Use of A level mathematics in
University Degrees and
in the workplace
Richard Lissaman
Stephen Lee
1
Abstract
• This session will present detailed
information on which aspects of A level
Mathematics are used in different degree
courses. This arose from a review of
modules offered by various departments
within a number of universities and
through MEI’s work with industry.
2
Session Overview
• Which university courses use A-level
Mathematics and Further Mathematics?
• How much do they use? Which topics?
• A detailed look at content for selected
university courses
• Useful examples for use in the classroom
• Mathematics in Industry
3
So…
• Which university courses use A-level
Mathematics and Further Mathematics?
• How much do they use? Which topics?
4
Review of University Courses
5
Review of University Courses
• Aim of document is to give overview, not
an all inclusive list
• Content as explicitly mentioned on module
contents
• Very much evident the wide range of uses
of A level Maths and Further Maths
material in university courses
6
Review of University Courses
• Heavy reliance on Stats for Actuarial work
• Large amount of Mechanics in
Engineering (Aeronautical, Mechanical),
Physics and Sports Science
• Considerable Decision maths in
Operational Research courses
• Note important nature of the APPLIED
modules, thus an AS in FM is very useful!
7
Mathematical content of selected
university courses
• We have chosen to look at Chemistry,
Economics, Biology, Geography and
Sports Science
• These have been chosen because their
relationship to Mathematics is perhaps
less obvious than for subjects such as
Physics and Engineering
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Chemistry
• From the Royal
Society of
Chemistry
Tutorial
Chemistry Text
Series
recommended
by the Higher
Education
Academy
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12
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Economics
• Recommended
text by a
Business
department for
first year study
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15
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Geography
• Recommended
text by a
Geography
department for
first year study
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20
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• Very descriptive
• Not ‘mathematically
Set out
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Human and Life
Sciences
• Recommended
text by a Human
Sciences
department for
first year study
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• Very ‘chatty’
style
• Different to
standard texts
found in A
levels
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• 2nd recommended text
• Clear indication of the attitude towards
statistics from such subject areas!
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Sports Science
• Recommended
text by a top
Sports Science
department for
first year study
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Contents
• 1 Biomechanics in Physical Education
• 2 Forms of motion
• 3 Linear Kinematics
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–
–
–
–
–
–
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Distance and displacement
Speed and velocity
Acceleration
Units in linear kinematics
Acceleration due to gravity
Vectors and scalars
Resultant vector
Vector components
Uniformly accelerating motion
Projectiles
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•
•
•
•
•
4 Angular Kinematics
5 Linear Kinetics
6 Angular Kinetics
7 Fluid Mechanics
8-15 Specific Sports analysis (including)
– Athletics
– Golf
– Gymnastics
– Swimming
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Examples from 1st Year
Undergraduate Chemistry
• Example 1 – Interaction of molecules
– Differentiation
• Example 2 – Wavefunction
– Integration
• Example 3 – Hydrogen ion concentration
– Differentiation
• Example 4 – Vapour pressure
– Differential Equations
31
Examples from Economics
• Example 5 – Model of trading nations
– Algebra and Matrix Manipulation
• Examples 6 and 7 – Utility Function
– Partial Differentiation
• Example 8 – Exports
– Matrices
• Example 9 – Maximum Profit
– Partial Differentiation
32
Examples from Geography
• Example 10 – Flooding
– Poisson Distribution
• Example 11 – Demographics
– Exponential Growth
33
Examples from Human and Life
Sciences
• Example 12 – Marine Biology
• Example 13 – Concentration of Drug
34
Examples from Sports Science
• Example 14 – Resting Energy Expenditure
• Example 15 – Pulse-Rate on a long slow
run
35
Our ideas for resources
• Examples 1 – 15 had the question and a
solution technique
• Exemplar 1 and exemplar 2 have the
question and Teachers notes
• University-endorsed Worksheet
36
Mathematics in Industry
• Currently in a golden age of numbers in
the real world - mainly due to computing.
• Relevant ‘real’ applications of mathematics
from these areas can be made accessible
to teenagers at a range of ability levels.
37
Maths in the real world
• Google worksheet
38
Industry Examples
• Through funding from the IMA a
competition is being held with YINI students
• There are being asked to give exemplars of
work that they are undertaking in their
placement, which uses and builds upon A
level Mathematics
– Example of stress fractures
39
Undergraduate Students’ Posters
• Examples found on FMN Site
– Curves of constant width
– Projective geometry
– Markovian spam filtering
40
New diplomas
The mathematics
(Level 3 Engineering diploma)
Support for teaching and learning
• MEI/FMN are in discussion with the Royal
Academy of Engineering – support might
involve:
– Online resources
– Exemplars using maths in context
– Professional development for teachers
41
Industry endorsement of
mathematical problems
• It could be possible to produce a resource
consisting of mathematical problems
endorsed by companies (e.g. Google,
IBM, Rolls Royce, Facebook) and make
them available through the MEI resources.
