Math Filter: What is Calculus?
July 3, 2014 12:49 PM   Subscribe

My son would like to know: Is there more to calculus than functions, examples of functions, various formats for presenting functions, and so on? That is, does calculus have unique to it other tools that are not just variations on functions?

I dropped out of calculus my first quarter in college at age 18. I am in over my head. Please hope me, serious math geeks.

posted by Michele in California to Science & Nature (32 answers total) 14 users marked this as a favorite
Can you be more specific? Most of mathematics could be described as "just functions".

More generally you have things like infinitesimals and the underpinnings of calculus- which probably won't get covered very deeply in an introductory class.
posted by dilaudid at 12:57 PM on July 3, 2014

I'm not sure what the term "variations on functions" means. Most of math in general is "just variations on functions."

Your son might be interested in the (ε, δ)-definition of limit (which motivates calculus in general). Somewhat out of the box in the calculus toolbox is is non-standard calculus.
posted by saeculorum at 12:59 PM on July 3, 2014

Honestly, the basics of functions are usually considered precalculus.

Calculus lets us study how quantities change, and it lets us generalize a lot of our basic physics formulas. For example, most of us learn pretty early on that distance = rate * time. That's all fine and good if you travel the same speed throughout the problem, but motion is rarely that simple. Integral calculus lets us (for example) find the distance covered if we have a more complicated function representing our speed.

We can also find areas and volumes of shapes way more interesting than the basics people memorize formulas for (areas of triangles/rectangles/trapezoids, volumes of boxes/spheres/pyramids/cones, etc).

Here's an example with differential calculus (usually the first semester). Kids learn to find the slopes of lines in middle school or high school. With calculus, you can find the "slope" of crazier functions, though it won't just be one slope -- you'll have different slopes at different points along your curve/function. If your function represents your position in space, the slope at a point is telling you your speed (actually velocity) at that instant.

There are TONS more examples, but these are a couple that should be fairly sensible to someone with a typical high school math background.
posted by ktkt at 1:06 PM on July 3, 2014 [2 favorites]

Response by poster: From son:
As far as I could tell from reading up on calculus, all it's about is functions, and all a function is goes like this: "If you know number A, you can know number B by plugging in calculation C". I'm basically wanting to know if that really is all there is to calculus. (Saying it's more complicated than that to mean that it includes cases like "If you know number A and also D, then you can know number B by plugging in calculation C" isn't really indicating that there's more to calculus than just this basic concept)

My son read the Cartoon Guide to Calculus and, I guess, the Anime Guide to Calculus. And he felt confused. Yesterday, he was saying to me something like that it looked to him like calculus is just defining a relationship between two things. I said "I am not sure but that could be correct. The book A Tour of the Calculus was filled with parabolas. So you might be right."

So that's what we are trying to figure out.
posted by Michele in California at 1:06 PM on July 3, 2014

I also am not sure what you mean by "more than functions." Calculus does focus on doing specific things with functions. I had some slightly incoherant thoughts on functions and their relation to math here, then I thought to check wikipedia, which has a really excellent summary that makes much more sense than mine:

Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit...Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

posted by pseudonick at 1:08 PM on July 3, 2014 [1 favorite]

Sounds like maybe he's learning about derivatives and integrals. A lot of that is about rearranging functions and matching them to certain standard forms that are easier integrate. Much of that ends up feeling like a long list of tricks, rather than a cohesive theory of some kind.
posted by ryanrs at 1:09 PM on July 3, 2014 [1 favorite]

I'm more like you than your son, so this is probably not what he means, but I didn't understand until I heard it explained on CarTalk that calculus is uniquely useful because it allows us to study the area under a curve- that is, it lets us understand the properties of a shape in which at least one side is not a straight line segment.

