You are Socrates, I am Meno's slave.

January 17, 2010 6:09 PM

Please explain what calculus is to a dummy.

I never learnt calculus at school, not even close, as I don't even know what it is. I've tried looking up some super simple explanation, but they say things like "it's about infintesimals!", "it's about change!", or try to tell me about integrative and difference functions...or something. I'm way past the point of trying to learn how to do calculus (so don't even try), but I would really like to know what it is and what it does - it's been bugging me for a while not knowing

So, in plain language and using really basic maths concepts, can you please describe calculus to me? at least to the point so I am aware of what I'm missing out on?



(For example: to me, geometry is about using universal ratios/known formulas to figure out the size of different parts of a shape in relation to others. That's the level of explanation I'm looking for.)

posted by Sova to science & nature (48 answers total)

Take a ball traveling over a given distance.

Using algebra, you can estimate its average velocity over time.

Using calculus you can determine its specific velocity at a particular instant during the time it is traveling.
posted by dfriedman at 6:13 PM on January 17, 2010 [4 favorites]

You can find an extended layperson's explanation by searching for "calculus" in John Allen Paulos's Beyond Numeracy on Amazon, and reading from page 27 onward.
posted by Jaltcoh at 6:17 PM on January 17, 2010

Calculus is used for measuring the rate of change. Consider a curved line on a graph. Now consider a straight line that touches that curve at one point. The slope of that line (m) where it touches at that one point (the tangent) is the derivative.

At that single point in time, the slope represents how much change is occurring. The curve could be describing motion in space, or any number of other things.

There are some very interesting intro-to-calculus lectures available on iTunes, under the iTunes U section.
posted by jquinby at 6:21 PM on January 17, 2010 [2 favorites]

Humans can look at a graph and tell where it's 'steep', where it's 'shallow', and whether the space under it is large or small.

Calculus is about making those sorts of distinctions with mathematical precision.
posted by LogicalDash at 6:23 PM on January 17, 2010 [14 favorites]

I would say there are two main concepts: derivatives and integrals. Both of these are operations you apply to a function to get another function.

The derivative of a function is another function that at each point, tells you how fast the first function is changing at that point. For example, say you have a function f(t) that gives the position of something at time t. So the derivative, written f'(t) (or, the ugly way, "df/dt"), is the speed of that thing time t. Likewise, the derivative of the speed function is acceleration.

Taking the integral of a function is the opposite of taking its a derivative. If v(t) is speed at time t, the integral of v(t) with respect to t is position. This is also the area under the curve that v(t) makes when plotted on a graph—I'll post again if I think of a good intuitive explanation for this.
posted by k. at 6:24 PM on January 17, 2010

It is about slopes and areas. You are driving along a road, and every second your speed is plotted on a graph -- it will look like this. Want to know your acceleration? Well, that is the instantaneous slope at any point on the graph (the green lines, but you could put one anywhere on the graph), and in calculus speak it is called the derivative. Want to know how far you've gone. That is the area under the curve (the yellow block), and it is called the integral.

That's it, except it works for everything in nature, not just speed.
posted by Chuckles at 6:26 PM on January 17, 2010

There are two sides of calculus - differentiation and integration. Differentiation has already been described here.

Let's say you have driven a car at, say, 60mph for 3 hours. If you want to find your total distance traveled, you can multiply 60 miles/hour * 3 hours = 180 miles.

Integration lets you do this when the velocity is not constant. Integration is a trick to add up the infinite number of different velocities that your car had at every second of your trip. This can be done if you are given a function for your velocity, but there are also tricks to do it if you have only velocity or acceleration data points.

This is what the Wiimote is doing every time it detects your motion - the sensor is an accelerometer, it has no way to directly measure position or velocity. So, it has to go from acceleration to velocity to position using calculus tricks.
posted by Earl the Polliwog at 6:29 PM on January 17, 2010

Trying to describe it in roughly the same terms as your description of geometry… I'd say calculus is about dealing with smoothly-varying functions, either to figure out what their overall cumulative effect(?) is (integration) or to understand their behavior in terms of how they change from moment to moment (differentiation).

