1. Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part 2)

2. Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park

3. Objectives Understand the relationship between differentiability and continuity.

4. If f is differentiable at x = c, then f is continuous at x = c. Differentiability implies continuity. If a function is NOT continuous at x=c, then it is NOT differentiable. Is the converse true? No Theorem 2.1

5. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

6. Graph with a Sharp Turn f '(2) DNE (the tangent line is not unique)

7. Graph with a Sharp Turn The graph is continuous at x=0, but f ' (0) DNE.

8. Most of the functions we study in calculus will be differentiable. So f ' (x) does NOT exist (or f is not differentiable) if the graph has a sharp corner or turn, a vertical tangent line, or a discontinuity.

9. Homework 2.1 (page 102) #33, 39-42 all, 61, 67-85 odd