Core details
| Lecturer in charge | Dr John Mackenzie |
|---|---|
| Other lecturing staff | Dr Oleg Davydov |
| Semester | 1/2 |
| Credits | 20 |
Class delivery (hours)
| Lecture hours | 48 |
|---|---|
| Tutorial hours | 24 |
| Laboratory hours | 0 |
| Assignment hours | 36 |
| Self study hours | 92 |
| Total hours | 200 |
Compulsory for Students
This class is compulsory for:
- Architectural Engineering
- Civil Engineering
- DMEM
Essential Prerequisites
SQA Higher Mathematics (grade B) or equivalent
Incompatible (Overlapping) Classes
This class can't be taken with:
- MM101
- MM103
- other MM11x
- MA101
- MA102
- MA107
- MA108
- MA11x
General Aims
To give a basic understanding of the concepts and applications of mathematical functions, differentiation, integration, matrices and vectors.
Syllabus
Mathematical Foundations
- Algebra - mathematical notation, number sets and inequalities, basic operations (+,-,×,÷), rules of precedence, use of brackets, expanding brackets, simplifying algebraic expressions, factorisation, common denominators, cancelling common factors, proportionality, modulus, factorial, binomial expansion, mathematical formulae and transposition, partial fractions.
- Functions - basic concepts and notation, graphs, domain and range (briefly); continuity and limits; composition of functions; inverses; linear and quadratic functions, completing the square; other commonly occurring functions (including polynomials, rational functions, exponentials, logarithms, hyperbolic functions).
- Solving equations - linear and quadratic equations; simultaneous equations in two unknowns.
- Trigonometry - definitions and graphs of sine, cosine and tangent; periodicity; radian measure; definitions of sec, cosec and cot, and of sin^{-1}x and tan^{-1}x; important trigonometric identities; solving trigonometric equations.
Introduction to Calculus
- Differentiation - motivation (e.g. velocity) and definition; notations dy/dx, f'(x), d(f(x))/dx; simple examples from first principles; graphical interpretation as slope of tangent to graph, increasing and decreasing functions, stationary points; higher derivatives (e.g. acceleration).
- Standard derivatives - including x^a and trigonometric, exponential and natural log functions.
- Rules of differentiation - linearity; product rule; quotient rule; chain rule.
- Indefinite integration - reversing differentiation; standard integrals; linearity.
- Definite integration - motivation: area under a curve;
Matrices, Vectors and Complex Numbers
- Matrices - definition; rows, columns, order, transpose, square, identity; multiplication by a constant; addition of matrices; product of matrices; non-commutativity; associativity; inverse of a square matrix; determinant of a 2 × 2 matrix and a 3 × 3 matrix; inverse of a 2 × 2 matrix; the notion of singular and non-singular matrices.
- Vectors - motivation as quantities having magnitude and direction, e.g. force, displacement, velocity, etc; vectors as directed line segments; vector algebra; orthogonal unit vectors; representation of vectors as number triples; scalar and vector products; triple products.
- Complex numbers - motivation and definition; roots of quadratic equations; real and imaginary parts; the arithmetic of complex numbers.
Further Calculus
- Implicit differentiation - first derivative; derivatives of inverse trigonometric functions.
- Parametric differentiation - first derivative.
- Applications - graph sketching; optimisation problems; linear approximation and error analysis; related rates of change.
- Methods of integration - linearity; substitution; integration by parts; integration of rational functions, up to (linear)/(quadratic); integrals of some trigonometric functions.
- Applications - area between two curves; volumes of revolution about x and y axes.
Learning Outcomes
On completion of this class, the student should:
- understand the concept of a mathematical function, its domain and its range
- be familiar with commonly occurring functions and their properties, and be able to manipulate and solve equations and inequalities involving them
- know the factorial and binomial coefficient notation, and be able to use the binomial theorem
- be able to convert a proper rational function into partial fractions
- be able to differentiate functions, via combinations of the various differentiation rules
- be able to integrate simple functions
- be able to differentiate functions defined either implicitly or parametrically
- be able to locate and classify stationary points, and find optimal values of a function of one variable
- be able to find definite and indefinite integrals using substitutions, partial fractions and integration by parts
- be able to use integration to calculate area between two curves and volumes of revolution
- be able to carry out simple matrix operations
- be familiar with the concept of a vector and the fundamental operations with vectors: addition, multiplication by a scalar, and scalar and vector products
- be able to manipulate complex numbers in Cartesian form.
Method of Assessment
Satisfactory submission of assignments and satisfactory attendance at tutorials is necessary. Three-hour degree examination in May/June with August resit.
Exemption from degree examination is possible based upon performance in class tests.
Reading list
-
**Mathematics for Engineers: A Modern Interactive Approach (3rd Ed.)
Croft, A. and Davison, R.
Harlow: Prentice Hall
ISBN: 0132051567
Library location: D510.2462 CRO