Further exercises; Chapter 9. Mean Values and Taylor Series; 9.1 The Mean Value Theorem; 9.2 Tests for extreme points; 9.3 L'Hôpital's Rules and the calculation of limits; 9.4 Differentiation of power series; 9.5 Taylor's Theorem and series expansions; Summary; Further exercises; Chapter 10. The Riemann Integral; 10.1 Primitives and the 'arbitrary constant'; 10.2 Partitions and step functions: the Riemann Integral; 10.3 Criteria for integrability; 10.4 Classes of integrable functions; 10.5 Properties of the integral; Summary; Further exercises; Chapter 11. Integration Techniques.
Building on the basic concepts through a careful discussion of covalence, (while adhering resolutely to sequences where possible), the main part of the book concerns the central topics of continuity, differentiation and integration of real functions. Throughout, the historical context in which the subject was developed is highlighted and particular attention is paid to showing how precision allows us to refine our geometric intuition. The intention is to stimulate the reader to reflect on the underlying concepts and ideas.