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

2. Calculus, Section 3.22

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

6. 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

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

8. 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.

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

10. Calculus, Section 3.210 What is not differentiable? Functions with corners Why?

11. Calculus, Section 3.211 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.

12. Calculus, Section 3.212 What is not differentiable? Functions with discontinuities Vertical tangents. Why? Vertical tangent have slopes that are undefined.

13. Calculus, Section 3.213 Th’m If f is differentiable, then it is continuous

14. Calculus, Section 3.214 Assignment Section 3.2, 1-29, odd