1. 3.2 Differentiability Arches National Park - Park Avenue
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

2. North Window Arch
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

3. Balanced Rock Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2010

4. Delicate Arch Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2010

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

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

7. Three theorems:
If f has a derivative at x = a, then f is continuous at x = a.
Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

8. Intermediate Value Theorem
If a function is continuous between a and b, then it takes on every value between and
Because the function is continuous, it must take on every y value between and

9. p Intermediate Value Theorem for Derivatives
If a and b are any two points in an interval on which f is differentiable, then takes on every value between and
Between a and b, must take on every value between and . p