If for all x, and , what can you conclude about ?" By MVT: for some . Since , we get , so . L'H pital's Rule: If is of the indeterminate form or , and if exists (or is ), then:

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