Section 3.2 The Derivative as a Function AP Calculus September 24, 2009 Berkley High School, D2B2.

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Presentation transcript:

Section 3.2 The Derivative as a Function AP Calculus September 24, 2009 Berkley High School, D2B2

Calculus, Section 3.22

3

4

5 Definition of Derivative We call this derivative of a function f. We use notation f’(x) (f prime of x)

Calculus, Section 3.26 Warning: Nasty Algebra Ahead Many of the homework problems require a great attention to detail. Many terms, many negative signs, and lots of places to make a mistake. Take your time, pay attention, use lots of paper, be patient

Calculus, Section 3.27 Example (Yesterday’s and Today’s)

Calculus, Section 3.28 What is differentiable? A function f is differentiable at a if f’(a) exists. (Remember f’(a) is really a limit.) A function f is differentiable on an open interval (a,b) if f is differentiable for every number in the interval.

Calculus, Section 3.29 What is not differentiable? Functions with corners Why?

Calculus, Section What is not differentiable? Functions with corners Why?

Calculus, Section What is not differentiable? Functions with corners Because the limit is undefined at the corner. The limit is undefined because the left side limit and right side limit don’t agree.

Calculus, Section What is not differentiable? Functions with discontinuities Vertical tangents. Why? Vertical tangent have slopes that are undefined.

Calculus, Section Th’m If f is differentiable, then it is continuous

Calculus, Section Assignment Section 3.2, 1-29, odd