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About this book

The best-selling introductory mathematics textbook for students on science and engineering degree and pre-degree courses. Sales stand at more than half a million copies world-wide.

Its unique programmed approach takes students through the mathematics they need in a step-by-step fashion with a wealth of examples and exercises. The text demands that students engage with it by asking them to complete steps that they should be able to manage from previous examples or knowledge they have acquired, while carefully introducing new steps. By working with the authors through the examples, students become proficient as they go. By the time they come to trying examples on their own, confidence is high.

Aimed at undergraduates on Foundation and First Year degree programmes in all Engineering disciplines and Science. The Foundation section covers mathematics from GCSE onwards to allow for revision and gap-filling, and so means the book can be used for a range of abilities and all levels of access.

Table of Contents

Foundation topics

Frontmatter

1. F.1 Arithmetic

Abstract
When you have completed this Programme you will be able to: Carry out the basic rules of arithmetic with integers Check the result of a calculation making use of rounding Write a whole as a product of prime numbers Find the highest common factor and lowest common multiple of two whole numbers Manipulate fractions, ratios and percentages Manipulate decimal numbers Manipulate powers Use standard or preferred standard form and complete a calculation to the required level of accuracy Understand the construction of various number systems and convert from one number system to another.
K.A. Stroud, Dexter Booth

2. F.2 Introduction to algebra

Abstract
Use alphabetic symbols to supplement the numerals and to combine these symbols using all the operations of arithmetic Simplify algebraic expressions by collecting like terms and by abstracting common factors from similar terms Remove brackets and so obtain alternative algebraic expressions Manipulate expressions involving powers and logarithms Multiply and divide algebraic expressions Manipulate algebraic fractions Factorize algebraic expressions using standard factorizations Factorize quadratic algebraic expressions
K.A. Stroud, Dexter Booth

3. F.3 Expressions and equations

Abstract
When you have completed this Programme you will be able to: Numerically evaluate an algebraic expression by substituting numbers for variables Recognize the different types of equation Evaluate an independent variable Change the subject of an equation by transposition Evaluate polynomial expressions by ‘nesting’ Use the remainder and factor theorems to factorize polynomials Factorize fourth-order polynomials
K.A. Stroud, Dexter Booth

4. F.4 Graphs

Abstract
When you have completed this Programme you will be able to: Construct a collection of ordered pairs of numbers from an equation Plot points associated with ordered pairs of numbers against Cartesian axes and generate graphs Appreciate the existence of asymptotes to curves and discontinuities Use an electronic spreadsheet to draw Cartesian graphs of equations Describe regions of the x–y plane that are represented by inequalities Draw graphs of and algebraically manipulate the absolute value or modulus function
K.A. Stroud, Dexter Booth

5. F.5 Linear equations

Abstract
When you have completed this Programme you will be able to: Solve any linear equation Solve simultaneous linear equations in two unknowns Solve simultaneous linear equations in three unknowns If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

6. F.6 Polynomial equations

Abstract
When you have completed this Programme you will be able to: Solve quadratic equations by factors, by completing the square and by formula Solve cubic equations with at least one linear factor Solve fourth-order equations with at least two linear factors If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

7. F.7 Binomials

Abstract
When you have completed this Programme you will be able to: Define n! and recognize that there are n! different arrangements of n different items Evaluate n! for moderately sized n using a calculator Manipulate expressions containing factorials Recognize simple properties of combinatorial coefficients Construct Pascal’s triangle Write down the binomial expansion for natural number powers Obtain specific terms in the binomial expansion using the general term Use the sigma notation Recognize and reproduce the expansion for ex where e is the exponential number If you already feel confident about these why not try the short quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

8. F.8 Partial fractions

Abstract
When you have completed this Programme you will be able to: Factorize the denominator of an algebraic fraction into its factors Separate an algebraic fraction into its partial fractions Recognize the rules of partial fractions If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

1. F.9 Trigonometry

Abstract
Convert angles measured in degrees, minutes and seconds into decimal degrees Convert degrees into radians and vice versa Use a calculator to determine the values of trigonometric ratios for any acute angle Verify trigonometric identities If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

