The Calculus Diaries - Jennifer Ouellette [20]
In a perfect world, I would have the patience of Job and would continue doing this unbelievably repetitive process at smaller and smaller intervals, thereby getting ever finer approximations of the likely distance traveled. The range becomes smaller and smaller, converging toward a single answer without ever reaching it exactly. In this case, the answer converges toward 172 miles, where the highway intersects with (I kid you not) Zzyzx Road. (Memo to road planners: Buy a vowel already.) Now it is a matter of subtracting the 110-mile mark—our last stop in Barstow—from 172. We traveled 62 miles since stopping in Barstow an hour ago.
I don’t determine a precise location via any single division of the interval of time; I get the answer via an infinite number of increasingly improved approximations. Although this exercise in precalculus merely gives me a series of approximations, at some point the intervals become so small that the difference between approximations becomes trivial. AAA can probably find us if we tell them we’re within five feet and ten feet of the intersection of I-15 and Zzyzx Road. Integral calculus can simplify matters greatly. Fully integrating speed over time using the velocity function would give me an exact answer for my position. Think of it as Eudoxus’s method without his exhaustion.
DERIVER’S ED
If we can closely approximate our instantaneous speed and position using the precalculus methods outlined above, it’s reasonable to ask why we need calculus at all. It all comes down to functions. Rather than performing an infinite series of calculations for every point along a given curve, the function gives us the value for each of those points all at once, saving us considerable effort and time. Functions confer tremendous predictive power. More important, functions are connected to each other in valuable ways: Velocity is the derivative of position, and acceleration is the derivative of velocity. We integrate acceleration over time to find the velocity function, and we integrate velocity over time to find our position function. These connections let us make inferences based on what we do know, to figure out what we don’t know.
In Zero, Charles Seife compares a standard equation to a machine in which you punch in a number and get another number back. That’s what functions do. Plug any number into a function, and it will give you a new number. Taking a derivative or an integral does the same thing, except you feed the machine a function and it sends back a new function. It’s just a higher level of abstraction. That is how, using calculus, we can transform one problem into another. “Nature doesn’t speak in ordinary equations. It speaks in differential equations, and calculus is the tool you need to pose and solve these differential equations,” Seife writes. “Plug in an equation that describes the conditions of the problem . . . and out pops the equation that encodes the answer.”
The “plug and chug” method might get you through high school geometry and algebra, but rote memorization of every function, along with its derivative and integral (if known), won’t be enough to succeed at calculus. At its core, calculus is about creating and solving logic problems—a most creative endeavor. In fact, constructing a calculus problem is akin to telling a story; we’re just doing it with numerical symbols instead of words.
Every narrative has a logical progression, and so does every calculus problem. You identify your central characters and sketch an outline of the plot to create a structural framework. Then you color in the details as you go. The story can be as simple and straightforward as The Cat in the Hat or as complicated as James Joyce’s Finnegan’s Wake, but in each case it evolves naturally from the starting point of setting the narrative parameters. Writers