Stewart's calculus, early transcendentals

chapter 1
A graphical representation of a function––here the number of hours of daylight as a function of the time of year at various latitudes–– is often the most natural and convenient way to represent the function.

Functions and Models

The fundamental objects that we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerningfunctions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of real-world phenomena. We also discuss the use of graphing calculators andgraphing software for computers.

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Four Ways to Represent a Function
Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A r 2. With each positive number r there is associated one value of A, and we say that A is afunction of r. B. The human population of the world P depends on the time t. The table gives estimates of the world population P t at time t, for certain years. For instance, P 1950 2,560,000,000

Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Population (millions) 1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080

But for each value of the time t there is a correspondingvalue of P, and we say that P is a function of t. C. The cost C of mailing a first-class letter depends on the weight w of the letter. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows agraph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a.
a {cm/s@} 100

50

5

10

15

20

25

30

t (seconds)

FIGURE 1

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Calif. Dept. of Mines and Geology

Vertical ground acceleration during the Northridge earthquake

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CHAPTER 1 FUNCTIONSAND MODELS

Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number ( A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number. A function f is a rule that assigns to each element x in a set A exactly one element, called f x , in a set B. We usually consider functions for which the sets A and B are sets of realnumbers. The set A is called the domain of the function. The number f x is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f x as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependentvariable. In Example A, for instance, r is the independent variable and A is the dependent variable. It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f x according to the rule of the function. Thus, we can think of the domain as the set of all possibleinputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or sx ) and enter the input x. If x 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will...