Sec 2 8 THE DERIVATIVE AS A FUNCTION

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 discont. at a f not diff. at a f discont. at a

Sec 2. 8: THE DERIVATIVE AS A FUNCTION f cont. at a f discont. at a f not diff. at a Example: f diff. at a f not diff. at a f discont. at a

Sec 2. 8: THE DERIVATIVE AS A FUNCTION HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?

Sec 2. 8: THE DERIVATIVE AS A FUNCTION Higher Derivative Note: velocity acceleration jerk

R

H R

H

H H

R R H

H H R H