Engineering Mathematics [8 ed.] 9781352010275, 9781352010282

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English Pages [1181] Year 2020

Engineering Mathematics [8 ed.]
9781352010275, 9781352010282

Author / Uploaded
Dexter Booth
Ken Stroud

Table of contents :
Summary of contents
Contents
Preface to the first edition
Preface to the second edition
Preface to the third edition
Preface to the fourth edition
Preface to the eighth edition
How to use this book
The companion website – www.macmillanihe.com/stroud
Student review panel
The Personal Tutor
Useful background information
Symbols used in the text
Useful mathematical information
1 Algebraic identities
2 Trigonometrical identities
3 Standard curves
4 Laws of mathematics
PART I Foundation topics
Programme F.1 Arithmetic
Learning outcomes
Quiz F.1
Types of number
The natural numbers
Numerals and place value
Points on a line and order
The integers
Brackets
Addition and subtraction
Multiplication and division
Brackets and precedence rules
Rounding
Basic laws of arithmetic
1 Commutativity
2 Associativity
3 Distributivity
Estimating
Review summary
Review exercise
Factors and prime numbers
Factors
Prime numbers
Prime factorization
Fundamental theorem of arithmetic
Highest common factor (HCF)
Lowest common multiple (LCM)
Review summary
Review exercise
Fractions, ratios and percentages
Division of integers
Multiplying fractions
Of
Equivalent fractions
Dividing fractions
Adding and subtracting fractions
Fractions on a calculator
Ratios
Percentages
Review summary
Review exercise
Decimal numbers
Division of integers
Rounding
Significant figures
Decimal places
Trailing zeros
Fractions as decimals
Decimals as fractions
Unending decimals
Unending decimals as fractions
Rational, irrational and real numbers
Review summary
Review exercise
Powers
Raising a number to a power
The laws of powers
Powers on a calculator
Fractional powers and roots
Surds
Multiplication and division by integer powers of 10
Precedence rules
Standard form
Working in standard form
Using a calculator
Preferred standard form
Checking calculations
Accuracy
Review summary
Review exercise
Number systems
Denary (or decimal) system
Binary system
Octal system (base 8)
Duodecimal system (base 12)
Hexadecimal system (base 16)
An alternative method
Review summary
Review exercise
Change of base from denary to a new base
Binary form
Octal form
Duodecimal form
A denary decimal in octal form
Use of octals as an intermediate step
Reverse method
Review summary
Review exercise
Can you?
Checklist F.1
Test exercise F.1
Further problems F.1
Programme F.2 Introduction to algebra
Learning outcomes
Quiz F.2
Algebraic expressions
Symbols other than numerals
Constants
Variables
Rules of algebra
Commutativity
Associativity
Distributivity
Rules of precedence
Terms and coefficients
Collecting like terms
Similar terms
Expanding brackets
Nested brackets
Review summary
Review exercise
Powers and logarithms
Powers
Rules of indices
Logarithms
Rules of logarithms
Base 10 and base
Change of base
Logarithmic equations
Review summary
Review exercise
Algebraic multiplication and division
Multiplication
Division
Review summary
Review exercise
Algebraic fractions
Addition and subtraction
Multiplication and division
Review summary
Review exercise
Factorization of algebraic expressions
Common factors
Common factors by grouping
Useful products of two simple factors
Quadratic expressions as the product of two factors
Review summary
Review exercise
Can you?
Checklist F.2
Test exercise F.2
Further problems F.2
Programme F.3 Expressions and equations
Learning outcomes
Quiz F.3
Expressions and equations
Evaluating expressions
Equations
Conditional equation
Identity
Defining equation
Assigning equation
Formula
Evaluating independent variables
Transposition of formulas
The evaluation process
Review summary
Polynomials
Polynomial expressions
Review exercise
Evaluation of polynomials
Evaluation of a polynomial by nesting
Remainder theorem
Factor theorem
The general quadratic equation
Factorization of fourth-order polynomials
Review summary
Review exercise
Can you?
Checklist F.3
Test exercise F.3
Further problems F.3
Programme F.4 Graphs
Learning outcomes
Quiz F.4
Graphs of equations
Equations
Ordered pairs of numbers
Cartesian axes
Drawing a graph
Review summary
Review exercise
Using a spreadsheet
Spreadsheets
Rows and columns
Text and number entry
Formulas
Clearing entries
Construction of a Cartesian graph
Displays
Review summary
Review exercise
Inequalities
Less than or greater than
Review summary
Review exercise
Absolute values
Modulus
Graphs
Inequalities
Less-than inequalities
Greater-than inequalities
Interaction
Review summary
Review exercise
Can you?
