342
Chapter 4 | Applications of Derivatives
ground) change to allow the camera to record the flight of the rocket as it heads upward? (See Example 4.3 .) A rocket launch involves two related quantities that change over time. Being able to solve this type of problem is just one application of derivatives introduced in this chapter. We also look at how derivatives are used to find maximum and minimum values of functions. As a result, we will be able to solve applied optimization problems, such as maximizing revenue and minimizing surface area. In addition, we examine how derivatives are used to evaluate complicated limits, to approximate roots of functions, and to provide accurate graphs of functions. 4.1 | Related Rates Learning Objectives 4.1.1 Express changing quantities in terms of derivatives. 4.1.2 Find relationships among the derivatives in a given problem. 4.1.3 Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities. We have seen that for quantities that are changing over time, the rates at which these quantities change are given by derivatives. If two related quantities are changing over time, the rates at which the quantities change are related. For example, if a balloon is being filled with air, both the radius of the balloon and the volume of the balloon are increasing. In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities. Setting up Related-Rates Problems In many real-world applications, related quantities are changing with respect to time. For example, if we consider the balloon example again, we can say that the rate of change in the volume, V , is related to the rate of change in the radius, r . In this case, we say that dV dt and dr dt are related rates because V is related to r . Here we study several examples of related quantities that are changing with respect to time and we look at how to calculate one rate of change given another rate of change. Example 4.1 Inflating a Balloon
A spherical balloon is being filled with air at the constant rate of 2cm 3 /sec ( Figure4.2 ). How fast is the radius increasing when the radius is 3cm?
Figure 4.2 As the balloon is being filled with air, both the radius and the volume are increasing with respect to time.
Solution
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online