This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture V Calculus of OneVariable Functions Let us first review some definitions in calculus on real numbers. In order to define limit on the real numbers we will use the concept of funnel functions . Definition 1 A function ( t ) on [0 , d ] is called a funnel function if it has the following properties: 1. It is strictly increasing. 2. For every a > , there exists t , 0 < t ≤ d , such that 0 < ( t ) < a . Definition 2 Let f ( x ) be a scalar function and let l be a real number. We say that the limit of f at c is l and we denote this by lim f ( x ) = l x c → if for some d > 0 there exists a funnel function ( t ) on [0 , d ] such that for every x with 0 < x − c ≤ d it follows that f ( x ) − l ( x − c ) .     ≤    f ( c + h ) − f ( c ) For a real function f and c, h real numbers, the fraction h is called difference quotient and is denoted by Δ f . Δ x f ( c + h ) − f ( c ) Definition 3 For a real function f , if lim h 0 h exists and has a real → df value, we call this value the derivative of f at c and denote it by f ( c ) or dt  c ....
View
Full
Document
This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.
 Two '04
 Duorg
 Math, Calculus, Real Numbers

Click to edit the document details