I probably mangled that badly enough that it will offend real mathematicians, but that explanation helped me understand why calculus was important and why you would want to figure out some method for understanding those shapes (or I guess really it's a curved line which forms a "shape" when plotted on a graph with Cartesian coordinates). A lot of times, I got lost in the formulaics of something and it was hard to slog through the boring (f)x= type stuff without the higher level understanding of what it was all for. Sorry of that's not really what you were asking for.
posted by Snarl Furillo at 1:09 PM on July 3, 2014 [1 favorite]

Some of the more concrete things I remember doing in calculus class were optimization and finding the volume of weird 3-D shapes. There's also all of physics.
posted by ryanrs at 1:14 PM on July 3, 2014 [1 favorite]

Best answer: Calculus is a part of a broader branch of mathematics called analysis. It was the first part of this branch to emerge, and was connected to "applications" in astronomy and physics (Newton and Leibniz have roughly equal claim to having discovered it first, and worked independently of one another, so the idea was very much "in the air"). The formal notion of a "function" was actually developed by Euler after Leibniz and Newton did their work.

So what is analysis all about? Well, to give a very loose high level description: the study of continuous, rather than discrete, mathematical constructions. A question like "How many 5 card hands can be drawn from a deck" has a very discrete flavor. A question like "If I throw this rock in the air, what point will it spin around? What's its center of gravity?" or "How will air flow around this car?" are paradigmatic examples of phenomena considered continuous.

saeculorum mentions the epsilon-delta definition of a limit. This the the modern formalization of what we mean by continuity (which Leibniz and Newton did not employ at all in their treatment!). It's very elegant though, and basically says: for a function (a rule that lets you map each number to another number) to be continuous, it has to be the case that we can define exactly how close the input needs to be to a particular value to ensure the output is within a given range. To put it another way, it says that when we're dealing with a continuous function, we can be arbitrarily precise, as precise as we want to be. This deals with the issue of an infinite regress: Where is the center of the gravity of the rock? Depends on how wide the rock is. Well, we can always measure the width more precisely. Well, for a given degree of precision of the width measurement, I can tell you how precisely we know the center of gravity.
posted by phrontist at 1:16 PM on July 3, 2014 [11 favorites]

Let's say your son goes for a walk with one of those fancy step counters. Every minute, he records the total number of steps he's taken and the time, so he has a chart that looks like this:
time | steps
  0  |   0
  1  |   5
  2  |  10
  3  |  19
  4  |  24
  5  |  30
Using what he learned in algebra and basic physics, he could say "well, distance = rate x time, and in 5 minutes I took 30 steps, so my average rate was 6 steps per minute." But what if instead of knowing the averge rate, he wants to know exactly how fast he was walking at minute #2? This was a question that absolutely plagued mathematicians, and led them to developing differential calculus.

Calculus also helps go to the other way. What if instead of a step counter, your son had a little velocity meter that recorded his walking speed, but he wanted to use that to calculate the total number of steps he took instead, so he could prove that he passed his school Fitness Challenge? Integral Calculus was developed to solve this very question.

I'm sure this is all discussed in the Cartoon Guide to calculus. Calculus is just a tool used to solve an unimaginably large set of math problems in an elegant way. To me, Calculus is more about developing the function you need to use to solve a problem than it is about the actual algebraic steps used to solve the problem ("plug and chug.") But it sounds like this book is starting from the point where it has to define what a function is before it can talk about how calculus the tool can be used to develop helpful problem-solving functions from other functions.
posted by muddgirl at 1:17 PM on July 3, 2014 [5 favorites]

calculus is just defining a relationship between two things.

Even if true, that is EVERYTHING. What is figuring out life and this world, really, other than defining the relationship between things? Finance, physics, computer programming, social sciences, etc. These all use calculus. Here's a halfway decent Tedx talk.

Has your son watched the new Cosmos with Neil DeGrasse Tyson? He should.
posted by melissasaurus at 1:34 PM on July 3, 2014 [1 favorite]

Not a mathematician: Calculus provides tools for transforming complex functions into simpler functions, which allow you to calculate new related values. So for example, in basic geometry, you're told that a sphere has the following properties:

volume = (4/3)πr3
area = 4πr2

(That's pi in those equations, not n.)

