3.2 Differentiability.

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Presentation transcript:

3.2 Differentiability

Differentiability A function is differentiable at point c if and only if the derivative from the left of c equals the derivative from the right of c. AND if c is in the domain of f’.

Differentiability Find the derivative of at x = 0. f is not differentiable at 0.

DIFFERENTIABILITY A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. No “sudden change” in slope.

DIFFERENTIABILITY Derivatives will fail to exist at: corner vertical tangent cusp any discontinuity

Using the calculator The numerical derivative of f at a point a can be found using NDER on the calculator. Syntax: NDER (f(x), a) Note: The calculator uses h = 0.001 to compute the numerical derivative, so it is a close approximation to the actual derivative. Example: Compute NDER of f(x) = x3 at x = 2.

differentiability THEOREM: Differentiability implies continuity. If f has a derivative at x = a, then f is continuous at x = a. Differentiability implies continuity.