A common intro calculus example is the trajectory of a thrown ball. Gravity acts on it constantly, and you can integrate that to learn about the ball's velocity. Its velocity "acts on" its position, in a manner of speaking; so you can integrate velocity to learn about position, or differentiate position to learn about velocity.

The clever bit, the bit that makes calculus useful, is that instead of actually going through and summing the always-changing velocity moment by moment to get the ball's position moment-by-moment, you can apply calculus to get an equation for position if you have an equation for velocity. (Adding up the velocity moment-by-moment is sometimes called "numerical integration".)
posted by hattifattener at 6:31 PM on January 17, 2010

In a very, very basic sense:

There are two parts to calculus: integrals and derivatives.

Given a curve, an integral is the area "under" that curve. Like this: That curve labelled "f(x)"? The integral of it, from "a" to "b", is the size of the area that's shaded grey.

Given a curve, the derivative of it (at some point on the curve) is the slope of the line that is "tangent" to the curve, which if you don't know what that means, is probably easier seen than described. Like this: The blue curve's derivative, at the point near the middle of the picture, is the slope of the red line.

These two concepts can be used in an astonishing variety of ways, to figure out many, many different things.
posted by Flunkie at 6:32 PM on January 17, 2010

I'm way past the point of trying to learn how to do calculus...

I'm not so sure about this - between the iTunes stuff I've looked at, perusal of my wife's college calculus textbook and poking around at various websites, a lot of the black magic aura of calculus has dissipated for me. I'm way out of practice insofar as the mechanics of solving/simplifying go, but the basic concepts don't seem nearly as opaque as they did to me a long time ago, when I was dumb, impatient, and failing pre-calc with a vengeance.
posted by jquinby at 6:32 PM on January 17, 2010 [1 favorite]

A fun book (no, seriously!) about calculus is A Tour of the Calculus by David Berlinski. It's hilarious in part because he tries to be funny and in part because his prose is so over-the-top, but I think he also does a very good job of intuitively conveying what calculus is about. I read it before I learned calculus for real, and I think it was good preparation.
posted by k. at 6:34 PM on January 17, 2010 [4 favorites]

There are two main applications of calculus.

One, derivatives, allows you to find the particular rate at which you're increasing/ decreasing. Say you know that your speed for a trip was 60 mph, but you need to know whether you were slowing down or speeding up at a particular second. You can calculate this with calculus.

The other, integrals, allows you to find the area of an irregular shape. For instance, you can use geometry to know that a rectangle has a=lw. But what if a shape has a funky edge, like y=3x^3 -14x^2 ? You can use calculus to calculate its area.
posted by estlin at 6:34 PM on January 17, 2010

Here's a really easy explanation of one thing calculus can do (one of the first thing one learns in calculus).

Imagine a shape with wavy sides. How do you determine its area? Do determine it very roughly, you could fill it with a rectangle that's pretty close to the wavy shape's area, then just figure out the area of the rectangle. But you'd have a lot of the shape that didn't get covered by that rectangle, so it's a rough estimate. Well, how about using two smaller rectangles to better fill that wavy shape? You'd have an even closer estimate of the area. How about three? Even closer... How about infinitely many rectangles? Calculus.
posted by dayintoday at 6:34 PM on January 17, 2010 [1 favorite]

Calculus is used for measuring the rate of change. Consider a curved line on a graph. Now consider a straight line that touches that curve at one point. The slope of that line (m) where it touches at that one point (the tangent) is the derivative.

At that single point in time, the slope represents how much change is occurring. The curve could be describing motion in space, or any number of other things.

I think I understand this (and the others which say the same thing). But how can you know a slope from a point? I mean, don't you need two points to make a slope?

