Calculus: The Devil is in the Details
Calculus, formerly known as "the" calculus, was invented in about 1700, give or take, by Newton, if you're a native English speaker, by Leibniz, if you're German, and by one or the other, if you're anyone else. People argue about who invented calculus as if it mattered. It doesn't. More important, in my view, is its relative youth. 1700 was only three hundred years ago. That means all those calculus texts burdening the shelves of all those thrift stores were written in the past 300 years.
For some reason, no one ever explains on the first day of Calc I how it works or what it's trying to accomplish. So, here's the unifying idea of differential calculus in one sentence: Differential calculus enables us to treat complicated nonlinear functions as if they were simple linear functions. A more sophisticated version might be: Differential calculus enables us to treat really, really complicated nonlinear functions as if they were simpler functions, possibly even linear.
Let's start with a review of straight lines. I'll assume you know from Algebra I how to graph a straight line: y = mx + b. If you persevered through Algebra II, you know that the big-league way to write this is f(x) = mx + b. The symbol f(x), pronounced "eff of ex" means "function" of x. A function is nothing but a rule. If f(x) = mx + b, the rule is "multiply x by m and then add b, and write down the result." If you do this for a few different values of x, and plot the points on a graph, you gradually catch on that the dots are all in a straight line. Connecting the dots accounts for all the values of x you didn't try.
You may recall, probably not, that when we write y = mx + b, we're using the "slope-intercept" form of the function. The slope of our straight line is "m," and the y-intercept is "b." (The y-intercept is relatively unimportant for this essay, but to jog your memory, I'll remind you that the y-intercept is the value of mx+b you get if x = 0. Its value is left as an exercise for the reader.)
"Slope" is a critical concept here. Surely you recall the mantra, "slope equals rise over run." It's true, and idea of slope = rise/run is important in real life. As you drive through the Rocky, Sierra, Bitterroot or other mountain ranges of the North American West, you'll see signs warning of, perhaps, a 5% grade. Sometimes they include a little pictogram of a runaway truck. A 5% grade is a grade with a slope of 5%; that is, it rises 5 feet for every 100 horizontal feet travelled. If you're going the other way, it also falls 5 feet for every 100 horizontal feet travelled. For an 18-wheeler, that's a lot. On steep slopes, trucks' brakes can fail, so it's a good idea to warn truckers at the top of the hill so they can use other techniques to keep their trucks under control. Hence the sign. Slope matters.
Look at your graph and identify the portion of f(x) = mx + b corresponding to 2 < x < 3.5. That interval, whose length is 1.5, is the "run." On your graph, or on your calculator, find f(2), i.e. the value of y when x = 2, and f(3.5). Let's say the values are f(2) = 3 and f(3.5) = 6. The value of y has risen by 3, from 3 to 6, as x ran from 2 to 3.5. We say the rise is 3, and the run is 1.5. The slope is the ratio 3/1.5 = 2. (Whoever uses this information to determine the value of b gets a gold star.)
Now, the slope of a straight line is the same everywhere, so the slope at y = 3m + b is 2. But wait. I just palmed a card. If the slope is the ratio of two intervals, rise over run, how can we even talk about the slope at a single point? We just did. You accepted it. If pressed, you might say something like, "By slope at x = 3' we mean if we examine the length of any interval (run) of x that includes x = 3, and the corresponding rise of y, and compute the ratio of the rise over run, we will always get the answer rise/run = 2.'"
Even if you might not have said such a thing, let us proceed as if you had, and let's focus on small intervals around x, x-h < x < x+h for some tiny positive number h. How small? Right now, it doesn't matter, but it will in the next paragraph. For now, merely note that any positive value of h gives us the run, 2h, and the rise, f(x+h)-f(x-h) (make sure you see why f(x+h)-f(x-h) the rise of the function for the interval x-h < x < x+h), and we are confident that this rise divided by this run continues to equal 2. It's easy enough to write out, even in ASCII: m=2, f(x+h) = m(x+h)+b=2(x+h)+b, and f(x-h) = 2(x-h) + b, so f(x+h)-f(x-h) = (2(x+h)+b)-(2(x-h) +b) = 2x + 2h + b 2x + 2h b = (2x-2x)+(b-b)+(2h+2h) = 4h. So the rise over the run is 4h/2h, or 2, just as we suspected it would be.
