WHAT IS CALCULUS? (Part II)
Last time on WHAT IS CALCULUS?, we met the monster, stared it square in the eye, put on clean underpants, and explored the integral. We learned the integral rules one half of the great realm of calculus, the other half being differential calculus with its favorite son the derivative. This fleecy beast is the quarry of today's hunt.
In most calculus books, where authors answer to editors and shareholders, the derivative is presented long before the integral. Not so in the wild ride that is LabKitty. Thanks to our unorthodox approach, you already have the integral under your belt, and if not you should probably turn to Part I before proceeding. Or don't, for who are we to quash a kindred rebel spirit? We dare say our story is accessible via alternate routes, though success in these sort of things depends upon how much whiskey was consumed during the writing (think of us as the Hemingway of calculus). Your mileage may vary.
So, without further ado, we give you WHAT IS CALCULUS? Part II: The Derivative.
Our journey begins, oddly enough, with a car trip.
A Car Trip
The speedometer on a car is a magical device. Every time you glance at it, it tells you how fast you are going. Without delay. Sofort!, you might say if you drive one of those schweet BMWs. Right now. Right now!
The point being: how is this possible? One can imagine a cartoon speedometer that might appear in a Disney film, perhaps with a delightful German accent, that tells you your speed only after the following conversation:
You vant to know how how fast zee are going? I vill count to ten and keep track of how far vee go. Then by dividing zee distance by 10, I vill know zee speed. Which I vill tell you in der dashboard dial.
For example, if we go 60 feet in 10 seconds, then our speed is 60/10 = 6 feet per second (since our speedometer is German, we should probably be using meters but whatever). Or, to use a longer example: if grandma lives 10 miles away and it took us an hour to get there, we drove at 10 miles per hour.
But that's not how a speedometer works. There's no waiting. There's no ASK button you push and the thing reports back 10 seconds (or any seconds) later. It's always working. You look down and -- bam! -- there's your speed. Like telling the speed of a spinning propeller from a picture. It's spooky, if you think about it. (Aside: if you are offended by our crude caricature of a German speedometer, take it up with Disney. LabKitty is just the messenger.)
Let's see if we can't figure out how this magic trick works.
Have a gander at the figure to the right (or left, depending on how your browser is formatting the page). We show here a car trip, with time (t) increasing to the right and distance-from-home (y) increasing upward. We've labeled some points of interest to add a little color commentary to the math. At one end of the trip: your house. At the other end: grandma's house. We labeled your house with a house. We don't have a symbol for "grandma," so we used a pirate.
Leaving home, we start off slow, say, making our way through city traffic. On this part of the curve, your finger moves to the right (time) along the curve without moving very far up (distance). Ergo: slow. However, we soon we hit the freeway. We are now moving faster, and the slope of the curve increases. We hit maximum speed right about midpoint. Mom is asleep riding shotgun, the kids are watching Pokemen vids in the back, and dad is making good time. This continues for awhile, then the curve levels off. What does a leveled-off curve mean? Again, drag a finger. Your finger is moving in time (rightward) but not much in position (up-movement is decreasing). The car is slowing. Why? Alas, the highway patrol was displeased with dad's automotive shenanigans and is pulling us over, where there will be much debate over dad's speed.
What we have learned so far is that given a plot of position vs time, we can read off our speed at any point as the slope of the curve at that point. In geometric terms, we require a line tangent to the curve. The slope of the tangent line is equal to our speed at that point. So, apparently, our speedometer is computing a tangent line, and then displaying the slope of it in the dashboard on a dial or perhaps a digital readout if you drive one of those schweet BMWs.
Alas, a speedometer does not have pen and paper, not even the speedometer in one of those schweet BMWs. It can't draw tangent lines. We need to devise a way to compute the tangent slope using numbers rather than pen and paper.
Do you know how fast you were going, Sir? Yes, officer, Dad replies, because I know CALCULUS!
Let's go to the whiteboard.
We would like to calculate dad's speed at the location indicated by the red star on the previous figure, a portion of which we reproduce bigger here. Inspired by our earlier success in integral calculus, we hit upon the idea to begin with an approximation. Speed is the slope of the curve -- by which we really mean the slope of the line tangent to the curve -- and we can approximate a tangent line with a secant line (to refresh your geometry: a tangent line touches a curve in one and only one place. A secant crosses it in two). To compute the slope of the secant line, we dust off that old sock from elementary school, the bane of the paste-eaters: the rise-over-run formula.
We start big and work smaller. The first approximation is labeled, creatively, first approximation. Indicating "rise" by the symbol Δy (math for "a little change in y") and the run by Δt ("a little change in t") we find the slope of the secant line to be Δy/Δt = 0.333 / 4.0 = 0.0833. (To get numbers, we had to invent a function describing the curve. We used: y(t) = t / (1+ t ).)
Unit check: "distance" divided by "time" indeed gives "speed" for example miles divided by hours is miles-per-hour. So far so good.
We can tell this is not a very accurate approximation by just eyeballing the graph and comparing this secant line to the tangent line (drawn in red). You probably see where this is headed. We attack the problem by taking smaller and smaller changes in Δt and correspondingly smaller and smaller changes in Δy, each time recomputing the slope of the new improved secant line. This process should generate better and better approximations to the tangent as we proceed, or so we hope.
Here, a reasonable person might say: That's redonkulos! If we keep dividing a smaller and smaller number by another smaller and smaller number, won't we eventually be dividing zero by zero? How can that result be anything other than zero? Or is it infinity? Whichever. It certainly isn't a regular number, no? Grrr!
