## 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 15% of the mark for the module. There is also a written exam which counts for the remaining 85%. 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.

Dr Paul Bilokon