Module overview
This module provides a bridge between A-level mathematics and university mathematics. Some of the material will be similar to that in A-level Maths and Further Maths but will be treated in more depth, and some of the material will be new. Topics of study include functions, limits, continuity, differentiation, integration and ordinary differential equations.
Aims and Objectives
Learning Outcomes
Learning Outcomes
Having successfully completed this module you will be able to:
- Be able to understand the concept of integration, and be able to apply different methods of integration to find areas under curves
- Be able to accurately handle inequalities for real numbers
- Be able to solve a range of first and second order ordinary differential equations
- Be able to understand the definition of limit, and be able to use the algebra of limits, l'Hopital's rule, etc., to determine limits of simple expressions
- Be able to understand the definitions of continuity and differentiability, and be able to differentiate continuous functions
- Be able to state and apply the Intermediate Value Theorem, Rolle's Theorem and the Mean Value Theorem
Syllabus
Functions
• Domain, range.
• Inverse functions.
• Injection, surjection, bijection.
Limits
• Informal and formal concepts of limit.
• Examples of definition of limit in terms of epsilon and delta: linear and scalar cases.
• Algebra of limits.
• Definition of continuity; sums, products and compositions of continuous functions.
• Intermediate Value Theorem.
Differentiation
• Slopes and tangent lines, formal definition in terms of a limit.
• Rules of differentiation: product rule, quotient rule and chain rule.
• Differentiation of trigonometric and hyperbolic trigonometric functions, inverse trigonometric functions, the exponential and logarithmic functions.
• Rolles' Theorem and the Mean Value Theorem.
• L’Hôpital’s rule.
• Maxima and minima, curve sketching.
• Taylor series.
Integration
• The concept of integration via Riemann sums.
• Fundamental Theorem of Calculus.
• Methods of integration: substitution, parts, partial fractions.
• Improper integrals.
Differential Equations
• First order differential equations: separable, linear, homogeneous, Bernoulli, Clairaut.
• Second order constant coefficient differential equations: complementary functions and particular integrals.
Learning and Teaching
Teaching and learning methods
Lectures, problem classes, workshops, private study
| Type | Hours |
|---|---|
| Teaching | 48 |
| Independent Study | 102 |
| Total study time | 150 |
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
| Method | Percentage contribution |
|---|---|
| Coursework | 40% |
| Exam | 60% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
| Method | Percentage contribution |
|---|---|
| Exam | 100% |
Repeat
An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.
| Method | Percentage contribution |
|---|---|
| Exam | 100% |