Using calculus, you can figure out how these two equations are related. From a mathematical perspective, these transformations can help reveal new ideas or properties of the objects/equations being studied. From a scientific perspective, these transformations can be useful for understanding certain relationships. For example, surface-area:volume ratios for different shapes are important in chemistry, engineering, and biology.
posted by CBrachyrhynchos at 1:37 PM on July 3, 2014

Mentally, my short hand way of thinking of Arithmetic vs Calculus is as Discrete vs Continous and/or Model World vs. Real World.

This is most likely a gross simplification of the relationship however it allows me to read popular science books and not have my mind blown too much by the abilities of mathematicians and physicists.

I took 13 math classes in college and am approaching 50 years old so I may be on target or way off base.
posted by dgeiser13 at 1:48 PM on July 3, 2014 [1 favorite]

I always thought the pat answer was "calculus is the study of change." The advantage calculus has is being able to look at change over an infinitessimally small period. If you have some function of time, say temperature throughout the day, it's easy to figure out how much the temperature changed the past hour but calculus lets you figure out how quickly the temperature changing right this second.

I would recommend the book Calculus Made Easy. It's a bit old fashioned but delightful. The first chapter is called "To Deliver You from the Preliminary Horrors" which is just awesome. Then it explains that the intimidating integral sign is really just an s, and stands for "sum" cause all integrals are doing is adding things up.

Functions are just how mathematicians like to describe the world and are not specific to calculus. I always thought of them as a machine that you stick some input into and the function spits out one output. So you can say that birthdays are a function of people, where you put a person into your machine and it tells you what their birthday is. Or as above, temperature is a function of time. Stick a time into that function and it will tell you the temperature at that time.
posted by carolr at 1:57 PM on July 3, 2014 [3 favorites]

Calculus is used to calculate (predict) the movements of planets and satellites in our Solar system. It is even used to calculate the behaviour of the universe itself (expansion of the universe after the Big Bang). It is used to make predictions about interactions of the smallest constituents of matter. It is used in a number of fields.

Yes, it is (in a way) all about functions, like he said. However, one could be dismissive of everything that we do and say that it is either about words or numbers. (Even pictures and sounds can be described uniquely using numbers.)
posted by aroberge at 2:23 PM on July 3, 2014

Response by poster: He is not being dismissive. He is pretty severely aspie-ish and has dyscalculia. He is trying to understand but the books have given him a lot of different specific examples and have not given him a big picture overview on terms that work for him.

Because he is really, really aspie and literal-minded, if you showed him, say, 50 chairs to explain "chair" to him this would overwhelm him. He would focus on all their differences ("But the wooden dining chair looks NOTHING like the upholstered living room chair. How can they both be called a chair? What IS a chair, mom? I just don't understand.") So he is looking for the essence of calculus (the way the essence of "chair" is "something designed for one person to sit on which has a back to lean against") so he has a context to put the details into. He just can't figure out how else to relate.

He is very into science-y subjects (and explained relativity to me when he was 13) but his math disability makes dealing with the math part very challenging. When he was little, he successfully lied to both me and one of his teachers all year long so he could sneak his classwork home, tell me it was homework and have me explain it -- even though we frequently fought about it for an hour and he often ended up in tears. He really wanted to understand. And I have a strong math background and was able to give him the kinds of explanations that worked for him (and then we homeschooled starting early in sixth grade). But I just don't know enough about calculus to really help him with this.

Thank you for the answers so far.
posted by Michele in California at 2:33 PM on July 3, 2014

Carolr: I always thought the pat answer was "calculus is the study of change."

Yes, this.

Derivatives measure the rate of change of something - the rate of change of your position as you drive a car, the rate of change of your speed as you step on the gas, the rate of change of the volume of the observable Universe as it expands. And yes, the slope of a parabola (y = a*x^2 + b*x + c) at a particular point (x=4).

