What's the Big Deal About Calculus?

Great Debates
1

I took calculus classes in the early 90’s, in the University of Michigan, Dearborn campus (a commuter college–the main campus is in Ann Arbor). And I don’t mean to brag. But I was a little unimpressed.

To me, it seemed to just be a continuation of high school geometry. I may be showing you all my ignorance when I say that. But it’s true. For example, instead of finding the slope of the line, you found the slope of the curve, at a point. Frankly, I don’t know why they didn’t just cover it in hs geometry–and call it that: geometry.

Secondly, why did it take so long to discover calculus? As I understand it, Isaac Newton discovered it, in the 17th or 18th century. For goodness sake, thousands of years ago, the ancients were building pyramids and other magnificent structures. Why didn’t they figure out then even? Anyways, that is my second question. Why did it take so long to discover, even for people who could build the pyramids?

I do have to confess though. Commuter colleges have a tendency to teach things in an assembly line way, making it as simple as possible, for the less gifted students. So doubtless, I probably got the Dummy’s Guide To… version too.

But back to my questions: Why don’t they just call it, and teach it, in geometry class? And why did it take so long to discover?

:):):):slight_smile:

2

Did geometry teach you how to do a geometric construction for something like y = 5 x ^ 4 + 3 x ^2? How about y = tan(x)? My geometry class did not. For that we needed algebra and trig. Which means it helps to have a coordinate system the definition of a function etc.

Now what can you do with the geometric construction of that function assuming you can do that without algebraic tools? How do you express an indefinite integral geometrically?

3

The Egyptians no doubt did have the intuitive “high school geometry” view of things. Archimedes certainly did, and much more, see

It’s notable that both Newton and Leibniz devised calculus independently, so in a sense you’re right that it was a natural progression from the intuitions of geometry. But these were two of the greatest minds in history, and their breakthrough with calculus was the rigorous treatment of infinitesimals. It’s far from obvious how to make that work.

4

It also helps to know what you are differentiating with respect to. Without a coordinate system and the idea of functions how do you differentiate geometrically let’s say a circle?

You construct a bunch of tangents to the curve? What’s that do for you? I’m just not seeing how you go from classical geometry to the development of calculus without the tools of analytical geometry.

I wonder if they still teach that in high school.

We did algebra, geometry, algebra ii, trig/analytical geometry, calculus at our high school.

5

You cannot properly learn physics without the understanding of calculus. It appears that you may have stopped at computing derivatives, let me assure you that integration is much more demanding and has many applications in other sciences.

6

Calculus describes the relationship between the state of a system and its rate of change. Which allows us to calculate the behavior of real-world objects. For example, if you understand how the speed (i.e. rate of change of position) of an object changes over time, you can calculate the trajectory (i.e. position). For example, you can analytically calculate the trajectory of a ball thrown into the air, because you know the initial speed and how the speed changes over time (it accelerates downwards at a constant rate). This is what Newton invented calculus for.

7

Long before Newton, Archimedes understood some of the ideas behind Calculus. The reason he didn’t go ahead and invent Calculus as such is that he lacked the background concepts and notation—some of the things described by octopus. A really rigorous Calculus class is going to use most of what the students have learned before, about algebra and trigonometry (and that’s what causes many Calc students to struggle).

Note also that the generic, small-c definition of “calculus” is “a particular method or system of calculation or reasoning.” Calculus isn’t just the ideas (of the slope of a curve, for example) but the system for finding them. And the geometry is just one interpretation: you could do Calculus without thinking of it in geometrical terms.

And you can’t claim to really have Calculus until you get to the Fundamental Theorem of Calculus, which ties together the two main branches of the subject, differential and integral calculus, and shows how they’re two sides of the same coin.

8

I dreamed of trigonometric substitution when I took differential equations.

9

I dreamed of Jeannie with the light brown cosine…

10

That’s not cos, that’s tan.

11

It is not a bragworthy achievement to be unimpressed by something that was taught to you in the most basic manner possible.

I was bored out of my mind in calculus class, because typically the professor would spend most of the lecture deriving equations. But learning calculus in physics class made it come together for me (somewhat).

Saying that calculus is nothing more than geometry is llike saying that geometry is nothing more than algebra. Geometry is algebra kicked up a notch. Calculus is geometry and trigonometry kicked up a notch.

12

My favorite part of precalculus was learning trigonometry and actually I LUSTED for Matrices

13

As BobLibDem suggests, it sounds like you’re think of calculus as calculating the derivative of a function: the slope of a curve at any given instant point on that curve.

There’s much more. Geometry can get you the volume of a cylinder. But imagine an area bounded by the curve y=x[sup]2[/sup] and the lines x=6 and y=0. Now rotate that area about the x-axis and find me the volume of the resulting solid. Geometry isn’t as helpful here.

Although to be fair, that process builds on geometry principles. We can certainly take the volume of the thin, flat cylinder cut out of this solid at the widest point, to the right: it has a radius of 36 and let’s say a height of 1. So its volume, by familiar principles of geometry, is 1296(pi). And we can take the volume of the next cylinder slice leftward: radius 25 and height 1=volume of 625(pi). And the next: radius 16 and height 1=volume 256(pi).

