Chapter 7
Differentiation Problems
IN THIS CHAPTER
Position, velocity, and acceleration — VROOOOM
Related rates — brace yourself
Standing in line for linear approximations
In this chapter I finally get around to showing you how to use calculus to solve some practical problems.
Optimization Problems
One practical use of differentiation is finding the maximum or minimum value of a real-world function: the maximum output of a factory or range of a missile, the minimum time to accomplish some task, and so on. Here’s an example.
The maximum area of a corral
A rancher can afford 300 feet of fencing to build a corral that’s divided into two equal rectangles. See Figure 7-1. What dimensions will maximize the corral’s area?
1.a. Express the thing you want maximized, the area, as a function of the two unknowns, x and y.
In some optimization problems, you can write the thing you’re trying to maximize or minimize as a function of one variable — which is always what you want. But here, the area is a function of two variables, so Step 1 has two additional substeps.
1.b. Use the given information ...
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