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Updated on December 21, 2014

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Thus far on WHAT IS CALCULUS?, we have explored the two great realms of calculus -- the integral and the differential -- and met their sovereigns. The Integral: he of areas and volumes and the finding thereof. The Derivative: she of rates of change, be they in time, in space, or in something else. Now it is time to wed. Time for these realms to make peace with one another and be joined into a glorious whole. We are not strands of hair; we are hair! George Washington beseeched his reluctant colonists to unify and become the great county of Washingtonland, a name ultimately rejected for something that looked better on the currency. But it's the thought that counts.

And so we now stand at base camp, our resolute faces turned toward the summit visible through the clouds in the brilliant morning sun. What had been for so long just a dream is now in our sights. It will soon be in our mitts.

We give you WHAT IS CALCULUS? (Part III): The Fundamental Theorem of Calculus.

dude looks like a lady
dude looks like a lady

Exciting Climax

Perhaps you've heard that calculus was invented by Newton. Not true. The concepts of the integral and the derivative had been kicking around for quite awhile when Isaac happened upon the scene. For example, the circle area thing goes all the way back to ancient Greece (although the Greeks screwed the pooch with the Zero Paradox which isn't true at all but that's only because they didn't know about convergent infinite series which is something we're not going to get into and why are we still typing). And if we have done our job, we have convinced you that the integral and the derivative are not difficult ideas. They are familiar geometric concepts (slope, area) that calculus flogs to death. Sure, there is a mountain of details we keep alluding to that can make the going unpleasant, as any freshman can attest. But if you have read Parts I & II, you now know the nitty-gritty, the marrow, the substance of the two great halves of calculus: differential and integral.

Is that all there is to it? Surely there must be something more that earns calculus it's fierce reputation. There must be more devil than mere details. Where's the beef? Clara Pellar was heard to exclaim.

Well, Ms. Pellar, here comes the beef.

What Newton (and some others) did was discover a relationship between differential and integral calculus and join them. Like Aragon reforging the Elf Sword of Berkenstock after it was broken (or whatever the heck he did and to what). The relationship is not obvious. It may even seem a little mysterious. Finding it required insight beyond circle tiling and secant drawing.

This piece of work goes by the name The Fundamental Theorem of Calculus, which sounds pretty darn pretentious. Still, lets see if we can't get this fish in the boat. As with the integral and the derivative, our explanation of the FToC begins with a story.

one stop shopping
one stop shopping

A Curious Shoppe

Imagine you run a calculus curio shoppe, where customers bring you functions and pay you to find the derivatives at places they point to (meh, it could happen. Humor us). You usually perform the (tedious) secant line thing to find these derivatives. However, your years in the differentiating biz have allowed you to compile a list of functions that are the derivatives of other functions. For example, you have noted that no matter where you point to on the sine function -- which we will write as sin(x) for "sine at x" -- if you go through the whole secant approximation limit argle-bargle, you get exactly the same number if you just look up (or punch into your calculator) the value of the cosine at x -- written: cos(x). A curious result, no doubt, but curious is your business (heck, it's right there in the name). What matters most is keeping the customers happy, and if the secret list saves time and impresses the punters, then who cares why it works.

One day, a customer enters your shoppe. You immediately sense this guy is trouble. The customer has brought a function, but he does not want you to differentiate the function, he wants you to tell him what function when differentiated produces the function he has brought. That is to say, he does not want a derivative, he wants (wait for it) an anti-derivative.

If he has brought you a function that happens to be on your secret list, all is well. You simply use your list backwards. For example, if the customer brought you cos(x), then you can immediately tell him the anti-derivative is sin(x), because you know that the derivative of sin(x) is cos(x).

But what if you are not so fortunate? What if he brings a function that is not on your list? One you have never seen before?

The customer jabs a finger at the function. At this location, he demands, what is the value of the anti-derivative?

Your mind reels, your brow sweats. If you fail this man, word of your incompetence will surely spread. Customers will stop coming! Your derivative business will go under! You will have to get a real job!

And then you remember the Fundamental Theorem of Calculus:

The value of the anti-derivative at any point on a curve is the area under the curve up to that point.

(There is a slight inaccuracy in this bold claim that we will correct in just a moment -- although it is a 100% true statement for the example we're about to show. Including all the fine print would have ruined our dramatic reveal.)

This result is so important we should draw a box around it, but we're pretty stupid when it comes to HTML, so you're just going to have to imagine a box drawn around it (hey, we're impressed we figured out red text). We learned way back in Part I that the area under a function is the integral. But, we claim, the integral is also the anti-derivative of the function. That is, integration gives you a function that, when differentiated, gives you the original function that you just computed the area under. This grand connection between integrals and anti-derivatives, between differentiated functions and the area under them, is called The Fundamental Theorem of Calculus.

