The Calculus Diaries - Jennifer Ouellette [21]
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