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Unformatted text preview: (5/17/07) Chapter 0. Mathematical Models: Functions and Graphs The story of calculus goes back thousands of years. Mathematicians of the ancient world, including Pythagoras (c. 580 BC), Euclid (c. 300 BC), Archimedes (c. 287212 BC), and Apollonius (c. 262190 BC), developed a great deal of the mathematics that is used in calculus. This includes the theory of proportions, most of what is now known as plane geometry, the theory of conic sections, and results concerning tangent lines and areas of curved regions. They did all this without using letters for known and unknown quantities or equations relating quantities as in modern algebra, without representing points by coordinates or describing curves and surfaces with equations as in analytic geometry, and without using the modern concept of function. Modern calculus, in contrast, relies heavily on algebra and analytic geometry and on properties of power, exponential, logarithmic, and trigonometric functions. These topics are reviewed in this precalculus chapter. Read it carefully, answer the questions in the discussions, study the examples, and be sure you can work all of the tune-up exercises and any problems in the regular problem sets that you are assigned. You might also want to work through the problem tutorials on the web page for this text. Section 0.1 deals with mathematical models, functions, and graphs. Notation and terminology for intervals of numbers are described and rules for solving inequalities are discussed in Section 0.2. The definitions, basic properties, and graphs of power functions, exponential functions, logarithms, and trigonometric functions are reviewed in Sections 0.3 through 0.5. Section 0.6 deals with sums, products, quotients, and compositions of these basic functions. Work through Sections 0.1 through 0.3 before or while you study Chapter 1, and work through Sections 0.4 through 0.6 before or while you study Chapter 2. Section 0.1 Measuring and modeling: variables and functions Overview: To solve a problem using calculus, the first step is to set up a mathematical model in which the quantities that vary are represented by variables and functions. The procedures of calculus are applied, using arithmetic and algebra as required. Then the solution is interpreted in the context of the application. The notation and terminology concerning functions that are used in this process are discussed in this section and illustrated with applications to a variety of fields. Topics: Variables, functions, and graphs Functions given by formulas, tables, and graphs Finding formulas for mathematical models When is a graph the graph of a function? Change, percent change, and relative change Variables, functions, and graphs Suppose we want to find the area of the 1978 silver dollar in Figure 1 and we know that its radius is 1.9 centimeters. We use the formula A = r 2 for r (1) for the area of a circle of radius r (Figure 2), and we conclude that the area of the coin is...
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