The FToC is by no means obvious. In fact, it's a little brain melty. It's going to take more than one sentence in cool red text to explain just what the heck is going on.

Let's go to the whiteboard.

The Fundamental Theorem of Calculus

Meet the FToC

At the top of the accompanying figure is the function brought to you by Mr. Pesky Customer. We assumed the function is cos(x), which you may remember from your trigonometry days. In the table to the right of the graph, we've picked a few representative x between 0 and π/2 and listed values of the function at these locations. (There is no profound reason for stopping at π/2 in the table except that figure space is tight.)

Remember, the customer does not want the derivative of cosine; he wants the function that when differentiated gives cosine.

In the middle panel we've given you the answer, which is the sine function or sin(x). First, let's verify that claim is true. In the table to the right, we've picked a few representative x between 0 and π/2 and listed the value of the derivative of sin(x) in the right-hand column (a vertical bar with an x at the bottom is standard notation for "the derivative at x"). These we obtained using the secant limit method described in Part II, an example of which is drawn on the graph for x = π/3. You may easily verify that the values shown here for d sin(x) / dx are indeed the same as the values of cos(x) listed in the top panel for corresponding values of x. That is, sin(x) is the function that when differentiated gives cos(x).

So far, so good.

We now return to the customer's function -- cos(x) -- in the bottom panel. In the table to the right, we've picked a few representative x between 0 and π/2 and listed values of the area under cos(x) between 0 and x in the center column. These values were obtained using the tiling limit method described in Part I, and are equal to the integral of cos(x) from 0 to x, which is how we've labeled the column (recall that when we write an integral, we indicate the left and right endpoints of the area on the bottom and top of the integral sign).

In the right column of the table we show the value of sin(x) at the same x. The values in the center and right columns are equal! What on Earth can this mean?

The bottom panel shows that the integral of cos(x) is sin(x). But the center/top panels show that the derivative of sin(x) is cos(x). We have found the anti-derivative via integration! That is, the integral gives us the function that when differentiated gives us the function we are integrating. You may want to re-read that a few times; the logic is a little like a snake eating its own tail.

It turns out this is not an isolated incident. Here we have seen d sin(x)/dx = cos(x) so ∫ cos(x) dx = sin(x). But it is also true that d cos(x)/dx = -sin(x) so ∫ sin(x) dx = -cos(x); d tan(x)/dx = sec2(x) so ∫ sec2(x) = tan(x); d log(x)/dx = 1/x so ∫ 1/x dx = log(x). And on and on. Heck, much of first-semester calculus is memorizing a big honkin' list of derivative/integral pairs. And once you know the trick, you can skip all the secant line flogging and rectangle tiling bashing. You just write down the answer.

This, then, is the Fundamental Theorem of Calculus. The relationship that ties together derivatives and integrals or, more to the point, integrals and anti-derivatives. The relationship that ties together rates-of-change and areas-under-curves. In a single sentence: the value of a function at x is the rate at which area is being added to its integral at x.


Minor Correction

When we sprung the FToC upon you in the last section (which we wrote in red and everything) we mentioned there was a slight inaccuracy in our dramatic reveal. Here is the inaccuracy: the area under the curve is not equal to the value of the anti-derivative at the right endpoint, but rather the value of the anti-derivative at the right endpoint minus the value of the anti-derivative at the left endpoint. The example above was craftily chosen so that (1) the anti-derivative was sin(x) and (2) the starting point was zero. Because sin(0) = 0, there is nothing to subtract.

the happy couple
the happy couple

Achievement Unlocked

And so we have arrived at the marriage of differential and integral calculus as promised. The derivative on one side of the coin and is engraved rate-of-change. The integral on the other side of the coin and is engraved area-under-the-curve. Flipping from heads to tails is differentiation. Flipping from tails to heads is integration, yes, but it is also anti-differentiation. That is to say, given a description of the rate of change of a thing, we can use integration to obtain the thing itself.

Here your mind should wander back to our interlude on Nature (witch, rhymes with). Integration is how we deal with natural laws expressed in terms of rates of change. It is how we solve differential equations; how we turn a description of how a thing is changing into the the thing itself. More than computing areas, more than finding volumes or lengths or whatever, this feature of integration is key. It gives mathematics the tool it needs to model the real world. The flaming hoop to Nature's tiger, the time-out to her rough-housing, the restraining order to her psychosis.

