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.” Definition of the Derivative of a Function (p. 119)

7. Graphical Representation

8. f(x) So, what’s the point?

9. f(x)

12. Notation and Terminology Terminology differentiation, differentiable, differentiable on an open interval (a,b) Differing Notation Representing “Derivative”

13. Example 1 (#2b)

14. Example 2

15. Example 3

18. Example 4 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. Theorem 3.1 Differentiability Implies Continuity

20. Example 5