Checklist F.4
Test exercise F.4
Further problems F.4
Programme F.5 Linear equations
Quiz F.5
Linear equations
Solution of simple equations
Simultaneous linear equations with two unknowns
1 Solution by substitution
2 Solution by elimination
Simultaneous linear equations with three unknowns
Pre-simplification
Review summary
Review exercise
Can you?
Checklist F.5
Test exercise F.5
Further problems F.5
Programme F.6 Polynomial equations
Learning outcomes
Quiz F.6
Polynomial equations
Quadratic equations
Cubic equations having at least one simple linear factor
Fourth-order equations having at least two linear factors
Review summary
Review exercise
Can you?
Checklist F.6
Test exercise F.6
Further problems F.6
Programme F.7 Binomials
Learning outcomes
Quiz F.7
Factorials and combinations
Factorials
Combinations
Three properties of combinatorial coefficients
Review summary
Review exercise
Binomial expansions
Pascal’s triangle
Binomial expansions
The general term of the binomial expansion
Review summary
Review exercise
The Σ (sigma) notation
General terms
The sum of the first n natural numbers
Rules for manipulating sums
The exponential number e
Review summary
Review exercise
Can you?
Checklist F.7
Test exercise F.7
Further problems F.7
Programme F.8 Partial fractions
Learning outcomes
Quiz F.8
Partial fractions
Review summary
Review exercise
Denominators with repeated and quadratic factors
Review summary
Review exercise
Can you?
Checklist F.8
Test exercise F.8
Further problems F.8
Programme F.9
Trigonometry
Learning outcomes
Quiz F.9
Angles
Rotation
Radians
Triangles
Trigonometric ratios
Reciprocal ratios
Pythagoras’ theorem
Special triangles
Half equilateral
Review summary
Review exercise
Trigonometric identities
The fundamental identity
Two more identities
Identities for compound angles
Trigonometric formulas
Sums and differences of angles
Double angles
Sums and differences of ratios
Products of ratios
Review summary
Review exercise
Can you?
Checklist F.9
Test exercise F.9
Further problems F.9
Programme F.10 Functions
Learning outcomes
Quiz F.10
Processing numbers
Functions are rules but not all rules are functions
Functions and the arithmetic operations
Inverses of functions
Graphs of inverses
The graph of y = x3
The graph of y = x1/3
The graphs of y = x3 and y = x1/3 plotted together
Review summary
Review exercise
Composition
Function of a function
Inverses of compositions
Review summary
Can you?
Checklist F.10
Review exercise
Test exercise F.10
Further problems F.10
Programme F.11 Trigonometric and exponential functions
Learning outcomes
Quiz F.11
Introduction
Trigonometric functions
Rotation
The tangent
Period
Amplitude
Phase difference
Inverse trigonometric functions
Trigonometric equations
Equations of the form acosx + bsinx = c
Review summary
Review exercise
Exponential and logarithmic functions
Exponential functions
Review summary
Review exercise
Indicial equations
Odd and even functions
Odd and even parts
Odd and even parts of the exponential function
Limits of functions
The rules of limits
Review summary
Review exercise
Can you?
Checklist F.11
Test exercise F.11
Further problems F.11
Programme F.12 Differentiation
Learning outcomes
Quiz F.12
Gradients
The gradient of a straight-line
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Two straight lines
Two curves
Differentiation of polynomials
Second and higher derivatives and an alternative notation
Review summary
Review exercise
Standard derivatives and rules
Limiting value of
Standard derivatives
Derivative of y = sinx
Derivative of y = cosx
Derivative of y = ex
Derivative of a product of functions
Derivative of a quotient of functions
Derivative of a function of a function
Derivative of y = ax
Review summary
Review exercise
Newton–Raphson iterative method
Notation
Tabular display of results
First approximations
Can you?
Checklist F.12
Review summary
Review exercise
Test exercise F.12
Further problems F.12
Programme F.13 Integration
Learning outcomes
Quiz F.13
Integration
Constant of integration
Standard integrals
Review summary
Review exercise
Integration of polynomial expressions
Functions of a linear function of x
Review summary
Review exercise
Integration by partial fractions
Review summary
Review exercise
Areas under curves
Review summary
Review exercise
Integration as a summation
The area between a curve and an intersecting line
Review summary
Review exercise
Can you?
