 \$23.55

# Maths for CAPE® Examinations Volume 1

By Dipchand Bahall
US\$ 23.55
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Book Description

• Cover
• Title Page
• 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
• 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 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
• 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
• 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
• 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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