Integration lets you do this when the velocity is not constant. Integration is a trick to add up the infinite number of different velocities that your car had at every second of your trip. This can be done if you are given a function for your velocity, but there are also tricks to do it if you have only velocity or acceleration data points.

Same here? I mean how can you work out an "infinite" number of velocities? I assume this is the magic of calculus right? And I'm thinking about it in the wrong way.
posted by Sova at 6:34 PM on January 17, 2010

("To" determine it... not "Do"...)
posted by dayintoday at 6:35 PM on January 17, 2010

I'd also agree that almost anyone without some sort of mental deficiency can learn calculus. I taught a Survey of Calculus course many times - it's extemely accessible and light on algebra - only what you need to know to say you understand calculus. I've also heard the course labeled as Business Calculus or Harvard Calculus - might be worth looking into.
posted by Earl the Polliwog at 6:36 PM on January 17, 2010

If you have a curve that shows how fast something has been moving (assuming that the velocity has been changing) then differential calculus will tell you how much acceleration it's been doing, and integral calculus can tell you how far it's moved.
posted by Chocolate Pickle at 6:36 PM on January 17, 2010

I'm not so sure about this - between the iTunes stuff I've looked at, perusal of my wife's college calculus textbook and poking around at various websites, a lot of the black magic aura of calculus has dissipated for me. I'm way out of practice insofar as the mechanics of solving/simplifying go, but the basic concepts don't seem nearly as opaque as they did to me a long time ago, when I was dumb, impatient, and failing pre-calc with a vengeance.

Last time I properly learnt maths was when I was 12.
posted by Sova at 6:37 PM on January 17, 2010

wikipedia has a pretty straight forward explanation of calculus on the simple english page.
posted by phil at 6:39 PM on January 17, 2010

But how can you know a slope from a point?

From a single point, you don't - but there is a line that will touch the curve at one point, and you'd need to find the slope of that. In Flunkle's second link, that'd be the slope of the red line. What you end up figuring out is a formula to describe that slope, into which you could plug in values to come up with the answer you're looking for - at point X in time, I was accelerating (or slowing down) at a rate of Y mph.

You're probably closer to understanding it than you think - the rest is just usin' the tools. :)
posted by jquinby at 6:40 PM on January 17, 2010

I think I understand this (and the others which say the same thing). But how can you know a slope from a point? I mean, don't you need two points to make a slope?
What calculus does is looks at what happens when you slide those two points together and keep measuring the slope. The values you get will get closer to the "true" value of the slope at that point. It's the slope of a line tangent to the function.

Same here? I mean how can you work out an "infinite" number of velocities? I assume this is the magic of calculus right? And I'm thinking about it in the wrong way.
Well, that's sort of the trick of calculus. It uses the idea of limits to do this. It's basically saying "if I repeat this process, and increase my accuracy each time, what number will my measurements approach?". Calculus lets you bypass the problems that would arise if you were trying to take this sort of infinitely precise measurement - dividing by zero, etc.
posted by Earl the Polliwog at 6:40 PM on January 17, 2010 [1 favorite]

I think I understand this (and the others which say the same thing). But how can you know a slope from a point? I mean, don't you need two points to make a slope?

You don't get the slope from a point; you get the slope from a line.

It's a special line, called the "tangent" to the curve at that point. Again, it's probably easier seen than described, and I am highly confident that you yourself can easily draw the tangent to a curve at a point without even knowing any mathematics whatsoever.

Again, take a look at this picture. The red line is the tangent to the blue curve (at the point that the meet).
posted by Flunkie at 6:41 PM on January 17, 2010

It's a special line, called the "tangent" to the curve at that point. Again, it's probably easier seen than described, and I am highly confident that you yourself can easily draw the tangent to a curve at a point without even knowing any mathematics whatsoever.

Sorry to labor this, but I assume from what you're saying is that there's only one possible line I could draw to touch that point? Isn't there lots?

