The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  The officers don’t have a radar gun, which measures velocity directly, but unfortunately for Sean, they are well versed in math. They do have a time-stamped photograph of the Prius at a similar intersection one minute before. So it’s a simple matter for the officers to show where we were at the traffic light—the two-minute mark—and subtract our position at the previous intersection (the one-minute mark) to determine how far the Prius traveled in that time: in this case, one mile. Then they can divide that by the time it took to travel that one mile, and this gives them the car’s average speed: one mile per minute, or 60 mph.

  Ah, but Sean doesn’t give up so easily; he has one more argument to make. The officers are assuming the Prius was moving at a constant speed. Yet every experienced driver knows that one’s speed is rarely constant. Just because our average speed was 1 mile per minute doesn’t necessarily mean that was our instantaneous speed at the moment we crossed the intersection.

  The officers remain undaunted. They don’t have access to the information recorded by our trusty speedometer and odometer, so they have supplemented this imaginary stretch of road with some pretty cutting-edge technology, dividing it into intervals at every possible distance and placing tiny nanosize traffic cameras at each and every interval. Call it willing suspension of disbelief, although at the rate nanotechnology research is currently progressing, a scenario quite close to this may one day become a reality.

  Thanks to our imaginary nanocameras, the officers have an infinite number of time-stamped still shots of our humble Prius, taken at infinitesimally small intervals along this extremely high-tech futuristic road. This is incontrovertible ocular proof 12 of the car’s position at every given point in time since we left home: In calculus terminology, this is our position function. We know the position of the Prius as a function of time. The cameras reveal that there was less time between equal intervals as the Prius approached the light—which means we were actually accelerating.

  The basic concept is the same whether we’re talking about driving down the imaginary highway at a constant rate or about a more complicated real-world scenario in which our speed is constantly changing. Even though the Prius is accelerating, it still has one specific speed at each instant, and I can use the same highly repetitive process of accumulating evidence to prove it, showing where the car was at all times. I run the same calculation outlined above over and over, for ever smaller intervals, to show how fast the car was going at any given moment in time.

  This time, there is a crucial difference: Instead of getting the same answer each time—as in the constant-speed scenario—I get slightly different answers each time. But as the intervals get shorter and shorter, those answers get closer to a point of convergence: 2 miles per minute. The answer is never exactly 2. But the answers are clearly converging toward a single answer, to a very close approximation. The limit rears its ugly head. I can safely conclude that the car’s instantaneous speed at the moment in question must be 2 miles per minute.

  Ingenious, isn’t it? Hats off to Newton, Leibniz, and untold mathematicians before and after them who repeated the same exact process of calculation, over and over again, until they’d compiled sufficient proof that the derivative formula works. Thanks to their collective effort, we can simplify this incredibly repetitive process by taking the derivative of our known position function, which will give us our velocity function.13 Then we can revert to basic algebra: We take the value for the point in time that we’re interested in, and we just plug it into that equation. That will give us our speed at that instant. Behold the power of calculus!

  None of this, alas, helps Sean avoid an imaginary traffic ticket. He grudgingly admits defeat. The mark of all good scientists is the willingness to abandon a pet hypothesis if the experimental evidence contradicts it—but that doesn’t mean they have to be happy about it.

  THE SUM OF ALL THINGS

  Taking a derivative is pretty straightforward. Finding the integral is trickier. Conceptually, it’s just the flip side of the derivative: With the derivative, I can figure out my car’s speed based on how its position changes over time. With the integral, I should be able to determine how far we’ve traveled in the Prius based only on measurements of its speed at given locations along our high-tech highway.

  Thanks to modern technology, I can just use the car’s odometer and built-in GPS system to find the answer. But what if the odometer is broken, the computer has malfunctioned, and we find ourselves stranded in the middle of nowhere, with no other cars in sight? These highfalutin hybrids with their onboard computers and hordes of sensors are pretty sensitive, after all.

