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/
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