Well, that's sort of the trick of calculus. It uses the idea of limits to do this. It's basically saying "if I repeat this process, and increase my accuracy each time, what number will my measurements approach?". Calculus lets you bypass the problems that would arise if you were trying to take this sort of infinitely precise measurement - dividing by zero, etc.

So it's an approximation of infinity?
posted by Sova at 6:47 PM on January 17, 2010

There are lots of lines that touch at that point. There's only one tangent line that touches at that point.

If you think of the curve as a roller coaster, and yourself as a passenger in a car, and you're always looking straight forward (relative to the car), the tangent at any point is the direction you're looking in.
posted by Flunkie at 6:49 PM on January 17, 2010 [1 favorite]

The "magic of calculus" that lets you calculate "infinite" things (infinitely small, infinitely close, infinitely many rectangles) is accomplished by means of "limits". The derivative of a function is a limit of the slope of a line approximating the function over an "infinitely short" distance (i.e. a single point). The integral is a limit of a sum of the areas of "infinitely many" rectangles under the curve.

Limits are usually pretty easy to understand intuitively, but the actual definition is quite complicated.
posted by k. at 6:51 PM on January 17, 2010 [2 favorites]

Calculus is, at its core, about understanding now quantities change. Let's say I release a ball from a building and let it fall. Your intuition tells you that the ball accelerates over time, starting slow and building up speed as it continues toward the ground. If you wanted to know how fast it was going at some particular point in time knowing just the balls position and time everywhere, how would you find it?

Well, if this were a car traveling at some constant speed, you could find the speed by asking how far the car travels in some known amount of time. If the car travels 30 miles over the course of an hour, you know the car's speed is the distance traveled divided by the time it took to travel that. In this case, 30 miles / hour. This ball doesn't have constant speed, though, so that approach doesn't quite work. If I ask how far the ball travels in a second, its velocity will have changed quite a bit and so distance traveled/time is a poor estimate of the velocity. But if I ask how far the ball travels in 1/1000th of a second, the velocity will have changed hardly at all and so this distance traveled/time is a good estimate. You can imagine that if I shrink the time interval more, the ball accelerates less, and distance traveled/time will become a better and better estimate. The derivative is the method of how to do that shrinking until the time interval is infinitely small.

The integral is the flip side of the derivative. Now let's say that I always know our ball's velocity as it's falling, but I want to know how far it has fallen. If the ball were that car going at a constant 30 miles/hour, I could just say that distance = speed * time. Two hours of travel at 30 mph will make one go 60 miles, for instance. But since the speed is changing, what do we do? Well, I can estimate the distance the ball has gone by breaking up the fall into small chunks of time. During 1/1000ths of a second, the velocity doesn't change much, so I can estimate that the distance travelled after 1/1000th of a second is the speed going into that time interval times the duration of the time interval (0.001 seconds). By adding up the distance travelled during each 1/1000th of a second, I can estimate the distance by knowing the velocity. You might imagine that similar to before, the shorter the time duration (and thus more chunks), the better that estimate will be. The integral is the method of how to do this shrinking so that you add up an infinite number of infinitely small chunks.

And that's calculus! Everything else is just a consequence of those things.
posted by Schismatic at 6:51 PM on January 17, 2010 [2 favorites]

If you think of the curve as a roller coaster, and yourself as a passenger in a car, and you're always looking straight forward (relative to the car), the tangent at any point is the direction you're looking in.

Maybe this would have been more clearly expressed as:

If you think of the curve as a roller coaster, the tangent at any point is the direction that the roller coaster's car's headlights are pointing when the car is at that point on the curve.

I know, roller coaster cars don't have headlights, but hey
posted by Flunkie at 6:53 PM on January 17, 2010

I think I understand this (and the others which say the same thing). But how can you know a slope from a point? I mean, don't you need two points to make a slope?

Imagine two points on a curve. If you move the points closer and closer together, then the 'overlap' of the line and the curve gets smaller and smaller, and you'll get a better and better approximation of what the slope is at any one point.

