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Chapter 3 Derivatives Section 3.2 Differentiability.

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Presentation on theme: "Chapter 3 Derivatives Section 3.2 Differentiability."— Presentation transcript:

1 Chapter 3 Derivatives Section 3.2 Differentiability

2 Quick Review

3 Quick Review

4 Quick Review Solutions

5 Quick Review Solutions

6 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.

7 How f ′(a) Might Fail to Exist

8 How f ′(a) Might Fail to Exist

9 How f ′(a) Might Fail to Exist

10 How f ′(a) Might Fail to Exist

11 How f ′(a) Might Fail to Exist

12 Example How f ′(a) Might Fail to Exist

13 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.

14 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.

15 Differentiability Implies Local Linearity

16 Numerical Derivatives on a Calculator

17 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

18 Example Derivatives on a Calculator

19 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.

20 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.

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

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