1. Drill Tell whether the limit could be used to define f’(a).

2. LESSON 3.2 HW: PAGE 114: 2, 4, 10-22 (EVEN), 31-35 Differentiability

3. Objectives Students will be able to  find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents.  approximate derivatives numerically and graphically.

4. Where Might a Derivative Not Exist? At a corner  Where the one-sided derivatives differ  Example: y = |x|; there is a corner at x = 0 At a cusp  Where the slopes of the secant lines approach ∞ from one side and -∞ from the other  Example: y = x 2/3, there is a cusp at x = 0

5. Where Might a Derivative Not Exist? At a vertical tangent  Where the slopes of the secant lines approach either ±∞ from both sides  Example: f(x) = x 1/3 At a discontinuity  Which will cause one or both of the one-sided derivatives be non- existent  Example: a step function

6. Compare the right-hand and left-hand derivatives to show that the function is NOT differentiable.

7. The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? (-3. 2) (2. -2)

8. The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? (-3. 0) (3. 0)

9. Derivatives on a Calculator For small values of h, the difference quotient is often a good numerical approximation of f’(a). However, using the same value of h will actually yield a much better approximating if we use the symmetric difference quotient: Numerical Derivative at f at some point a: NDER

10. Example: Computing a Numerical Derivative Compute NDER (x 2, 3). Using h = 0.001 and

11. Numerical Derivatives on the TIs MATH nDeriv(function, variable, number)

12. Example 2 More Numerical Derivatives Compute each numerical derivative.

13. Theorems Theorem 1: If f has a derivative at x = a, then f is continuous at x = a Theorem 2: 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)  Example: Does any function have the Unit Step Function as its derivative?  No, choose some a = -.5 and some b =.5, Then U(a) = -1 and U(b) = 1, but U does NOT take on any value between -1 and 1.