Add up all the cylinder slice volumes and we have a fair approximation of the total volume of the shape. Right? We’re just missing those little areas where the curve is a little longer than our straight cylinder edges.

But what if we doubled the number of slices? We could make each cylinder 1/2 high, and we’d reduce the missed space at the edges. Still an approximation, to be sure, but a much better one.

What if we made each slice .001 wide? Then we’d have a really really good approximation. But not exact.

Now… what happens as the slice width approaches zero?

As the slice approaches zero, the approximation error approaches zero.

Calculus allows us to calculate the exact volume, by accurately calculating the sum of infinitesimal slices.

Both Newton and Leibnitz were inventors of this method, but a guy named Bernhard Riemann was the first to create a rigorous definition of the process applied over a continuous interval.

That’s the step that isn’t part of geometry: the summation of infinitesimal quantities to achieve an exact result. That’s the beginnings of calculus.

14

My high school calculus teacher was, in fact, fond of saying, “My students NEVER fail calculus. (Pause) They fail algebra first.”

15

Calculus describes how things change. Since lots of things change, turns out Calculus is a big deal. Now, why is it considered such a difficult subject? My theory is learning something new is usually tricky and most people just begin to learn about it’s foundations before giving up learning math for good when they graduate.

It’s a shame proofs aren’t covered earlier and more thoroughly. For me, that was the beginning of creativity in the subject which made it far more interesting.

16

The way that they usually teach calculus is HORRIBLE. They start off with the fundamental theorem of calculus and all of this difficult math without relating it to anything which makes it very difficult to understand. I was a couple of weeks into my first calculus class (many many years ago) and started relating it to geometry and then everything clicked, and after that it was simple. Once you understand what you are doing, calculus is easy to understand.

My son is taking calculus this year in high school. I didn’t know if they had come up with a better way of teaching it in the past 30+ years or so since I took my first class but apparently they were doing the same old crap. I explained to him that the derivative is just figuring out the slope of a curve and an integral is just the area under the curve, and once he got the basic concepts he was like wow, this actually is pretty simple.

It’s not just all about geometry though. Calculus shows up all over the place in basic physics. Acceleration, velocity, and position all have an integral/differential relationship, for example. And in electronics, if you have a simple coil of wire, the current and voltage also have an integral/differential type of relationship (v = L di/dt). And a capacitor is similar (i = C dv/dt). So simple electronics 101 type stuff dives right into calculus. When you get into more advanced things like feedback control, PID loops are a very common type of control. PID is Proportional Integral and Differential. Basically, in the feedback loop you measure the proportional error, then you add in an integral of the error to correct for any long-term errors in your control loop, and add in some differential error to make the system respond better to quick changes.

So yeah, the point is, it’s not just geometry. But if you relate it to geometry that makes it very easy to understand (IMHO).

As for why it took so long to develop calculus, the answer is that for a lot of stuff you don’t need it. You don’t need calculus to build pyramids. In fact, you can actually tell that they used very simple geometry to build the pyramids because the ratio of the length of the sides to the heights includes a factor of pi in it. This makes ancient astronaut theorists go off on strange ideas about aliens and advanced mathematics, but all it really means is that they took a wheel and attached a stick to it to mark off distances along the ground (google “measuring wheel” if you can’t picture what I mean here). If your wheel is one cubit in diameter, and you put a mark on one point on that wheel, then all you do is pace off the distance and count how many times that mark comes around. In other words, if you run your little wheel along the ground and you count ten ticks as the mark goes by, then the distance you traveled will end up being 10 x pi cubits. It’s just simple geometry. No advanced mathematics or aliens required.

Simple geometry only gets you so far, though. When you try to describe things that depend on the rate of change of other things, calculus makes your life a lot easier. The ancient Greeks actually came up with a lot of the concepts, but it wasn’t all put together into the modern form that we use until the 17th century. Newton and Leibniz get credit for inventing modern calculus, but they built their work on the foundations laid by previous mathematicians. Both Newton and Leibniz were trying to solve problems that couldn’t be adequately expressed using classical geometry type mathematics. Modern calculus allowed them to solve their problems.

17

Many years ago there was a thread here on the SDMB asking “Is Calculus hard?” I haven’t been able to find the thread, and it may be one that got lost, but I did copy down some of the comments from that thread. Among them:

18

Thudlow’s thread.

19

This discussion reminds me. My dad was a math professor specializing in differential equations (I think). One year he went on an exchange program to the math department at a university in Japan and he said he couldn’t figure out what they called calculus. When he would ask them they didn’t know what he was asking. Apparently they taught all the components of calculus but didn’t have a name for them all together.

20

What did you actually learn, and did you study it to a sufficient depth that you can use it for something practical? As others have mentioned basic derivation of polynomials is easy if one understands math and functions. Integration is trickier. Functions where you need the chain rule and product rule are trickier still. And judging how hard it was to figure those out from first principles after being taught the results of centuries of advance collected in a single volume, which is what you’re trying to do, is trickier still.