Department:Mathematics and Statistics
Course Convenor:Professor G Blower
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Curriculum Design: Outline Syllabusback to top
Arithmetic of complex numbers;
Rational functions and partial fractions;
Exponential and hyperbolic functions;
Compositions and inverses;
Sequences and limits;
Product and Chain rules;
Maxima and minima;
Complex exponentials and trigonometric functions;
Definite integral as areas;
Fundamental theorem of calculus;
Integration by parts and by substitution;
Curriculum Design: Pre-requisites/Co-requisites/Exclusionsback to top
A-level Mathematics at A-grade or above, or equivalent.
Educational Aims: Subject Specific: Knowledge, Understanding and Skillsback to top
To provide the student with an understanding of functions, limits, and series, and a knowledge of the basic techniques of differentiation and integration.
The purpose of this course is to study functions of a single real variable. Some of the topics will be familiar from A-level, others will be studied more thoroughly in subsequent courses. The module begins by introducing examples of functions and their graphs, and techniques for building new functions from old. We consider rational functions and the exponential function. We then consider the notion of a limit, sequences and series and then introduce the main tools of calculus. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The rules for calculating derivatives are obtained form the definition of the derivative as a rate of change. Taylor series are calculated for functions such as sin, cos and the hyperbolic functions. We then introduce the integral and review techniques for calculating integrals. We learn how to add, multiply and divide polynomials and introduce rational functions and their partial fractions. Rational functions are important in calculations, and we learn how to integrate rational functions systematically. The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrize geometrical curves.
Educational Aims: General: Knowledge, Understanding and Skillsback to top
The purpose of the course is to present the calculus of functions of a single real variable in a systematic manner, that develops intuitive concepts from A level courses and leads the way to subsequent rigorous courses in analysis. The emphasis of the course is on practical calculation, instead of theory or applications. The module aims to encourage students to think of functions in terms of graphs, and introduces the notion of derivative via the gradient of the tangent to a graph.
The module develops intutitive ideas such as rate of change, monotonicity, continuity, maxima and minima in the context of graphs. Limits are introduced in the context of simple examples sequences which will appear as fundamental examples in subsequent courses in analysis. Whereas the course does not develop applications, some basic terminology from kinetics is used to
make calculus more intuitive.
Learning Outcomes: Subject Specific: Knowledge, Understanding and Skillsback to top
On successful conclusion of this module, the student should be able to: carry out arithmetic for complex numbers and for rational functions; sketch graphs of real rational functions, with asymptotes; sketch graphs of monotone functions with their inverses, and identify the domains and ranges; factorize real polynomials as products of real linear factors and irreducible quadratics; calculuate partial fractions of real rational functions; divide real polynomials; verfiy given formulas for inverse functions and functional identities; manipulate expressions involving the exponential functiona nd natural logarithm; verify given identities depending upon an integral variable by induction; use partial fractions to compute the partial sums of telescoping series; take limits of simple algebraic expressions; differentiate products and quotients of functions; differentiate compound functions by the chain rule; differentiate inverse functions including inverse trigonometric and hyperbolic functions; compute Taylor series of given functions; apply the general binomial theorem for real exponents; obtain the general binomial theorem as a special case of Maclaurin's theorem; use the derivative test to locate and determine the nature of stationary points; understand the statement of the fundamantal theorem of calculus; understand the respective roles of definite versus indefinite integrals; integrate functions by parts; apply repeated integration by parts and recurrence relations to obtain families of trigonometric integrals; understand the formulas for integration by susbstution of indefinite end definite integrals, and change limits when making substitutions; apply standard substitutions for quadratics and quadratic surds; integrate real rational functions by systematic substitutions.
Learning Outcomes: General: Knowledge, Understanding and Skillsback to top
Students should learn how to solve problems in calculus of functions of a single real variable, and present their solutions in an orderly and coherent manner. The solutions may involve several steps, each depending upon the correct solution of previous steps. They should also be able to interpret calculations in terms of graphs, and conversely. Whereas most results in the course are not proved in detail, students should understand the notion of a proof and how it justifies certain calculations. They should also be able to prove simple algebraic identities by formal induction arguments. They should appreciate the importance of precise terminology and learn to use the traditional language that is used to describe problems in calculus. In multiple choice tests, they should be able to distinguish between correct and incorrect statements, possibly after extensive reductions and calculations.
Assessment: Details of Assessmentback to top
Assessment will be through:
(i) weekly coursework, aimed at testing and consolidating understanding of the basic elements of the course;
(ii) a test at the end of the module which assesses the students' understanding through short questions;
(iii) an examination in the Summer which assesses more fully the students' understanding and summative knowledge of the topics.
Curriculum Design: Select Bibliographyback to top
GILBERT, J. & JORDAN, C. (2002) Guide to Mathematical Methods, Second Edition. Palgrave-Macmillan.
EDWARDS, C.H. & PENNEY, D.E. (2002) Calculus, Sixth Edition. Prentice Hall.