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.
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