Cover -- Copyright -- Dedication -- Contents -- Preface -- 1 Fundamentals -- 1.1 Why Mathematics -- 1.2 What's it all about? -- 1.2.1 x? -- 1.2.2 Mathematics? -- 1.2.3 Functions and Equations -- 1.2.4 Relationships -- 1.2.5 Why don't we speak mathematics all the time? -- 1.2.6..and why do I need to understand it? -- 1.3 Working with Equations -- 1.3.1 Rearranging Equations -- 1.3.2 Order of Evaluating Algebraic Equations -- Example - Evaluate 3 £ (2 + 3)2 -- Example - Evaluate 3 £ 2 + 32 -- 1.3.3 Some Useful Algebraic Relationships -- 1.3.4 A word about Calculators -- 1.4 Preliminary Calculations - check the problem -- 1.4.1 Dimension Analysis -- 1.4.2 An example calculation -- 1.4.3 And the back of the envelope -- Exercises -- Answers -- 2 Numbers -- 2.1 Decimal Number Representation -- 2.1.1 Significant Figures and Decimal Places -- 2.1.2 Scientific Notation -- 2.2 Binary and Hexadecimal numbers -- 2.2.1 Binary Numbers -- 2.2.2 Conversion from Decimal to Binary -- 2.2.3 Hexadecimal Numbers -- 2.2.4 Conversion from Decimal to Hexadecimal -- 2.2.5 Binary-Hex conversion -- Exercises -- Answers -- 3 Powers and Logarithms -- 3.1 Powers and Indices -- 3.1.1 Some general rules of powers and indices -- 3.1.2 Rules of Powers and Indices - Summary -- 3.2 Logarithms -- 3.2.1 What are logarithms? -- 3.2.2 Definition -- Common Logarithms -- Natural Logarithms -- e - an Interesting Number -- 3.2.3 Mathematical Derivation of the Rules of Logarithms . 27 -- Example - Calculate log10(10 x 100) -- Example - Calculate log2(2 x 16) -- Example - Calculate log4(163) -- Example - Find the logarithm of -- 3.2.4 Calculating logarithms to a different base -- Example - Suppose we know ln(2) but need log(2) -- 3.2.5 Rules of Logarithms - Summary -- 3.3 Exponential Functions -- Population Dynamics -- Exercises -- Answers -- 4 Calculations and Applications.

4.1 Convert miles/hour (mph or miles hour¡1) to m s¡1 -- 4.2 The pH of a solution -- 4.3 How many microbes? The Viable Count Method -- 4.4 Surface Area of Humans -- 4.5 Fluid Flow - Poiseuille's Formula -- 4.6 The Growth of a Bacterial Population -- 4.7 The Beer-Lambert Law -- 4.8 A River Pollution Incident -- 4.9 Linear Regression -- 4.9.1 Notation for sums of sequences -- 4.9.2 Fitting the best Straight Line -- 4.10 The Michaelis-Menton equation -- 4.10.1 The Lineweaver-Burke transformation -- 4.10.2 The Eadie-Hofsee transformation -- 4.10.3 Fitting the parameters the Modern Way -- 4.11 Graphs and Functions -- 4.11.1 Plotting Graphs -- 4.11.2 Shapes of some useful functions -- 5 Neat Tricks and Useful Solutions -- 5.1 The Difference of Two Squares -- 5.2 Mathematical Induction -- 5.3 Pythagoras' Theorem -- 5.4 Pythagoras' Theorem revisited -- 5.5 Limits -- Example - The limiting value of y = -- 5.6 Trigonometry - angles with a difference -- 5.6.1 Radians and Degrees -- 5.7 Trigonometric Ratios -- 6 Differential Calculus -- 6.1 Introduction -- 6.1.1 What is differentiation? -- 6.2 Distance and Velocity -- 6.2.1 Average Velocity -- 6.2.2 Instantaneous Velocity -- 6.3 The Differential Coefficient of any function -- Example - The differential coefficient of the function y = 3x -- Example - The derivative of the function y = x2+3x+2 66 -- Example - the derivative of x -- 6.3.1 Differentiability -- 6.4 Differentials involving two Functions -- 6.4.1 The derivative of a sum u(x) + v(x) -- Example - the derivative of x1=2 + 3x -- 6.4.2 The derivative of a product u(x)v(x) -- Example - The derivative of x2 -- Example - The derivative of x3 -- 6.4.3 The derivative of a quotient u(x)=v(x) -- 6.5 Some important derivatives -- 6.5.1 The derivative of a Constant -- 6.5.2 The derivative of xn -- 6.5.3 The derivative of sin x.

6.5.4 The derivative of a constant times a function of x -- Example - The derivative of 3 sin x -- Example - The derivative of tan x -- Example - The derivative of x sin x -- Example - The derivative of 4x2 -- 6.5.5 The derivative of ex -- 6.5.6 The derivatives of ln x and ax -- 6.5.7 The Chain Rule -- Example - The derivative of sin(x2) -- Example - The derivative of (x2 + 3x + 1)4 -- Example - the derivative of ekt -- 6.6 Optimum values - maxima and minima -- Example - How fast should a fish swim? -- 6.7 Small Errors -- Example - Estimation of Errors -- 6.8 Summary Notes on Differentiation -- 6.8.1 Standard Derivatives -- 6.8.2 Rules for Differentiation -- 6.8.3 Maxima and Minima -- 6.9 Applications -- Equation for Radio-active Decay -- Half-life -- Linear Regression - The Method of Least Squares -- Cylinder of Minimum Surface Area -- Exercises -- Answers -- 7 Integral Calculus -- 7.1 Introduction -- 7.2 Integration as the Area under a Curve -- Example - Area under the curve y = x -- Example - The area bound by y = cos x for ¼ 2 · x · ¼ 92 R -- Example - -- 7.2.1 Area of a Circle 1 -- 7.2.2 Area of a Circle 2 -- 7.3 Techniques of Integration -- 7.3.1 The Chain Rule -- Example - sin2 x cos x dx -- Example - R 2 0 e¡x2 x dx -- 7.3.2 Integration by Parts -- Example - xex dx -- Example - x2 sin x dx -- Example - The factorial function -- 7.4 Summary Notes on Integration -- 7.4.1 Standard Integrals -- 7.4.2 Techniques -- 7.5 Applications -- Mean Value -- Example - The mean value of y = x for a · x · b -- Surfaces and Volumes of Revolution -- Equations of Motion -- Pollution of a Lake -- Exercises -- Answers -- 8 Matrix Algebra -- 8.1 Introduction -- 8.2 What is a Matrix -- 8.2.1 Equality of Matrices -- 8.2.2 Addition of Matrices -- 8.2.3 Subtraction of Matrices -- 8.2.4 Zero or Null Matrix -- 8.2.5 Identity Matrix -- 8.2.6 Multiplication by a scalar.

8.2.7 Multiplication of Matrices -- 8.2.8 Using matrix multiplication to rotate coordinates -- 8.2.9 Transpose -- 8.3 Determinants -- 8.3.1 The determinant of a 3 £ 3 matrix -- 8.3.2 Minors and Cofactors -- 8.3.3 Area of a Triangle -- 8.3.4 Some properties of Determinants -- 8.4 The Inverse Matrix -- 8.4.1 Solution of Linear Simultaneous Equations -- 8.5 Applications -- 8.5.1 Application to Population Dynamics -- 8.5.2 Eigenvalues and eigenvectors -- 9 The End of the Beginning -- 9.1 Further Reading.