# 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 rate of change, standard derivatives; product, quotient, function of a function rules; implicit, parametric and logarithmic differentiation; maxima / minima, curve sketching; Maclaurin's and Taylor's series.

Integration: standard integrals, definite integrals, area under a curve; using substitution, partial fractions and by parts; applications (eg volumes, r.m.s. values).

Ordinary differential equations: Solution by direct integration. Solution of first order equations by separation of variables and use of an integrating factor. Solution of homogeneous and non-homogeneous second order equations with constant coefficients.

Functions of several variables: partial differentiation, Taylor's series in two variables, total first order change, analysis of errors, total rate of change, change of variables; stationary points, maxima / minima / saddle points of functions of two variables.

Method of Lagrange Multipliers: constrained maxima / minima, classification of stationary points.

Multiple integrals: double and triple integrals, change of order of integration, use of polar coordinates, simple applications.

### 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. You will attend seminars throughout the academic year, during which you will work through problems to 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 problem solve in groups with your peers. The mathematical rigour associated with this module naturally increases students’ employability and is a highly transferable skill.

The first summative assessment will be an in-class test (worth 20% of the module mark) early in the first semester to assess fundamentals of calculus whilst a formal closed book written examination (worth 80% of the module mark) at the end of the year will allow you to apply higher level skills including multivariable calculus to mathematical problems. The in-class test and examination requires 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 and relevant multimedia materials.

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. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk

### What will I be expected to achieve?

Knowledge & Understanding:
1. Differentiate functions of a single variable and apply to problems associated with maxima/minima and MacLaurin's series. Integrate functions of a single variable and use with applications such as area under a curve, volume of revolution, etc. Solve elementary first and second order ordinary differential equations (see KU1 from Programme Learning Outcomes).
2. Partially differentiate functions of several variables and apply to, for example, the analysis of errors, change of variables and maxima / minima / saddle points. Use the method of Lagrange Multipliers. Evaluate multiple integrals, including the use of change of order and polar coordinates (see KU1 from Programme Learning Outcomes).

Intellectual / Professional skills & abilities:
3. Construct rigorous mathematical arguments to produce 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. In-class test (20%) – 1, 3
2. Examination (80%) – 1, 2, 3, 4, 5

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

Feedback is provided to students individually and in a plenary format both written and verbally to help students improve and promote dialogue around the assessment.

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

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

None

None

### Module abstract

Calculus will enable you to formulate and solve mathematical problems using the concepts of derivatives, integrals and differential equations. It will also develop your basic problem solving skills. The module will follow a combination of lectures and exercise classes. During the exercise classes, you will work through problems to develop your knowledge of the subject. You will be assessed by an in-class test and a final examination designed to put forward your newly developed skills and techniques. You will receive constructive feedback during exercise classes throughout the year, and eLearning Portal will serve as a point of contact, information and discussion with the tutor.
The concepts and skills that you will have learned in this module will form a solid foundation for your further studies and will enhance your future employability.

### 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 rate of change, standard derivatives; product, quotient, function of a function rules; implicit, parametric and logarithmic differentiation; maxima / minima, curve sketching; Maclaurin's and Taylor's series.

Integration: standard integrals, definite integrals, area under a curve; using substitution, partial fractions and by parts; applications (eg volumes, r.m.s. values).

Ordinary differential equations: Solution by direct integration. Solution of first order equations by separation of variables and use of an integrating factor. Solution of homogeneous and non-homogeneous second order equations with constant coefficients.

Functions of several variables: partial differentiation, Taylor's series in two variables, total first order change, analysis of errors, total rate of change, change of variables; stationary points, maxima / minima / saddle points of functions of two variables.

Method of Lagrange Multipliers: constrained maxima / minima, classification of stationary points.

Multiple integrals: double and triple integrals, change of order of integration, use of polar coordinates, simple applications.

### Course info

UCAS Code G101

Credits 20

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 2020 or September 2021

## Mathematics MMath (Hons)

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