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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 souls, 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.

Once we’ve identified the integral we need, we don’t have to resort to the tedious process of dividing up the area under the curve into tiny pieces and multiplying and adding ad nauseam. Instead, we just subtract the value of the integral at the end of the curve from the value at the beginning of the curve to get the answer. Let’s say we want to take the integral