1. 3.2 Differentiability Objectives Students will be able to: 1)Find where a function is not differentiable 2)Distinguish between corners, cusps, discontinuities, and vertical tangents 3)Approximate derivatives numerically and graphically

2. Recall:

3. Cases Where a Function is Not Differentiable 1)A corner 2) A cusp 3) A vertical tangent 4) A discontinuity

4. A Corner A corner is a location where the one-sided derivatives differ. Example: Absolute Value 1 Absolute Value 2

5. A Cusp A cusp is where the slopes of the secant lines approach infinity from one side and negative infinity from the other side. It is an extreme case of a corner. Example: Cusp

6. A Vertical Tangent A vertical tangent occurs when the slopes of the secant lines approach either infinity or negative infinity from both sides. Example: Vertical Tangent

7. A Discontinuity A discontinuity will cause one or both of the one-sided derivatives to be nonexistent. Example: Jump Discontinuity (FF to 3:30)

8. Review: Types of Discontinuity Types of Discontinuity Another type of discontinuity is oscillating discontinuity. Oscillating discontinuity occurs when the function oscillates and the values of the function appear to be approaching two or more values simultaneously. Example: f(x)=sin(1/x)

9. Local Linearity Differentiability implies local linearity. A function that is differentiable at a point (a,f(a)) is locally linear at that point, meaning that as you “zoom” in on that point, the function will resemble a straight line. Local Linearity

10. Derivatives on a Calculator Luckily, the nSpire can calculate derivatives, as well as a derivative at a point. Derivatives on the nSpire

11. Try a few to see if you got it! – Find the following derivatives on your nSpire:

12. Continuity Very important note: Differentiability implies continuity. Another important note: the converse is NOT true!!! A function may be continuous at x=a, but not differentiable there. Example: the absolute value function is continuous at x=0 but it is not differentiable there.

13. Intermediate Value Theorem for Derivatives If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b). What does that even mean??? – Intermediate Value Theorem Intermediate Value Theorem