# Calculus and Differential Equations

Fulford
Title Calculus and Differential Equations
Edition 1st
ISBN 9781442532502
ISBN 10 1442532505
Published 14/12/2009
Pages
Format Paperback
Available on demand

Total Price \$67.95 Add to Cart

Calculus and Differential Equations has been written with the needs of Australian students in mind. The book introduces differential equations much earlier than is done in more traditional calculus texts because it is one of the most important topics in calculus.  The material has been graded into core (important and fundamental material) through to extensions which are more conceptual and finally harder more advanced material. The exercises are similarly graded. This will enable students to first focus on and master the basic ideas before tackling the harder stuff.

Print Colour:
Black & White

Authors:
Glenn Fulford and Dann Mallet
Queensland University of Technology

Course Code / Course Name:

MM0401 – Calculus

Preface                                                                                               v
Introduction for students                                                                   vii

1 Differential calculus                                                                        1
1.1 Review of important concepts in differential calculus . . . . . . . . . . . 2
1.2 Functions: basic ideas and hyperbolic functions . . . . . . . . . . . . . . 4
1.3 Implicit derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Indeterminate limits and L’Hopital’s rule . . . . . . . . . . . . . . . . . . 20
1.5 *Further indeterminate forms . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6 +Continuity and differentiability . . . . . . . . . . . . . . . . . . . . . . 32
1.7 *Some further theoretical ideas . . . . . . . . . . . . . . . . . . . . . . . . 37
1.8 Examinable skills checklist . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.9 Exercise problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Differential equations and indefinite integration                        43
2.1 Introduction to differential equations . . . . . . . . . . . . . . . . . . . . 44
2.2 Antiderivatives (indefinite integrals) . . . . . . . . . . . . . . . . . . . . 48
2.3 Separable differential equations . . . . . . . . . . . . . . . . . . . . . . . 56
2.4 +Recognising 1st-order separable differential equations . . . . . . . . . . 65
2.5 Integration by parts — a review . . . . . . . . . . . . . . . . . . . . . . . 66
2.6 Integration using partial fractions . . . . . . . . . . . . . . . . . . . . . . 72
2.7 *Partial fractions with higher order factors . . . . . . . . . . . . . . . . . 78
2.8 *Solving the DEs for some models . . . . . . . . . . . . . . . . . . . . . 80
2.9 Examinable skills checklist . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.10 Exercise problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3 Definite integrals 87
3.1 Review of fundamental ideas . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3 +Applications of improper integrals . . . . . . . . . . . . . . . . . . . . . 97
3.4 *Improper integrals of type II . . . . . . . . . . . . . . . . . . . . . . . . 102
3.5 +Differentiating integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.6 Volumes, arc length and surface area . . . . . . . . . . . . . . . . . . . . . 107
3.7 *Further volume calculations . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.8 Examinable skills checklist . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.9 Exercise problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4 Further differential equations                                            127
4.1 Some applications of 2nd-order differential equations . . . . . . . . . . . 128
4.2 Homogeneous constant coefficient differential equations . . . . . . . . . 129
4.3 Nonhomogeneous constant coefficient differential equations . . . . . . . 133
4.4 +The integrating factor method for 1st-order DEs . . . . . . . . . . . . . 140
4.5 *Laplace transforms for solving differential equations . . . . . . . . . . . 143
4.6 *Numerical solution by Euler’s method . . . . . . . . . . . . . . . . . . 145
4.7 Examinable skills checklist . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.8 Exercise problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Approximating functions — Taylor and Fourier series 157
5.1 Introduction to approximating functions . . . . . . . . . . . . . . . . . . 158
5.2 Taylor polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.3 +Convergence of Taylor series . . . . . . . . . . . . . . . . . . . . . . . 165
5.4 Fourier series for periodic functions . . . . . . . . . . . . . . . . . . . . . 171
5.5 Examinable skills checklist . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.6 Exercise problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A Answers to exercise problems                               187
B The fundamental theorem of calculus                199
C Tables                                                                      201
C.1 The Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
C.2 Table of derivatives and indefinite integrals . . . . . . . . . . . . . . . . 202

Index                                                               203