Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other similar courses offered by the Department of Mathematics, University of Hong Kong, from the first semester of the academic year 1998-1999 through the second semester of 2006-2007. It can be used as a textbook or a reference book for an introductory course on one variable calculus. In this book, much emphasis is put on explanations of concepts and solutions to examples. By reading the book carefully, students should be able to understand the concepts introduced and know how to answer questions with justification. At the end of each section (except the last few), there is an exercise. Students are advised to do as many questions as possible. Most of the exercises are simple drills. Such exercises may not help students understand the concepts; however, without practices, students may find it difficult to continue reading the subsequent sections. Chapter 0 is written for students who have forgotten the materials that they have learnt for HKCEE Mathe- matics. Students who are familiar with the materials may skip this chapter. Chapter 1 is on sets, real numbers and inequalities. Since the concept of sets is new to most students, detail explanations and elaborations are given. For the real number system, notations and terminologies that will be used in the rest of the book are introduced. For solving polynomial inequalities, the method will be used later when we consider where a function is increasing or decreasing as well as where a function is convex or concave. Students should note that there is a shortcut for solving inequalities, using the Intermediate Value Theorem discussed in Chapter 3. Chapter 2 is on functions and graphs. Some materials are covered by HKCEE Mathematics. New concepts introduced include domain and range (which are fundamental concepts related to functions); composition of functions (which will be needed when we consider the Chain Rule for differentiation) and inverse functions (which will be needed when we consider exponential functions and logarithmic functions). In Chapter 3, intuitive idea of limit is introduced. Limit is a fundamental concept in calculus. It is used when we consider differentiation (to define derivatives) and integration (to define definite integrals). There are many types of limits. Students should notice that their definitions are similar. To help students understand such similarities, a summary is given at the end of the section on two-sided limits. The section of continuous functions is rather conceptual. Students should understand the statements of the Intermediate Value Theorem (several versions) and the Extreme Value Theorem. In Chapters 4 and 5, basic concepts and applications of differentiation are discussed. Students who know how to work on limits of functions at a point should be able to apply definition to find derivatives of “simple” functions. For more complicated ones (polynomial and rational functions), students are advised not to use definition; instead, they can use rules for differentiation. For application to curve sketching, related concepts like critical numbers, local extremizers, convex or concave functions etc. are introduced. There are many easily confused terminologies. Students should distinguish whether a concept or terminology is related to a function, to the x-coordinate of a point or to a point in the coordinate plane. For applied extremum problems, students ii should note that the questions ask for global extremum. In most of the examples for such problems, more than one solutions are given. In Chapter 6, basic concepts and applications of integration are discussed. We use limit of sums in a specific form to define the definite integral of a continuous function over a closed and bounded interval. This is to make the definition easier to handle (compared with the more subtle concept of “limit” of Riemann sums). Since definite integrals work on closed intervals and indefinite integrals work on open intervals, we give different definitions for primitives and antiderivatives. Students should notice how we can obtain antiderivatives from primitives and vice versa. The Fundamental Theorem of Calculus (several versions) tells that differentiation and integration are reverse process of each other. Using rules for integration, students should be able to find indefinite integrals of polynomials as well as to evaluate definite integrals of polynomials over closed and bounded intervals. Chapters 7 and 8 give more formulas for differentiation. More specifically, formulas for the derivatives of the sine, cosine and tangent functions as well as that of the logarithmic and exponential functions are given. For that, revision of properties of the functions together with relevant limit results are discussed. Chapter 9 is on the Chain Rule which is the most important rule for differentiation. To make the rule easier to handle, formulas obtained from combining the rule with simple differentiation formulas are given. Students should notice that the Chain Rule is used in the process of logarithmic differentiation as well as that of implicit differentiation. To close the discussion on differentiation, more examples on curve sketching and applied extremum problems are given. Chapter 10 is on formulas and techniques of integration. First, a list of formulas for integration is given. Students should notice that they are obtained from the corresponding formulas for differentiation. Next, several techniques of integration are discussed. The substitution method for integration corresponds to the Chain Rule for differentiation. Since the method is used very often, detail discussions are given. The method of Integration by Parts corresponds to the Product Rule for differentiation. For integration of rational functions, only some special cases are discussed. Complete discussion for the general case is rather complicated. Since Integration by Parts and integration of rational functions are not covered in the course Basic Calculus, the discussion on these two techniques are brief and exercises are not given. Students who want to know more about techniques of integration may consult other books on calculus. To close the discussion on integration, application of definite integrals to probability (which is a vast field in mathematics) is given. Students should bear in mind that the main purpose of learning calculus is not just knowing how to perform differentiation and integration but also knowing how to apply differentiation and integration to solve problems. For that, one must understand the concepts. To perform calculation, we can use calculators or computer soft- wares, like Mathematica, Maple or Matlab. Accompanying the pdf file of this book is a set of Mathematica notebook files (with extension .nb, one for each chapter) which give the answers to most of the questions in the exercises. Information on how to read the notebook files as well as trial version of Mathematica can be found at http://www.wolfram.com . Contents 0 Revision 1 0.1 Exponents ................................................... 1 0.2 Algebraic Identities and Algebraic Expressions ............................... 2 0.3 Solving Linear Equations ........................................... 4 0.4 Solving Quadratic Equations ......................................... 6 0.5 Remainder Theorem and Factor Theorem .................................. 8 0.6 Solving Linear Inequalities .......................................... 10 0.7 Lines ..................................................... 12 0.8 Pythagoras Theorem, Distance Formula and Circles ............................. 17 0.9 Parabola .................................................... 19 0.10 Systems of Equations ............................................. 20 1 Sets, Real Numbers and Inequalities 23 1.1 Sets ...................................................... 23 1.1.1 Introduction .............................................. 23 1.1.2 Set Operations ............................................ 28 1.2 Real Numbers ................................................. 32 1.2.1 The Number Systems ......................................... 32 1.2.2 Radicals ................................................ 34 1.3 Solving Inequalities .............................................. 37 1.3.1 Quadratic Inequalities ........................................ 38 1.3.2 Polynomial Inequalities with degrees ≥ 3 .............................. 39 2 Functions and Graphs 43 2.1 Functions ................................................... 43 2.2 Domains and Ranges of Functions ...................................... 45 2.3 Graphs of Equations ............................................. 49 2.4 Graphs of Functions .............................................. 53 2.5 Compositions of Functions .......................................... 64 2.6 Inverse Functions ............................................... 66 2.7 More on Solving Equations .......................................... 69 3 Limits 73 3.1 Introduction .................................................. 73 3.2 Limits of Sequences .............................................. 75 3.3 Limits of Functions at Infinity ........................................ 80 3.4 One-sided Limits ............................................... 86 3.5 Two-sided Limits ............................................... 89 3.6 Continuous Functions ............................................. 94 4