Integrals measure the cumulative effects of a change - they're the inverse of derivatives, in that sense. How far you've gone while driving at a given speed, how fast you're moving after accelerating for a minute from that stop light, the volume of the the Universe after it has expanded at an accelerating rate for 13.6 billion years. And also the area under a curve between two points (the area enclosed by y = a*x^2 + b*x + c and y=0 between x=0 and x=4). (Which is the same thing as the previous examples, if you think about it: your position on the highway is y = acceleration*t^2 + speed*t + starting position.)
posted by RedOrGreen at 2:34 PM on July 3, 2014 [1 favorite]

Calculus is the path to differential equations, which are used to model, explore, and predict the behaviors of the visible and invisible aspects of the macroscopic analog universe, both in motion and at rest.
posted by the Real Dan at 2:40 PM on July 3, 2014

The explanation that my physics and calculus teachers both went with as their elevator pitch is that "Physics is calculus and vice versa," which is pretty true as far as classical physics goes. The continuous and relational aspects might help him understand as well. I tend to think of algebra as sufficient for when you have one thing changing, but when you have two things changing at once, that relationship generally requires calculus to be robust.
posted by klangklangston at 2:42 PM on July 3, 2014 [1 favorite]

I had physics first (non-calc-based), and when I took calculus it felt immediately obvious why it had been invented. So maybe just trying to do a tiny bit of basic intro-textbook, first-few-chapters physics (kinematics) would help him understand why calculus was designed in the first place.
posted by you're a kitty! at 2:42 PM on July 3, 2014 [5 favorites]

My father had a college professor who said that if, in 25 years, his students remembered that differential calculus had to do with rates and integral calculus had to do with sums, he'd be happy.
posted by SemiSalt at 3:59 PM on July 3, 2014 [2 favorites]

Math is a language we use to describe the world. Functions tell you about how things are related. If you can buy two apples for the price of one peach, we can say 2*(price of apple) = (price of peach). If you're walking at 4 miles per hour, we can say (distance in miles) = 4 * (time in hours).

As you start trying to say different things about the world, you need more complex functions. If you're walking at a variable speed then (distance) = integral(velocity with respect to time, over the time you were walking). So at some level it's "all functions"; but that's like saying that literature and children's books and science textbooks are "all words". The value is in what the different functions let you do.

You can use non-calculus math to describe a lot of static situations. But if you want to be able to talk about something that's changing - like a ball thrown through the air - you need calculus, or you need someone else who's done the calculus to give you the equation (this is how non-calculus physics classes work).

It's sort of a problem in the way we teach math that people can be successful through years of schooling without ever realizing that math describes the world. But if he gets interested in figuring out how to express things - scientific things, observations, whatever - using math then he'll probably find it all much more sensible. Much like learning any other new language, memorizing vocabulary and grammar patterns is pointless and frustrating, but learning how to say new things is exciting.
posted by Lady Li at 4:06 PM on July 3, 2014 [3 favorites]

If he hasn't started using it, I think a simplified way to look at it is:

Calculus is a set of mathematical tools that help to simplify the solving of certain kinds of complicated algebra problems with multiple variables. Without it many of these problems would otherwise require brute force techniques to solve.
posted by mzurer at 4:26 PM on July 3, 2014

This is a really interesting question, and not at all a trivial one. I'm "good at math" in the sense that I took and passed high-level math in school and have a knack for a few narrow party tricks, but I don't feel like I really "got" calculus until my first semester of grad school. I could apply the techniques, and I understood the relationship between derivatives and integrals and change, but for me, these felt like discrete concepts and tools that just happened to fit under the same umbrella of calculus, much like your son is describing.

I think you're a kitty!'s idea is a really good one--sophomore-level college calculus (the one that's aimed at all STEM types, not the one that's taught to people who are studying mathematics itself) is very focused on the mechanistic aspects of how to evaluate derivatives, integrals, etc., and so it winds up feeling more like Algebra Plus (If you know number A, you can know number B by plugging in calculation C) than anything else. If he's reading books that are aimed at people who are trying to pass those classes, he's going to get a very narrow view of calculus (like seeing only leg of a chair). You might want to see if your local library has any books on "recreational mathematics" (since they're aimed at people who are learning about this for fun, they tend to be freer to go off on tangents and try problems that have something interesting or weird about them, rather than ones designed to illustrate how to use this or that particular method). If you can't find any rec. math books, an entry-level calc-based-physics textbook is a fine and widely available substitute.

