Calculus 1: Single Variable

EMMS096S4 (30 credits)

Aims

This module aims to introduce the methods of calculus and some of its applications, together with essential algebraic methods required throughout the programme. You will also learn about methods of approximation, and their limitations. Knowledge of a mathematical computer package will be developed and then used to investigate and solve problems covered in the module.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term. There will be a mix of lectures, tutorials and computer sessions.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework  will consist of short, problem based assignments. You will have around three weeks to complete each one.

The examination in June has two sections. Section A (worth 40%) consists of compulsory short questions. Section B (also worth 40%) contains several longer questions of which you must answer two.

Syllabus

Algebraic Methods: polynomials, the Factor Theorem, polynomial equations, exact solutions to linear and quadratic equations, completing the squares, the Binomial Theorem, partial fractions, inequalities, arithmetic of complex numbers, complex numbers as the roots of polynomials.

Coordinate Geometry: straight lines, finding the equation of a straight lines, perpendicular lines, circles, tangent to a point on a circle, equation of a circle, finding the centre and radius of a given circle.

Real Functions: the properties and graphs of exponential, logarithmic and trigonometric functions, inverses, trigonometric identities

Sequences & Series: definitions, intuitive idea of a limit of a sequence, sigma notation, sum of i, i2 and i3 (i=1..n), arithmetic and geometric progressions.

Differentiation: derivatives of standard functions, the chain rule, the product rule, the quotient rule, the inverse function rule, implicit differentiation, logarithmic differentiation.

Integration: integrals of standard functions, definite integration and the area under a curve, integration by substitution, integration by parts, integration of rational functions.

Methods of Approximation: the bisection method, the Newton-Raphson method, the Trapezium rule, Simpson’s rule, Maclaurin and Taylor approximations, power series of standard functions

Applications of Calculus: tangents, stationary points, maxima, minima and points of inflexion, curve sketching, rates of change, motion in a straight line, arc length, volumes of revolution, first order ODEs: variables separable and integrating factors.

Computing: Introduction to Maple V and the use of this package to investigate and solve problems covered in the module. 

Learning Outcomes

Subject Specific

Knowledge and understanding of, and the ability to use, mathematical and/or statistical methods and techniques.

In particular:

• Ability to solve polynomial equations and simple inequalities.
• Ability to express a rational function in partial fractions.
• Knowledge of the methods of differentiation and integration, and the ability to differentiate and integrate functions of one variable.
• Knowledge of arithmetic and geometric progressions.

Knowledge and understanding of a range of results in mathematics and/or statistics.

In particular:

• Knowledge of the basic notation and terminology of calculus.
• Knowledge of the Binomial Theorem
• Knowledge of the properties of standard functions of one variable.

Knowledge and understanding of the processes and limitations of mathematical approximation and computational mathematics.

In particular:

• Knowledge of, and the ability to use, simple numerical methods for solving equations and for evaluating definite integrals.
• Ability to express a function of one variable as a power series, and use the power series as an approximation for the function.
• Awareness of the importance of convergence to the solution of a problem when using a numerical method, and the fact that, in some cases, such a method may fail to produce a valid solution.

Ability to use a modern mathematical and/or statistical computer package with a programming facility, together with knowledge of other suitable packages.

In particular:

• Use a mathematical computer package to investigate and find solutions to the problems considered in the module.

Intellectual

Ability to comprehend conceptual and abstract material.

Develop a logical and systematic approach to problem solving.

Practical

Ability to use a range of software packages including word processing and spreadsheets.

Problem-solving skills, including the ability to assess problems logically and approach them analytically.

Highly developed quantitative skills.

Personal and Social

Ability to work independently with patience and persistence.

Time-management skills and organizational skills.

General IT skills, including word processing and spreadsheets.

Good communication skills, including the ability to write coherently.

Ability to complete work in a limited time period.

Recommended Books:

• D. J. Booth, Foundation Mathematics, Addison-Wesley.
• D. W. Jordan & P. Smith, Mathematical Techniques, Oxford.