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Unformatted text preview: What is a derivative? Can you define it by defining and using the word “slope?” Yes, but how do you start from just the picture of the slope as the tangent of some angle and show that the derivative of x 2 with respect to x is 2 x ? It can be done, but you’ve probably never seen such a derivation. Try it if you like. The standard way that gets around this problem and that lets you apply it to many situations is df dx ( x ) = lim Δ x → f ( x + Δ x )- f ( x ) Δ x When you do a derivative, you will either recognize the answer because you have memorized it or you will use one of a few standard tools: product rule, chain rule, linearity and that’s about it. What is an integral? You’ve been using them for some time. I’ve been using the concept as if it’s something that everyone knows. But what is it? If your answer is something like “the function whose derivative is the given function” or “the area under a curve” then No. Both of these answers express an aspect of the subject but neither is a complete answer. The first actually refers to the fundamental theorem of calculus, and I’ll describe that shortly. The second is a good picture that applies to some special cases, but it won’t tell you how to compute it and it won’t allow you to generalize the idea to the many other subjects in which it is needed. There are several different definitions of the integral, and everysubjects in which it is needed....
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This note was uploaded on 01/08/2012 for the course PHYSICS 205 taught by Professor Galeazzi during the Fall '11 term at University of Miami.
- Fall '11