Unit information: Calculus 1 in 2012/13

Unit name	Calculus 1
Unit code	MATH11007
Credit points	20
Level of study	C/4
Teaching block(s)	Teaching Block 4 (weeks 1-24)
Unit director	Dr. Tourigny
Open unit status	Not open
Pre-requisites	Normally, an A in A-level mathematics or equivalent
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Calculus is based on the calculus learnt at school. It will develop a deeper understanding, a stronger grasp of the techniques, and further topics that are not included in A level syllabuses. This includes in particular the extension of methods of calculus to functions of two or more variables. The course also introduces symbolic computing via the software package MAPLE as a helpful tool to solve mathematical problems. Applications of MAPLE in Calculus will be stressed throughout. The logical foundations of calculus are not included in this unit; they are developed in the companion unit Analysis. The course will concentrate more on general ideas and methods, rather than rigorous logical development.

Aims:

• To provide some basic tools and concepts for mathematics at the undergraduate level, by

• developing and extending the calculus skills introduced at A level

• linking the material taught in Calculus to that of Analysis, Linear Algebra and Mechanics

Syllabus

Section I: Basics (Weeks 1-6)

• Before Calculus: a review of some elementary functions.
• Limits, continuity, derivative.
• Curve sketching.
• Antiderivatives.
• The definite integral.
• The Fundamental Theorem of Calculus.
• Tricks of the trade: integration by parts and substitutions.
• Recurrence relations, sequences.
• Infinite series

Section II - Ordinary Differential and Difference Equations and Dynamical Systems (Weeks 7-14)

• Differential equations of the first order.
• Differential equations of the second order.
• Discrete Dynamical Systems
• Continuous Dynamical Systems

Section III Multivariable Calculus (Weeks 15-24)

• Taylor's Theorem.
• Numerical methods.
• Parametric representation of curves.
• Curvilinear motion and polar coordinates.
• Partial derivatives.
• Surface and curves in space.
• Directional derivatives.
• Vector differentiation and integration.
• Double and iterated integrals.
• Further applications and developments.

Relation to Other Units

The material taught in this unit is linked to that of the following other units

• MATH11005 (linear Algebra and Geometry) develops an abstract framework which

emcompasses some of the objects developed in Calculus. For instance, one may viewfunctions or gradients as "vectors", and solution sets of some differential equations as "vector spaces".

• MATH11006 (Analysis) is a completely rigorous treatment of (some of) the

material presented in the Calculus unit.

• MATH11009 (Mechanics), as the unit that deals with

some of the consequences of Newton's laws of motion, provides countless applications of Calculus.

• MATH12001 (Computational Mathematics) which discusses the implementation on computers of many techniques developed in calculus.

• MATH11300 (Probability) uses the tools of Calculus (e.g. integration,

power series etc.) to study discrete and continuous random variables.

Intended Learning Outcomes

After taking this unit, students should

• be able to evaluate and manipulate derivatives and integrals with ease;

• be able to solve some simple first and second order differential equations;

• be able to use partial derivatives and the gradient vector;

• be familiar with vectors in 2 and 3 dimensions;

• be familiar with some standard curves and surfaces, and be able to work with them;

• be able to evaluate line integrals;

• understand the connection between Calculus on the one hand and Analysis, Probability and Mechanics on the other.

Transferable Skills

Problem-solving skills.

Teaching Information

Lectures, exercises to be done by students, tutorials.

Assessment Information

The final assessment mark for the unit is constructed from two unseen written examinations: a January mid-sessional examination (counting 10%) and a May/June examination (counting 90%). Calculators and notes are NOT permitted in these examinations.

• The mid-sessional examination in January lasts one hour. There are two parts, A and B. Part A consists of 4 shorter questions, ALL of which will be used for assessment. Part B consists of three longer questions, of which the best TWO will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%.

• The summer examination in May/June lasts two-and-a-half hours. There are again two parts, A and B. Part A consists of 10 shorter questions, ALL of which will be used for assessment. Part B consists of five longer questions, of which the best FOUR will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%.

Reading and References

The recommended text is:

• Schaum's Outline of Calculus (Fourth Edition), by Frank Ayres Jr and Elliott

• Mendelson. Schaum's Outline Series,

• McGraw-Hill, 1999. ISBN 0-07-041973-6