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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
whole manuscript and made helpful comments. I firmly believe that there is always a
scope for improvement which is never ending. The comments and suggestions from
readers are always welcome.



                                                              Yeung Ka Hung
                                                            B.Sc.(Hons.),Dip.Ed.,FRSS.
               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
     more easily readable and comprehensible.

    Lastly, I would like to express my sincere thinks to all students who have
proofread the new manuscript and made helpful comments, especially for Mr.
Yeung Sing Yiu.


                                                                    K.H. Yeung

				
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