In this module you will study functions of real numbers in a rigorous mathematical way, continuing in your introduction to using definitions and learning the language in which mathematicians write their ideas and convince others that their results are true. You will study a range of concepts including limits, continuity of functions, derivatives and the definite integral. Along the way you will derive many of the computational rules already used at A-level.

Particular attention will be given to theorems for continuous functions (e.g. the Intermediate Value Theorem and the Weierstrass Theorem), differentiable functions (e.g. the Mean Value Theorem) and their application to the broader study of functions and their graphs. You will be introduced to the Taylor expansion, which allows us to approximate most mathematical functions by polynomials, and   you will also study the general properties of the definite integral, including the Fundamental Theorem of Calculus.  This is a central result of the module which makes the connection between differentiation and integration

MLO.1. Demonstrate foundational knowledge and understanding of the core areas of the syllabus.

MLO.2. Know and apply the definitions of limit, continuity and differentiability for real-valued functions of a real variable.

MLO.3. Use theorems for continuous functions and differentiable functions.

MLO.4. Understand analytic and geometrical properties of the definite integral.

MLO.5. Find and write short proofs of basic logical mathematical statements.

MLO.6. Accurately communicate basic mathematical ideas to a subject specialist, using technical subject-specific terminology and notation.

This module will be delivered via in-person lectures and problem classes, which will cover all the content of the module as well as reviewing solutions to formative exercises. Electronic resources to support your learning (e.g. lecture notes, exercise sheets, outline solutions, and other resources) will be available through Learning Central.

You are also expected to undertake at least 100 hours of self-guided study throughout the duration of the module, including attempting formative exercises. This will provide opportunity to develop your understanding of key concepts, practice problem-solving, reflect on the feedback provided and assess your progress.

Subject-specific skills:

• Demonstrate foundational knowledge and understanding of the core areas of the syllabus; 
• Read and write proofs of mathematical statements within the syllabus, reasoning using logical arguments, applying concepts and principles in well-defined contexts; 
• Apply concepts and principles to solve foundational problems, showing judgement in the selection and application of concepts and techniques;

Professional & practical skills:

• Accurately communicate basic mathematical ideas to a subject specialist, using technical subject-specific terminology and notation;

Transferable/employability skills:

• Apply skills and knowledge gained in other areas of mathematics to foundational problems;
• Write ideas clearly and unambiguously in a formal, professional way.

Summative Assessment

• Exam (80%): Written exam covering theoretical understanding and application of the ideas introduced during the module.
• Coursework (20%): Continuous assessment of material introduced during the module.

Formative Assessment

Formative assessment is provided throughout the module to support your learning. This involves completion of regular exercises related to the topics introduced during the module in order to develop your understanding of key concepts, practice problem-solving, and identify areas for improvement. Feedback is provided through review of solutions, discussion, and outline solutions. Regular engagement with such exercises is essential and is aligned to the corresponding summative assessment for the module. Module leads are also available during office hours to discuss any individual queries.

The opportunity for reassessment in this module

The opportunity for reassessment will be provided in line with the relevant resit and repeat rules adopted by your programme and you will be notified of your eligibility for reassessment after the Examining Board has met:

• if the amount of credit you have failed is within the threshold set for the relevant resit rule, you will be given the opportunity to resit your assessment in the resit exam period, prior to the start of the following academic year
• if the amount of credit you have failed is more than the amount permitted by the relevant resit rule and within the threshold set by the relevant repeat rule, you will be given the opportunity to repeat your assessment during the following academic year
• if the amount of credit you have failed is more than the amount permitted by the relevant repeat rule, you will be required to withdraw from the programme

Reassessment

If you do not meet the required standard in the original assessment a reassessment for the module is typically offered during the designated resit period. The reassessment will take the same form as the original assessment and is designed to cover the same learning outcomes. Reassessments are usually capped at the pass mark.

Functions and limits

• Functions. Injectivity, surjectivity and bijectivity. Some common functions, their inverses and graphs.
• Definition of limits for functions (including the case of infinite limits and limits at infinity).
• Sequential Criteria, Squeeze Theorem, algebra of limits and other properties.
• One-sided limits.

Continuity

• Definition of continuity for real functions.
• Algebra of continuous functions. Continuity of elementary functions.
• Behaviour of continuous functions over open and closed intervals.
• The Intermediate Value Theorem.
• The Weierstrass Extreme Value Theorem.

Differentiation

• Definition of derivative. Illustrative examples of the use of the definition. Geometric interpretation. Continuity of a differentiable function.
• Derivatives of elementary functions. Algebra of derivatives: sums, products and quotients. Chain rule.  Derivatives of inverse functions.
• Stationary points, local and global maximum/minimum points.
• Theorems for differentiable functions. Mean Value Theorem.
• Monotone functions
• Higher derivatives. Stationary values and classification of local maxima and minima, convexity, concavity, inflection points.

Taylor expansion

• Definition of Taylor’s polynomial and Taylor’s expansion
• Taylor’s Theorem with applications.

Definite integral

• Definition of the definite integral. Illustrative examples of the use of the definition. Geometric interpretation.
• Fundamental Theorem of Calculus and applications.
• Properties of the definite integral: Integration by substitution, integration by parts, integration of piecewise functions
• Improper integrals.