1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.1: Linear Approximation and Applications

2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In this chapter, we learn to use derivatives to solve optimization problems. The honey comb in a beehive is designed to minimize the amount of wax needed, probably without the use of calculus

3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In some cases, we are interested in how a small change in the independent variable will affect the dependent variable. In many of these case, a linear approximation will suffice.

4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated below, the linear approximation uses the slope of the tangent line and the change in x to calculate the approximate change in f. As can be seen below, the large Δ x becomes the larger the error.

5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Suppose, instead of a pizza, Figure 3 represented the cross- section of the center of a sphere. What would be the loss or gain of volume and surface area if the diameter varied by 0.4 in?

6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 4, the tangent line gives a good approximation of the curve in a small neighbor of the point of tangency. This permits us to estimate values of a function proximate to the point of tangency linearization.

7. Example, Page 217 Estimate the quantity using Linear Approximation and find the error using a calculator. 16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Sometimes, the difference between the estimate and the actual value as a percentage of the actual is more important than the magnitude of the difference. In these cases, calculate the percentage error using the following:

9. Example, Page 217 Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error. 10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Example, Page 217 Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error. 14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company We may us the following equation to quantify the maximum possible error in a linearization or the error bound. In the equation, E is the error bound, and K is the maximum value of f ″ on the interval (a, a + h) where h = Δx. We can see from the equation that the smaller h is, the small the error bound.

13. Homework Homework Assignment #22 Read Section 4.2 Page 217, Exercises: 1 – 65 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company