Well, let's find out. We put all this on a computer (we used MATLAB, but you can do this on a spreadsheet or some other program) and calculated some slopes.. Here's results for our first dozen or so secant lines:
The numbers neither shrink to zero nor run off to infinity. In fact, they are approaching 0.25. The (true) slope of the tangent at t = 1 is 0.25. (Unfortunately this is not some easily-recognizable number (like the area of the circle was in our integration example from last time). You’ll just have to trust us.)
The point is this: our secant line slopes are converging on something. Logic tells us that “something” is the slope of the tangent line.
Meet the Derivative
Of course you can't play the secant game forever, even on a computer. But in a calculus course you learn techniques for finding out what the answer WOULD be if you DID. The quantity you get if you spend the rest of eternity taking smaller and changes in time and distance -- until the secant for all intents and purposes becomes equal to the tangent -- is known as THE DERIVATIVE.
Here, then, is the defining beast of differential calculus. The derivative gives us an incredibly convoluted way to find the slope of a tangent line (hooray!). The usefulness of the derivative is that the same game can be applied to find the rate of change of any graph we like. Any something that is a function of something else. The position of your car on the highway as a function of time. Your child's height as a function of age. Temperature as a function of latitude. The amount of fur your cat sheds as a function of temperature. Some of these rates-of-change have familiar names like "speed." Some do not. But as long as we can describe the relationship using numbers (in math lingo: we have an equation that describes the curve) we can find the derivative.
And remember - the answer is not just an approximation. It's the real deal. The true value of the rate of change. The end result of taking smaller and smaller changes divided by shorter and shorter intervals, forever. The end result may seem like dividing a change we don't let happen by an interval that didn't change. The magic of calculus lets you do this.
Here's the notation used for derivative: dy/dt. It should remind you of the rise-over-run formula - a little change in y over a little change in t. We trade in pointy Δ's for round d's. This is to remind you that the derivative is the end result of playing a game of repeated secant approximations as described above.
Not surprisingly, there's more to the derivative and we fill-in a few details below. But you now know the crux, the nub, the kernel of differential calculus. Crazy!
Here's a few bonus points-of-interest for the derivative. These will help you begin to grok calculus material as you bump into it elsewhere.
Bonus Knowledge #1: Calculus jocks call the process of repeating a calculation over and over using a smaller and smaller change "taking a limit." This is usually abbreviated "lim" and the abbreviation usually includes an indication of what is being made small. So, for example: lim Δt → 0 indicates the limit as Δt is squished to zero. So if we were to dress up the definition of the derivative in a full formal gown, she would look like this: dy/dt = lim Δt → 0 Δy/Δt. The notation is a reminder of the game we play to get the result.
Bonus Knowledge #2: Notation, notation, notation. You can write down any kind of derivative you like: dy/dt, say, or dy/dx or dB/dw. In general d(argle)/d(bargle) is just the rate of change of argle with respect to bargle. If you want to emphasize the derivative at a particular place rather than just the general concept, we add a little vertical bar with subscript indicating the place of interest: For example, dy/dt |a is dy/dt evaluated at t = a.
Warning: mathematicians, being jerks, sometimes use other notations for the derivative. Some use a prime, like this: y'. Some use a dot over the variable. Some use a big D, like this: Dy. FYI.
Bonus Knowledge #3: There is no reason we have to stop at the first derivative. We can take a derivative of a derivate, in effect asking: how is the change changing? If that's confusing, here's an everyday example: the change in position with time is velocity. The change in velocity with time is acceleration. Acceleration is what you feel when you stomp on the gas or brake. But if your speed is not changing, you feel no acceleration (because there isn't any). When we write these "higher" derivatives we include a superscript that indicates the order: d2y/dt2, d3y/dt3, and so on.
Bonus Knowledge #4: If the changing quantity depends on more than one thing, calculus jocks will write the derivative using a little hungover "d." It looks like this ∂ and is called (wait for it) a partial derivative. Example: stand in a room with a fireplace. If you walk away from the fireplace the temperature decreases. But the temperature also decreases if you stand still and wait for the fire to die down. The first change we could write as ∂T/∂x where T is the room temperature and x is distance from the fireplace. The second change we could write as ∂T/∂t where T is the room temperature and t is time.
Bonus Knowledge #5: There is a "derivative" in economics that has nothing to do with the derivative used in calculus. Derivatives (in the economic sense) are what destroyed the economy in 2008.
Achievement Unlocked: Here's the Wikipedia page on the derivative. You should be able to go there and recognize the notation and basic descriptions. As we mentioned in the integral bonus section, don't sweat the details.
Calculus on Amazon
Half math, half history, and half Alice in Wonderland, Berlinsky recounts the development of calculus as Newton might have seen it, from dark months spent in alone with quill and parchment, to rambunctious breakthrough, to eventual triumph (which gave Newton the means to carry out petty vendettas on just about everyone). Berlinkski mixes the tale with priceless interludes chronicling his experiences as faculty in the math department of an unnamed California university, where he is tormented by a vinegary colleague who began as a student of the legendary Russian mathematician Kolmogorov and now finds himself teaching calculus to valley girls in night school.
Next time on What is Calculus?
Next time on WHAT IS CALCULUS?: a wedding! Witness The Fundamental Theorem of Calculus join the integral and differential realms into a single whole, surely to bring forever peace to all the land. If it worked for Henry and Anne Boleyn, it can work for calculus.
For as we claimed in Part I, rates of change and areas are two sides of the same coin. A bold claim, no doubt, but one we will soon demonstrate. Before that can happen, however, the integral makes a return, like prodigal son bursting through the front door with a trollop on each arm and stinking of gin.
Joins us next time for the exciting climax of our story. Don't pay for a whole seat, because you'll only need the edge!
Darwin Airlines Saab 2000 by Juergen Lehle (website) and appears under the terms of the GNU Free Documentation License. Image of Plya the cat by David Corby and appears under the terms of the GNU Free Documentation License.
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