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(MATH024)Notes.pdf
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Calculus
Rigor, Concision, Clarity

Department of Mathematics The Hong Kong University of Science and Technology
Copyright c
.2005 by Department of Mathematics, HKUST


Contents
1 Real Numbers and Functions 1
1.1 Real Number System and Inequalities 2
1.1.1 Real Number System 2
1.1.2 Inequalities 3


1.2 Functions 6
1.3 Coordinate Plane and Graphs of Functions 12
1.4 Summary 18
2 Limits 21
2.1 Limits of Sequences 22
2.1.1 The Concept of Limits and Properties 22
2.1.2 Rigorous De.nition of Limits 28


2.2 Limits of Functions 35
2.2.1 Rigorous De.nition of Limits 39
2.2.2 Relations With Limits of Sequences 45


2.3 Continuous Functions 47
2.4 Summary 57
3 Di.erentiation 59
3.1 Derivatives 60
iii

3.2 Properties of Di.erentiation 72
3.3 Mean-Value Theorem and LHospitals Rule 85
3.3.1 The Mean Value Theorem 86
3.3.2 LHospitals Rule 88


3.4 Summary 96
4 Curve Sketching and Optimization 99
4.1 Monotonicity and Concavity 101
4.1.1 Increasing and Decreasing 101
4.1.2 Concavity and In.ection Point 103


4.2 Maximum and Minimum Values 110
4.3 Curve Sketching 118
4.4 Optimization Problems 122
4.5 Summary 128
5 Integration 131
5.1 De.nite Integrals 131
5.2 Numerical Integration 142
5.3 Properties of De.nite Integrals 150
5.4 Summary 160
6 Methods of Integration 165
6.1 Inde.nite Integrals 165
6.1.1 Integration by Substitution (or Change of Variable) 167
6.1.2 Integration by Parts 170


6.2 Integrals of Rational Functions 175
6.2.1 Integrals Involving Some Radicals 179
6.2.2 Rational functions of sin x and cos x. 182
6.2.3 Micellaneous Techniques 183


6.3 Summary 187
7 Applications of Integration 189
7.1 Applications in Physics 190
7.1.1 Arc Length 190
7.1.2 Volume of Revolution 195
7.1.3 Area of Surface of Revolution 198
7.1.4 Area Bounded by Polar Curves 200
7.1.5 Center of Mass 203
CONTENTS v

7.1.6 Work 208


7.2 Improper Integrals 213
7.2.1 De.nite Integrals over Unbounded Intervals 213
7.2.2 Integration of Unbounded Functions 222


7.3 Summary 226
8 In.nite Series 231
8.1 In.nite Series 231
8.1.1 Non-negative Series 234
8.1.2 Absolute Convergence and Conditional Convergence 243


8.2 Uniform Convergence 249
8.3 Power Series 262
8.3.1 Radius of Convergence and Interval of Convergence 262
8.3.2 Term by Term Di.erentiation and Integration 264
8.3.3 Multiplying Power Series 267


8.4 Taylors Formula 269
8.4.1 Little-o and Big-O Notations 270
8.4.2 Taylor Expansion 273


8.5 Fourier Series 279
8.5.1 Fourier Series Expansion 280
8.5.2 Bessel Inequality 285
8.5.3 Fourier Convergence Theorem 286
8.5.4 Di.erentiation and Integration 288


8.6 Summary 290
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Real Numbers and Functions
In this chapter, the topics we study are preparations for the rest of the course.
We start with real numbers. This is the number set that calculus works wit