tailieunhanh - 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. | 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. Four Ways to Represent a Function Functions arise whenever one quantity depends on another. Consider the following four situations. Year Population millions 1900 1650 1910 1750 1920 1860 1930 2070 1940 2300 1950 2560 1960 3040 1970 3710 1980 4450 1990 5280 2000 6080 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 irr2. 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 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 generated by seismic activity during the .