普林斯顿大学的统计学教材 Princeton calculus精品.pdfVIP

(copyright by Scott M. Lynch, February 2003) Brief Calculus Review (Soc 504) For the most part, calculus can be divided into two halves: differential and integral calculus. In a nutshell, differential calculus is concerned with slopes of curves and rates of change. Integral calculus, on the other hand, is concerned with area under and between curves. 1 Limits Central to both branches of calculus is the notion of ‘limits,’ which we will develop briefly here. A ‘limit’ is the result a function or expression returns as some quantity in the function approaches some value. In calculus, we often refer to the “limit of an expression as x approaches infinity.” For example, the limit of 1 as x approaches infinity is 0. In calculus x notation we write this expression as: lim 1 = 0 (1) x → ∞ x The concept of a limit is fairly intuitive. If we imagine, in this example that x = 1, then the function value is 1. If x = 2, then the function value is smaller, at 1 . As x gets larger, 2 the fraction gets smaller, so that when x is as large as possible, the fraction approaches 0. Evaluating limits can be quite difficult, and a good part of a first course in calculus is spent learning how to determine limits when equations are more complex. For example, imagine if the limit in the above expression were taken as x → 0. 2 Differential Calculus With the basic concept of a limit defined, the two main branches of calculus can be developed. We turn first