It's kind of the difference between teaching kids how to count and and when they start to really grasp what a number line represents, how fractions can be interpolated between them, what negative numbers "mean", how basic arithmetic operations (+ - * /) relate to one another, etc. There's a certain method to all those things, but there's also an abstract, higher level overhead to it. You can sort of help along that development of the abstract sense by doing arbitrary problems, whether they're 2 x 3 = 6 or int(1/x) = lnx, but nearly everyone needs some more concrete (there's a few people I know who are basically unicorns of high-level mathematical abstract thinking and can just get the abstract implications really quickly, but it's rare.) For young kids, it's pretty easy to come by those opportunities, in part because counting and arithmetic are tasks that come up frequently, in part because adults can easily recognize and point them out. For adults trying to learn calculus, it usually comes in the form of studying other scientific disciplines. (A lot of more modern approaches to early math education is about trying to introduce kids to some of the basic concepts of calculus earlier, so that they can have a better foundation for the abstract/numerical sense underpinnings of it if they decide to study it more later on.) Once you really start to get it, it really does change how you think about quantity and relationships between different variables and how things work, in much the way really starting to understand numbers did. That's what makes it kind of magical.

One concept that helped things click together for me that no one has really talked about is related to the "area under a curve" concept, but it's about probability. Probability distributions (bell curves are probably some of the most familiar ones, so let's use that), are normalized--that is, you assume that if you were to graph the curve and add up all of the area underneath, it would add up to 1 (or 100 if you want to think about it as a percentage, or N if you know how many total outcomes you're evaluating). If you know that a particular outcome falls far away from the center--out on one of the edges of the bell curve--you lop off that part and add up (integrate) under just that. It's why the probability of being further from the center drops faster and faster the further away you go--it has to do with how the shape of the curve changes the area underneath
posted by kagredon at 4:38 PM on July 3, 2014 [1 favorite]

well, you can think of calculus as the study of "functions of functions" with the Derivative and Integral mapping functions to functions (simplification). D[x^2]=2x or I[x^2]=x^3/3+C. there's a lot more to calculus (applications, implications, etc), but a lot centers on these two operators
posted by youchirren at 5:30 PM on July 3, 2014 [1 favorite]

He is very into science-y subjects (and explained relativity to me when he was 13) but his math disability makes dealing with the math part very challenging.
You can't really get physics unless you get calculus. Some people get physics better than math because there's a more obvious reason/purpose. If your son gets relativity, check out MTW's Gravitation, it goes over the calculus needed for relativity. That might help your son pair up the concepts.

More jumping off points can be found in the wikipedia article the mathematics of relativity.

Others have mentioned it already, calculus is about rates of change and summations. If he's getting caught up in "these are just all functions", one response could be ...
... and all these letters are just variables and while they all seem to be manipulated in ways that vary slightly from one to the next, combined, these are all mathematical tools. Just like a mechanic might have several hex drivers in metric and another set that looks very similar but is SAE. The mechanic may also have philips screw drivers, slotted drivers, tri-wing, torn, etc. While they all may be similar in either type or size, if a 6mm hex driver is needed, a 1/4 inch hex driver would be just as useless as a 6mm slotted driver. You use the right tool for the right job and we're lucky to have so many tools already developed for us, letting us all stand on the shoulders of giants."
posted by Brian Puccio at 5:42 PM on July 3, 2014 [1 favorite]

To me, the basic question of calculus is this:

If you have some function f(x), how does its value change as x changes?

As x gets higher, does f(x) increase rapidly? Get closer and closer to some limit? Oscillate between high and low values?

Whatever the answer is, that's called the derivative of f(x), and is another equation. You can also take derivatives of derivatives and so on. And you can invert derivatives and that's called the integral. You can also think of the derivative as the slope of the curve at every point of the graph of f(x), and the integral as the total area under a curve.

