## Description

This student friendly workbook addresses mathematical topics using SONG - a combination of Symbolic, Oral, Numerical and Graphical approaches. The text helps to develop key skills, communication both written and oral, the use of information technology, problem solving and mathematical modelling. The overall structure aims to help students take responsibility for their own learning, by emphasizing the use of self-assessment, thereby enabling them to become critical, reflective and continuing learners – an essential skill in this fast-changing world.

The material in this book has been successfully used by the authors over many years of teaching the subject at Sheffield Hallam University. Their SONG approach is somewhat broader than the traditionally symbolic based approach and readers will find it more in the same vein as the Calculus Reform movement in the USA.

## Key Features

- Addresses mathematical topics using SONG - a combination of Symbolic, Oral, Numerical and Graphical approaches
- Helps to develop key skills, communication both written and oral, the use of information technology, problem solving and mathematical modelling
- Encourages students to take responsibility for their own learning by emphasizing the use of self-assessment

## Readership

University students

## Table of Contents

- About the Authors
- Foreword
- 1: Numbers, Graphics and Algebra
- 1.1 NUMBERS, GRAPHICS AND ALGEBRA
- 1.2 WHAT NUMBERS ARE
- 1.3 HOW NUMBERS (AND LETTERS) BEHAVE
- 1.4 FRACTIONS, DECIMALS AND SCIENTIFIC NOTATION
- 1.5 POWERS OR INDICES
- 1.6 ANGLE AND LENGTH - GEOMETRY AND TRIGONOMETRY
- END OF CHAPTER 1 - CALCULATOR ACTIVITIES – DO THESE NOW!

- 2: Linking Algebra and Graphics 1
- 2.1 ALGEBRA AND PICTURES
- 2.2 NUMBERS, LETTERS AND BRACKETS
- 2.3 “SPEAKING” ALGEBRA
- 2.4 ALGEBRAIC FRACTIONS
- 2.5 SOLVING SIMPLE EQUATIONS
- 2.6 CONNECTING STRAIGHT LINES AND LINEAR EXPRESSIONS
- 2.7 SOLVING LINEAR EQUATIONS GRAPHICALLY
- 2.8 TRANSPOSING FORMULAE
- 2.9 STRAIGHT LINES IN ENGINEERING
- 2.10 STRATEGIES FOR HANDLING LINEAR EQUATIONS AND GRAPHS
- END OF CHAPTER 2 - MIXED ACTIVITIES – DO THESE NOW!

- 3: Linking Algebra and Graphics 2
- 3.1 MORE ON CONNECTING ALGEBRA TO GRAPHS
- 3.2 QUADRATIC FUNCTIONS
- 3.3 SOLVING QUADRATIC EQUATIONS
- 3.4 AN ALGEBRAIC TRICK - COMPLETING THE SQUARE
- 3.5 A DIVERSION - MATCH THE GRAPHS WITH THE FUNCTIONS
- 3.6 STRATEGIES FOR HANDLING QUADRATIC FUNCTIONS
- 3.7 WHERE NEXT WITH POLYNOMIALS?
- END OF CHAPTER 3 ACTIVITY - DO THIS NOW!

- 4: Other Essential Functions
- 4.1 ESSENTIAL ENGINEERING FUNCTIONS
- 4.2 THE BASICS OF EXPONENTIALS AND LOGARITHMS
- 4.3 HOW THE EXPONENTIAL FUNCTION BERAYES
- 4.4 HOW THE LOGARITHM FUNCTION BEHAVES
- 4.5 THE BASICS OF TRIGONOMETRIC FUNCTIONS
- 4.6 INVERSE FUNCTIONS AND TRIGONOMETRIC EQUATIONS
- END OF CHAPTER 4 - MIXED ACTIVITIES!

- 5: Combining and Applying Mathematical Tools
- 5.1 USING YOUR TOOLBOX
- 5.2 THE MOST BASIC FUNCTION – THE STRAIGHT LINE
- 5.3 TRANSFORMATIONS OF GRAPHS
- 5.4 DECAYING OSCILLATIONS
- 5.5 A FOGGY FUNCTION
- 5.6 HEAT LOSS IN BUILDINGS – A MATHEMATICAL MODEL

- 6: Complex Numbers
- 6.1 THE NEED FOR COMPLEX NUMBERS
- 6.2 THE j NOTATION AND COMPLEX NUMBERS
- 6.3 ARITHMETIC WITH COMPLEX NUMBERS
- 6.4 GEOMETRY WITH COMPLEX NUMBERS: THE ARGAND DIAGRAM
- 6.5 CARTESIAN AND POLAR FORM, MODULUS AND ARGUMENT
- 6.6 EULER’S RELATIONSHIP AND EXPONENTIAL FORM
- 6.7 SOME USES OF POLAR AND EXPONENTIAL FORM
- 6.8 COMPLEX ALGEBRA
- 6.9 ROOTS OF COMPLEX NUMBERS
- 6.10 MINI CASE STUDY
- END OF CHAPTER 6 – MIXED EXERCISES – DO ALL THESE NOW!