10. F.10 Functions

Abstract
When you have completed this Programme you will be able to: Identify a function as a rule and recognize rules that are not functions Determine the domain and range of a function Construct the inverse of a function and draw its graph Construct compositions of functions and de-construct them into their component functions If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

11. F.11 Trigonometric and exponential functions

Abstract
When you have completed this Programme you will be able to: Develop the trigonometric functions from the trigonometric ratios Find the period, amplitude and phase of a periodic function Distinguish between the inverse of a trigonometric function and the inverse trigonometric function Solve trigonometric equations using the inverse trigonometric functions and trigonometric identities Recognize that the exponential function and the natural logarithmic function are mutual inverses and solve indicial and logarithmic equations Find the even and odd parts of a function when they exist Construct the hyperbolic functions from the odd and even parts of the exponential function Evaluate limits of simple functions If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

12. F.12 Arithmetic

Abstract
When you have completed this Programme you will be able to: Determine the gradient of a straight-line graph Evaluate from first principles the gradient at a point on a quadratic curve Differentiate powers of x and polynomials Evaluate second derivatives and use tables of standard derivatives Differentiate products and quotients of expressions Differentiate using the chain rule for a ‘function of a function’ Use the Newton–Raphson method to obtain a numerical solution to an equation If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

13. F.13 Integration

Abstract
When you have completed this Programme you will be able to: Appreciate that integration is the reverse process of differentiation Recognize the need for a constant of integration Evaluate indefinite integrals of standard forms Evaluate indefinite integrals of polynomials Evaluate indefinite integrals of ‘functions of a linear function of x’ Integrate by partial fractions Appreciate that a definite integral is a measure of the area under a curve Evaluate definite integrals of standard forms Use the definite integral to find areas between a curve and the horizontal axis Use the definite integral to find areas between a curve and a given straight line If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
K.A. Stroud, Dexter Booth

Foundation topics

Frontmatter

14. Complex numbers 1

Abstract
When you have completed this Programme you will be able to: Add, subtract and multiply complex numbers Find the complex conjugate of a complex number Divide complex numbers State the conditions for the equality of two complex numbers Draw complex numbers and recognize the parallelogram law of addition Convert a complex number from Cartesian to polar form and vice versa Write a complex number in its exponential form Obtain the logarithm of a complex number
K.A. Stroud, Dexter Booth

15. Complex numbers 2

Abstract
When you have completed this Programme you will be able to: Use the shorthand form for a complex number in polar form Write complex numbers in polar form using negative angles Multiply and divide complex numbers in polar form Use DeMoivre’s theorem Find the roots of a complex number Demonstrate trigonometric identities of multiple angles using complex numbers Solve loci problems using complex numbers
K.A. Stroud, Dexter Booth

16. Hyperbolic functions

Abstract
When you have completed this Programme you will be able to: Define the hyperbolic functions in terms of the exponential function Express the hyperbolic functions as power series Recognize the graphs of the hyperbolic functions Evaluate hyperbolic functions and their inverses Determine the logarithmic form of the inverse hyperbolic functions Prove hyperbolic trigonometric identities Understand the relationship between the circular and the hyperbolic trigonometric functions
K.A. Stroud, Dexter Booth

17. Determinants

Abstract
When you have completed this Programme you will be able to: Expand a 2 × 2 determinant Solve pairs of simultaneous linear equations in two variables using 2 × 2 determinants Expand a 3 × 3 determinant Solve three simultaneous linear equations in three variables using 3 × 3 determinants Determine the consistency of sets of simultaneous linear equations Use the properties of determinants to solve equations written in determinant form
K.A. Stroud, Dexter Booth

18. Matrices

Abstract
When you have completed this Programme you will be able to: Define a matrix Understand what is meant by the equality of two matrices Add and subtract two matrices Multiply a matrix by a scalar and multiply two matrices together Obtain the transpose of a matrix Recognize special types of matrix Obtain the determinant, cofactors and adjoint of a square matrix Obtain the inverse of a non-singular matrix Use matrices to solve a set of linear equations using inverse matrices Use the Gaussian elimination method to solve a set of linear equations Evaluate eigenvalues and eigenvectors
K.A. Stroud, Dexter Booth

