1. AP CALCULUS AB Chapter 3: Derivatives Section 3.2: Differentiability

2. What you’ll learn about  How f’(a) Might Fail to Exist  Differentiability Implies Local Linearity  Derivatives on a Calculator  Differentiability Implies Continuity  Intermediate Value Theorem for Derivatives … and why Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists.

3. How f’(a) Might Fail to Exist

8. Example How f’(a) Might Fail to Exist

9. How f’(a) Might Fail to Exist Most of the functions we encounter in calculus are differentiable wherever they are defined, which means they will not have corners, cusps, vertical tangent lines or points of discontinuity within their domains. Their graphs will be unbroken and smooth, with a well-defined slope at each point.

10. Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability.

11. Differentiability Implies Local Linearity

12. Section 3.2 - Differentiability  You try: Find all points in the domain of f(x) where f is not differentiable: 1. f(x) = |x – 3| + 4 2.

13. Derivatives on a Calculator

14. Section 3.2 - Differentiability  Derivatives on a Calculator: 1. On the TI-83, TI-83+ or TI-84 Use Nderiv (expression, x, x-value) Example: 2. On the TI-89 1. Use d (expression, x, order)| x= x-value Example: 2. Use Nderiv (expression, x) | x = x-value Example:

15. Example Derivatives on a Calculator

16. Derivatives on a Calculator Because of the method used internally by the calculator, you will sometimes get a derivative value at a nondifferentiable point. This is a case of where you must be “smarter” than the calculator.

17. Section 3.2 - Differentiability  You try: Compute each numerical derivative: 1. NDERIV 2. NDERIV

18. Differentiability Implies Continuity The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point.

19. Intermediate Value Theorem for Derivatives Not every function can be a derivative. If f is differentiable on the interval (a, b) and a < c < b, then f’(a) < f’(c) < f’(b).