Calculus is about making those two points infinitely close so that they are effectively at the same place. Using calculus you take the equation of the curve and convert it into an equation that tells you the slope of that tangent line at any point.

Integration works the same way. Lets say you want to know the area under the curve, but you don't know calculus. So you find the height of the curve at some number of points, say 10. Then you calculate the area of a rectangle as high as the curve is at each point, and wide enough so that they all touch.

Lets say you want a closer approximation, so rather then 10 strips, you use 100. But that's a lot of work.

The but, using integration you can make the strips infinitely thin. And you end up with an equation that will tell you the area under the curve between negative infinity and any point you want.

So basically the trick of calculus is converting equations into their derivative (for finding slopes) and integral (for finding area) forms. And the cool thing is that derivatives and integrals are opposites. So you convert a function into it's integral, the original function is the derivative of the new one. And if you find the derivative of a function, the original is an integral of new one.



---

The concepts of limits plays into this. Lets say you have a function like 6/x. You know if x = 0, then the function is undefined. But what if you want to know 6/0.0001 or 6/0.0000001. You can keep doing that you'll get larger and larger numbers. So you know the limit of 6/x as x approaches infinity is also infinity.

Calculus is all about finding the limit of these functions, for finding the area and for finding the tangent. Except you ordinarily don't need to think about the limits, you just apply the formulas and transformations to convert derivative -> integral and vise versa.
posted by delmoi at 6:53 PM on January 17, 2010

but I assume from what you're saying is that there's only one possible line I could draw to touch that point? Isn't there lots?

For any curve that's "smooth enough", there is only one line that is tangent to the curve at that point. Basically, "tangent" means it locally touches only one point and no others, and if it only touches one point then it is a unique solution. There are math-y proofs for this.

There are some functions or curves that have points on them where you could draw lots of lines that touch that point and no others. Think about a function that's got a right angle in it - you could obviously draw lots of lines that touch only that point. We call those functions/curves "un-differentiable." So basically, differentiation only works for a certain subset of curves.
posted by muddgirl at 6:54 PM on January 17, 2010

Sorry, I should have linked Limit of a function. The basic idea is the value of a function f(x) as x gets really close to some point, but not exactly on it, or (for limits at infinity) when x gets really big but not technically infinite.
posted by k. at 6:54 PM on January 17, 2010

A recent In Our Time offers a nice historical discussion of the problems that calculus was designed to solve in the late 1600s, and the rivalry between Newton and Leibniz over its origins.

The Newtonian strand is geometric -- it's basically about how to find a tangent of any given point on a curve. (That's derivatives.) The Leibnizian one is algebraic -- an method of calculating and expressing infinite series (e.g. Achilles and the Tortoise) and that gets you integration.
posted by holgate at 6:57 PM on January 17, 2010

Calculus is sort of trigonometry and algebra + another dimension. The volume of a curve rotated about the x axis, rates of change, etc.

Calculus is super easy, if you know the algebra and trig that it builds on. It is magic if you don't.
posted by gjc at 6:58 PM on January 17, 2010

Thanks everybody! I feel I've kinda abused AskMe for a quick maths lesson, but I think I get what it's about now. That's really great of you lot to dumb it down. Thanks again!


Calculus is super easy, if you know the algebra and trig that it builds on. It is magic if you don't.

Oh man, you don't know. It seems pretty magical from where I'm sitting.
posted by Sova at 7:01 PM on January 17, 2010

But how can you know a slope from a point? I mean, don't you need two points to make a slope?

By the way, this, right here, strikes to the very heart of a mathematical/philosophical argument that raged for a really long time at the dawn of calculus, and indicates to me that your basic intuitive grasp of the relevant questions is completely sound.
posted by escabeche at 7:11 PM on January 17, 2010 [2 favorites]

Oh man, you don't know. It seems pretty magical from where I'm sitting.

