Example 044 if x 2 0 2 and y 5 0 1 use the triangle

This preview shows page 19 - 23 out of 228 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Calculus
The document you are viewing contains questions related to this textbook.
Chapter 6 / Exercise 6
Calculus
Stewart
Expert Verified
11 Example 0.4.4 If | x - 2 | < 0 . 2 and | y - 5 | < 0 . 1 , use the triangle inequality to estimate | ( x + y ) - 7 | .
Example 0.5.1 Find all the read numbers x R such that the following is true, | 2 x - 4 | ≥ 5 .
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Calculus
The document you are viewing contains questions related to this textbook.
Chapter 6 / Exercise 6
Calculus
Stewart
Expert Verified
12 Chapter 0. Introduction
CHAPTER ONE Functions This section will focus on what is a function, different functions and how they can be used. 1.1 FOUR WAYS TO REPRESENT A FUNCTION See section 1.1 of the textbook In high school mathematics we saw linear functions (also called linear mappings). We also saw quadratic functions (mappings) as well as a bunch of other examples. Here, we take a step back and define very precisely what we mean by a function. Then we will build up our understanding of them by developing a rather extensive toolbox. From the textbook we have the following definition. Definition 1.1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f ( x ) , in a set E . This can be described as, f : D E This says that f is a function, or a mapping , from the set D to the set E . In this course we will assume that both D and E are sets of real numbers. We can describe f as a real-valued function of a single real variable. There are many different interpretations of functions. The function f can be seen as a machine that takes an input, x , and spits out an output, f ( x ) . Can picture a function as an arrow diagram that maps every point in the set D to a unique point in the set E . 13
14 Chapter 1. Functions Example 1.1.1 If f ( x ) = x 2 + 1 and h = 0 compute, f ( a + h ) - f ( a ) h . Solution. We substitute not the expression and simplify, f ( a + h ) - f ( a ) h = ( a + h ) 2 + 1 - ( a 2 + 1) h , = a 2 + 2 ah + h 2 - a 2 h , = 2 ah + h 2 h , = 2 a + h. J The graph of the function f is the set of ordered pais, { ( x, f ( x )) | x D } . When discussing functions two important ideas are the domain and the range. Definition 1.1.2. The domain of the function f is all the points x R where f ( x ) is defined. The domain can be the whole real line but it need not be. That is to say functions need not be define everywhere, for example 1 /x . Definition 1.1.3. The range of the function is the set of all values f ( x ) where it is assumed that x is in the domain of the function. These ideas extend nicely to higher dimensions, as you will see in subsequent courses.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture