# CALCULUS - 2025/6

Module code: MAT1030

## Module Overview

This module introduces students to the most important techniques in Calculus. In particular, the module leads to a deeper understanding of the concepts of differentiation and integration. These concepts provide fundamental tools for quantitative descriptions of the real world across the entirety of applied mathematics. Tools and methods for differentiation and integration will be presented in detail. In addition, simple first and second order ordinary differential equations will be studied. Such equations have important applications for interpreting and understanding the world around us.

### Module provider

Mathematics & Physics

### Module Leader

GUTOWSKI Jan (Maths & Phys)

## Overall student workload

Independent Learning Hours: 47

Lecture Hours: 44

Seminar Hours: 5

Guided Learning: 10

Captured Content: 44

Semester 1

None

## Module content

Indicative content includes:

• Functions: Properties and types of functions. Inverse, parametric and implicit functions. Composition of functions. Exponential, logarithmic and trigonometric functions.

• Limits, graphs and equations: Limits, limit rules and the Squeeze Theorem. Curve sketching. Standard transformations of graphs. Plane polar coordinates.

• Differentiation: Definition of the derivative, standard derivatives and rules of differentiation.   Parametric, implicit and logarithmic differentiation. L’Hôpital’s rule.

• Series: Power series. Taylor and Maclaurin series.

• Integration: Definite and indefinite integrals. Fundamental Theorem of Calculus. Standard indefinite integrals. Integration by substitution and integration by parts. Reduction formulae. Applications of integration, including mean values, arc length, areas of surfaces and volumes of revolution.

• Complex numbers: The modulus, argument and exponential form of complex numbers. De Moivre’s theorem. Hyperbolic functions. Roots of complex numbers.

• First order ordinary differential equations: Separation of variables. Integrating factor method for solving linear equations. Homogeneous equations. Bernoulli equations. Applications to real-world problems.

• Second order linear ordinary differential equations: Homogeneous equations with constant coefficients. Inhomogeneous equations.

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework Assessed Coursework 1 5
Online Scheduled Summative Class Test In-Semester Test 15
Coursework Assessed Coursework 2 5
Examination End-of-Semester Examination (2 hours) 75

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions and results in Calculus.

• The ability to identify and use the appropriate methods to solve unseen problems relating to functions, differentiation, integration, complex numbers and ordinary differential equations.

Thus, the summative assessment for this module consists of:

• Two assessed courseworks, each worth 5% of the module mark. The first coursework will correspond to Learning Outcome 1. The second coursework will correspond to Learning Outcomes 1 to 5.

• One in-semester test (50 minutes), run using Mobius software in an invigilated computer laboratory, worth 15% of the module mark and corresponding to Learning Outcomes 1 to 3.

• A synoptic examination (2 hours), worth 75% of the module mark, corresponding to all Learning Outcomes 1 to 6.

Formative assessment

Students will receive formative feedback at biweekly seminars on problem sheets, provided to students in advance and designed to consolidate student learning.

Feedback

Students will receive feedback on the two assessed courseworks and the invigilated Mobius in-semester test. The feedback is timed such that the feedback from the first assessed coursework assists students with preparation for the in-semester test. The feedback from both assessed courseworks and the in-semester test assists students with preparation for the synoptic examination. This feedback is complemented by verbal feedback at biweekly seminars and at office hours.

## Module aims

• Develop students' understanding of a variety of functions.
• Provide students with a firm foundation in Calculus, including techniques in differentiation and integration, building on and extending the A-level syllabus.
• Introduce students to and provide students with practice manipulating complex numbers.
• Introduce students to methods of solving simple first and second order ordinary differential equations.

## Learning outcomes

 Attributes Developed 002 Students will differentiate functions by applying standard rules and techniques. KC 003 Students will be able to determine Taylor and Maclaurin series expansions of functions. KC 004 Students will evaluate integrals by means of substitution, integration by parts and other techniques. KC 005 Students will be able to manipulate complex numbers and solve problems involving complex numbers. KC 006 Students will be able to solve first and second order ordinary differential equations arising in real-world problems. KCT 001 Students will be able to manipulate functions and sketch the graphs of functions in a systematic way. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Provide students with a detailed introduction to functions, differentiation, integration, complex numbers and ordinary differential equations.

• Provide students with experience of methods used to understand and solve problems in Calculus and interpret the results.

The learning and teaching methods include:

• Four one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five seminars for guided discussion of solutions to problem sheets (provided to students in advance) to reinforce their understanding and guide their learning.

• Two assessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT1030

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MAT1030 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing. Students also complete a digital assessment run in Mobius software which enables them to further develop their digital capabilities.

Employability: The module MAT1030 equips students with skills which significantly enhance their employability. The mathematical proficiency gained hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

Global and Cultural Capabilities: Student enrolled in MAT1030 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in seminars and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT1030 is a module which demands the analytical ability to perform unseen calculations in Calculus accurately. Students will gain skills in analysing problems and lateral thinking, and will complete assessments which challenge them and build resilience.

Sustainability: Students will learn to model real-world scenarios relating to sustainability, such as population models, by ordinary differential equations. They will solve these differential equations using techniques in Calculus, and interpret and evaluate their results for sustainable decision-making.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Data Science BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.