Elementary Analysis

Preface1. Number Systems Number Scales Denary Scales Binary Scales Rational Numbers Geometrical Representation Nested Intervals Recurring Decimals Irrational Numbers Square Roots Real Numbers Geometrical Representation Binary Operations Axioms Deductive Methods Order Inequalities2. Sets Definition of a Set Equal Sets Subsets Empty Set Inclusion Operations on Sets Algebra of Sets Union and Intersection Tables De Morgan's Rule Counting Sets Applications3. Vectors and Congruences Directed Lengths Ordered Pairs Coordinates Vectors-Addition and Subtraction Multiplication by a Scalar Mid-points The Section Formula Homothetic Figures Three Dimensions Congruences4. Functions Relations Graphical Representation Mapping Functions Inverse Functions Locus and Graph Equation and Graph5. Exponential and Logarithmic Function Notation The Exponential Function Index Laws The Logarithmic Function An Inverse Function The Laws of Logarithms Logarithms to Base 10 Logarithms to Any Base Napierian Logarithms The Slide Rule Graphical Solution of Equations6. The Straight Line Projections The Distance Formula Gradient Positive and Negative Gradients Equation of the Straight Line Perpendicular Lines Distance of a Point from a Line Intersection of Two Lines7. The Linear Function Polynomials The Linear Polynomial ax + b Identity Ordered Triples in 3-Space Parametric Equations of a Line in 3-Space Linear Polynomial in Two Variables The Sign of ax + by + c Linear Programming8. The Quadratic Function Graph of Quadratic Function The Sign of(x - a)(x - b) Graphical Solution of Inequalities The Function y = ax2 + bx + c The Quadratic Equation Roots and Coefficients of a Quadratic Equation Complex Numbers The Algebra of Complex Numbers9. Sequences, Series, Limits Sequences The Arithmetic Sequence The Arithmetic Mean The Geometric Sequence The Geometric Mean The Geometric Series Limit of a Sequence Limit of a Function Formal Definition of a Limit10. Mathematical Induction and Applications The Principle of Mathematical Induction Proof of the Principle A Warning Summation of Series The Method of Differences The Sigma Notation The Limit Sum of a Series11. Differentiation Gradient of a Chord Gradient of a Tangent Gradient at a Point on a Curve Gradient Formula Derived Function Differentiation of Powers of x The Derivative of xn The Chain Rule for Differentiation Implicit Differentiation12. Applications of Differentiation and the Inverse Process Rate of Change Stationary Values of a Function Small Changes Differential Equations Velocity and Acceleration13. Further Differentiation and the Applications Differentiation of a Product Differentiation of a Quotient Implicit Differentiation The Second Derivative Acceleration The Leibniz Notation14. Integration and Applications An Important Limit The Definite Integral An Example of an Integral Area by Integration Area Beneath a Curve A Function Defined as a Definite Integral The Fundamental Theorem Square Bracket Notation Applications of Integration Volume of a Solid of Revolution Mean Values Centre of Mass Numerical Integration The Trapezium Rule Simpson's RuleAnswersIndex