Oxford IB Diploma Programme: Mathematics Higher Level: Calculus Course Companion

1. Patterns to infinity ; 1.1 From limits of sequences to limits of functions ; 1.2 Squeeze theorem and the algebra of limits of convergent sequences ; 1.3 Divergent sequences: indeterminate forms and evaluation of limits ; 1.4 From limits of sequences to limits of functions ; 2. Smoothness in mathematics ; 2.1 Continuity and differentiability on an interval ; 2.2 Theorems about continuous functions ; 2.3 Differentiable functions: Rolle's Theorem and Mean Value Theorem ; 2.4 Limits at a point, indeterminate forms, and L'Hopital's rule ; 2.5 What are smooth graphs of functions? ; 2.6 Limits of functions and limits of sequences ; 3. Modeling dynamic phenomena ; 3.1 Classifications of differential equations and their solutions ; 3.2 Differential Equations with separated variables ; 3.3 Separable variables differential Separable variables differential ; 3.4 Modeling of growth and decay phenomena ; 3.5 First order exact equations and integrating factors ; 3.6 Homogeneous differential equations and substitution methods ; 3.7 Euler Method for first order differential equations ; 4. The finite in the infinite ; 4.1 Series and convergence ; 4.2 Introduction to convergence tests for series ; 4.3 Improper Integrals ; 4.4 Integral test for convergence ; 4.5 The p-series test ; 4.6 Comparison test for convergence ; 4.7 Limit comparison test for convergence ; 4.8 Ratio test for convergence ; 4.9 Absolute convergence of series ; 4.10 Conditional convergence of series ; 5. Everything polynomic ; 5.1 Representing Functions by Power Series 1 ; 5.2 Representing Power Series as Functions ; 5.3 Representing Functions by Power Series 2 ; 5.4 Taylor Polynomials ; 5.5 Taylor and Maclaurin Series ; 5.6 Using Taylor Series to approximate functions ; 5.7 Useful applications of power series ; 6. Answers