Checklist F.13
Test exercise F.13
Further problems F.13
PART II
Programme 1 Complex numbers 1
Learning outcomes
Introduction
Ideas and symbols
The symbol j
Quadratic equations
Powers of j
Positive integer powers
Negative integer powers
Complex numbers
Addition and subtraction
Multiplication
Division
Equal complex numbers
Review exercise
Graphical representation of a complex number
Argand diagram
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Review summary
Can you?
Checklist 1
Test exercise 1
Further problems 1
Programme 2 Complex numbers 2
Learning outcomes
Polar-form calculations
Review exercise
Roots of a complex number
Expansions
Expansions of sinnθ nd cosnθ
Expansions for cosnθ and sinnθ
Loci problems
Review summary
Can you?
Checklist 2
Test exercise 2
Further problems 2
Programme 3 Hyperbolic functions
Learning outcomes
Introduction
Graphs of hyperbolic functions
Review exercise
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
Review summary
Can you?
Checklist 3
Determinants of the third order
Test exercise 3
Further problems 3
Programme 4 Determinants
Learning outcomes
Determinants
Evaluation of a third-order determinant
Simultaneous equations in three unknowns
Review exercise
Consistency of a set of equations
Properties of determinants
Review summary
Can you?
Checklist 4
Test exercise 4
Further problems 4
Programme 5 Matrices
Learning outcomes
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Scalar multiplication
Multiplication of two matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Cofactors
Adjoint of a square matrix
Inverse of a square matrix
Product of a square matrix and its inverse
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
Eigenvectors and eigenvalues
Eigenvalues
Eigenvectors
Cayley–Hamilton theorem
Inverse matrices
Raising a matrix to a whole number power
Review summary
Can you?
Checklist 5
Test exercise 5
Further problems 5
Programme 6 Vectors
Learning outcomes
Introduction: scalar and vector quantities
Vector representation
Types of vector
Addition of vectors
The sum of a number of vectors a+b+c+d + . . .
Components of a given vector
Components of a vector in terms of unit vectors
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
Review summary
Can you?
Checklist 6
Test exercise 6
Further problems 6
Programme 7 Differentiation
Learning outcomes
Standard derivatives
Functions of a function
Products
Quotients
Logarithmic differentiation
Review exercise
Implicit functions
Parametric equations
Review summary
Can you?
Checklist 7
Test exercise 7
Further problems 7
Programme 8 Differentiation applications
Learning outcomes
Differentiation of inverse trigonometric functions
Review exercise
Derivatives of inverse hyperbolic functions
Review exercise
Maximum and minimum values
Points of inflexion
Review summary
Can you?
Checklist 8
Test exercise 8
Further problems 8
Programme 9 Tangents, normals and curvature
Learning outcomes
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
Centre of curvature
Review summary
Can you?
Checklist 9
Test exercise 9
Further problems 9
Programme 10 Sequences
Learning outcomes
Functions with integer input
Sequences
Graphs of sequences
Arithmetic sequence
Geometric sequence
Harmonic sequence
Recursive prescriptions
Other sequences
Review summary
Review exercise
Difference equations
Solving difference equations
Second-order, homogeneous equations
Equal roots of the characteristic equation
Review summary
Review exercise
Limits of sequences
Infinity
Limits
Infinite limits
Rules of limits
Multiplication by a constant
Sums and differences
Products and quotients
Indeterminate limits
Review summary
Review exercise
Can you?
Checklist 10
Test exercise 10
Further problems 10
Programme 11 Series 1
Learning outcomes
Series
Arithmetic series
Arithmetic mean
Geometric series
Geometric mean
Series of powers of the natural numbers
Sum of natural numbers
Sum of squares
Sum of cubes
Infinite series
Limiting values
Convergent and divergent series
Tests for convergence
Test 1. A series
Test 2. The comparison test
Test 3: D’Alembert’s ratio test for positive terms
Absolute convergence
Review summary
Can you?
Checklist 11
Test exercise 11
Further problems 11
Programme 12 Series 2
Learning outcomes
Power series
Introduction
Maclaurin’s series
Series for tan x
Standard series
Review summary
The binomial series
Taylor’s series
Limiting values – indeterminate forms
L’Hȏpital’s rule for finding limiting values
Approximate values
Review summary
Can you?
Checklist 12
Test exercise 12
Further problems 12
Programme 13 Curves and curve fitting
Learning outcomes
Introduction
Standard curves
Straight line
Second-degree curves
Third-degree curves
Circle
Ellipse
Hyperbola
Logarithmic curves
Exponential curves
Hyperbolic curves
Trigonometrical curves
Tangent curve
Asymptotes
Determination of an asymptote
Asymptotes parallel to the x- and y-axes
Systematic curve sketching, given the equation of the curve
Symmetry
Intersection with the axes
Change of origin
Asymptotes
Large and small values of x and y
Stationary points
Limitations
Curve fitting
Straight-line law
Graphs of the form y = axn, where a and n are constants
Graphs of the form y = aenx
Method of least squares
Fitting a straight-line graph
Using a spreadsheet
Correlation
Correlation
Measures of correlation
The Pearson product-moment correlation coefficient
Spearman’s rank correlation coefficient
Review summary
Can you?
