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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.
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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 any discontinuity cusp
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) = x3at x = 2.
differentiability • THEOREM: • If f has a derivative at x = a, then f is continuous at x = a. Differentiability implies continuity.
