Cálculo - james stewart

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 concerning functions, 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 and graphing 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 a function 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 corresponding value 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 a graph