The module will introduce the students to the concepts at the heart of modern mathematics, the derivative and the Riemann integral. These will be looked at from two perspectives: intuitive, geometric, and mathematically rigorous, furnished by the idea of a continuous limit.

On this foundation further calculus tools, indispensable in the study of mathematics, will be explored: the general properties of the derivative and the Riemann integral will be studied in detail, followed by the demonstration of the techniques of integration.

In the course of this the students will be thoroughly exposed to the class of elementary functions, based on the fundamental concept of the exponent.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• Understand the concept of the derivative as a continuous limit, its geometric and mechanical interpretations, know and be able to apply its properties.
• Understand the concept of the Riemann integral, its geometric meaning, know and be able to use its properties
• Understand the importance of the fundamental theorem of calculus as the link between the concepts of the derivative and the integral, and be able to apply it in specific examples.
• Be proficient in differentiating and integrating standard elementary functions.
• Be proficient in evaluating integrals of rational functions and integrals reducible to rational functions.

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:
Evaluating derivatives and integrals of elementary functions.

Transferable Skills:
The fluency in handling derivatives and integrals that is fundamental for all branches of mathematics.

Ability to understand, manipulate and formulate abstract mathematical concepts.

Critical thinking and problem solving skills.

Ability to present work in a scholarly manner.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes. The formative assessment will also include a class test during the first half of the course.

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• The definition of a continuous limit.
  • The concepts of continuity, piecewise continuity and differentiability.
  • The geometric and mechanical meaning of the derivative.

• Rules for differentiation
  • table of standard derivatives, including hyperbolic and trigonometric functions and their inverses.

• The concept of the Riemann integral;
  • Examples of Riemann sums.
  • Geometric meaning of the Riemann integral.
  • Existence of the Riemann integral for a piecewise continuous function (without proof).

• The fundamental theorem of calculus.
  • Differentiation of an integral with a variable upper limit.
  • Properties of the Riemann integral, including substitution and integration by parts.

• Techniques of integration.
  • Table of standard Integrals
  • Integrals of rational functions and integrals reducible to rational functions.

• The definitions and basic properties of improper integrals.
  • Comparison test for the convergence of improper integrals of non-negative functions.

Calculus, Finney, R. L., & Thomas, G. B., Addison-Wesley

Calculus (6th Ed), Edwards, C. H., & Penney, D. E., Prentice Hill (2002)

Calculus (5th Ed), Stewart J, Thompson (2003)

Calculus, Finney, R. L., & Thomas, G. B., Addison-Wesley

Calculus (6th Ed), Edwards, C. H., & Penney, D. E., Prentice Hill (2002)

Calculus (5th Ed), Stewart J, Thompson (2003)