1. Section 3.1 The Derivative and the Tangent Line Problem

2. Remember what the notion of limits allows us to do . . .

4. Tangency

5. Instantaneous Rate of Change

6. The Notion of a Derivative Derivative • The instantaneous rate of change of a function. • Think “slope of the tangent line.”

7. Graphical Representation

8. So, what’s the point? f(x)

9. f(x)

10. f(x)

11. f(x)

12. Notation and Terminology

13. Example 1 (#2b) Estimate the slope of the graph at the points and .

14. Example 2 Find the derivative by the limit process (a.k.a. the formal definition).

15. Example 3 Find an equation of the tangent line to the graph of at the given point.

16. Graphs of and

17. Graphs of and (cont.)

18. Example 4 Use the alternative form of the derivative.

19. When is a function differentiable? • Functions are not differentiable . . . • at sharp turns (v’s in the function), • when the tangent line is vertical, and • where a function is discontinuous.

20. Example 5 Describe the -values at which is differentiable.