The Fundamental Theorem of Calculus literally changed the world. It is the difference between the mush-head physics of the ancient Greeks where falling things seek their center and modern physics that puts numbers and predictive power in our hands. The ancient medicine of demonic possession and humors that became modern medicine of titration and robotics. Ancient yearnings of flapping flight that became modern engineering that builds machines that fly, and many things besides. More progress happened in the two centuries after Newton then in the 20 that came before.

It's strange to think of the Fundamental Theorem of Calculus just sitting there for thousands of years, waiting to be discovered. If we take the founding of Plato's Academy to be day one of math class, his students would be getting to the FToC around their 4000th semester. These days it's taught somewhere in Calc I or perhaps at the beginning of Calc II. It's in a bazillion lectures on YouTube and iTunes and probably even turns up in a TED talk or two. The discovery did not require billion-dollar machines or multinational laboratories. Quill, parchment, candle, and -- if we know mathematicians -- a bottle of absinthe and all was put in light. Newton released the Krakken, raised the floodgates. The night busted open, this two-lane will take us anywhere.

One can't help but wonder what might still be out there lurking unseen in the shadows.

Calculus on Amazon

Schaum's Outline of Understanding Calculus Concepts
Schaum's Outline of Understanding Calculus Concepts

If you're looking to take your first real steps into calculus, you could not find a better guide. More of a textbook than the standard Schaum's format, and one that that is brief and superbly written (and cheap!). It's all here: differentiation, integration, infinite series, and applications. Throughout, Passow employs the approximation-refinement-limit approach we used as a conceptual thread that ties everything together. It doesn't quite work as a formal textbook (Passow omits eleventy gazillion integration tricks, most of analytic geometry, and all of vector calculus) but it can't be beat for getting your foundations right or as supplementary material to explain what your course textbook isn't.

that woman is a woman!
that woman is a woman!


This very day thousands of students sit in calculus classrooms around the globe, hair combed, shoes shined, their freshly-scrubbed faces eager to suckle learning goodness from the bosom of our Harsh Mistress. It is a formidable task, transforming their little mathematical garter snake arms into mighty problem-solving pythons. (A task increasingly assigned to a clown car of adjunct faculty and graduate students for reasons beyond our ken. But we digress.) What exactly goes on behind the closed doors of these hallowed chambers?

A traditional calculus course mostly teaches you clever ways to compute derivatives and integrals. Charming though it may be, calculus jocks don't break out little computer programs every time they need to find a derivative or integral. There's memorization of some basic results, and of some basic rules, after which application of the rules to the results generates a cornucopia of derivative and integrals to last a lifetime. There are forays into geometry, and also into the study of infinite sums of numbers (a topic we skipped completely) which is how functions are implemented on the buttons of your calculator. Along the way, you learn to apply calculus to solve real-world problems in physics, chemistry, engineering, biology, ecology, epidemiology, economics, probability, statistics.

And we lied a little. When the going gets really tough, we do break out the computers.There's entire industries, cottage and otherwise, devoted to solving calculus problems on computers. Warning! Buzzwords ahead! Finite difference method. Finite element method. Boundary value method. Newton-Raphson. Quadrature. Numerical integration. Runge Kutta. Example usage in a sentence: I couldn't figure out the integral on the homework so I slapped that bad boy with a second order Newton-Raphson and BAM! in five minutes I was sipping mai tais on the veranda with Miranda. The implication being, yes, people will pay you to do calculus.

Freeman Dyson famously quipped that Nature gives a clear answer when a scientist asks a clear question. You now know that Nature does not give you an answer, she gives you a differential equation. And you now know that calculus is the tool used to turn those differential equations into answers. The true golden lamp that lit the doorway, calculus is. The beacon that guided our tempest-tossed forbearers from vague seas to certain shore, where huddled-masses yearning to breath free finally made landfall, where foamy whitecaps of muddled thinking were traded for a rock of sound footing.

Upon that rock, we have built the world.

Image Credits

The Matterhorn by Zacharie Grossen and appears under the terms of the Creative Commons Attribution-Share Alike 3.0 Unported license. Image of Wenlock Bookstore by Michael Maggs and appears under terms of the Creative Commons Atribution-Share Alike 3.0 Unported license. Wedding of Queen Victoria and Prince Albert engraved by S. Reynolds after F. Lock and is in the public domain. Woman teaching geometry from Wikimedia Commons and is in the public domain.

Images may have been cropped, resized, color adjusted, or otherwise modified from their originals.


All other weirdness (c) 2013-14 LabKitty Design

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