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Calculus 1 (MATH 11007)

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Contents of this document:

Administrative Information

  1. Unit number and title: MATH 11007 Calculus 1
  2. Level: C/4 (Honours)
  3. Credit point value: 20 credit points
  4. Year: 11/12
  5. First Given in this form: 07/08; similar to parts of previous unit Core A
  6. Unit Organiser: Yves Tourigny
  7. Lecturer: Yves Tourigny and Noah Linden
  8. Teaching block: 1 and 2
  9. Prerequisites: A at A-level Mathematics or equivalent

Unit 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

General Description of the Unit

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 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.

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 view
    functions 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.

Teaching Methods

Lectures, exercises to be done by students, tutorials.

Learning Objectives

Learning Objectives

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.

Assessment Methods

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%.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

The assessment mark is calculated as described in the Assessment section above. Details of the university's common criteria for the award of credit points are set out in the Regulations and Code of Practice for Taught Programmes at http://www.bristol.ac.uk/esu/assessment/codeonline.html

Note that for this unit:

  • first year students are expected to attend all the relevant tutorials,
  • all students are expected to hand in attempts to the weekly exercises set.

Transferable Skills

Problem-solving skills.

Texts

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

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.

Unit web page

http://www.maths.bris.ac.uk/~mayt/MATH11007