As preparation for the Level 2 entry Maths Summer School (MSS)
participants should revise and familiarise themselves with the following
topics which will have appeared in their earlier mathematics learning.
While many of these topics may be revisited during the first week of the
Summer School, some level of familiarity will be assumed since these
topics are first encountered much earlier in mathematics education.
This list is broken into two halves, the first half covers
materials from the Engineering Maths 1 & 2 units, and are specific
for students applying for Year 2 direct entry. The second half covers
typical mathematics topics that appear across all of engineering.
Links have been provided to online resources to aid you with any
topics of which you may like to refresh your knowledge. For the later
general materials you will find many other resources online, for the
specific Year 2 direct entry materials more tailored links are provided
to assist you. The links are found in the footnotes and are accessible
via the numbered superscripts in the main text.
Year 2 Entry Engineering pre-requisites
Exponentials and logarithms
- Definition of logarithms
- Definition of exponentials
- Appreciate the relationship between exponentials and logarithms
- Use the three logarithm laws
- Sketch and identify basic logarithm and exponential graphs
Vectors in 2D & 3D
- Definition of a vector
- Work with vectors in 2D and 3D including the topics of:
- magnitude
- scalar product
- angles between vectors
- Convert from Cartesian [Rectangular] form to Polar form and vice
versa
Complex numbers
- Basic algebra of complex numbers, like addition, subtraction,
multiplication and division
- Conversion between Cartesian (rectangular) format and polar format
- Perform mutliplication and division of complex numbers in polar
format
- Represent complex numbers on an Argand diagram
Differentiation
- Differentiation of standard functions, like \(ax^n, \sin(ax+b), \ln(ax+b)\) and \(e^{ax+b}\)
- Use the chain rule for differentiation
- Use differentiation to find stationary/turning points of
functions
- Calculate second derivatives to classify stationary/turning points
- Use differentiation in real-world problems to determine rates of
change, or optimize a variable.
Integration
- Appreciation of integration as anti-differentiation
- Integrate standard functions, like \(ax^n\), \(\sin(ax+b)\), \(\frac{a}{ax+b}\) and \(e^{ax+b}\)
- Calculate definite integrals with given limits
- Appreciate usage of integration to find areas under graphs, to solve
real-world problems
General engineering maths pre-requisites
Numerical skills
- Calculating percentages
- Reversing percentage calculations, e.g. calculating original
quantities after percentage reductions
- Orders of operations: i.e. bodmas or
bidmas or pedmas (depending on when you learnt it)
Algebra skills
- Working with brackets
- Expanding brackets, e.g. \((x+1)(ax^2+bx+c)\)
- Factorising into brackets, spotting common factors
- Using algebraic formulae, like Pythagoras’ theorem
- Simplifying by grouping ‘like’ terms
Fractions
- Reducing fractions to lowest forms
- Multiplication, division, addition and subtraction of fractions
- Relationship between fractions and negative powers (see Powers)
Linear equations
- Identifying linear equations
- Re-arranging to make a variable the subject
- Solving linear equations
More general equations
- Rules for re-arranging equations: changing sides, dividing through,
squaring and square-rooting
- Substitution methods
- Making a variable the subject
- Simultaneous linear equations
- Formulating simultaneous equations from text questions
- Solving simultaneous equations
Powers
- Simplifying powers and rationalising denominators of fractions
- The standard power laws
- Multiplication and division using positive, negative and fractional
indices
- Expanding brackets with powers, e.g. \((ab)^n=a^nb^n\)
- Powers of powers, e.g. \((a^m)^n =
a^{mn}\)
- Fractional powers as roots, \(a^{m/n}=\sqrt[n]{a^m}\)
Simple graphs
- Equations of straight lines
- Graphs of quadratics
- Recognising maxima and minima of graphs
Quadratics
- Working with quadratics
- Completing the square
- Factorising quadratics
- Using the ‘quadratic formula’
- Identifying graphs of quadratics (see also Simple graphs)
Trigonometry
- Basic graphs of sine and cosine
- Amplitude
- Translations vertically and horizontally
- The \(\sin^2(x)+\cos^2(x)=1\)
identity
- The \(\tan(x)=\frac{\sin(x)}{\cos(x)}\)
identity
- Using sine, cosine and tangent in right-angled triangles, i.e. soh-cah-toa
Trigonometric equations & hyperbolics functions
- Solve problems of the form \(A
\sin(mx+a)=b\) and similar
- The reciprocal trignometric functions (\(\mathrm{cosec}, \sec, \cot\))
- Working with the compound angle formulae, e.g. \(\sin(A+B)\)
- Using trigonometric identities, like \(\cos(2A)=2\cos^2(A)-1\)
- (Nice to know, but optional) Hyperbolic functions:
- Evaluate \(\sinh\), \(\cosh\) and \(\tanh\) functions
- Use hyperbolic identities
Links to revision resources
If you would like to look up some revision notes on any of the above
topics then you should find a very wide range of resources, especially
videos online. For the latter general engineering mathematics there are
many very high quality resources.
Furthermore, since many of these topics will be revisited in the
first week of the Maths Summer School (MSS) you will find a lot of the
topics have at least summary notes available here, in the MSS Week 1 algebra notes. This is
the default advice for initial revision.
Some topics above do not explicitly appear in the Week 1 notes, for
those you will find some useful links provided via the
superscripts/footnotes embedded in this document.
As a final bonus, here are a few of the most popular websites which
specifically offer maths learning resources (though YouTube is also
excellent):