A Brief Overview of Calculus

Date: 01/16/2001 at 08:37:12
From: Pat
Subject: How can I understand Calculus?

I am trying to get into another college, and they want me to take a 
calculus course for my math requirement. I have never had any 
experience with it, and I have no clue what it entails. Could you give 
me a brief overview on what it is all about? Also, what I should 
mainly concentrate on?

Thanks.

Date: 01/16/2001 at 09:55:24
From: Doctor Ian
Subject: Re: How can I understand Calculus?

Hi Pat,

Briefly, there are two parts to the calculus: differential and 
integral.

Differential calculus works this way: Imagine that there is some 
curve, say, y = x^2. You can choose any two points on the curve, and 
there will be some line that connects them. As the second point gets 
closer and closer to the first, the line should eventually become the 
line that is tangent (parallel) to the curve at the first point.

In general, you can always do this to find the slope of the tangent 
line at a given point:

           f(x+h) - f(x)
     limit -------------
      h->0       h

If you look at this, you'll see that this is a 'rise over run' 
equation - the numerator is the change in y, and the denominator is 
the change in x.

It's a mess to work out for any given function, but it turns out that 
for lots of common functions, there are shortcuts that you can use to 
find an equation for the tangent without having to take any limits. 
For example, if you have an equation like:

     y = ax^n

it turns out that the slope of the tangent - which is called the 
'derivative' - at any point is:

     y' = anx^(n-1)

So the derivative of

     y = x^2
is
     y' = 2x

Note that at the origin (x = 0), the derivative is 0, meaning that the 
tangent line is horizontal. At x = 1, the derivative is 2, at x = 2 
the derivative is 4. As x increases, the slope of the tangent gets 
steeper and steeper. If you draw a picture of the function, you can 
see that this is true.

Anyway, that's differential calculus in a nutshell. You learn what a 
derivative _is_, and then you learn about 47,000 separate tricks for 
computing derivatives in various situations, along with some 
applications.

Integral calculus works this way: Imagine that there is some curve, 
say y = x^2, and you'd like to know the area under the curve for some 
interval, say from x = 1 to x = 3.

You can divide the area between the curve and the x-axis into very 
thin rectangles, and use the y-value of the curve at various values of 
x to estimate the height of each rectangle. Then you set up a sum:

     Sum (f(x) * delta(x))

where f(x) and delta(x) tell you the height and width of each 
rectangle respectively. You choose a standard width for the 
rectangles, and as before, you take the limit of the sum as the width 
goes to zero.

As with differentiation, this is a mess. But fortunately, it turns out 
that there is a nice relation between differentiation and integration: 
If you can guess the function (F) whose derivative is the function (f) 
that you're integrating, then you can just evaluate F at the endpoints 
of the interval, and subtract to get the area. Very nice... except 
it's really hard to guess F for a given f.

So as people find F:f pairs, they write them down in 'integral 
tables'. Also very nice.

Except calculus students are generally prohibited from using integral 
tables on their tests. They're asked to memorize about 47,000 F:f 
pairs, so that they can promptly forget them at the conclusion of the 
course.

So, that's what you have to look forward to.

What you should concentrate on depends on why you're taking the 
course. If you're going to be a physicist, for example, you would 
actually _use_ calculus on a day-to-day basis, in which case it's 
worth actually memorizing various formulas for derivatives and 
integrals on a long-term basis. If you're just supposed to get an 
'appreciation' for calculus, then you should make sure that you 
understand all the definitions, and that you can set up integrals, and 
you should ask around to try to find a professor like the one _I_ 
had, who didn't require us to memorize a lot of stuff. (On the first 
day of class he announced that he wasn't going to ask us to memorize 
anything that he hadn't memorized, which consisted of the following 
formula: sin^2 + cos^2 = 1.)

That's the two-minute tour. I hope it helps. Let me know if you'd like 
to talk about this some more, or if you have any other questions.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/