Differential Calculus: From Practice to Theory

While it would not be correct to say that Calculus was invented solely to solve optimization problems, it’s pretty close. Once it was invented, of course, Calculus turned out to be a very useful tool for lots of other problems as well but it began with optimization.

Recall that one of the fundamental tenets of modern science is that nature is lazy. Light follows the fastest (minimal) path between two points. Soap bubbles are spherical because that is the shape that encloses the most (maximal) volume for a given surface area. Alternatively, a soap bubble uses the least (minimal) surface area to enclose a particular volume. Because it is lazy, nature optimizes.

Mathematically speaking all optimization starts in the same place: We have some quantity (revenue, tax liability, temperature, resource consumption, travel time, potential energy, volume, surface area, whatever), which depends on one or more variables. The variables are (usually) constrained in some way, and we need to find conditions that make the quantity as large, or as small as possible within those constraints. The “thing to be optimized” is called the objective function and the constraints on the variables are called . . . well, they’re called the constraints.

When we looked at Fermat’s Method of Adequality in Section 3.4 we observed that if \(f(x)\) is a (continuous) function then near an optimal point \(f(x+h)\approx f(x)\) as long as \(h\) is not too big. Loosely speaking, this says that near an extremum the graph of the function is practically horizontal. More precisely (and more formally) we have:

In view of Fermat’s Theorem it appears that a likely strategy for attacking optimization problems is to find an expression, say \(f(x)\text{,}\) for the objective function and then solve the equation \(\dfdx{f}{x} = 0\text{.}\) In fact, this is precisely the strategy we used in Section 5.4 to find the maximal height attained by a tossed ball. If we toss a ball upward we know that it will reach a maximal height. At the top of its flight, when the ball transitions from upward movement to downward movement, there is a moment when its velocity is zero. Thus we saw that if \(p(t)\) is a formula that gives the height of the ball at any time, setting \(\dfdx{p}{t}=0\) and solving for \(t\) will yield the time when the ball is at the top of its flight.

At their simplest, optimization problems really are just that straightforward. But “straightforward” does not necessarily mean easy. There are several nuances that, if not clearly understood, can cause much confusion and difficulty. We will proceed carefully.

To begin, take careful note of what Fermat’s Theorem says and, just as important, what it does not say: It says that if \((a, f(a))\) is a point on the graph of a differentiable function \(f(x)\text{,}\) and if \(f(a)\) is known to be an extremum (either a maximum or a minimum) then \(\dfdxat{f}{x}{a}=0\text{.}\)

Fermat’s Theorem does not say that if \(\dfdxat{f}{x}{a}=0\) then \(f(a)\) must be an extremum. In fact, that is not necessarily true. For example, if \(f(x)=x^3\) then \(\dfdxat{f}{x}{0}=0\) but the graph has neither a maximum or a minimum at \((0, f(0))\text{.}\)

You have a lifetime of experience with tossed balls. You know that a tossed ball will reach a maximum height. Since Fermat’s Theorem 9.1.1 tells us that whenever that maximum height is attained, \(\dfdx{p}{t}\) will be zero all we have to do for this problem is find the one and only time when \(\dfdx{p}{t}=0\text{.}\) This is the time when the maximum height is reached.