Checklist 13
Test exercise 13
Further problems 13
Programme 14 Partial differentiation 1
Learning outcomes
Partial differentiation
Review summary
Review exercise
Small increments
Can you?
Checklist 14
Test exercise 14
Review summary
Further problems 14
Programme 15 Partial differentiation 2
Learning outcomes
Partial differentiation
Rate-of-change problems
Change of variables
Can you?
Checklist 15
Review summary
Test exercise 15
Further problems 15
Programme 16 Integration 1
Learning outcomes
Introduction
Standard integrals
Functions of a linear function of x
Integrals of the forms
Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions
Review summary
Can you?
Checklist 16
Test exercise 16
Further problems 16
Programme 17 Integration 2
Learning outcomes
Review summary
Can you?
Checklist 17
Test exercise 17
Further problems 17
Programme 18 Reduction formulas
Learning outcomes
Review summary
Can You?
Checklist 18
Test exercise 18
Further problems 18
Programme 19 Integration applications 1
Learning outcomes
Basic applications
Areas under curves
Definite integrals
Parametric equations
Mean values
Root mean square (rms) value
Review summary
Can you?
Checklist 19
Test exercise 19
Further problems 19
Programme 20 Integration applications 2
Learning outcomes
Introduction
Volume of a solid of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Length of a curve
Parametric equations
Surface of revolution
Parametric equations
Rules of Pappus
Review summary
Can you?
Checklist 20
Test exercise 20
Further problems 20
Programme 21 Integration applications 3
Learning outcomes
Moments of inertia
Radius of gyration
Parallel axes theorem
Perpendicular axes theorem (for thin plates)
Useful standard results
Second moment of area
Composite figures
Centre of pressure
Pressure at a point P, depth z below the surface of a liquid
Total thrust on a vertical plate immersed in liquid
Depth of the centre of pressure
Can you?
Checklist 21
Review summary
Test exercise 21
Further problems 21
Programme 22 Approximate integration
Learning outcomes
Introduction
Approximate integration
Method 1: By series
Method 2: By Simpson’s rule
Proof of Simpson’s rule
Review summary
Can you?
Checklist 22
Test exercise 22
Further problems 22
Programme 23 Polar coordinate systems
Learning outcomes
Introduction to polar coordinates
Polar curves
Standard polar curves
Applications
Review summary
Can you?
Checklist 23
Test exercise 23
Further problems 23
Programme 24 Multiple integrals
Learning outcomes
Summation in two directions
Double integrals
Triple integrals
Applications
Review exercise
Alternative notation
Determination of areas by multiple integrals
Determination of volumes by multiple integrals
Review summary
Can you?
Checklist 24
Test exercise 24
Further problems 24
Programme 25 First-order differential equations
Learning outcomes
Introduction
Formation of differential equations
Solution of differential equations
Method 1: By direct integration
Method 2: By separating the variables
Review exercise
Method 3: Homogeneous equations – by substituting y = vx
Review exercise
Method 4: Linear equations – use of integrating factor
Review exercise
Bernoulli’s equation
Review summary
Can you?
Checklist 25
Further problems 25
Programme 26 Second-order differential equations
Learning outcomes
Homogeneous equations
Test exercise 25
Review exercise
Inhomogeneous equations
Particular solution
Can you?
Checklist 26
Review summary
Test exercise 26
Further problems 26
Programme 27 Introduction to Laplace transforms
Learning outcomes
The Laplace transform
The inverse Laplace transform
Table of Laplace transforms
Review summary
Review exercise
Laplace transform of a derivative
Two properties of Laplace transforms
Table of Laplace transforms
Review summary
Review exercise
Generating new transforms
Laplace transforms of higher derivatives
Table of Laplace transforms
Linear, constant-coefficient, inhomogeneous differential equations
Review summary
Review exercise
Can you?
