Associated Topics || Dr. Math Home || Search Dr. Math

### 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
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

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search