Chapter 12

# Your Problems Are Solved: Differentiation to the Rescue!

In This Chapter

Getting the most bang for your buck — optimization problems

Position, velocity, and acceleration — VROOOOM

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.

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