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Preface
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It will provide encouragement and reinforcement as needed, and diagnostic exercises will help the student to measure his or her progress A comprehensive exam at the end of the book will help the student to assess his mastery of the subject, and will point to areas that require further work We expect this book to be the cornerstone of a series of elementary mathematics books of the same tenor and utility Steven G Krantz St Louis, Missouri
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Basics
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10 Introductory Remarks
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Calculus is one of the most important parts of mathematics It is fundamental to all of modern science How could one part of mathematics be of such central importance It is because calculus gives us the tools to study rates of change and motion All analytical subjects, from biology to physics to chemistry to engineering to mathematics, involve studying quantities that are growing or shrinking or moving in other words, they are changing Astronomers study the motions of the planets, chemists study the interaction of substances, physicists study the interactions of physical objects All of these involve change and motion In order to study calculus effectively, you must be familiar with cartesian geometry, with trigonometry, and with functions We will spend this rst chapter reviewing the essential ideas Some readers will study this chapter selectively, merely reviewing selected sections Others will, for completeness, wish to review all the material The main point is to get started on calculus ( 2)
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The number systems that we use in calculus are the natural numbers, the integers, the rational numbers, and the real numbers Let us describe each of these: The natural numbers are the system of positive counting numbers 1, 2, 3, We denote the set of all natural numbers by N
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Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use
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The integers are the positive and negative whole numbers and zero: , 3, 2, 1, 0, 1, 2, 3, We denote the set of all integers by Z The rational numbers are quotients of integers Any number of the form p/q, with p, q Z and q = 0, is a rational number We say that p/q and r/s represent the same rational number precisely when ps = qr Of course you know that in displayed mathematics we write fractions in this way: 1 2 7 + = 2 3 6 The real numbers are the set of all decimals, both terminating and nonterminating This set is rather sophisticated, and bears a little discussion A decimal number of the form x = 316792 is actually a rational number, for it represents x = 316792 = A decimal number of the form m = 427519191919 , with a group of digits that repeats itself interminably, is also a rational number To see this, notice that 100 m = 427519191919 and therefore we may subtract: 100m = 427519191919 m = 4275191919 Subtracting, we see that 99m = 423244 or 423244 99000 So, as we asserted, m is a rational number or quotient of integers The third kind of decimal number is one which has a non-terminating decimal expansion that does not keep repeating An example is 314159265 This is the decimal expansion for the number that we ordinarily call Such a number is irrational, that is, it cannot be expressed as the quotient of two integers m= 316792 100000
CHAPTER 1 Basics
In summary: There are three types of real numbers: (i) terminating decimals, (ii) non-terminating decimals that repeat, (iii) non-terminating decimals that do not repeat Types (i) and (ii) are rational numbers Type (iii) are irrational numbers You Try It: What type of real number is 341287548754875 Can you express this number in more compact form
Coordinates in One Dimension
We envision the real numbers as laid out on a line, and we locate real numbers from left to right on this line If a < b are real numbers then a will lie to the left of b on this line See Fig 11
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