Higher Engineering Mathematics

FM-H8152.tex 19/7/2006 18: 59 Page v

Contents

Preface xv

Syllabus guidance xvii

Section A: Number and Algebra 1

1 Algebra 1

1.1 Introduction 1

1.2 Revision of basic laws 1

1.3 Revision of equations 3

1.4 Polynomial division 6

1.5 The factor theorem 8

1.6 The remainder theorem 10

2 Inequalities 12

2.1 Introduction to inequalities 12

2.2 Simple inequalities 12

2.3 Inequalities involving a modulus 13

2.4 Inequalities involving quotients 14

2.5 Inequalities involving square

functions 15

2.6 Quadratic inequalities 16

3 Partial fractions 18

3.1 Introduction to partial fractions 18

3.2 Worked problems on partial fractions

with linear factors 18

3.3 Worked problems on partial fractions

with repeated linear factors 21

3.4 Worked problems on partial fractions

with quadratic factors 22

4 Logarithms and exponential functions 24

4.1 Introduction to logarithms 24

4.2 Laws of logarithms 24

4.3 Indicial equations 26

4.4 Graphs of logarithmic functions 27

4.5 The exponential function 28

4.6 The power series for ex 29

4.7 Graphs of exponential functions 31

4.8 Napierian logarithms 33

4.9 Laws of growth and decay 35

4.10 Reduction of exponential laws to

linear form 38

5 Hyperbolic functions 41

5.1 Introduction to hyperbolic functions 41

5.2 Graphs of hyperbolic functions 43

5.3 Hyperbolic identities 44

5.4 Solving equations involving

hyperbolic functions 47

5.5 Series expansions for coshxand

sinhx 48

Assignment 1 50

6 Arithmetic and geometric progressions 51

6.1 Arithmetic progressions 51

6.2 Worked problems on arithmetic

progressions 51

6.3 Further worked problems on

arithmetic progressions 52

6.4 Geometric progressions 54

6.5 Worked problems on geometric

progressions 55

6.6 Further worked problems on

geometric progressions 56

7 The binomial series 58

7.1 Pascal’s triangle 58

7.2 The binomial series 59

7.3 Worked problems on the binomial

series 59

7.4 Further worked problems on the

binomial series 61

7.5 Practical problems involving the

binomial theorem 64

8 Maclaurin’s series 67

8.1 Introduction 67

8.2 Derivation of Maclaurin’s theorem 67

8.3 Conditions of Maclaurin’s series 67

8.4 Worked problems on Maclaurin’s

series 68

8.5 Numerical integration using

Maclaurin’s series 71

8.6 Limiting values 72

Assignment 2 75