Checklist 27
Test exercise 27
Further problems 27
Programme 28 Data handling and statistics
Learning outcomes
Introduction
Arrangement of data
Tally diagram
Grouped data
Grouping with continuous data
Relative frequency
Rounding off data
Class boundaries
Histograms
Frequency histogram
Relative frequency histogram
Measures of central tendency
Mean
Coding for calculating the mean
Decoding
Coding with a grouped frequency distribution
Mode
Mode with grouped data
Median
Median with grouped data
Measures of dispersion
Mean deviation
Range
Standard deviation
Alternative formula for the standard deviation
Distribution curves
Frequency polygons
Frequency curves
Normal distribution curve
Standardized normal curve
Review summary
Can you?
Checklist 28
Test exercise 28
Further problems 28
Programme 29 Probability
Learning outcomes
Probability
Random experiments
Events
Sequences of random experiments
Combining events
Events and probabilities
Probability
Assigning probabilities
Review summary
Probabilities of combined events
Or
Non-mutually exclusive events
And – multiplication rule
Dependent events
Independent events
Probability trees
Conditional probability
Review summary
Probability distributions
Random variables
Expectation
Variance and standard deviation
Bernoulli trials
Binomial probability distribution
Expectation and standard deviation
The Poisson probability distribution
Binomial and Poisson compared
Continuous probability distributions
Normal distribution curve (or normal curve)
Standard normal curve
Review summary
Can you?
Checklist 29
Test exercise 29
Further problems 29
Answers
Part I
Quiz F.1 (page 4)
Test exercise F.1 (page 59)
Further problems F.1 (page 60)
Quiz F.2 (page 64)
Test exercise F.2 (page 94)
Further problems F.2 (page 95)
Quiz F.3 (page 98)
Test exercise F.3 (page 121)
Further problems F.3 (page 121)
Quiz F.4 (page 124)
Test exercise F.4 (page 154)
Further problems F.4 (page 154)
Quiz F.5 (page 158)
Test exercise F.5 (page 1704)
Further problems F.5 (page 170)
Quiz F.6 (page 174)
Test exercise F.6 (page 185)
Further problems F.6 (page 185)
Quiz F.7 (page 188)
Test exercise F.7 (page 212)
Further problems F.7 (page 212)
Quiz F.8 (page 216)
Test exercise F.8 (page 232)
Further problems F.8 (page 232)
Quiz F.9 (page 236)
Test exercise F.9 (page 256)
Further problems F.9 (page 257)
Quiz F.10 (page 260)
Test exercise F.10 (page 277)
Further problems F.10 (page 277)
Quiz F.11 (page 280)
Test exercise F.11 (page 305)
Further problems F.11 (page 306)
Quiz F.12 (page 310)
Test exercise F.12 (page 345)
Further problems F.12 (page 346)
Quiz F.13 (page 348)
Test exercise F.13 (page 374)
Further problems F.13 (page 375)
Part II
Test exercise 1 (page 403)
Further problems 1 (page 404)
Test exercise 2 (page 428)
Further problems 2 (page 429)
Test exercise 3 (page 451)
Further problems 3 (page 452)
Test exercise 4 (page 480)
Further problems 4 (page 481)
Test exercise 5 (page 515)
Further problems 5 (page 516)
Test exercise 6 (page 541)
Further problems 6 (page 542)
Test exercise 7 (page 560)
Further problems 7 (page 561)
Test exercise 8 (page 582)
Further problems 8 (page 582)
Test exercise 9 (page 604)
Further problems 9 (page 605)
Test exercise 10 (page 638)
Further problems 10 (page 639)
Test exercise 11 (page 663)
Further problems 11 (page 664)
Test exercise 12 (page 689)
Further problems 12 (page 689)
Test exercise 13 (page 732)
Further problems 13 (page 733)
Test exercise 14 (page 754)
Further problems 14 (page 755)
Test exercise 15 (page 771)
Further problems 15 (page 771)
Test exercise 16 (page 797)
Further problems 16 (page 797)
Test exercise 17 (page 826)
Further problems 17 (page 826)
Test exercise 18 (page 838)
Further problems 18 (page 839)
Test exercise 19 (page 856)
Further problems 19 (page 857)
Test exercise 20 (page 878)
Further problems 20 (page 878)
Test exercise 21 (page 907)
Further problems 21 (page 908)
Test exercise 22 (page 926)
Further problems 22 (page 927)
Test exercise 23 (page 949)
Further problems 23 (page 950)
Test exercise 24 (page 974)
Further problems 24 (page 975)
Test exercise 25 (page 1009)
Further problems 25 (page 1009)
Test exercise 26 (page 1034)
Further problems 26 (page 1034)
Test exercise 27 (page 1052)
Further problems 27 (page 1052)
Test exercise 28 (page 1081)
Further problems 28 (page 1082)
Test exercise 29 (page 1122)
Further problems 29 (page 1123)
Index