Sec 2.8: THE DERIVATIVE AS A FUNCTION

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Presentation transcript:

Sec 2.8: THE DERIVATIVE AS A FUNCTION replace a by x

Sec 2.8: THE DERIVATIVE AS A FUNCTION Slopes : 0 + -

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION

Sec 2.8: THE DERIVATIVE AS A FUNCTION Definition: A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval.

Sec 2.8: THE DERIVATIVE AS A FUNCTION Continuity VS Differentiability Continuity: Measure if the function continues Differentiability: Measure if the function smooth Example:

Sec 2.8: THE DERIVATIVE AS A FUNCTION continuity 2 properties differentiability Proof: Remark: f cont. at a f diff. at a Remark: f discont. at a f not diff. at a Remark: f not diff. at a f discont. at a

Sec 2.8: THE DERIVATIVE AS A FUNCTION f cont. at a f diff. at a f discont. at a f not diff. at a f not diff. at a f discont. at a Example:

HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE? Sec 2.8: THE DERIVATIVE AS A FUNCTION HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?

Higher Derivative Sec 2.8: THE DERIVATIVE AS A FUNCTION Note: velocity acceleration jerk

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