19. Vectors

Abstract
When you have completed this Programme you will be able to: Define a vector Represent a vector by a directed straight line Add vectors Write a vector in terms of component vectors Write a vector in terms of component unit vectors Set up a coordinate system for representing vectors Obtain the direction cosines of a vector Calculate the scalar product of two vectors Calculate the vector product of two vectors Determine the angle between two vectors Evaluate the direction ratios of a vector
K.A. Stroud, Dexter Booth

20. Differentiation

Abstract
When you have completed this Programme you will be able to: Differentiate by using a list of standard derivatives Apply the chain rule Apply the product and quotient rules Perform logarithmic differentiation Differentiate implicit functions Differentiate parametric equations
K.A. Stroud, Dexter Booth

21. Differentiation applications

Abstract
When you have completed this Programme you will be able to: Differentiate the inverse trigonometric functions Differentiate the inverse hyperbolic functions Identify and locate a maximum and a minimum Identify and locate a point of inflexion
K.A. Stroud, Dexter Booth

22. Tangents, normals and curvature

Abstract
When you have completed this Programme you will be able to: Evaluate the gradient of a straight line Recognize the relationship satisfied by two mutually perpendicular straight lines Derive the equations of a tangent and a normal to a curve Evaluate the curvature and radius of curvature at a point on a curve Locate the centre of curvature for a point on a curve
K.A. Stroud, Dexter Booth

23. Sequences

Abstract
When you have completed this Programme you will be able to: Recognize a sequence as a function Plot the graphs of sequences Recognize specific types of sequence Recognize arithmetic, geometric and harmonic sequences Convert the descriptive prescription of the output from arithmetic and geometric sequences into a recursive description and recognize the importance of initial terms Generate the recursive prescription of a sequence from a given sequence of numbers Determine the order and generate the terms of a difference equation Obtain the solution to an homogeneous, linear difference equation Derive the limit of a sequence using the rules of limits
K.A. Stroud, Dexter Booth

24. Series 1

Abstract
When you have completed this Programme you will be able to: Manipulate arithmetic and geometric series Manipulate series of powers of the natural numbers Determine the limiting values of arithmetic and geometric series Determine the limiting values of simple indeterminate forms Apply various convergence tests to infinite series Distinguish between absolute and conditional convergence
K.A. Stroud, Dexter Booth

25. Series 2

Abstract
When you have completed this Programme you will be able to: Derive the power series for sin x Use Maclaurin’s series to derive series of common functions Use Maclaurin’s series to derive the binomial series Derive power series expansions of miscellaneous functions using known expansions of common functions Use power series expansions in numerical approximations Extend Maclaurins series to Taylor’s series
K.A. Stroud, Dexter Booth

26. Curves and curve fitting

Abstract
When you have completed this Programme you will be able to: Draw sketch graphs of standard curves Determine the equations of asymptotes parallel to the x- and y-axes Sketch the graphs of curves with asymptotes, stationary points and other features Fit graphs to data using the ‘straight-line’ forms Fit graphs to data using the method of least squares Understand what is meant by the correlation of two variables Calculate the Pearson product-moment coefficient Calculate Spearman’s rank correlation coefficient
K.A. Stroud, Dexter Booth

27. Partial differentiation 1

Abstract
When you have completed this Programme you will be able to: Find the first partial derivatives of a function of two real variables Find second-order partial derivatives of a function of two real variables Calculate errors using partial differentiation
K.A. Stroud, Dexter Booth

28. Partial differentiation 2

Abstract
When you have completed this Programme you will be able to: Derive the first- and second-order partial derivatives of a function of two real variables Apply partial differentiation to rate-of-change problems Apply partial differentiation to change-of-variable problems
K.A. Stroud, Dexter Booth

29. Integration 1

Abstract
When you have completed this Programme you will be able to: Integrate standard expressions using a table of standard forms Integrate functions of a linear form Integrate by parts Integrate by partial fractions Integrate trigonometric functions
K.A. Stroud, Dexter Booth

30. Integration 2

Abstract
This is the first of nine standard results which we are going to establish in this Programme. They are useful to remember since the standard results will remove the need to work each example in detail, as you will see.
K.A. Stroud, Dexter Booth

