# calc

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```					                                      CONTENTS

UNIT 1   SEQUENCE AND SERIES                                     .......................                             1
Sequence and series       ........................................                                          1
Finite series       ..............................................                                          2
Recurrent sequence        ........................................                                          7
Exercise 1        ................................................                                         14
Chapter summary          .........................................                                         17

UNIT 2   LIMITS OF SEQUENCE AND SERIES                                                  ............                19
Introduction            .............................................                                      19
Intuitive concept of limit        . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .   19
Properties of convergent sequence              .............................                               22
Some important results of limits of sequences                    ....................                      28
Operations of limits of sequences            ..............................                                31
Operations of convergent sequences and sequences divergent
to infinity         ...............................................                                        33
Exercise 2(a)         ..............................................                                       37
Finding limit of recurrent sequence             .............. .............                               39
Exercise 2(b)         ..............................................                                       47
Limit of series        ............................... .............                                       52
Exercise 2(c)         ..............................................                                       59
Chapter summary             .........................................                                      62
Revision exercise 2         .........................................                                      66

UNIT 3   LIMIT OF FUNCTION                                  .........................                               73
Introduction          ..............................................                                       73
Limit of function       ....................................... ...                                        73
Limit of function and limit of sequence       .........................                                    76
Evaluate of limit of function       ..................... ............                                     77
Two standard results of limits of functions      ......................                                    81
Exercise 3         ................................................                                        85
Chapter summary             .........................................                                      88
UNIT 4   CONTINUITY AND DIFFERENTIABILITY                                                           .......            91
Introduction         ..............................................                                           91
Discontinuity of a function          ..................................                                       92
Operations of continuous functions            ...........................                                     94
Properties of continuous functions           ............................                                     97
Exercise 4(a)        ..............................................                                          100
Derivative of a function         .....................................                                       104
Continuity and differentiability of a function       ....................                                    106
Exercise 4(b)        ..............................................                                          113
Chapter summary            .........................................                                         116
Revision exercise 4        .........................................                                         118

UNIT 5   RULES OF DIFFERENTIATION                                              .................                      123
Introduction           ..............................................                                        123
Rules of differentiation        . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .   123
Differentiation of trigonometric functions                     ......................                        125
Differentiation of inverse functions               ............................                              128
Parametric differentiation           ...................................                                     130
Implicit differentiation       ......................................                                        132
Logarithmic differentiation            ..................................                                    135
Summary of standard results of derivatives of elementary functions                                   ...     139
Exercise 5          ................................................                                         140
Chapter summary             .........................................                                        142

UNIT 6   HIGHER ORDER DERIVATIVES                                                ................                     145
Successive differentiation        ....................................                                       145
The nth order derivative          ....................................                                       149
Infinite Maclaurin’s series*       ...................................                                       156
Exercise 6         ................................................                                          160
Chapter summary              .........................................                                       164
Revision exercise 6          .........................................                                       167

UNIT 7   APPLICATIONS OF DIFFERENTIATION                                                        .........             171
Introduction        ..............................................                                           171
Indeterminate forms        .......................................                                           171
Properties of differentiable fucntions      ..........................      175
Functional inequalities          .....................................      181
Relative extremum and absolute extremum          .....................      182
Exerecise 7(a)         .............................................        189
Concavity and point of inflexion         .............................      193
Asymptotes            ..............................................        196
Curve sketching            ..........................................       199
Exercise 7(b)         ..............................................        208
Chapter summary             .........................................       212
Revision exercise 7(a)         .......................................      216
Revision exercise 7(b)         .......................................      222

UNIT 8   INDEFINITE INTEGRALS                    ......................              227
Introduction          ..............................................        227
Meaning of indefinite integral         ................................     227
Some standard integration formulae          ...........................     228
Rules of indefinite integration        ................................     228
Techniques of indefinite integration        ...........................     230
Integration of polynomials and power functions       ..................     230
Integration of exponential and logarithmic functions     ..............     231
Integration of rational functions        ..............................     233
Integration of irrational functions*       ............................     235
Integration of trigonometric functions        .........................     237
Further rule of indefinite integration       ..........................     241
Exercise 8          ................................................        244
Chapter summary              .........................................      246

UNIT 9   DEFINITE INTEGRALS                   ........................               249
The Riemann definition of definite integral         .....................   249
Properties of definite integrals      ................................      251
Mean value theorem of integral calculus        ........................     252
Fundamental theorem of calculus          .............................      253
Evaluation of definite integral       ................................      253
Further method of finding harder definite integrals      ................   258
Improper integrals            ........................................      268
Exercise 9         ................................................         272
Chapter summary             .........................................       280
Revision exercise 9         .........................................       283
UNIT 10   MORE ABOUT DIFFERENTIAL AND INTEGRAL
CALCULUS    ..................................                             287
Introduction         ..............................................        287
Infinite sum by definite integral     ...............................      287
Property of primitive function        ...............................      289
Comparison property of definite integral      .......................      295
More about limits of sequences and series     .......................      303
Exercise 10         ...............................................        316
Chapter summary             .........................................      322
Revision exercise 10         ........................................      323

