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

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

3. Objectives
Understand the relationship between differentiability and continuity.

4. Theorem 2.1
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

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

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. 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.
Most of the functions we study in calculus will be differentiable.

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