《Lecture 1 Single-Variable Calculus》.pdf

Postgraduate Maths Part I: The Basics of Single-Variable Calculus As an introduction to this topic, Blume and Simon (1994) write: “A central goal of economic theory is to express and understand relationships between economic variables. These relationships are described mathemati- cally by functions. If we are interested in the e↵ect of one economic variable (like government spending) on one other economic variable (like gross na- tional product), we are led to the study of functions of a single variable - a natural place to begin our mathematical analysis”. In what follows, I will briefly discuss the basic concepts in single-variable calculus that will form the basis for the Postgraduate Maths Course. Terminology • R ⌘ R1 denotes the set of real numbers • R+ is the set of nonnegative real numbers, and is defined R+ = {x 2 R |x 0}. • By a function we mean a mapping that assigns to a given number in R another number in R. We denote such a mapping by f , and it assigns to some number x 2 R the value y 2 R. We write f : R ! R, x 7! f (x) = y .We call x the exogenous variable, and y the endogenous variable. • Not all functions are defined on the whole set R of real numbers, but on a proper subset of R. Given a function f , the set of numbers x at 1 which ( ) is defined is called the of . Consider the function f x domain f ( ) = 1 . As division by zero is undefined, this function is not f x /x defined at x = 0. Since it is defined everywhere else, its domain is := 0 (read ”the set of real numbers without zero). As is not D R\ { } D identical t