Read this section to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1-3.
Practice 1: The surface area of the cylinder is . From the Example, we know that and , and we want to know how fast the surface area is changing when and .
Practice 2: The volume of the cylinder is area of the bottom) (height . We are told that , and that , and
Practice 3: Fig. 23 represents the situation described in this problem. We are told that The variable represents the distance of the fish from the angler, and we are asked to find , the rate of change of when .
Fortunately, the problem contains a right triangle so there is a formula (the Pythagorean formula) connecting and so
We could also find ' implicitly: so, differentiating each side,
Then we could use the given values for and ' and value of (found using the Pythagorean formula) evaluate .