Understanding Basic Calculus

1 UnderstandingBasic ChungDedicated to all the people who have helped me in my book is a revised and expanded version of the lecture notes forBasic Calculusand other similar coursesoffered by the Department of Mathematics, University of Hong Kong, from the first semester of the academicyear 1998-1999 through the second semester of 2006-2007. It can be used as a textbook or a reference bookfor an introductory course on one variable this book, much emphasis is put on explanations of concepts and solutions to examples. By readingthe book carefully, students should be able to understand the concepts introduced and know how to answerquestions with justification. At the end of each section (except the last few), there is an exercise. Studentsare advised to do as many questions as possible. Most of the exercises are simple drills.

2 Such exercises maynot help students understand the concepts; however, without practices, students may find it difficult to continuereading the subsequent 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 1 is on sets, real numbers and inequalities. Since the concept of sets is new to most students, detailexplanations and elaborations are given. For the real number system, notations and terminologies that willbe used in the rest of the book are introduced. For solving polynomial inequalities, the method will be usedlater when we consider where a function is increasing or decreasing as well as where a function is convex orconcave. Students should note that there is a shortcut for solving inequalities, using the Intermediate ValueTheorem discussed in Chapter 2 is on functions and graphs.

3 Some materials are covered by HKCEE Mathematics. New conceptsintroduced include domain and range (which are fundamental concepts related to functions); composition offunctions (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 usedwhen we consider differentiation (to define derivatives) and integration (to define definite integrals). There aremany types of limits. Students should notice that their definitions are similar. To help students understandsuch similarities, a summary is given at the end of the section on two-sided limits. The section of continuousfunctions is rather conceptual.

4 Students should understand the statements of the Intermediate Value Theorem(several versions) and the Extreme Value Chapters 4 and 5, Basic concepts and applications of differentiation are discussed. Students who knowhow 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 usedefinition; instead, they can use rules for differentiation. For application to curve sketching, related conceptslike critical numbers, local extremizers, convex or concave functions etc. are introduced. There are many easilyconfused terminologies. Students should distinguish whether a concept or terminology is related to a function,to thex-coordinate of a point or to a point in the coordinate plane.

5 For applied extremum problems, studentsiishould note that the questions ask for global extremum. In most of the examples for such problems, more thanone solutions are Chapter 6, Basic concepts and applications of integration are discussed. We use limit of sums in a specificform to define the definite integral of a continuous function over a closed and bounded interval. This is to makethe definition easier to handle (compared with the more subtle concept of limit of Riemann sums). Sincedefinite integrals work on closed intervals and indefinite integrals work on open intervals, we give differentdefinitions for primitives and antiderivatives. Students should notice how we can obtain antiderivatives fromprimitives and vice versa. The Fundamental Theorem of Calculus (several versions) tells that differentiationand integration are reverse process of each other.

6 Using rules for integration, students should be able to findindefinite integrals of polynomials as well as to evaluate definite integrals of polynomials over closed andbounded 7 and 8 give more formulas for differentiation. More specifically, formulas for the derivatives ofthe sine, cosine and tangent functions as well as that of the logarithmic and exponential functions are that, revision of properties of the functions together with relevant limit results are 9 is on the Chain Rule which is the most important rule for differentiation. To make the ruleeasier to handle, formulas obtained from combining the rule with simple differentiation formulas are should notice that the Chain Rule is used in the process of logarithmic differentiation as well as thatof implicit differentiation. To close the discussion on differentiation, more examples on curve sketching andapplied extremum problems are 10 is on formulas and techniques of integration.

7 First, a list of formulas for integration is should notice that they are obtained from the corresponding formulas for differentiation. Next, severaltechniques of integration are discussed. The substitution method for integration corresponds to the Chain Rulefor differentiation. Since the method is used very often, detail discussions are given. The method of Integrationby Parts corresponds to the Product Rule for differentiation. For integration of rational functions, only somespecial cases are discussed. Complete discussion for the general case is rather complicated. Since Integrationby Parts and integration of rational functions are not covered in the courseBasic Calculus , the discussion onthese two techniques are brief and exercises are not given. Students who want to know more about techniques ofintegration may consult other books on Calculus .

8 To close the discussion on integration, application of definiteintegrals to probability (which is a vast field in mathematics) is should bear in mind that the main purpose of learning Calculus is not just knowing how to performdifferentiation and integration but also knowing how to apply differentiation and integration to solve that, one must understand the concepts. To perform calculation, we can use calculators or computer soft-wares, likeMathematica,MapleorMatlab. Accompanying the pdf file of this book is a set of Mathematicanotebook files (with extension .nb, one for each chapter) which give the answers to most of the questions in theexercises. Information on how to read the notebook files as well as trial version of Mathematica can be .. Identities and Algebraic Expressions.. Linear Equations.

9 Quadratic Equations.. Theorem and Factor Theorem.. Linear Inequalities.. Theorem, Distance Formula and Circles.. Systems of Equations..201 Sets, Real Numbers and .. Operations.. Numbers.. Number Systems.. Inequalities.. Inequalities.. Inequalities with degrees 3..392 Functions and .. and Ranges of Functions.. of Equations.. of Functions.. of Functions.. Functions.. on Solving Equations..693 .. of Sequences.. of Functions at Infinity.. Limits.. Limits.. Functions..944 .. for Differentiation.. Derivatives..123ivCONTENTS5 Applications of Sketching.. and Decreasing Functions.. Extrema.. Sketching.. Extremum Problems.. Extrema.. Maxima and Minima.. to Economics..1536 Integrals.. Theorem of Calculus .. Integrals.. of Integration..1737 Trigonometric.

10 Functions.. of Trigonometric Functions..1878 Exponential and Logarithmic Functions.. Functions.. of Exp and Log Functions..2019 More rule.. Differentiation.. Curve Sketching.. Extremum Problems..22210 More More Formulas.. Substitution Method.. Integration of Rational Functions.. Integration by Parts.. More Applications of Definite Integrals..248A Answers255B Supplementary Mathematical Induction.. Binomial Theorem.. Mean Value Theorem.. Fundamental Theorem of Calculus ..276 Chapter ExponentsDefinition(1)Letnbe a positive integer and letabe a real number. We defineanto be the real number given byan=a a a nfactors.(2)Letnbe a negative integern, that is,n= kwherekis a positive integer, and letabe a real numberdifferent from 0. We definea kto be the real number given bya k=1ak.