Sure. But that's why it's really quite fun to learn - you start with the basics, and work your way up, and suddenly you can do magic.
posted by Tomorrowful at 7:28 PM on January 17, 2010 [3 favorites]

I think I understand this (and the others which say the same thing). But how can you know a slope from a point? I mean, don't you need two points to make a slope?


When you look at your car's speedometer, and it says 50 miles per hour, what does it mean? If speed is distance divided by time, how can you have a distance with one point and a time at one instant? Your intuitive idea of what is meant by a speedometer reading is what is made mathematically rigorous by calculus.
posted by Obscure Reference at 7:28 PM on January 17, 2010 [1 favorite]

Here it is in plain English

Calculus is the study of change and rates. Algebra is how you model something, calculus is how that system 'changes' and how to predict that.
posted by townster at 7:30 PM on January 17, 2010 [2 favorites]

When I first learned calculus, and I spent a lot of time thinking about it and putting it in to my own words, I came up with a metaphor for thinking about it. I still remember the moment the idea came to me - it was pretty cool.

Let's say it's raining. You are able to tell by looking whether it's pouring, or just barely sprinkling, or somewhere in between, right? You are able to tell the rate at which rain is falling at any particular moment in time. Not with huge precision - you probably can't say, "oh, it's raining at a rate of 2 inches per hour" - but you'll have a qualitative sense of the rate of rainfall.

This is like knowing a derivative. As you have pointed out in your comments, it's somewhat meaningless to say that in one single moment it's raining at a rate of 2 inches per hour - it's not a very physically relevant thing to say. But what we really MEAN is: if it kept raining exactly as hard as it is right now, for an hour, 2 inches of rain would fall. That's what it means. That's slope, too, with just a little more imagination. Think of a graph, with a line on it representing how much rain has fallen so far. Let's say the rain was first very light, then heavy, then light again. The graph would rise up slowly at first, then very steeply, then level off again - it would look sort of like the profile of a step in a staircase, leaning to the side. As time marches on, at every moment, the line is going a certain direction. If at a certain moment we were to fix the direction of the line and have it continue forever in the exact same direction, we'd get a perfectly straight line that is the slope of the original curve at the point of interest. There is a slope, representing the momentary direction, for every point on your line.

Now, here's the other side of calculus: if you stand outside in the rain with a little tube, you'll be able to tell how much rain has fallen. At the beginning, your tube will be empty, and at the end, it'll have a certain amount in it, however much rain has fallen overall. In between, as the tube fills up, if you were feeling industrious, you might mark down, say, every 5 minutes, how much rain is in the tube. If, as described before, the rain is first light, then heavy, then light again, after the first 15 minutes you might have only 0.1 inch, but then after 30 minutes total you might have 1 inch, meaning that 0.1 inches of rain fell in the first 15 minutes, but 0.9 inches fell in the second 15 minutes. You could make a graph representing the rising level in your tube as time goes on. If you think about it, your graph will look just like the "step" graph I attempted to describe above!

Or, you could make a different graph - one showing the rate of rainfall, not the cumulative amount. So, for the first 15 minutes, the rate would be low, right? But then as the downpour arrives, the rate would rise steeply, and hold relatively steady for a while, then taper off as the downpour ends. Right?

And here's where the integral comes in. At the end of the rainstorm, you're holding your tube full of water and you want to know where on your new, second graph that final quantity of water is represented. Well, as it turns out, it's the space under the step. The area under the stair. On the x-axis, we have time, and on the y-axis, we have amount of rainfall per time. You know that if you want to find the area of a rectangle, you multiply one side by the other to get area. Same here - we want to know the area under the curve, so we multiply time by rainfall per time, and we end up with rainfall! Only, and here's where the actual math comes in, when the graphs get complex and curvy, you can't use regular multiplication to do this, and you must use integration. Integration is basically just an even fancier form of addition than multiplication is (multiplication itself, remember, is just adding a number to itself a specified number of times).

