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. Example, Page 217 Use the Linear Approximation to estimate Δf = f (3.02) – f (3) for the given function. 2. f (x) = x 4 How accurate is the estimate? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Example, Page 217 34. If the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a bus company takes in a monthly revenue of R(x) = 1.5x – 0.01x 2 (in thousands of dollars). (a) Estimate the change in revenue if the price rises from $50 to $53. (b) Suppose x = 80. How will revenue be affected by a small change in price? Explain using Linear Approximation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The accuracy of the cable transducer is affected by temperature changes. For temperatures below those at which the properties of the cable change, a linear relationship may be assumed. If the cable is measured at an ambient temperature of 120ºF, estimate the increase in length of an 85-in throttle cable at 123ºF, if the coefficient of thermal expansion is 9.6 x 10 –6 ºF –1.

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

9. Example, Page 217 32. A spherical balloon has a radius of 6-in. Estimate the change in volume and surface area if the radius increases by 0.3-in. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company