- 7: Differential Calculus 1
- 7.1 THE NEED FOR DIFFERENTIAL CALCULUS
- 7.2 DIFFERENTIAL CALCULUS IN USE
- 7.3 WHAT DIFFERENTIATION MEANS GRAPHICALLY
- 7.4 VARIOUS WAYS OF FINDING DERIVATIVES
- 7.5 NUMERICAL DIFFERENTIATION
- 7.6 PAPER AND PENCIL APPROACHES TO DIFFERENTIATION
- 7.7 COMPUTER ALGEBRA SYSTEMS OR SYMBOL MANIPULATORS

- 8: Differential Calculus 2
- 8.1 DIFFERENTIAL CALCULUS: TAKING THE IDEAS FURTHER
- 8.2 SOLVING THE EXAMPLES FROM CHAPTER 7
- 8.3 HIGHER ORDER DERIVATIVES AND THEIR MEANING
- 8.4 FINDING MAXIMUM AND MINIMUM POINTS
- 8.5 PARAMETRIC DIFFERENTIATION
- 8.6 IMPLICIT DIFFERENTIATION
- 8.7 PARTIAL DIFFERENTIATION
- 8.8 AN ENGINEERING CASE STUDY
- END OF CHAPTER 8 EXERCISES – DO ALL THESE NOW!

- 9: Integral Calculus 1
- 9.1 THE NEED FOR INTEGRAL CALCULUS
- 9.2 INDEFINITE INTEGRATION AND THE ARBITRARY CONSTANT
- 9.3 USING A COMPUTER ALGEBRA SYSTEM TO MAKE A TABLE OF INTEGRALS
- 9.4 DEFINITE INTEGRATION AND AREAS
- 9.5 USING AREAS TO ESTIMATE INTEGRALS
- 9.6 APPROXIMATE INTEGRATION - THE TRAPEZIUM RULE AND SIMPSON’S RULE
- 9.7 PAPER AND PENCIL APPROACHES TO INTEGRATION
- END OF CHAPTER 9 - MIXED EXERCISES – DO ALL THESE NOW!

- 10: Integral Calculus 2
- 10.1 INTRODUCTION
- 10.2 INTEGRATION AS SUMMATION: MEAN AND RMS
- 10.3 INTEGRATION AS SUMMATION: CHARGE ACCUMULATION
- 10.4 INTEGRATION AS SUMMATION: VOLUME AND SURFACE AREA
- 10.5 A FIRST LOOK AT DIFFERENTIAL EQUATIONS
- END OF CHAPTER 10 EXERCISES - DO ALL OF THESE NOW!

- 11: Linear Simultaneous Equations
- 11.1 THE NEED FOR LINEAR SIMULTANEOUS EQUATIONS
- 11.2 WHERE SIMULTANEOUS EQUATIONS OCCUR – TWO EXAMPLES
- 11.3 SOLVING SIMULTANEOUS EQUATIONS GRAPHICALLY
- 11.4 SOLVING SIMULTANEOUS EQUATIONS WITH SIMPLE NUMBERS
- 11.5 SOLVING SIMULTANEOUS EQUATIONS ALGEBRAICALLY
- 11.6 SOLVING SIMULTANEOUS EQUATIONS USING TECHNOLOGY
- 11.7 EQUATIONS WITH NO UNIQUE SOLUTION - SINGULAR EQUATIONS
- 11.8 ILL CONDITIONED EQUATIONS
- 11.9 SOLVING THE “REAL” PROBLEMS
- END OF CHAPTER 11 - MIXED EXERCISES – DO THESE NOW!

- 12: Matrices
- 12.1 MATRICES: WHAT ARE THEY, AND WHY DO YOU NEED THEM?
- 12.2 ARITHMETIC AND ALGEBRAIC OPERATIONS WITH MATRICES
- 12.3 MATRICES AND TECHNOLOGY
- 12.4 MATRICES AND SIMULTANEOUS EQUATIONS – THE MATRIX INVERSE
- 12.5 MATRICES AND GEOMETRICAL TRANSFORMATIONS
- END OF CHAPTER 12 - MIXED EXERCISES - DO ALL THESE NOW!

- 13: More Linear Simultaneous Equations
- 13.1 LARGER SETS OF SIMULTANEOUS EQUATIONS
- 13.2 ELIMINATION METHODS: GAUSSIAN ELIMINATION
- 13.3 ITERATIVE METHODS
- 13.4 THE GAUSS-JORDAN METHOD
- 13.5 FINDING A MATRIX INVERSE BY THE GAUSS-JORDAN METHOD
- 13.6 ENGINEERING CASE STUDY: HEATING AND COOLING

- 14: Vectors
- 14.1 INTRODUCTION
- 14.2 REPRESENTING VECTORS
- 14.3 THE ALGEBRA OF VECTORS
- 14.4 PRODUCTS OF VECTORS
- END OF CHAPTER 14 EXERCISES – DO ALL OF THESE NOW!

