Topic 2.3: Estimating Derivatives of a Function at a Point

Night court

This is an empty basketball arena at night, and that flash you keep seeing is a strobe light, firing ten times every second and freezing the ball wherever it happens to be. Let’s let it bounce a few more times and see what all those flashes leave behind. From the side, it turns into one of the most famous pictures in physics books.

What I love about it is that it’s secretly a graph, because the ball drifts across the court at a steady speed, so left to right is time ticking by, and up is height. Every frozen ball is a measurement, a time and a height, and from nothing but those you can work out how fast the ball was falling at almost any instant.

Almost any instant. Look at the top of each hop, where the ball turns around smoothly, and then look down at the floor, where it reverses in an instant and leaves a sharp corner. Right at that snap, in our model of the bounce, the math refuses to give you one velocity, and a calculator will hand you one anyway, with total confidence.

By the end of this lesson, you’ll be able to read how fast something is rising or falling off a strobe photo, or your phone’s slow-motion video, at almost any instant. You’ll also know exactly where a derivative can’t exist, even when your calculator swears it can. Let’s start with the flashes themselves, because a picture like this is really a table of numbers in disguise.

A strobe photo is a table

This is the first fall up close. The strobe fires every tenth of a second, so each ghost gives us a time and the height of the ball above the floor at that moment. At zero point three seconds the ball is two point oh six meters up, and what I’d love to know is how fast it’s falling right at that moment.

One frozen ball can’t tell you how fast it’s falling, because falling means the height changes, and a single snapshot can’t show a change. Let’s use the flashes on either side of it instead, at zero point two and zero point four seconds, and take the slope of the line between them.

Let’s put those flashes on the board as a table, the way an AP question would hand them to you. We want h prime of zero point three. The closest readings on either side are at zero point two and zero point four, so the estimate is the change in height between them divided by the change in time.

That’s one point seven two minus two point three, which is negative zero point five eight meters, over zero point two seconds, and it comes out to negative two point nine meters per second. The units come straight from the table, meters over seconds, and the negative sign says the height is going down, which is exactly what a falling ball should do.

On the exam, a number by itself doesn’t finish the job, so get in the habit of saying what it means. At zero point three seconds, the ball’s height is decreasing at about two point nine meters per second.

Let’s try estimating the rate at a time that falls between two rows of the same table. Using this table, estimate h prime of zero point six five.

Real data: the burn

That ball was the friendly version, so let’s try the same move on something a lot bigger, the propellant inside a Falcon 9’s first stage. If you took precalc with us, you’ve been inside these tanks before.

This time the rate display is dead, and all we have is a log from our model of the flight, the tonnes of propellant left at a handful of moments. If we call the propellant P, the question is P prime of sixty, how fast it’s draining at exactly sixty seconds after liftoff, and notice that sixty isn’t even in the log.

Tap the two readings you’d use to estimate P prime of sixty, and the tanks will show you exactly what your choice means. Now let’s switch the rate display back on, and look at that, the model’s true rate at sixty seconds is negative one point nine tonnes per second, exactly what our two readings gave us.

Our flight model turns the engines down to about seventy percent as the air pushes hardest against the rocket, and it holds that throttle across these ten seconds, so the propellant drains at a steady rate. A wider pair like forty and eighty would reach into the full-throttle burn on both sides and give about negative two point two, so the closest pair really is the best one.

This is the kind of question the AP exam asks, and full credit takes a few things written down. First you write the setup with the table’s actual numbers plugged in, because the formula with letters alone doesn’t earn the point. Then you give the value, negative one point nine, with its units, tonnes per second.

If they ask what it means, you answer in a full sentence with the time, the quantity, which way it’s changing, how fast, and the units. At sixty seconds, the propellant in the stage is decreasing at about one point nine tonnes per second. Notice it says decreasing at one point nine, not decreasing at negative one point nine, because the word decreasing already carries the minus sign, and saying both turns it into a double negative.

Off a graph, out of a machine

Sometimes the exam doesn’t give you a table, it gives you a graph, like this smooth curve of the ball’s first fall, height against time. To estimate h prime of zero point five, picture a ruler laid against the curve at that point so it just touches it and leans the same way the curve does, and that’s the tangent line.

Pick two points on that line where it crosses the grid nicely, far apart, and notice they’re on the line, not on the curve. The line falls about two and a half meters over half a second, so the slope is about negative five meters per second. The exact value is negative four point nine, so the graph gets us close.

Try this one the way it’d look on the exam. The line drawn in is tangent to the graph of f at x equals three, so what’s your best estimate of f prime of three?

On the calculator part of the exam, you’ll meet functions you can’t differentiate by hand yet, and the calculator can estimate a derivative at a point in a blink. A lot of calculators, the TI-84 included, get a derivative at a point the way we used the strobe, with a centered difference, two points just a thousandth on either side.

In this lesson’s calculator, switch to Graph mode, type the function, and use the derivative tool at the x you want. On a TI-84, the same tool is called nDeriv, and it lives in the math menu.

Use the calculator’s derivative-at-a-point tool to find f prime of two for f of x equals e to the sine of x, rounded to three decimal places. The TI-84’s own guidebook warns that nDeriv can give a false answer where the derivative doesn’t exist.