Calculus and Analysis 1
Module code: MA1015
The branches of Mathematical Analysis, and differential and integral calculus in particular, form an essential part of the toolbox of any pure or applied mathematician, scientist or engineer. They lead on to the important topics of Differential Equations and of Complex Analysis, and form the foundation for Mathematical Modelling of real-life problems. The history of analysis in Europe began in the 17th century, with the work of Newton, L’Hopital, and Leibniz, continued through the 18th with the work of Euler, of Lagrange, and of Taylor, and into the 19th with Fourier, Cauchy, Riemann, Bolzano, Weierstrass. Work of most of these pioneering analysts will be explained in the 1st and 2nd semester Calculus and Analysis courses.
The central (but hard!) idea behind all of Calculus and Analysis is the idea of a limit. We start the course with this. The idea of limit then gives us a specific and rigorous way to understand lots of intuitive ideas you might already know: what is meant by saying that a function is continuous, and what is meant by the gradient of a tangent line, what precisely is an integral, and so on. The fundamental theorems of calculus form the bridge between differentiation and integration, unifying two central ideas of the subject. In this course, we will learn to apply these theorems to rigorously compute definite integrals. We will determine integrability of functions of one variable by exploring when a function is Riemann integrable, by examining whether its Riemann sums approach a common limit as the partition gets finer capturing the precise idea of area under a curve through limits. A brief discussion about solving first order separable differential equations will be part of this module.