Calculus Starting Points

Date: 10/23/2001 at 23:59:29
From: Robert Best
Subject: Where should I start if I want to learn about calculus?

I'm in grade 11 and I want to learn calculus because I have heard that 
it is important for applied math in engineering and I want to be an 
engineer. What should I learn, and in what order? For example: before 
I could learn exponents I had to know all about multiplication and 
addition.

Right now I have no knowledge of log, e, ln or any of that "as x 
approaches infinity" stuff. But what I do know is that they are 
related to calculus.

My goal is to have a firm grasp on these functions (log, e, ln) and 
other basic calculus things by the end of this year. I will probably 
use your site as my tool. But what I need to know is what to learn and 
when to learn it. I am asking this because my math class is too easy 
and I want to learn extra.

Date: 10/24/2001 at 17:14:25
From: Doctor Ian
Subject: Re: Where should I start if I want to learn about calculus?

Hi Robert,

Congratulations on your decision to take charge of your own education 
in mathematics. You can start learning about the calculus by taking 
the following 'two-minute tour',

   A Brief Overview of Calculus
   http://mathforum.org/library/drmath/view/52121.html   

which should give you some idea of what you're getting yourself into.  

You might also benefit from reading this:

   Why Should I Study Calculus?
   http://mathforum.org/library/drmath/view/52362.html   


although it looks like you've already figured out that calculus will 
be easier for you if you become familiar with the functions for which 
derivatives and integrals are commonly computed. 

For each class of functions (polynomials, conic sections, trig 
functions, exponents, logarithms), you should concentrate on the 
following fundamentals:

  1. How is the function defined?  How can you tell whether
     or not a given function belongs to this class?

  2. What are the properties shared by all the functions in 
     this class?  Are there special cases of the functions that 
     are particularly easy, or particularly difficult, to work 
     with?  

  3. What do the graphs of the functions in this class look like?
     What tricks are known for sketching them quickly (e.g., finding
     intercepts and asymptotes)?  

  4. What forms can be used to write them down?  What techniques
     can be used to move among these competing representations?
     Which representations are especially good (or especially bad)
     for particular purposes? 

  5. What kind of behavior arises from combining these functions
     with functions of other classes? 

  6. Can functions in this class be approximated with functions from
     other classes?  Under what conditions?  With what kind of 
     accuracy?

In addition to being able to chug through pages of calculations, your 
goal should be to get a feel for how the functions behave in various 
circumstances. Ideally, before you set out to compute a derivative or 
integral, you'd like to be able to guess what the answer should look 
like when you're finished. That can help you make the right choices in 
your computations. It can also help you catch yourself when you make 
errors. 

The master at this was Richard Feynman:

  A prospective student at Caltech once asked whether they taught
  Feynman's problem-solving methods there. Another Nobel Prize 
  winner, Murray Gell-Mann, said that they did not. The student 
  asked why not. Gell-Mann replied: "Here is Feynman's method. 
  First, you write down the problem." Squeezing his eyes closed 
  and holding his fists against his forehead, he continued:  
  "Second, you think _really_ hard." Opening his eyes, he 
  concluded: "Third, you write down the answer."

Of course, this isn't a skill that Feynman was born with. He worked at 
developing it. Anyway, if you haven't already read _Surely You're 
Joking, Mr. Feynman_, you should probably do that as soon as possible.  

You should also try to get comfortable with other coordinate systems 
(e.g., polar and cylindrical coordinates). Often the correct choice of 
coordinates can make the difference between an easy solution and a 
horribly complicated one. Whenever you detect some kind of symmetry in 
a function, it might be a good idea to spend a few minutes thinking 
about whether it could be expressed more naturally in another 
coordinate system. 

One other thing you'll want to get used to is solving multi-step 
problems. In calculus, especially integral calculus, most of the work 
usually goes into setting up equations, and often that requires 
finding an expression for something in terms of something else, which 
requires finding an expression for the second thing in terms of a 
third, and so on.  Actually solving the equations can be almost 
anti-climactic. 

I hope this helps.  Write back 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/   

Date: 10/24/2001 at 23:52:21
From: Robert Best
Subject: Where should I start if I want to learn about calculus?

I would like to thank you for replying to my question and say that you 
have inspired me. Math is now more fun, interesting and accessible 
thanks to your site.

Richard Feynman sounds like an interesting man and the story you told 
me about his problem-solving method is very interesting.