Here's where Newton and Leibniz (Newniz?) had their great insight. Suppose we're interested in some other function, g(x) (pronounced "gee of ex," but nevertheless, g(x) is a function, not a gunction), that is not linear? What is its slope at x = 3? As with a straight line, they had to revise their query to, "What is the slope of g(x) for x-h < x < x+h?" This time it was important to clarify the length of the "run," because the one thing we know for sure about a line that is not straight is that it does not have the same slope everywhere along its length. To which Newton and Liebniz said, "So what? We can still calculate g(x-h), and g(x+h), plot the points (x-h,g(x-h)), (x+h,g(x+h)), and connect the points with a straight line. Then the rise (remember, rise = g(x+h)-g(x-h)) divided by the run (still 2h) is the slope of the line we just drew."
Skeptics, and there were many, said, "Okay, but who cares about that straight line? We care about gee of ex, not that line you just drew." To which Newton and Leibniz replied, "Ah, but look here! We can do this for any value of aitch, no matter how small. (FYI: "aitch" is the official spelling of the letter "h") One one-millionth? No problem. One one-billionth? Still no problem. And as the aitches get smaller and smaller, the discrepancy between our line and the true function, g(x), also gets smaller and smaller. In fact," they continued, "you tell us how accurate you want our approximation to be, and we'll find a value of aitch that puts our estimate of the slope of g(x) within your margin of error."
The next step was the big one, even though it was infinitesimally small. As h gets smaller and smaller, it approaches zero. (Well, duh.) You can't divide by zero. If you try, your calculator will chasten you with some well-chosen words of cyber-profanity. So as h approaches zero, what happens to the ratio (g(x+h)-g(x-h))/2h? It usually approaches some number; let's call it, arbitrarily, L. That is, for smaller and smaller values of h, we can calculate (g(x+h)-g(x-h))/2h and usually can figure out what L would be. Ideally, we can use the rules of algebra to get h out of the denominator altogether. L is called the "limit" of (g(x+h)-g(x-h))/2h "as h goes to zero." (In truth, h doesn't go to zero; we're looking at a sequence of ever-smaller h's. But let that pass.)
The "limit" is really the defining concept of calculus, but you're not quite there. Not yet. This particular example, "the limit as h goes to zero of (g(x+h)-g(x-h))/2h" is called the "derivative" of g(x) at the point x; usually written g'(x) (pronounced "gee prime of ex") or dg/dx (pronounced "dee gee dee ex", don't say "over"). It's called the derivative because it's derived from g(x), all the other quantities that can be derived from g(x) being also-rans. "Derivative," by the way, is Leibniz's word for it. Newton preferred "fluxion." Two points for Leibniz. The calculation itself, for some reason, is called "taking" the derivative of g(x). Taking where? From whom? Beats me.
The value of g'(x) at x=3 can be written g'(x when x=3). This is not the common notation, but it's the best we can do in ASCII. Now, we can treat g(x) in the vicinity of x=3 as if it were a straight line, y = kx + c, with k = g'(x when x=3). so y = xg'(x when x=3) + c. (I moved the x in front of g'(x) so you'd see it) We're making an error: in truth g(x) does not exactly equal xg'(x when x=3)) + c. But the linear version is much easier to work with, especially in the era of slide rules, and we know that the error is small. If it strikes you as slipshod and inaccurate, you're correct, technically, but when it's put to use, we get astronauts on the moon, the Trans-Siberian Railway, and Bonnie & Clyde's getaway car. The proof, as they say, is in the pudding.
Calculating the derivative can be a drag. As ever in mathematics, the devil resides in the details. But if you keep in mind the idea of why the derivative might matter, sort of like navigating by the North Star, you should be able to keep your bearings and separate the really trivial from the less trivial. And that's a start.
Next lesson: Integration.