40016

Calculus

Module aims

In this module you will study the calculus needed for many applications of Computing, such as graphics, vision, robotics, operations research and statsitical machine learning. It also provides the background for follow-on mathematics modules that lay the foundations for advanced electives in the above topics and other mathematically-focused areas such as optimisation and finance.

Learning outcomes

Upon successful completion of this module you will be able to:
- Establish convergence or divergence of sequences and series and determine limit values
- Derive Maclaurin and Taylor series and determine radius of convergence
- Find approximations from below and above to the value of an integral
- Find the minima, maxima and saddle points of multivariate functions
- Find approximations to roots of multivariate functions
- Compute integrals with respect to cylindrical and spherical coordinate systems

Module syllabus

• Ordering and supremum of real numbers
• Sequences, series and convergence
• Limits of functions and continuity, intermediate value theorem, uniform continuity
• Differentiation and its properties
• Mean value theorem
• Riemann integral and its properties
• Trapezium rule
• Uniform convergence and power series
• Metric spaces
• Contraction mapping theorem
• Multivariate Differential Calculus
• Partial derivatives
• Hessian and the Taylor series formula
• Extrema of scalar fields
• Newton’s method
• Vector valued functions
• Jacobian matrix
• Cylindrical and spherical coordinate systems

Teaching methods

The material will be taught through lectures that mix problem solving exercises and taught content.

There are regular unassessed, i.e. formative, tutorial exercises that are submitted for marking and feedback as part of separate Maths Methods Tutorials (MMTs), which run throughout the  Autumn and Spring terms. These tutorials encourage group discussions and group problem solving designed to reinforce your understanding of key topics in  calculus and linear algebra.

An online service will be used as a discussion forum for the module.

Assessments

There is one assessed coursework which counts 20% of the mark for the module. There is also a written exam which counts for the remaining 80%. Formative assessment is via weekly mathematics methods tutorials (MMTs) which address topics in this module and Linear Algebra. Approximately half of the MMT exercises will be dedicated to Calculus.

A combination of written and verbal feedback will be provided for the MMT tutorial exercises and assessed coursework.

Reading list

Reading list

• Analysis for Computer Scientists : Foundations, Methods, and Algorithms

  Oberguggenberger, Michael; Ostermann, Alexander

  Second edition, Springer

• Mathematical analysis : a straightforward approach

  Binmore, K. G.

  2nd ed., Cambridge University Press

• Set theory and metric spaces

  Kaplansky, Irving,

  Second edition., American Mathematical Society

• How to think about analysis

  Alcock, Lara,

  Oxford University Press

• Set theory and metric spaces

  Kaplansky, Irving.

  Allyn and Bacon

• Calculus

  Spivak, Michael.

  3rd ed., Publish or Perish, Inc

• Calculus

  Apostol, Tom M.

  2nd ed., Wiley

• Calculus

  Ayres, Frank, 1901-1994, author.

  6th edition., McGraw-Hill

• The foundations of analysis : a straightforward introduction / Bk. 2, Topological ideas

  Binmore, K. G.

  Cambridge University Press.

• Newton's Principia, first book, sections I., II., III. : with notes and illustrations, and a collection of problems principally intended as examples of Newton's methods

  Newton, Isaac, Sir, 1642-1727.

  Macmillan

• Calculus : [concepts and methods]

  Binmore, K. G. (Kenneth George), 1940-

  Cambridge University Press

• Introduction to topology

  Mendelson, Bert, 1926-1988.

  3rd ed., Dover

• How to prove it : a structured approach

  Velleman, Daniel J., author.

  Third edition., Cambridge University Press

• Metric Spaces / [electronic resource]

  Copson, E. T.,

  Cambridge University Press

• Metric spaces

  Copson, E. T.

  Cambridge University Press

• How to solve it : a new aspect of mathematical method

  Polya, George, 1887-1985.

  [2nd ed.]., Princeton University Press

• Naive set theory

  Halmos, Paul R. 1914-

  Springer-Verlag

• Advanced calculus

  Wrede, Robert C., author.

  Third edition, McGraw-Hill

• Never at rest : a biography of Isaac Newton

  Westfall, Richard S. (Richard Samuel).

  Cambridge University Press

• Metric Spaces : Iteration and Application

  Bryant, Victor, author.

  Cambridge University Press

• Leibniz: An Intellectual Biography: Antognazza, Maria Rosa: New York: Cambridge University Press, 652 pp., Publication Date: October 2008

  Hight, Marc A.

  History (Washington) Taylor & Francis Group

• The foundations of analysis : a straightforward introduction / Bk. 1, Logic, sets and number

  Binmore, K. G. (Kenneth George), 1940-

  Cambridge University Press.

• The History of the Priority Di∫pute between Newton and Leibniz Mathematics in History and Culture

  Sonar, Thomas. author.

  1st ed. 2018., Springer International Publishing

Module leaders

Professor Abbas Edalat