- 15: First Order Ordinary Differential Equations
- 15.1 INTRODUCTION
- 15.2 OVERVIEW
- 15.3 DIRECT INTEGRATION REVISITED
- 15.4 SOLUTION BY SEPARATION OF VARIABLES
- 15.5 ENGINEERING CASE STUDIES
- 15.6 NUMERICAL SOLUTION METHODS – THE EULER METHOD
- 15.7 EXPLORING THE PARAMETERS
- END OF CHAPTER 15 MIXED EXERCISES – DO ALL OF THESE NOW!

- 16: Second Order Ordinary Differential Equations
- 16.1 INTRODUCTION
- 16.2 SECOND ORDER DIFFERENTIAL EQUATIONS
- 16.3 CASE STUDY: A SUSPENSION SYSTEM
- 16.4 THE SOLUTION OF LINEAR SECOND ORDER O.D.E.S
- 16.5 THE COMPLEMENTARY FUNCTION/PARTICULAR INTEGRAL APPROACH
- 16.6 THE CF - FINDING OUT ABOUT UNFORCED CHANGE
- 16.7 FINDING THE ARBITRARY CONSTANTS BY USING INITIAL CONDITIONS
- 16.8 FINDING THE P.I. - THE EFFECT OF FORCING CHANGE
- 16.9 TECHNOLOGICAL SOLVERS
- 16.10 SOME FINAL EXAMPLES
- END OF CHAPTER 16 MIXED EXERCISES - DO ALL OF THESE NOW!

- 17: Laplace Transforms And Ordinary Differential Equations
- 17.1 THE USEFULNESS OF THE LAPLACE TRANSFORM
- 17.2 WHAT IS THE LAPLACE TRANSFORM?
- 17.3 THE LAPLACE TRANSFORM IN ACTION: FIRST ORDER ODES
- 17.4 USING THE LAPLACE TRANSFORM WITH SECOND ORDER ODEs
- 17.5 A FINAL SPECIAL CASE – RESONANCE
- END OF CHAPTER 17 EXERCISES – DO ALL THESE NOW!

- 18: Taylor Series
- 18.1 THE ESSENTIAL ROLE OF TAYLOR SERIES
- 18.2 LINEARISATION
- 18.3 MACLAURIN SERIES
- 18.4 GETTING AWAY FROM
*x*= 0: TAYLOR SERIES - 18.5 USES OF TAYLOR AND MACLAURIN SERIES
- END OF CHAPTER 18 PROBLEMS – DO ALL THESE NOW!

- 19: Statistics And Data Handling
- 19.1 WHY ENGINEERS NEED DATA HANDLING SKILLS
- 19.2 PRESENTING DATA IN PICTURES
- 19.3 SUMMARISING DATA SETS IN A FEW NUMBERS
- 19.4 FITTING LAWS TO EXPERIMENTAL DATA

- 20: Probability
- 20.1 WHAT IS PROBABILITY?
- 20.2 SIMPLE EXAMPLES – COMPLETE ENUMERATION
- 20.3 MORE COMPLEX SITVATIONS – THE LAWS OF PROBABILITY
- 20.4 TREE DIAGRAMS
- 20.5 SOME MORE PROBABILITY PROBLEMS
- 20.6 WHERE NEXT WITH PROBABILITY?

- Glossary
- G1 GREEK ALPHABET
- G2 SI UNITS
- G3 COMMON GRAPHS TO NOTE
- G4 POWER SERIES
- G5 COMMON NOTATION
- G6 TABLE OF TRIGONOMETRIC FUNCTION FORMULAE

## Details

- No. of pages:
- 288

- Language:
- English

- Copyright:
- © Woodhead Publishing 2008

- Published:
- 1st January 2008

- Imprint:
- Woodhead Publishing

- eBook ISBN:
- 9780857099396

- Paperback ISBN:
- 9781898563655

## About the Author

### N Challis

*Neil Challis* was born in Cambridge, UK. He studied mathematics at the University of Bristol and subsequently worked for some years as a mathematician in the British Gas Engineering Research Station at Killingworth. Since 1977, he has worked in the Mathematics Group, Sheffield Hallam University, UK and is currently head of that group. He obtained a PhD in mathematics from the University of Sheffield in 1988 and has taught mathematics to a wide variety of students, across the spectrum from first year engineers and other non-mathematicians who need access to mathematical ideas, techniques and thinking, to final year single honours mathematics students.

### H Gretton

*Harry Gretton* was born in Leicester, UK. He studied mathematics at the University of Sheffield, obtaining his PhD from there in 1970. He has taught mathematics sciences since then, both at Sheffield University and Sheffield Hallam University, and has been a tutor with the Open University since it was conceived. He has taught many varied students on many varied mathematically-related courses. In recent years he has developed a particular interest in the impact of technology on the way mathematics is taught, practiced and assessed.

### Affiliations and Expertise

Sheffield Hallam University, UK