Maths for CAPE® Examinations Volume 1

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Description

Contents

Reviews

Language

English

ISBN

9780230465756

Cover

Title Page

Copyright

Contents

INTRODUCTION

MATHEMATICAL MODELLING

MODULE 1 BASIC ALGEBRA AND FUNCTIONS

CHAPTER 1 REASONING AND LOGIC

Notation

Simple statement

Negation

Truth tables

Compound statements

Connectives

Conjunction

Disjunction (‘or’)

Conditional statements

Interpretation of p → q

The contrapositive

Converse

Inverse

Equivalent propositions

Biconditional statements

Tautology and contradiction

Algebra of propositions

CHAPTER 2 THE REAL NUMBER SYSTEM

Subsets of rational numbers

Real numbers

Operations

Binary operations

Closure

Commutativity

Associativity

Distributivity

Identity

Inverse

Constructing simple proofs in mathematics

Proof by exhaustion

Direct proof

Proof by contradiction

Proof by counter example

CHAPTER 3 PRINCIPLE OF MATHEMATICAL INDUCTION

Sequences and series

Finding the general term of a series

Sigma notation

Expansion of a series

Standard results

Summation results

Mathematical induction

Divisibility tests and mathematical induction

CHAPTER 4 POLYNOMIALS

Review of polynomials

Degree or order of polynomials

Algebra of polynomials

Evaluating polynomials

Rational expressions

Comparing polynomials

Remainder theorem

The factor theorem

Factorising polynomials and solving equations

Factorising xn - yn

CHAPTER 5 INDICES, SURDS AND LOGARITHMS

Indices

Laws of indices

Surds

Rules of surds

Simplifying surds

Conjugate surds

Rationalising the denominator

Exponential functions

Graphs of exponential functions

The number e

Exponential equations

Logarithmic functions

Converting exponential expressions to logarithmic expressions

Changing logarithms to exponents using the definition of logarithm

Properties of logarithms

Solving logarithmic equations

Equations involving exponents

Change of base formula (change to base b from base a)

Logarithms and exponents in simultaneous equations

Application problems

Compound interest

Continuous compound interest

CHAPTER 6 FUNCTIONS

Relations and functions

Describing a function

The vertical line test

One-to-one function (injective function)

Onto function (surjective function)

Bijective functions

Inverse functions

Graphs of inverse functions

Odd and even functions

Odd functions

Even functions

Periodic functions

The modulus function

Graph of the modulus function

Composite functions

Relationship between inverse functions

Increasing and decreasing functions

Increasing functions

Decreasing functions

Transformations of graphs

Vertical translation

Horizontal translation

Horizontal stretch

Vertical stretch

Reflection in the x-axis

Reflection in the y-axis

Graphs of simple rational functions

Piecewise defined functions

CHAPTER 7 CUBIC POLYNOMIALS

Review: Roots of a quadratic and the coefficient of the quadratic

Cubic equations

Notation

Finding a3 + ß3 + y3, using a formula

Finding a cubic equation, given the roots of the equation

CHAPTER 8 INEQUALITIES AND THE MODULUS FUNCTION

Theorems of inequalities

Quadratic inequalities

Sign table

Rational functions and inequalities

General results about the absolute value function

Square root of x2

The triangle inequality

Applications problems for inequalities

MODULE 1 TESTS

MODULE 2 TRIGONOMETRY AND PLANE GEOMETRY

CHAPTER 9 TRIGONOMETRY

Inverse trigonometric functions and graphs

Inverse sine function

Inverse cosine function

Inverse tangent function

Solving simple trigonometric equations

Graphical solution of sin x = k

Graphical solution of cos x = k

Graphical solution of tan x = k

Trigonometrical identities

Reciprocal identities

Pythagorean identities

Proving identities

Solving trigonometric equations

Further trigonometrical identities

Expansion of sin (A ± B)

Expansion of cos (A ± B)

Expansion of tan (A + B)

Double-angle formulae

Half-angle formulae

Proving identities using the addition theorems and the double-angle formulae

The form a cos Ɵ + b sin Ɵ

Solving equations of the form a cos Ɵ + b sin Ɵ = c

Equations involving double-angle or half-angle formulae

Products as sums and differences

Converting sums and differences to products

Solving equations using the sums and differences as products

CHAPTER 10 COORDINATE GEOMETRY

Review of coordinate geometry

The equation of a circle

Equation of a circle with centre (a, b) and radius r

General equation of the circle

Intersection of a line and a circle

Intersection of two circles

Intersection of two curves

Parametric representation of a curve

Cartesian equation of a curve given its parametric form

Parametric equations in trigonometric form

Parametric equations of a circle

Conic sections

Ellipses

Equation of an ellipse

Equation of an ellipse with centre (h, k)

