Warm-Up: October 19, 2012

1. Warm-Up: October 19, 2012 • Given the graph of f(x), graph f’(x)

2. Differentiability Section 3.2

3. No Derivative • The derivative does not exist for any of the following: • Corner • Cusp • Vertical Tangent • Discontinuity

4. Corner • When the one-sided derivatives differ. • Example: f(x) = |x| at x=0

5. Cusp • Extreme case of a corner where one sided derivatives approach ±∞ (one side positive, one side negative) • Example: f(x) = x2/3 at x=0

6. Vertical Tangent • Both one-sided derivatives approach ∞ or both one-sided derivatives approach -∞ • Example: at x=0

7. Discontinuity • Any point of discontinuity is not differentiable

8. Local Linearity • Local linearity means that in a small interval, the graph is close to a straight line. • On a graphing calculator, zoom in repeatedly to check local linearity. • If a graph is locally linear near a point, then it is differentiable at that point.

9. Derivatives on TI-83 • [MATH] [8:nDeriv(] • nDeriv(f(x), x, a) • NOT ALWAYS CORRECT

10. Example 1 • Find the following using TI-83:

11. Symmetric Difference Quotient • TI-83 calculates numerical derivatives using a symmetric difference quotient: • This is the same as f’(a) when the derivative exists. • TI-83 uses h=0.001

12. Graphing Derivatives • Y1=nDeriv(_______,X,X) • Graph the derivatives of the following functions and try to identify the equation of the derivative:

13. Differentiability Implies Continuity • Theorem 1: • If f has a derivative at x=a, then f is continuous at x=a. • Proof on page 110.

14. Intermediate Value Theorem • Theorem 2 – 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).

15. Assignment • Read Section 3.2 (pages 105-110) • Page 111 Exercises #1-22 all • Read Section 3.3 (pages 112-119)

17. Warm-Up: October 22, 2012 • Calculate the following derivatives, or state that they do not exist. You may use your graphing calculator.

18. Homework Questions?

19. Exploration Activity • Is either of these functions differentiable at x=0? • Graph f and g together in a standard viewing window. How do they compare? • Turn off the graph of g. Graph f and zoom in on (0,1) several times. Does the graph show signs of straightening out? • Turn off the graph of f and turn on the graph of g. Return to a standard viewing window, and then zoom in on (0.1). • How many zooms until glooks horizontal?