Further calculus (half course)

Описание мероприятия

Описание программы

This half course provides students with useful techniques and methods of calculus and enables students to understand why these techniques work. Throughout, the emphasis is on the theory as well as the methods.

The objectives specifically include:

• enable students to acquire further skills in the techniques of calculus,
• enable understanding of the principles underlying the subject of calculus,
• prepare students for further courses in mathematics and/or related disciplines (e.g. economics, actuarial science).

Assessment

This course is assessed by a two-hour unseen written examination.

Учебный план:

Functions of one variable: Limits; continuity; differentiability; Taylor’s Theorem; L’Hôpital’s rule.

The Riemann integral: The definition of the Riemann integral; the Fundamental Theorem of Calculus.

Improper integrals: The definition of an improper integral; tests for the convergence of an improper integral with a positive integrand (including the direct comparison test and the limit comparison test); absolute convergence of improper integrals with an integrand of variable sign.

Double integrals: Double integrals; repeated integrals; change of variable techniques.

Manipulation of integrals: Joint continuity and the manipulation of proper integrals; dominated convergence and the manipulation of improper integrals; the Leibniz rule for differentiating an integral.

Laplace transforms: The definition of the Laplace transform; functions of at most exponential growth; standard Laplace transforms; properties of the Laplace transform; the Gamma function; using Laplace transforms to solve differential equations; convolutions and the Convolution Theorem; the Beta function.

Результат обучения:

At the end of the course and having completed the essential reading and activities students should be able to:

• demonstrate knowledge of the subject matter, terminology, techniques and conventions covered in the subject,
• demonstrate an understanding of the underlying principles of the subject,
• demonstrate the ability to solve problems involving an understanding of the concepts.

Требования к поступающим:

If taken as part of a BSc degree, courses which must be passed before this course may be attempted:

• MT1174 Calculus

Exclusion

This half course may not be taken with MT3095 Further mathematics for economists.