31. Reduction formulas

Abstract
When you have completed this Programme you will be able to: Integrate by parts and generate a reduction formula Integrate by parts using a reduction formula
K.A. Stroud, Dexter Booth

32. Integration applications 1

Abstract
When you have completed this Programme you will be able to: Evaluate the area beneath a curve Evaluate the area beneath a curve given in parametric form Determine the mean value of a function between two points Evaluate the root mean square (rms) value of a function
K.A. Stroud, Dexter Booth

33. Integration applications 2

Abstract
When you have completed this Programme you will be able to: Calculate volumes of revolution Locate the centroid of a plane figure Locate the centre of gravity of a solid of revolution Determine the lengths of curves Determine the lengths of curves given by parametric equations Calculate surfaces of revolution Calculate surfaces of revolution using parametric equations Use the two rules of Pappus
K.A. Stroud, Dexter Booth

34. Integration applications 3

Abstract
When you have completed this Programme you will be able to: Determine moments of inertia Determine the radius of gyration Use the parallel axes theorem Use the perpendicular axes theorem for thin plates Determine moments of inertia using standard results Determine second moments of area Determine centres of pressure
K.A. Stroud, Dexter Booth

35. Approximate integration

Abstract
When you have completed this Programme you will be able to: Recognize when an integral cannot be evaluated directly Approximate integrals using series expansions Use Simpson’s rule to approximate the area beneath a curve 902
K.A. Stroud, Dexter Booth

36. Polar coordinate systems

Abstract
When you have completed this Programme you will be able to: Convert expressions from Cartesian coordinates to polar coordinates and vice versa Plot the graphs of polar curves Recognize equations of standard polar curves Evaluate the areas enclosed by polar curves Evaluate the volumes of revolution generated by polar curves Evaluate the lengths of polar curves Evaluate the surface of revolution generated by polar curves
K.A. Stroud, Dexter Booth

37. Multiple integrals

Abstract
When you have completed this Programme you will be able to: Determine the area of a rectangle using a double integral Evaluate double integrals over general areas Evaluate triple integrals over general volumes Apply double integrals to find areas and second moments of area Apply triple integrals to find volumes
K.A. Stroud, Dexter Booth

38. First-order differential equations

Abstract
When you have completed this Programme you will be able to: Recognize the order of a differential equation Appreciate that a differential equation of order n can be derived from a function containing n arbitrary constants Solve certain first-order differential equations by direct integration Solve certain first-order differential equations by separating the variables Solve certain first-order homogeneous differential equations by an appropriate substitution Solve certain first-order differential equations by using an integrating factor Solve Bernoulli’s equation
K.A. Stroud, Dexter Booth

39. Second-order differential equations

Abstract
When you have completed this Programme you will be able to: Use the auxiliary equation to solve certain second-order homogeneous equations Use the complementary function and the particular integral to solve certain second-order inhomogeneous equations
K.A. Stroud, Dexter Booth

40. Introduction to Laplace transforms

Abstract
When you have completed this Programme you will be able to: Derive the Laplace transform of an expression by using the integral definition Obtain inverse Laplace transforms with the help of a Table of Laplace transforms Derive the Laplace transform of the derivative of an expression Solve first-order, constant-coefficient, inhomogeneous differential equations using the Laplace transform Derive further Laplace transforms from known transforms Use the Laplace transform to obtain the solution to linear, constant-coefficient, inhomogeneous differential equations of second and higher order
K.A. Stroud, Dexter Booth

41. Data handling and statistics

Abstract
When you have completed this Programme you will be able to: Distinguish between discrete and continuous data Construct frequency and relative frequency tables for grouped and ungrouped discrete data Determine class boundaries, class intervals and central values for discrete and continuous data Construct a histogram and a frequency polygon Determine the mean, median and mode of grouped and ungrouped data Determine the range, variance and standard deviation of discrete data Measure the dispersion of data using the normal and standard normal curves
K.A. Stroud, Dexter Booth

42. Probability

Abstract
When you have completed this Programme you will be able to: Understand what is meant by a random experiment Distinguish between the result and an outcome of a random experiment Recognize that, whilst outcomes are mutually exclusive, events may not be Combine events and construct an outcome tree for a sequence of random experiments
K.A. Stroud, Dexter Booth
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