  Assuming our cell phones still work, we can call AAA, but we need to be able to tell them our precise position. There are no obvious landmarks. “Third tumbleweed on the left next to the giant boulder” isn’t going to narrow things down sufficiently. We know we haven’t passed Baker. Even if you missed the Mad Greek Cafe—despite the fact that it is gaily painted with the colors of the Greek flag and adorned with plaster replicas of naked Greek statues out front—you’d certainly notice Baker’s other main attraction: the World’s Tallest Thermometer. Baker is located at the 188-mile mark between our Los Angeles loft and the Luxor in Vegas. Let’s say that an hour before we got stranded, we stopped for coffee in Barstow, which is at the 110-mile mark. So I know we are somewhere between 110 miles and 188 miles from our home in Los Angeles.

  Had our speed been perfectly constant, this would be a simple task, and we would have no need for calculus. Assuming a constant speed of 60 mph, for instance, and knowing that exactly one hour has elapsed since we left home, I can multiply our speed by the time and conclude that we have gone 60 miles. It’s probably a pretty good approximation. But that doesn’t reflect actual driving conditions; a car’s speed is constantly changing, even more so if there are spots of heavy traffic, and if my lead-footed spouse drives faster than 60 mph to make up for lost time whenever traffic clears.

  The only concrete information I have about our velocity is from monitoring the speedometer. Fortunately that’s all I need to figure out how far we’ve traveled and thus pinpoint our location for AAA. The speedometer has displayed our speed at every instant along our journey; taken together, this gives me our velocity function. So I should know exactly how fast I was going at any given moment.

  How do I take the variation in speed into account? I set boundaries around the correct answer to get a workable range for determining the distance. First, I do a series of calculations based on the slowest (starting) speed—in this case, at the point where we left Barstow—breaking that journey into smaller and smaller increments of time and adding up the pieces to arrive at a close approximation to the total distance traveled. But this will be an underestimate. So I also need to do the same labor-intensive process for the fastest speed the car was traveling over our entire one-hour journey. The resulting approximation will be an overestimate of how far we went, but at least I know that the correct distance is somewhere in between those two values. I then go through the same process for different speeds within the minimum and maximum to further narrow the range. The shorter the intervals of time that I choose to employ, the better, because the speed is less likely to vary by much over tiny times and distances.

  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 appr
oximations. 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 and physicists alike spend a great deal of time staring at a blank sheet of paper (or computer screen), waiting for inspiration to strike. This phenomenon can be witnessed firsthand on any given night at our house.

  Let’s revisit our two idealized scenarios from the perspective of a narrative. Who is my main character? In the first example (Sean attempting to avoid an imaginary traffic ticket), it would be position, because that is the accumulation of data available to us—what we already know. At every point in time, our Prius has a position on the road; all those points taken together comprise the position function (position as a function of time), which we can represent algebraically as p(t), where p stands for position,14 and t stands for time. Note that I picked p because it’s easy to remember; I could have called it x or q or even Sally, and it would still stand for the exact same thing in this context: position. It is the context that gives a particular variable its meaning.

  We can graph every single value for p as a point on a Cartesian grid and connect the dots to get a curve. Now we have a “face” for our main character, the position function. That means we can plug different values into this equation to find where we are at any point in time using basic algebra.

  Sean admits that usually, collecting data from the real world doesn’t give us a simple function, “but as physicists we often find it useful to approximate the messy real world by some simple function that we can write down cleanly.” Fair enough: Plenty of writers take liberties with narratives, too, if it makes for a better story.

  What is the main character’s ultimate goal? Given the “clues” about our known position, we want to figure out how fast we are traveling at a particular point on our trip. There is even a central conflict: How does the main character reach that goal? It’s a process of deduction, using the clues we’ve been given: namely, our position function. We can take the derivative of the position function—a process of subtraction and division—to find the corresponding velocity function, which we can use to determine our instantaneous speed at any given point. To do this, we start with our current position (p), take our position a tiny bit into the future, then subtract the two to find out how far we went. Then we divide the distance traveled (Δp) by the small change in time (Δt) and we get the average velocity during that short interval.

