## MAS140 Mathematics (Chemical)

 Both semesters, 2021/22 20 Credits Lecturer: Prof Elizabeth Winstanley Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

This module aims to reinforce students' previous knowledge and to develop new basic mathematical techniques needed to support the engineering subjects taken at Levels 1 and 2. It also provides a foundation for the Level 2 mathematics courses in the appropriate engineering department. The module is delivered via online lectures, reinforced with 2 weekly interactive problem classes.

Prerequisites: A-level mathematics (or equivalent)
No other modules have this module as a prerequisite.

## Aims

1. To reinforce previous mathematical knowledge.
2. To develop new mathematical concepts needed to support engineering at Level 1.
3. To provide a foundation for the Level 2 mathematics courses for engineers.

## Learning outcomes

Semester 1:
1. Ability to sketch functions and evaluate simple limits using algebraic techniques.
2. Ability to differentiate, find maxima and minima and apply this technique to curve sketching.
3. Ability to find 1st and 2nd order partial derivatives.
4. An understanding of hyperbolic functions.
5. Ability to apply l'Hôpital's rule.
6. Ability to manipulate complex numbers.
7. Knowledge of the basic properties of vectors.

Semester 2:
1. Ability to evaluate indefinite and definite integrals using the techniques of substitution and integration by parts.
2. Ability to manipulate matrices, evaluate determinants and find the inverse of a non-singular square matrix.
3. Ability to apply matrix methods to the solution of systems of simultaneous linear equations.
4. Ability to find eigenvalues and corresponding eigenvectors of a square matrix
5. Ability to solve first order ordinary differential equations which are (i) variables separable, (ii) linear,
6. Ability to solve second order linear homogeneous ordinary differential equations with constant coefficients.
7. Ability to solve second order linear inhomogeneous ordinary differential equations with constant coefficients, using a trial technique for the particular integral.
8. Ability to apply Laplace Transforms and use them to solve linear differential equations.

## Teaching methods

Online video lectures, online tests, problem classes, problem solving.

5 lectures, 40 tutorials

## Assessment

One three-hour exam at the end of the year (90%). Online tests (10%).

## Full syllabus

Semester 1:
1. Functions of a real variable and limits:
The concept of a function and simple limits, continuity.
2. Differentiation:
Basic rules of differentiation: maxima, minima and curve sketching. Inverse functions.
3. Partial differentiation:
1st and 2nd derivatives, geometrical interpretation.
4. Hyperbolic functions:
Definitions and derivatives of hyperbolic functions and their inverses.
5. Series:
Taylor and Maclaurin series, L'Hôpital's rule.
6. Complex numbers:
basic manipulation, Argand diagram, de Moivre's theorem, Euler's relation.
7. Vectors:
Vector algebra, dot and cross products, differentiation.

Semester 2:
1. Integration:
Indefinite integrals of simple functions. Simple substitutions. Standard forms involving inverse trigonometric and inverse hyperbolic functions. Examples using completing the square and partial fractions. Integration by parts. Definite integrals: properties, evaluation, application to area.
2. Matrices and linear equations:
Definition of an m ×n matrix. Special matrices (identity, zero, square etc.). Matrix algebra. Transpose. Symmetric and skew-symmetric matrices, and the decomposition of square matrices. Determinants. Inverse of a non-singular matrix. Use of matrices to solve systems of linear equations (homogeneous and nonhomogeneous). Gaussian elimination. Eigenvalues and eigenvectors.
3. Ordinary differential equations:
First order differential equations: variables separable, linear with integrating factor, general solution, solution satisfying given initial conditions. Second order linear differential equations with constant coefficients: auxiliary equation, complementary function. Particular integral for polynomials, exponentials, trigonometric functions and products of polynomials and exponential/trigonometric functions on right-hand side. Laplace Transforms and their use in the solution of linear differential equations.