I hope this helps!
posted by Cygnet at 7:31 PM on January 17, 2010 [2 favorites]

For tangents, it's helpful to think of setting a basketball down on the ground and placing a board on top of it. Where the board and ball meet is your point, to which the board is tangent.

Now push down on one side of the board to change the slope (without moving the ball. You'll find that the point of contact between the ball and the board is in a different spot.

In fact, if you pick a point on the ball, there's only one angle that the board can be at while touching that spot (in two dimensions). Likewise, for any point on a curve, there is only one tangent line.
posted by chrisamiller at 7:36 PM on January 17, 2010 [1 favorite]

Correction (sorry):

I said: Well, as it turns out, it's the space under the step. The area under the stair.

But the graph this part refers to is not step-shaped, it's sort of "outline-of-top-hat-shaped".

Everything else still applies.

(Sorry!!)
posted by Cygnet at 7:38 PM on January 17, 2010

I agree with pretty much everyone here in that Calculus is nothing without integrals and derivatives, describing Calculus only by derivatives and integrals is incomplete. They are the most important RESULTS from calculus, but don't necessarily DEFINE it.

Socrates: what is the definition of Calculus?

Meno: Calculus is the study of the infinitessimal.

Socrates: Ok, now, what the hell does that mean?

Meno: An infinitessimal is a really, really, really small number. So small, in fact, that for any number you guess, an infinitessimal is smaller. We can say then that Calculus is the study of really, really, really small shit.

Socrates: Ok, great! ... but what's it good for?

Meno: Well, I make better approximations of things like slope and area by breaking down a large, unwieldy function into smaller, more manageable bits... what would happen if i broke those bits down even further and turn them into infinitessimals?

Socrates: Here's the interesting results, and the point of Calculus: the smaller the bits into which we break up a function, the more accurate we will be able to calculate it's slope or area (for example). Hence, the idea is to break up the function up into the smallest bits we can think of (infinitessimals), and find the most exact slope or area possible.

Meno: So, Calculus is a way to exact an approximation?

Socrates: YES!
posted by chicago2penn at 9:10 PM on January 17, 2010

But how can you know a slope from a point? I mean, don't you need two points to make a slope?

Nope! That's the genius of it.

Take a straight line. It should be easy to determine the slope of a straight line. You subtract the Y coordinates and the X coordinates and divide the two.

With a curve, this naturally doesn't work, because you could take any two random points from any part of the curve and get completely different numbers. But here's the cool part: the closer those two points are to each other, the more accurate the slope. So, if you could take the slope of two points that were infinitely close to each other, you'd have the exact value of the slope at a specific point. This picture explains it best.
posted by Civil_Disobedient at 9:22 PM on January 17, 2010 [3 favorites]

I would second k.'s recommendation of Berlinski's A Tour of the Calculus. People who love math seem to find plenty of fault with it, but I found that it was easy to read and keep up with the concepts, and when I was done I had a much deeper understanding of what Calculus actually means.
posted by dreadpiratesully at 9:46 PM on January 17, 2010

Others have done a nice job of explaining what derivatives, integrals, and limits are.

Those are the three fundamental concepts you'll learn in calculus.

Here is how those three concepts relate:

1. The central insight of calculus is that the derivative and integral are inverses of each other. That is, they have a similar relation as do addition/subtraction, multiplication/division, or exponentiation/logarithms.

However the neat thing is that addition, multiplication, & exponentiation (and their inverses) work on numbers--whereas derivatives and integrals work on **functions**.

So if you have certain number and add two, then subtract two, you have the same number you started with. But if you have a certain **function**, say x^2, and you take the integral of it and then the derivative of that, you end up with x^2 again.

2. The idea of a limit is the idea that links derivatives and integrals together. The definitions of both derivatives and integrals make vital use of the idea of limits.

The idea of a limit is pretty simple: What happens when one number gets close to another?

