# KC4009 - Calculus

## What will I learn on this module?

The module is designed to introduce you to the principles, techniques, and applications of Calculus. The fundamentals of differentiation and integration are extended to include differential equations and multivariable calculus. On this module you will learn:
• Differentiation: derivative as slope and its relation to limits; standard derivatives; product, quotient, and chain rules; implicit, parametric, and logarithmic differentiation; maxima / minima, curve sketching; Taylor and Maclaurin series; L’Hopital’s rule.
• Integration: standard integrals, definite integrals, area under a curve; integration using substitutions, partial fractions decomposition and integration by parts; calculation of solid volumes.
• Functions of several variables: partial differentiation and gradients; change of coordinate systems; stationary points, maxima / minima / saddle points of functions of two variables; method of Lagrange multipliers (constrained maxima / minima).
• Double integrals: standard integrals, change of order of integration.
• Ordinary differential equations: First-order differential equations solved by direct integration, separation of variables, and integrating factor. Second-order differential equations with constant coefficients solved by the method of undetermined coefficients.

### How will I learn on this module?

A wide range of learning and teaching approaches are used in this module. The module will be delivered using a combination of lectures and seminars in which you will be able to obtain help with problems associated with the module. Lectures allow you to experience and understand the formalism of the relevant mathematical techniques and include relevant examples. Seminars throughout the semester will allow you to work through problems and develop your knowledge and skills, with the support of the tutor. Consequently, you will have an opportunity to enhance your understanding of the subject through seminars which promote independent learning and tackle relevant problems. You will be provided with formative feedback to problems in seminars and have the opportunity to solve problems in groups with your peers. The mathematical rigour associated with this module naturally increases students’ employability and is a highly transferable skill to other disciplines.

The first summative assessment will be a coursework (worth 30% of the module mark) to assess fundamentals of calculus whilst a formal closed book written examination (worth 70% of the module mark) at the end of the semester will allow you to apply higher-level skills including multivariable calculus to mathematical problems. Both assessments require you to analyse and solve problems associated with the module. The closed-book written examination assesses all Module Learning Outcomes.

### How will I be supported academically on this module?

In addition to direct contact with the module team during lectures and seminars, students are encouraged to develop their curiosity by making direct contact with the module team either via email or the open-door policy operated throughout the programme. Students will also be regularly referred to supporting resources including relevant texts, multimedia materials, and out-of-the-box MATLAB codes.

References to these resources will be made available through the e-learning portal and in lectures and seminars.

### What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. Online reading lists (provided after enrolment) give you access to your reading material for your modules. The Library works in partnership with your module tutors to ensure you have access to the material that you need.

### What will I be expected to achieve?

Knowledge & Understanding:
1. Differentiation of single-variable functions and its applications to L’Hopital’s rule, Taylor and Maclaurin series expansion, and computation of maxima/minima. Integration of single-variable functions with a variety of techniques and applications to areas, volumes, and solids of revolution. (see KU1 from Programme Learning Outcomes).
2. Partial differentiation of many-variable functions and applications to gradients, change of coordinate system, the determination of maxima / minima / saddle points, maxima / minima subject to constraints by use of the Lagrange Multipliers method. Evaluate double integrals, including change of integration order (see KU1 from Programme Learning Outcomes).

Intellectual / Professional skills & abilities:
3. Construct rigorous mathematical arguments to perform relatively complex calculations, understanding their effectiveness and range of applicability (see IPSA1 from Programme Learning Outcomes).
4. Select and apply appropriate exact and analytical methods to solve standard calculus problems (see IPSA2 IPSA3 from Programme Learning Outcomes).

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Demonstrate critical enquiry and the ability to learn independently (see PVA1 from Programme Learning Outcomes).

### How will I be assessed?

SUMMATIVE
1. Coursework (30%) – 1, 3
2. Examination (70%) – 2, 3, 4, 5

FORMATIVE
Seminar problems – 1, 2, 3, 4, 5

Feedback is provided to students throughout the semester both in written and verbal formats. During lectures, public feedback will promote dialogue between the students and the tutor. The open-door policy will allow for individual feedback.

Informal feedback on work in progress is given continuously during seminars.

Formal feedback will be given directly after the coursework and the exam.

None

None

### Module abstract

Calculus will enable you to formulate and solve mathematical and physical problems using the concepts of differentiation and integration. You will also develop problem-solving skills applied to diverse topics such as sketching functions, determining volumes and areas, and solving differential equations. The module will follow a combination of lectures and seminars. During the seminars, you will work through problems to develop your knowledge of the subject. You will be assessed by coursework and a final examination designed to put forward your newly developed skills and techniques. You will receive constructive feedback during seminars throughout the semester, and eLearning Portal will serve as a point of contact, information, and discussion with the tutor. The concepts and skills learned in this module will be foundational for your further studies in Mathematics or Physics, and will enhance your future employability.

### Course info

UCAS Code F2W4

Credits 20

Level of Study Undergraduate

Mode of Study 4 years Full Time or 5 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2024 or September 2025

## Physics with Astrophysics MPhys (Hons)

All information is accurate at the time of sharing.

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