Elementary Analysis

PrefaceIntroduction: A summary of Basic Ideas from Volume 115. Trigonometric Functions of the General Angle Angles Vectors Trigonometrical Ratios Special Angles The General Angle Periodic Functions Graphs General Formula, cos2¿+ in2¿=1 Inequalities16. Functions of Compound Angles Unit Vectors Compound Angles Multiple Angles Sums and Products Even and Odd Functions Parameters Inverse Functions The Circle17. Differentiation and Integration of the Trigonometric Functions Gradient of the Sine Function Radians and Degrees Differentiation of a Vector Motion in a Circle Inequalities Derivative of Tan x Integration Substitution Polar Coordinates Area of a Sector18. Applications of the Trigonometric Functions Dot Product of Vectors Compound Angle Formula Triangle Formula and Problems Simple Harmonic Motion Approximations Three Dimensions Projections19. Polynomials The Remainder Theorem The Factor Theorem The Identity Theorem Synthetic Division Evaluation of a Polynomial Repeated Factors20. Symmetric Functions of the Roots of a Polynomial Equation Complex Roots Symmetric Functions Sums of Powers of Roots Approximation for the Greatest Root21. The Binomial Theorem The Expansion of (a + x)n The Binomial Theorem Approximations Proof by Induction22. The Binomial Series Limit Sum The Binomial Series Partial Fractions Expansions Approximations23. Numerical Solution of Equations Linear Interpolation Newton's Method Trigonometric Equations Recurrence Relations and Iterative Methods Expansions24. Analytical Geometry Vectors-A Recapitulation Coordinates as Vectors Transformations Change of Axes Polar Coordinates The Parabola, Hyperbola, Circle and Ellipse Orthogonal Projection25. The Argand Diagram Lengths in the Argand Diagram Geometrical Illustrations Inversion Complex Equation of a Line and Circle Similarity Conformal Transformations The Equation xn=126. Matrices and Determinants Successive Transformations Definition of a Matrix Addition Product Applications Unit Matrix Solution of Linear Equations Elementary Row Operations The Inverse Matrix Determinants Cofactors Singular Matrices Quadratic Forms27. The Exponential and Logarithmic Functions Limit of a Function Continuous Functions Definite Integrals The Mean Value Theorem for Integrals The Fundamental Theorem The Logarithmic Function The Logarithmic Series The Exponential Function and Series Logarithmic Differentiation Integration of Rational, Algebraic and Trigonometric Functions Integration by Parts28. Introduction to Probability Theory The Concept of Probability A Sample Space Alternative or Disjoint Events Combined Events Permutations Combinations Set Theory and Probability Independent Events Tree Diagrams Binomial Distribution The Value of an ExpectationAnswersIndex