1. Curve Sketching with Radical Functions
Today we will use number line studies of the first and second derivatives to sketch graphs of radical functions.

2. Continuity
The intervals for continuity are all x-values where there are no “breaks”.
The intervals for continuity for radical functions will be the same as the intervals for the domain.

3. Discontinuity
The place we will see discontinuity occurring most often is where a function is undefined.
Radical functions often have infinite areas of discontinuity.
To help determine these intervals, we look at the sign study for f(x).

4. Differentiability
A function is differentiable if its derivative exists.
A function is not differentiable if it’s first derivative is not defined at the point (or over the interval).
For radical functions, intervals or points of non-differentiability happen at discontinuities, vertical tangents or cusps.
To help determine intervals of differentiability, look at the sign study for f ′(x)

5. Most Common Points of Non-Differentiability
Cusps
Vertical Tangents
Points of Discontinuity
*Look at the first derivative sign study and take out any values (or intervals) where the first derivative is undefined.

6. Cusps and Vertical Tangents for Radical Functions
A cusp occurs when f ′(x) is undefined and the first derivative sign study shows opposite signs around the undefined point.
A vertical tangent occurs when f ′(x) is undefined and the first derivative sign study shows either the same sign on both sides, or is undefined on one side, of the point.

7. Generalizations Continuous is not necessarily differentiable.
Differentiable is always continuous.

8. Using sign studies to sketch graphs of radical functions – from scratch . . . .

9. Using sign studies to sketch graphs of radical functions.

11. Using sign studies to sketch graphs of radical functions – from scratch . . . .

12. Using sign studies to sketch graphs of radical functions.

14. Using sign studies to sketch graphs of radical functions – from scratch . . . .

16. Assignment
A 3.11
Day 2: #1-5 parts h-k