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

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

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)

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

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Numerically x f(x)f(x) f ' (x) Compute Difference Quotients

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

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

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

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

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

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

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

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

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

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 ).

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

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 ).