Focus–directrix property of an ellipse

Parametric equations of ellipses

Equations of tangents and normals to an ellipse

Parabolas

Equation of a parabola

Parametric equations of parabolas

Equations of tangents and normals to a parabola

CHAPTER 11 VECTORS IN THREE DIMENSIONS (R3)

Vectors in 3D

Plotting a point in three dimensions

Algebra of vectors

Addition of vectors

Subtraction of vectors

Multiplication by a scalar

Equality of vectors

Magnitude of a vector

Displacement vectors

Unit vectors

Special unit vectors

Scalar product or dot product

Properties of the scalar product

Angle between two vectors

Perpendicular and parallel vectors

Perpendicular vectors

Parallel vectors

Equation of a line

Finding the equation of a line given a point on a line and the direction of the line

Finding the equation of a line given two points on the line

Vector equation of a line

Parametric equation of a line

Cartesian equation of a line

Finding the angle between two lines, given the equations of the lines

Skew lines

Equation of a plane

Equation of a plane, given the distance from the origin to the plane and a unit vector perpendicular to the plane

Equation of a plane, given a point on the plane and a normal to the plane

Cartesian equation of a plane

MODULE 2 TESTS

MODULE 3 CALCULUS I

CHAPTER 12 LIMITS AND CONTINUITY

Limits

The existence of a limit

Limit laws

Evaluating limits

Direct substitution

Factorising method

Conjugate method

Tending to infinity

Limits at infinity

Special limits

Continuity

Types of discontinuity

Infinite discontinuity

Point discontinuity

Jump discontinuity

Removable and non-removable discontinuity

CHAPTER 13 DIFFERENTIATION 1

Differentiation

The difference quotient

Existence of a derivative

Notation for derivatives

Interpretations of derivatives

Finding derivatives using first principles

Differentiation of ag(x) where a is a constant

Differentiation of sums and differences of functions

First principle and sums and differences of functions of x

Rate of change

Chain rule

Product rule

Quotient rule

Differentiation of trigonometric functions

Higher derivatives

CHAPTER 14 APPLICATIONS OF DIFFERENTIATION

Tangents and normals

Equations of tangents and normals

Increasing and decreasing functions

Stationary points/second derivatives

Maximum and minimum values

Stationary points

Classification of turning points

First derivative test

Second derivative test

Inflexion points

Practical maximum and minimum problems

Parametric differentiation

Rate of change

Curve sketching

Polynomials, rational functions, trigonometric functions

Graph of a polynomial

Graphs of functions of the form f(x) = xn where n is an even integer

Graphs of functions of the form f(x) = xn where n is an odd integer greater than 1

Graphs of polynomials

Zeros of a polynomial

Graphing functions

Graphing functions with a table of values

Solving simultaneous equations graphically

Solving inequalities graphically

Review of trigonometry

Sine, cosine and tangent of 45°, 30° and 60°

Graph of cosec x

Graph of sec x

Graph of cot x

Properties and graphs of trigonometric functions

Transformations of trigonometric functions

y = a sin (bx) + c and y = a cos (bx) + c

y = a tan (bx) + c

Graphs of rational functions

Vertical asymptotes

Horizontal asymptotes

Sketching graphs of rational functions

Shape of a curve for large values of the independent variable

CHAPTER 15 INTEGRATION

Anti-derivatives (integrations)

The constant of integration

Integrals of the form axn

Integration theorems

Integration of polynomial functions

Integration of a function involving a linear factor

Integration of trigonometric functions

Integration of more trigonometric functions

Integrating sin2 x and cos2 x

Integration of products of sines and cosines

The definite integral

Integration by substitution

Substituting with limits

The equation of a curve

CHAPTER 16 APPLICATIONS OF INTEGRATION

Approximating the area under a curve, using rectangles

Estimating the area under a curve using n rectangles of equal width

Using integration to find the area under a curve

Area between two curves

Area below the x-axis

Area between the curve and the y-axis

Volume of solids of revolution

Rotation about the x-axis

Rotation about the y-axis

Volume generated by the region bounded by two curves

CHAPTER 17 DIFFERENTIAL EQUATIONS

Families of curves

Classifying differential equations

Linear versus non-linear differential equations

Practical applications of differential equations

First order differential equations

Solutions of variable-separable differential equations

Modelling problems

Second order differential equations

MODULE 3 TESTS

UNIT 1—MULTIPLE CHOICE TESTS

INDEX

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