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

2. The honey comb in a beehive is designed to minimize the amount of
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
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

3. variable will affect the dependent variable. In many of these case, a
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.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

4. 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.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

5. would be the loss or gain of volume and surface area if the diameter
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?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

6. us to estimate values of a function proximate to the point of tangency
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.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

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. 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

9. 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:
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. 10.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

11. 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

12. 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

13. 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

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

15. 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.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

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