How close? Well, as close as you need--as close as you want--closer and closer and closer.

With two numbers that are close, you can always get closer yet.

The idea of limits is to examine what happens when you do that--start out with two numbers that are close and then keep getting closer and closer and closer.

So for example the sequence 1, 1/2, 1/4, 1/8, 1/16, etc. gets closer & closer to zero. So we say the limit of that sequence is zero.

The idea of limits gets more interesting if we apply it to a function.

* What happens to the function 1/x when x gets closer and closer to zero? (This is a particularly interesting example, because 1/0 is "undefined" but that doesn't stop us from trying numbers closer and closer to zero to see what happens to them when we put them in the function.)

* What happens to the function x/x when x gets closer & closer to zero?

* What happens to the function sin (1/x) when x gets closer & closer to zero?

* You can get trickier, like what happens to the function ((x + d)^ 2- x^2 )/d as d gets closer and closer to zero? (Use a bit of algebra tp multiply out the squares, then simplify a bit and you'll soon see.)

FWIW, these are very typical situations where the limit is very useful--because you can't divide by zero, but it is still useful to know what happens in those situations where the number in the denominator is getting very, very close to zero.

(For bonus points, think about each of the * problems above as approaching zero from above (like 1, 1/2, 1/4, 1/8, 1/16 etc) and also from below (like -1, -1/2, -1/4, -1/8, -1/16, etc.) Do you get the same answer, or different, when approaching from above vs from below? If you can figure out things like that you are doing a bit of calculus already.)
posted by flug at 12:01 AM on January 18, 2010

In addition to what everyone's already said, calculus gives you a tool to solve a class of problems usually referred to as "minimax" or "optimization". These problems crop up when you have an equation describing some quantity and you want to find out where the maximum or minimum value of that property occurs, e.g. "how many widgets should I produce to maximize profit?", "at what point is this thing moving the fastest?", etc. Sure, you could just plot the function and pan/zoom until you find the spot you're looking for and get an approximate idea, but calculus gives you to the tools to find it exactly without any guesswork or plotting. This derives from the idea that when something hits a minimum or maximum its derivative will cross zero at that point, i.e. the roller coaster will be completely horizontal at the very apex of its climb right before it's about to rush down the hill.
posted by Rhomboid at 1:05 AM on January 18, 2010

I feel I'm a bit late to this party but I have to champion Better Explained -- intuitive explanations of maths that are just, well, obvious.

For instance, barely anybody seems to understand why radians are a natural method of measuring angles. Degrees are terrible!
posted by katrielalex at 8:09 AM on January 18, 2010

There are some very interesting intro-to-calculus lectures available on iTunes, under the iTunes U section.

Salman Khan's massive collection of math videos is very popular on YouTube, and from what I've seen of it, pretty good. His material on calculus can be found here (114 videos).

As for your specific slope questions, try watching Derivatives 1 and Derivatives 2.
posted by effbot at 10:12 AM on January 18, 2010

A lot of people are talking about limits, which can be a mildly confusing concept. I'd just say you shouldn't get too hung up on it. In the early years of calculus there was no other rigorous way of approaching the issue of how calculus worked (because it clearly did) and the problems that arose if you thought about it in terms of doing relatively simple arithmetic on infinitesimal numbers. It actually took something like 150 years for the idea of limits to come along and make calculus mathematically 'acceptable'.

It's now known that you can use something called 'nonstandard analysis', developed in the 1960s, to make the infinitesimal approach work, so calculus can now again be thought of as more like arithmetic and algebra applied to infinitesimal numbers in order to get finite answers. Integral calculus is then simply how to sum up a lot of infinitesimals to get a finite answer, and derivatives are how to take ratios of two infinitesimal quantities. For largely historical reasons though, it's still usually taught with this extra layer of limiting processes, which can be a confusing formality for many.
posted by edd at 3:22 AM on January 19, 2010 [1 favorite]