When does a derivative exist? For a function, to be Differentiable at x=c: Left hand limit = Right hand limit of BOTH AND the function MUST be continuous.

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Presentation transcript:

When does a derivative exist? For a function, to be Differentiable at x=c: Left hand limit = Right hand limit of BOTH AND the function MUST be continuous at x = c Differentiability Implies Continuity If a function is differentiable at x=c then it MUST be continuous at x=c BUT if a function is continuous it does not have to be differentiable. S. Evans

Differentiability Ex. 1: For the following piecewise function determine if the function is differentiable at a) x=6 b) x= 4 c) x= 2 d) x=0 S. Evans

Differentiability a) Since for x=6.. Therefore, is differentiable at x = 6. S. Evans

Differentiability b) At x = 4.. BUT is NOT continuous at x = 4 Therefore, is NOT differentiable at x = 4. S. Evans

Differentiability c) At x = 2.. Therefore, is NOT differentiable at x = 2. d) At x = 0 Therefore, is NOT differentiable at x = 2. S. Evans