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ch00_01 - 2 CHAPTER 0 Preliminaries 0-2 y 35 30 25 20 15 10...

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2 CHAPTER 0 . . Preliminaries 0-2 12345678 x y 5 10 15 20 25 30 35 0 FIGURE 0.1 The Fibonacci sequence these is the attempt to ﬁnd patterns to help us better describe the world. The other theme is the interplay between graphs and functions. By connecting the powerful equation-solving techniques of algebra with the visual images provided by graphs, you will signiﬁcantly improve your ability to make use of your mathematical skills in solving real-world problems. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Although mathematics is far more than just a study of numbers, our journey into calculus begins with the real number system. While this may seem to be a fairly mundane starting place, we want to give you the opportunity to brush up on those properties that are of particular interest for calculus. The most familiar set of numbers is the set of integers, consisting of the whole numbers and their additive inverses: 0, ± 1 , ± 2 , ± 3 ,.... A rational number is any number of the form p q , where p and q are integers and q ²= 0 . Forexample, 2 3 , 7 3 and 27 125 are all rational numbers. Notice that every integer n is also a rational number, since we can write it as the quotient of two integers: n = n 1 . The irrational numbers are all those real numbers that cannot be written in the form p q , where p and q are integers. Recall that rational numbers have decimal expansions that either terminate or repeat. For instance, 1 2 = 0 . 5 , 1 3 = 0 . 3333 ¯ 3 , 1 8 = 0 . 125 and 1 6 = 0 . 16666 ¯ 6 are all rational numbers. By contrast, irrational numbers have decimal expansions that do not repeat or terminate. For instance, three familiar irrational numbers and their decimal expansions are 2 = 1 . 41421 35623 ..., π = 3 . 14159 26535 ... and e = 2 . 71828 18284 .... We picture the real numbers arranged along the number line displayed in Figure 0.2 (the real line ). The set of real numbers is denoted by the symbol R .

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0-3 SECTION 0.1 . . Polynomials and Rational Functions 3 05 4 3 2 1 ± 1 ± 2 ± 3 ± 4 ± 5 ± ± 2 ± 3 p e FIGURE 0.2 The real line For real numbers a and b , where a < b , we deﬁne the closed interval [ a , b ]tobethe set of numbers between a and b , including a and b (the endpoints ), that is, [ a , b ] ={ x R | a x b } , as illustrated in Figure 0.3, where the solid circles indicate that a and b are included in [ a , b ]. a b FIGURE 0.3 A closed interval a b FIGURE 0.4 An open interval Similarly, the open interval ( a , b )is the set of numbers between a and b , but not including the endpoints a and b , that is, ( a , b ) x R | a < x < b } , as illustrated in Figure 0.4, where the open circles indicate that a and b are not included in ( a , b ). You should already be very familiar with the following properties of real numbers. THEOREM 1.1 If a and b are real numbers and a < b , then (i) For any real number c , a + c < b + c . (ii) For real numbers c and d ,if c < d , then a + c < b + d .
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ch00_01 - 2 CHAPTER 0 Preliminaries 0-2 y 35 30 25 20 15 10...

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