  We can approach the same question geometrically. Remember that the derivative also gives us the slope of the tangent line on a curve. If our curve represents the position of the Prius at every point in time, then the slope of the tangent line to that curve at a specific point will tell us how fast the Prius was traveling then: the instantaneous speed. If the car is moving forward, that motion will be represented on the graph by a tangent line slanting upward; if the car is moving backward, the tangent line will slope downward. The steeper that line, the faster the car is traveling. The minimum or maximum of the graph has slope 0, which means the car is stopped.

  How do you find the exact slope of the tangent line? You draw a straight line between two points on the graph and then look at how much that line rises or falls (the y axis) over that set distance (the x axis) between two points. We get the derivative by looking at ratios—for example, a difference in the position of a moving car at two separate times—so the slope of that line is the fraction of the change in position divided by the change in time. You do the same thing again with two closer points; and so on, until all those straight lines converge to a tangent line whose slope is equivalent to our instantaneous speed. The closer those points are to one another, the closer we can approximate the slope; we have the exact answer when there is no distance between those two points. This is a visualization of the limit: the difference in height goes to zero and so does the distance between the two points.

  Now let’s revisit the integral via the second example: figuring out how far we have driven based on our velocity. We are telling the same story from another character’s point of view, and it changes the “narrative” in some crucial ways. In this case, our main character is the velocity function. We don’t know the position; we know the velocity, and we want to deduce our position from that. We’ve seen that it is possible to figure out how far we’ve driven knowing just the velocity of the Prius at each instant along the road—the velocity function—using that tediously time-consuming precalculus method. Since we have a “face” for our function, we can determine the area under that curve between the two points of interest via our old friend Eudoxus and the method of exhaustion.

  There is a shortcut: I would get exactly the same answer if I simply subtracted our beginning position from our ending position. Of course, I don’t know our exact ending position, which complicates matters. All I have is the velocity function and my known starting position. My myriad calculus books assure me that all I need to do is figure out which position function generates the known velocity function by taking an integral, then use that position function to determine where we are when our Prius has its hypothetical breakdown.

  How do physicists find the integral they need in the real world? They usually look it up. Seriously. A lot of this work has already been done by the generations of mathematicians who came before us, bless their detail-oriented sou
ls, so why waste valuable time recrunching all those numbers? Most standard calculus textbooks contain tables of known functions for both derivatives and integrals to assist beleaguered students—or their teachers provide them with formula sheets. Sean ditched his calculus textbook long ago; instead, he has a big blue book called Standard Mathematical Tables, filled with nothing but a bit of explanatory text and lots of incomprehensible notations. It’s now also possible to download calculus apps for your iPhone. The problem is that it is impossible to list every single integral. Even Standard Mathematical Tables soberly admits its own shortcomings: “No matter how extensive the integral table, it is a fairly uncommon occurrence to find in the table the exact integral desired.”15

  Occasional patterns do emerge. For instance, there is a mathy trick we can use to help us find the desired derivatives and integrals for any constant times x. Remember that the derivative and integral are opposite processes: Each undoes the work of the other. The integral is a process of multiplication and addition. If we are given the function 2x (2 is the constant, meaning it is unchanging), an integral of 2x is the function x2. Because the derivative is a process of subtraction and division, that means that the derivative of x2 is 2x. Similarly, for 2x, the derivative is the function 2. Indeed, Sean explains that this is pretty much a universal rule.†

  I know what you’re thinking: I thought 2 was a constant. How can it also be a function? That confused me, too, at first. Sean explained that in the above example, 2 plays different roles, depending on the context. It plays a constant in the function 2x. But then we took a derivative, an operation that gives us a new function back: Now 2 is playing the role of a function. Technically, it’s the dependent variable (generically represented by y). Plug in any random number (x, or the independent variable), and the function will send that number to 2. Think of it as an ordered pair (x, 2), where x can be any random number. The point is, in this particular scenario (a constant times x), whenever we have a derivative formula, we can automatically find an integral formula.