Calculus and Geometry - 2019 entry

Calculus has its origins in the study of planetary motion by Newton and his contemporaries. Calculus is fundamental to mathematics and provides tools for analysing a diverse range of problems across the physical, engineering, life and environmental sciences. You will develop your ability to think logically, to analyse complex relationships, and to recognize underlying simple ideas and structures common to almost all problems in mathematics – these are the skills required to do mathematics. You will renew your knowledge about tools and concepts, which you have already been in contact with in school in a more rigorous way, and study further links to more complex fundamental ideas, which will enable you to quickly and accurately perform calculus on functions using a variety of standard techniques.

The module gives an informal treatment of theorems from analysis. You will learn how to accurately sketch the graphs of functions. It will also teach you how to reason using abstract ideas, formulate and solve problems and communicate reasoning and solutions effectively in writing and oral presentation. As with the other modules, it will develop your self-management and time-management skills and broaden your use of learning resources including the use of IT.

You will work on weekly exercise sheets; you will be asked to form study groups with fellow students to solve these problems.

For the assessed coursework, you will take short in-class tests every 2-3 weeks during the term, which will assess understanding of basic concepts and methods, and consist of questions similar to those seen on the weekly exercise sheets. This should encourage you to work out solutions for all problems from the weekly exercise sheets (this can happen in small groups, but the in class assessment must be carried out individually).

The aim is to develop knowledge and skills in calculus and introduce some formal definitions and statements of theorems from analysis, but with only limited formal proofs. The module is about developing methods and skills for calculus-related manipulation of the mathematical objects that form the basis of much of an undergraduate course in mathematics, and subsequent applications in science and engineering.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 demonstrate an understanding of basic concepts concerning sequences, series, functions and limits;

2 perform accurate calculus manipulations using a variety of standard techniques; sketch the graphs of a variety of functions of one variable; determine and categorise extrema.

Discipline Specific Skills and Knowledge:

3 demonstrate a basic knowledge and understanding of fundamental concepts of functions, sequences, series, limits and differential calculus necessary for progression to successful further studies in the mathematical sciences;

4 understand how to read mathematical definitions, theorems and proofs; understand the logic behind different methods of proof, the mechanics of using them and gain experience in applying them; develop an appreciation of the role of rigorous proof in mathematics.

Personal and Key Transferable / Employment Skills and Knowledge:

5 reason using abstract ideas, formulate and solve problems and communicate reasoning and solutions effectively in writing and oral presentation;

6 work in groups to solve in-depth problems effectively; and learn to analyse/assess other solutions for problems;

7 acquire ability for self-criticism of your work;

8 demonstrate appropriate use of learning resources;

9 demonstrate self management and time management skills.

- “Calculus and Geometry” and “Vectors and matrices” fundamentals (see also ECM1902): Introduction to sets, number systems, logic and proof;inequalities and intervals [4 hours (+4 hours in “Vectors and Matrices”)];

- Functions: dependent and independent variables; injections, surjections and bijections, image and preimage. Real functions: graphs, even and odd functions, sums, differences, products, quotients, composition; inverse functions; trigonometric and hyperbolic functions [4 hours];

- Analytic geometry: Cartesian coordinates, parametric equations, conic sections, normals and tangents [4 hours];

- Sequences: finite and infinite sequences; geometric and arithmetic progressions; convergence, limits of sequences, algebra of limits [4 hours];

- Series: partial sums and series; algebraic operations on series; absolute convergence, convergence tests; power series [4 hours];

- Mid-term “Calculus and Geometry” and “Vectors and matrices” revision and mastery (see also ECM1902): focus on logic, rigorous analysis, mathematical proofs and in-depth problems [4 hours (+4 hours in “Vectors and Matrices”)];

- Continuity of functions: definition of a limit, sequential definition, one-sided limits; continuity; connectivity and the Intermediate Value Theorem [4 hours];

- Differentiation in one variable: definition of the derivative; rules for differentiation; differentiation techniques; higher order derivatives, Leibniz's rule [4 hours];

- Applications of differentiation: maxima and minima of functions; Mean Value Theorems; L'Hopital's rule [4 hours];

- Taylor polynomials and series [4 hours];

- Revision and mastery [4 hours].

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment. If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic reading:

ELE: http://vle.exeter.ac.uk

Other Resources:

Reading list for this module: