Calculus and Functions in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is largely quotation.]

Calculus aims primarily to analyze change (e.g. motion) and content (e.g. the computation of area and volume.) These problems are fundamental because we live in a world of ceaseless change, filled with bodies in motion and phenomena of ebb and flow.

Functions:

Most applications of calculus involve the use of real numbers or variables to describe changing quantities. The key to the mathematical analysis of a geometric or scientific situation is typically the recognition of relationships among the variables that describe the situation. Such a relationship may be a formula that expresses one variable as a function of another. For example:

The area A of a circle of radius r is given by

The volume V and surface area S of a sphere of radius r are given by:

Definition: Function:

A real-valued function f defined on a set D of real numbers is a rule that assigns to each number x in D exactly one real number, denoted by f (x).

The set D of all numbers for which f (x) is defined is called the domain (or domain of definition) of the function f. The number f (x), read "f of x," is called the value of the function f at the number (or point) x. The set of all values y = f (x) is called the range of f . That is, the range of f is the set

{y: y = f (x) for some x in D).

Example 1: The squaring function defined by:

assigns to each real number x its square

Because every real number can be squared, the domain of f is the set R of all real numbers. But only nonnegative numbers are squares. Moreover, if

so a is a square. Hence the range of the squaring function f is the set

of all nonnegative real numbers.

Functions can be described in various ways. A symbolic description of the function f is provided by a formula that specifies how to compute the number f (x) in terms of the number x. thus the symbol f ( ) may be regarded as an operation that is to be performed whenever a number or expression is inserted between the parentheses.

Example 2:

The formula

defines a function f whose domain is the entire real line R. Some typical values of f are f (-2) = -1, f (0) = -3, and f (3) = 9. Some other values of the function f are

When we describe the function f by writing a formula y = f (x), we call x the independent variable and y the dependent variable because the value of y depends – through f – upon the choice of x. As the independent variable x changes, or varies, then so does the dependent variable y. The way that y varies is determined by the rule of the function f. For example, if f is the function this equation

then y = -1 when x = -2, y = -3 when x = 0, and y = 9 when x = 3.

We will visualize the dependence of the value y = f (x) on x by thinking of the function f as a kind of machine that accepts as input a number x and then produces as output the number f (x), perhaps displayed or printed.

One such machine is the square root key of a simple pocket calculator. When a nonnegative number x is entered and this key is pressed, the calculator displays (an approximation to) the number

Note that the domain of this square root function

of all nonnegative real numbers, because no negative number has a real square root. The range of f is also the set of all nonnegative real numbers, because the symbol

always denotes the nonnegative square root of x. The calculator illustrates its "knowledge" of the domain by displaying an error message if we ask it to calculate the square root of a negative number.

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from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.2a-3c.