This is a second module in real analysis following on from the Convergence and Continuity module. We will explore in a rigorous way the main concepts, methods and results from the calculus of derivatives and integrals. 

• We formalize the definitions of differentiability and integrability and prove their basic algebraic properties. 
• We then explore some important results in real analysis and calculus such as the Mean Value Theorem and the Fundamental Theorem of Calculus. 
• We discuss Taylor’s Theorem and Improper integrals, widely used in probability and statistics, and close with the introduction of sequences of functions and their convergence.