1. Chapter 3
Derivatives
Differentiability

2. Bell Ringer Review

3. What you’ll learn about
Why f ′(a) might fail to exist at x = a
Differentiability implies local linearity
Numerical 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.

4. How f ′(a) Might Fail to Exist

5. How f ′(a) Might Fail to Exist

6. How f ′(a) Might Fail to Exist

7. How f ′(a) Might Fail to Exist

8. How f ′(a) Might Fail to Exist

9. Example How f ′(a) Might Fail to Exist

11. Solutions

13. Solution

15. Solution

17. 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.

18. Differentiability Implies Continuity
Also note that…
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.

20. Solutions

21. Multiple Choice

22. Solution

23. 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.

24. Differentiability Implies Local Linearity

25. Numerical Derivatives on a Calculator

27. Numerical Derivatives on a Calculator
The numerical derivative of f at a, which we will denote NDER is the number
The numerical derivative of f, which we will denote NDER is the function

28. Example Derivatives on a Calculator

29. TI-89 Calculator

30. Practice

31. Solution

32. Solution

33. 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.

34. Intermediate Value Theorem for Derivatives
Not every function can be a derivative.

35. More to Know…

36. Solution

37. Solution