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In Section 3.1 we learned about extreme values — the largest and smallest values a function attains on an interval. We motivated our interest in such values by discussing how it made sense to want to know the highest/lowest values of a stock, or the fastest/slowest an object was moving. In this section we apply the concepts of extreme values to solve “word problems,” i.e., problems stated in terms of situations that require us to create the appropriate mathematical framework in which to solve the problem.

We start with a classic example which is followed by a discussion of the topic of optimization.

Example4.3.1Optimization: perimeter and area

A man has \(100\) feet of fencing, a large yard, and a small dog. He wants to create a rectangular enclosure for his dog with the fencing that provides the maximal area. What dimensions provide the maximal area?


This example is very simplistic and a bit contrived. (After all, most people create a design then buy fencing to meet their needs, and not buy fencing and plan later.) But it models well the necessary process: create equations that describe a situation, reduce an equation to a single variable, then find the needed extreme value.

In “real life,” problems are much more complex. The equations are often not reducible to a single variable (hence multi-variable calculus is needed) and the equations themselves may be difficult to form. Understanding the principles here will provide a good foundation for the mathematics you will likely encounter later.

We outline here the basic process of solving these optimization problems.

Key Idea4.3.3Solving Optimization Problems

  1. Understand the problem. Clearly identify what quantity is to be maximized or minimized. Make a sketch if helpful.

  2. Create equations relevant to the context of the problem, using the information given. (One of these should describe the quantity to be optimized. We'll call this the fundamental equation.)

  3. If the fundamental equation defines the quantity to be optimized as a function of more than one variable, reduce it to a single variable function using substitutions derived from the other equations (we'll call these constraint equations).

  4. Identify the domain of this function, keeping in mind the context of the problem.

  5. Find the extreme values of this function on the determined domain.

  6. Identify the values of all relevant quantities of the problem.

We will use Key Idea 4.3.3 in a variety of examples.

Example4.3.4Optimization: perimeter and area

Here is another classic calculus problem: A woman has a 100 feet of fencing, a small dog, and a large yard that contains a stream (that is mostly straight). She wants to create a rectangular enclosure with maximal area that uses the stream as one side. (Apparently her dog won't swim away.) What dimensions provide the maximal area?


Keep in mind as we do these problems that we are practicing a process; that is, we are learning to turn a situation into a system of equations. These equations allow us to write a certain quantity as a function of one variable, which we then optimize.

Example4.3.6Optimization: minimizing cost

A power line needs to be run from an power station located on the beach to an offshore facility. Figure 4.3.7 shows the distances between the power station to the facility.

It costs $50/ft. to run a power line along the land, and $130/ft. to run a power line under water. How much of the power line should be run along the land to minimize the overall cost? What is the minimal cost?

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Figure4.3.7Running a power line from the power station to an offshore facility with minimal cost in Example 4.3.6.

In the exercises you will see a variety of situations that require you to combine problem-solving skills with calculus. Focus on the process; learn how to form equations from situations that can be manipulated into what you need. Eschew memorizing how to do “this kind of problem” as opposed to “that kind of problem.” Learning a process will benefit one far longer than memorizing a specific technique.

The Section 4.4 introduces our final application of the derivative: differentials. Given \(y=f(x)\text{,}\) they offer a method of approximating the change in \(y\) after \(x\) changes by a small amount.


Terms and Concepts