## MATH1000 Core Mathematics

### 40 creditsClass Size: 475

Module manager: Dr P. Walker
Email: P.Walker@leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2024/25

### Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

Module replaces

MATH1005, MATH1025, MATH1050, MATH1055, MATH1060, MATH1331, MATH1400

This module is not approved as a discovery module

### Module summary

This module introduces students to fundamental topics of mathematics. Foundational concepts of function, number and proof are introduced, equipping students with the language and skills to tackle their mathematical studies. This module consolidates basic calculus, extending it to more advanced techniques, such as functions of several variables. These techniques lead to methods for solving simple ordinary differential equations. Linear algebra provides a basis for wide areas of mathematics and this module provides the essential foundation.

### Objectives

This module will equip students with foundational mathematical concepts and skills necessary for a degree involving mathematics.

Learning outcomes
On completion of this module, students should be able to:
1. Connect technical descriptions of function properties to concrete examples;
2. Perform basic calculations with complex numbers;
3. Calculate the derivatives and integrals of elementary functions;
4. Construct a Taylor polynomial using given data;
5. Calculate partial derivatives of any order;
6. Perform calculations with vectors;
7. Apply Gaussian elimination to solve systems of linear equations in two and three variables;
8. Perform elementary matrix algebra;
9. Compute the determinant of a 3 x 3 matrix;
10. Solve first-order ODEs of various types;
11. State the classification of systems of two first-order, linear ODEs;
12. Identify the parts of a word problem that relate to a mathematical model;
13. Reflect on their own learning during the module.

Skills Learning Outcomes
SLO1. Communicate through written work, technical information and reasoning.
SLO2. Work in groups to solve mathematical problems.
SLO3. Reflect on and improve your own studies and work habits.
SLO4. Identify and develop your key learning strengths.
SLO5. Use technology appropriately in your work and studies.

### Syllabus

The following topics will be covered:
Numbers and functions
1.Functions: injectivity, surjectivity, composition, inverses, odd/even
2. Sets: intersection, union, complement, power sets
3. Size of sets and (un)countability
4. Proof by induction
5. Complex numbers: modulus, argument; de Moivre's Theorem; geometry of the complex plane; complex roots, roots of unity.

Calculus
6. Derivatives of a range of functions (e.g. hyperbolic trig)
7. Graph sketching
8. Derivatives and some theorems (e.g. Mean Value Theorem)
9. Taylor series (radius of convergence mentioned but not covered in detail)
10. Limits of functions (informal)
11. Integrals and range of functions (e.g. hyperbolic trig) + methods
12. Basic multi-dimensional integration (just Cartesian coordinates)

ODEs
13. Solution of first order differential equations by separation, by use of integrating factor
14. Second order linear differential equations with constant coefficients
15. Systems of differential equations; classification of fixed points
basic ideas of modelling with DEs

Linear Algebra
16. Vectors in 2 and 3 dimensions. Dot and cross products.
17. Geometrical interpretation.
18. Systems of equations. Gaussian elimination.
19. Vectors and matrices. Inverses. Transposes.
20. Determinants. Computation.
21. Eigenvalues and eigenvectors. Computation.
22. Linear maps and bases
23. Matrix of a linear map and change of basis
24. Diagonalisation.
25. Subspaces, and dimensions.

Methods of Assessment

We are currently refreshing our modules to make sure students have the best possible experience. Full assessment details for this module are not available before the start of the academic year, at which time details of the assessment(s) will be provided.

Assessment for this module will consist of;

2 x Coursework
2 x OTLA
2 x In-person open book exam

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 2.00 88.00 Practical 8 1.00 8.00 Seminar 20 1.00 20.00 Independent online learning hours 94.00 Private study hours 190.00 Total Contact hours 116.00 Total hours (100hr per 10 credits) 400.00

### Opportunities for Formative Feedback

Weekly tutorials. Examples sheets marked and returned with feedback.