That's basically all calculus is, at its core. Everything else is just kind of expanding on that basic concept (working with multiple variables, etc).
posted by empath at 4:13 AM on July 4, 2014 [1 favorite]

For a concrete example, consider the function:

f(x) = x

That gives you a line with a slope of 1 at every point, which is to say, that if x increases by 1, f(x) increases by 1.

So the derivative of f(x) [which is notated f'(x)] is


Let's look at something slightly more interesting:

f(x) = x^2

Note that f(x) starts very high at -infinity, goes down to 0 at 0 and rapidly goes towards infinity as x goes to +infinity.

So you start with a very high negative slope, goes to 0 slope (flat) at 0, and then goes to a very high positive slope at the end of the graph.

The way you actually calculate this gets into the interesting part of calculus, but it turns out that the function that defines the slope at every point on the graph of x^2 is


That's the very, absolute basics of what calculus is about, but if you think of it as trying to find derivatives and integrals for increasingly difficult and complicated f(x)'s you wouldn't be far off.
posted by empath at 4:23 AM on July 4, 2014 [1 favorite]

Along the lines of the suggestion above to check out some "general audience/math is beautiful" math books, another suggestion is to read about how (and more importantly why) calculus was developed in the first place. Amazon seems to have a whole "History of Mathematics" category that might yield some interesting books, and I'd think that biographies of Newton or Leibniz would also go into the subject.
posted by trig at 8:28 AM on July 4, 2014

Better explained is a great site for learning the intuition behind mathematical concepts.
posted by greytape at 12:19 PM on July 4, 2014 [2 favorites]

I'm definitely not a math expert, but am now out of the standard STEM calculus series. I am a very visual learner and blocks of symbols on a page do nothing for me.

If I had to break it down to its most basic level, calculus is all about infinities and infinitesimals.

The first thing you do (after limits) is derivatives. As stated above by others, the derivative is the instantaneous rate of change of a function at a point. This is really useful for a ton of real world applications (like velocity and acceleration to start). The proof for how to find this mechanistically by performing a series of changes on the original function is all rather symbol heavy and boring (to me).

Instead, draw a curve (this would also work with a straight line, but isn't as interesting). A parabola would work, but you don't want to get bogged down with specifics. Now, pick any arbitrary point on the curve and then ask the question, "What is the slope of the curve at this point?". This is the instantaneous rate of change of the function at that point. Seems like a tough question to answer though. You could probably try to tease it out geometrically and eventually you'd figure out that the slope is the tangent line to the curve at that point. What calculus does is give you a way to reliably find that tangent line every time (well for continuous functions). It does this by basically "zooming" in on the curve mathematically. The more you "zoom in" on the curve at the point, the less it looks like a curve and the more it looks like a straight line. Eventually, by zooming in on an infinitesimally small piece it will be indistinguishable from (will have a limit of) a straight line and that's also your tangent line. That's my intuition of how derivatives in calculus "work". The rest of it is just rules for how you can pretty quickly find the equation for the slope of the tangent line for any point. Multiply each term by the exponent and then reduce the exponent by one. What's the slope of the tangent line for x^2 anywhere? BLAM it's 2x! What about x^3? BLAM it's 3x^2!

The second part of calculus I learned is integrals. You can take that same curve you drew and then try to figure out the area underneath it. Why would you want to do this? Well, it has a bunch of real world applications but for now let's just say you want to know just because. One way to figure an approximate value for this area would be to draw a bunch of rectangles of equal width whose height is the height of the function. But, if we want an exact answer, we can imagine making the width of those rectangles infinitesimally small and adding them all up. Seems like that wouldn't work because then you have to add infinity little rectangles together and that would give you infinity. But, it turns out, that if you do this the value you get actually approaches a limit (well for most cases you're given in a calc class). Calculus! Again, there's a bunch of rules about how to manipulate the original function to figure out this area under the curve thing, but the main idea is that we can add up a bunch of infinitely small things to get a finite number. Pretty cool.
posted by runcibleshaw at 12:05 AM on July 6, 2014

Response by poster: Thank you so much to everyone who participated. It was helpful to my son. He feels he got what he needed out of it.
posted by Michele in California at 1:07 PM on August 3, 2014

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