UNIT 11   GEOMETRIC APPLICATIONS OF DEFINITE
INTEGRALS   ..................................                             331
Introduction          ..............................................       331
Plane areas by definite integral       ................................    331
Length of curve of function          ..................................    334
Volume of solid of revolution          ................................    337
Area of surface of revolution         .................................    343
Plane curve in parametric equations          ...........................   345
Mensuration formulae for plane curve in parametric equations     .......   346
Plane curve in polar coordinates         ..............................    352
Conversion between rectangular coordinates and poalr coordiantes     ...   353
Tangent to polar curve*            ....................................    354
Some standard polar curves           ..................................    356
Area of polar curve          ........................................      359
Length of poalr curve            ......................................    360
Area of surface of revolution of polar curve       .....................   361
Exercise 11         ...............................................        364
Chapter summary             .........................................      368
Revision exercise 11         ........................................      372

UNIT 12   ANALYTICAL GEOMETRY (I) STRAIGHT LINES
AND CIRCLES   ...............................                              377
Introduction        ..............................................         377
Basic knowledge in analytical geometry      .........................      377
Family of lines       ............................................         382
Simple concept of locus         .....................................      383
Equation of circle           .........................................       386
Point(s) of intersection of a line and a circle  .....................       386
Tangents to a circle         .........................................       386
Family of circles          ..........................................        391
Orthogonal circles           .........................................       395
Exercise 12          ...............................................         397
Chapter summary              .........................................       400

UNIT 13   ANALYTICAL GEOMETRY (II) CONIC SECTION                                 ..    403
Introduction          .............................................          403
Parabola in standard position         .................................      404
Parametric equations of parabola           .............................     404
Chords, tangents and normals of parabola          ......................     404
Exercise 13(a)        .............................................          413
Ellipse in standard position         ..................................      415
Parametric equations of ellipse          ...............................     416
Chords, tangents and normals of ellipse         ........................     416
Diameters of ellipse*         .......................................        419
Exercise 13(b)        .............................................          426
Hyperbola in standard position           ...............................     428
Parametric equations of hyperbola           ............................     429
Chords, tangents and normals of hyperbola          .....................     430
Diameters of hyperbola*           ....................................       433
Exercise 13(c)        .............................................          441
General conics and their classifictions*       .........................     443
Degenerated conic (a pair of straight lines)*      .....................     445
Equation of the pair of asymptotes of hyperbola*          ................   446
Equation of the pair of tangents from an external point to the conic*   ..   447
Exercise 13(d)*         ............................................         452
Chapter summary             .........................................        453
Revision exercise 13         ........................................        458

ANSWERS TO PROBLEM SETS                   ............................                 463

CHINESE TRANSLATION TERMS                     .........................                481
PREFACE TO THE FIRST EDITION

Calculus is a deductive science and a branch of pure mathematics. At the same time,
calculus has strong roots in physical problems and that it derives much of its power and
beauty from the wide range of its applications. It is possible to combine a strong
theoretical development with sound training in technique; this book represents an
attempt to strike a sensible balance between the two.

Analytical geometry is the modern methods to study the geometry of plane curves
by the properties of real numbers. The primitive concepts of the analytical geometry are
algebraically deduced in the coordinate plane. Elementary mensuration formulae for
plane curves by calculus will be highlighted.

The overall objectives of this course book are
1.   to develop students’ understanding of basic mathematical concepts required in
scientific and technological studies in the first year degree course in University;
2.   to extend students’ mathematical skills in problem solving;
3.   to further develop students’ ability to use the logic to argue and reason.

The main features of the book include:
1.   A complete course book, not a study guide;
2.   Intuitive and conceptual approach to the subject matters;
3.   Arguments and solutions are presented in logical symbols of language of
mathematics to enhance reasoning;
4.   Sufficient demonstrative examples for motivation;
5.   Abundant exercises in each chapter for consolidation;
6.   Revision exercises for miscellaneous review;
7.   Chapter summary for quick and effective revisions.

I would like to express my sincere thanks to all students who have proofread the
scope for improvement which is never ending. The comments and suggestions from

Yeung Ka Hung
PREFACE TO THE SECOND EDITION

The first edition of this textbook has published for five years and has
reprinted some times in the subsequent years. Since then, various changes of
teaching approaches to calculus have been introduced to cope with the
learning abilities of students. This is the main reason to revise the contents of
this textbook so as to make it more compatible with the Y2K trend.

In this new edition, the following modifications have been made:

1.   Some chapters are resolved into some sub-topics for beginner to
consolidate the pre-requisites for the next demanding chapters.

2.   Many topics are re-written with emphasis of theory and application.

3.   Many new and graded exercises are included for different abilities of
students:
Level 1           Short questions are of elementary level to be tackled
by direct applications of mathematical formulae and
simple concepts.
Level 2          Questions are of higher level to be tacked by applying
inter-related concepts of mathematical analysis.
Level 3         Structural questions are of advanced level to be
tackled by deductive thinking of some inter-related
concepts in depth.

4.   Revision exercises consist of questions mixing with different levels of
difficulties for examination preparation.

5.   For completeness of the chapter, some topics marked with ‘*’ are included
for interest and reference only.

6.   The typographical arrangements have been reset to make the textbook

Lastly, I would like to express my sincere thinks to all students who have