1. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 2.2 The Derivative Function

2. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Estimating Derivatives Graphically Numerically Analytically

3. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Rate of Change Average rate of change is a difference quotient. If y = f (x)

4. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Figure 2.2Figure 2.3

5. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Numerically x05101520 f(x)f(x)10070554640 f ' (x) Compute Difference Quotients -6 -4.5-2.4-1.5 -1.2

6. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Analytically Estimate f ′ (2)

7. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Graphically

8. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Graphically Problem 1

9. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Graphically Problem 2

10. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Graphically Problem 3

11. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Graphically Problem 4

12. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Box on page 107 Terminology

13. Which of the following graphs (a)-(d) could represent the slope at every point of the function graphed in Figure 2.6? Example

14. Which of the following graphs (a)-(d) could represent the slope at every point of the function graphed in Figure 2.8? Example

15. What can the derivative tell us? Below is a graph of the derivative of f (x) Example 1.Where is f ( x ) increasing? 2.Where is f ( x ) decreasing? 3.If f (0)=0, sketch a graph of f ( x ).

16. What can the derivative tell us? Below is a graph of the derivative of f (x) Example

17. What can the derivative tell us? Below is a graph of the derivative of g (x) 1.Where is g ( x ) increasing? 2.Where is g ( x ) decreasing? 3.If g (0)=0, sketch a graph of g ( x ).

18. Example What can the derivative tell us? Below is a graph of the derivative of g (x) 1.Where is g ( x ) increasing? 2.Where is g ( x ) decreasing? 3.If g (0)=0, sketch a graph of g ( x ).