Chapter 12

Your Problems Are Solved: Differentiation to the Rescue!

In This Chapter

arrow Getting the most bang for your buck — optimization problems

arrow Position, velocity, and acceleration — VROOOOM

arrow Related rates — brace yourself

In the Introduction, I argue that calculus has changed the world in countless ways, that its impact is not limited to Ivory Tower mathematics, but is all around us in down-to-earth things like microwave ovens, cell phones, and cars. Well, it’s now Chapter 12, and I’m finally ready to show you how to use calculus to solve some practical problems.

Getting the Most (or Least) Out of Life: Optimization Problems

One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function: the maximum output of a factory, the maximum strength of a beam, the minimum time to accomplish some task, the maximum range of a missile, and so on. Let’s get started with a couple standard geometry examples.

The maximum volume of a box

A box with no top is to be manufactured from a 30-inch-by-30-inch piece of cardboard by cutting and folding it, as shown in Figure 12-1.

Figure 12-1: The box is made from a 30"-by-30" piece of cardboard by cutting ...

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