• Company endorsed worksheet
42
Discussion
• What type of resource would be useful to
you?
43
Example Questions of Mathematics in Other Degrees
Contained within this document are example mathematics questions from
university textbooks in subjects other than Mathematics, Engineering and
Physics. These are nearly all from material presented in first year courses.
Example 1 – CHEMISTRY
The potential energy V arising from the forces of interaction between two molecules is
often written in the form
⎛ ⎛ σ ⎞12 ⎛ σ ⎞6 ⎞
V = 4ε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟
⎜⎝ r ⎠ ⎝ r ⎠ ⎟
⎝
⎠
in which ε and σ are constants and r is the distance between the molecules. By
differentiating this expression with respect to r, show that the potential has a minimum
1
when r = 2 6 σ .
Solution Technique
Requires partial differentiation, algebra and indices.
Example 2 – CHEMISTRY
It is shown in elementary treatments of quantum mechanics that a particle that is
constrained to move along a portion of the x-axis between x = 0 and x = a can be
described by a wavefunction that has form
⎛ nπ x ⎞
Ψ ( x) = A sin ⎜
⎟
⎝ a ⎠
a
In this case the function is said to be normalised if ∫ Ψ 2 dx = 1 .
0
Show that this will be the case if A =
2
.
a
Solution Technique
Requires integration of a relatively difficult function involving sin2 and unknown
constants.
1
Example 3 – CHEMISTRY
The pH of a solution in which the hydrogen ion concentration is [H+] is defined by
pH = – log10[H+]
Show that the change in pH, ΔpH that results from a small change in [H+] of Δ[H+] is
proportional to the ration Δ[H+]/ [H+].
Solution Technique
Involves differentiation of log and use of local tangent approximation to the function.
Example 4 – CHEMISTRY
When a liquid is in equilibrium with its vapour, the vapour pressure p is related to the
temperature T by the Clausius-Clapeyron equation
d (ln p) ΔH e
=
dT
RT 2
In which ΔH e is the enthalpy of vapourisation and R is the gas constant. If the vapour
pressure is p0 when the temperature is T0 show that
⎛ ΔH e ⎛ 1 1 ⎞ ⎞
p = p0 exp ⎜⎜
⎜ − ⎟ ⎟⎟
⎝ R ⎝ T0 T ⎠ ⎠
Solution Technique
Separation of variables technique for solving differential equations, handling of initial
conditions and the ln/exp relationship.
Example 5 – ECONOMICS
The equations defining a model of two trading nations are given by
Y1 = C1 + I1* + X 1 − M 1
C1 = 0.8Y1 + 200
Y2 = C2 + I 2* + X 2 − M 2
C2 = 0.9Y2 + 100
M 1 = 0.2Y1
M 2 = 0.1Y1
Express this system in matrix form and hence write Y1 in terms of I1* and I2*.
Solution Techniques
First eliminate all variable other than Y1, Y2, I1* and I2* using elementary algebra. The two
remaining equations can then be written as a matrix equation and Cramer’s rule is used to
get the answer.
2
Example 6 – ECONOMICS
An individual’s utility function is given by U = 260 x1 + 310 x2 + 5 x1 x2 − 10 x12 − x2 2
Where x1 is the amount of leisure measures in hours per week and x2 is the earned income
measured in dollars per week. Find the values of x1 and x2 which maximize U. What is
the corresponding hourly rate of pay?
Solution Techniques
Needs partial differentiation with respect to the two variables and linear algebra to
identify the turning points. Tests for the nature of the turning points of functions of two
variables are also needed.
Example 7 – ECONOMICS
Given the utility function:
U = x10.25 x2 0.75
Determine the values of the marginal utilities:
∂U
∂U
and
∂x1
∂x2
Hence estimate the change in utility when x1 decreases from 100 to 99 and x2 increased
from 200 to 201.
Solution Technique
Partial differentiation and use of tangent plane to approximate the utility function locally.
Example 8 – ECONOMICS
A monopolistic producer of two goods G1 and G2 has a joint total cost function
TC = 10Q1 + Q1Q2 + 10Q2
Where Q1 and Q2 denote the quantities of G1 and G2 respectively. If P1 and P2 denote the
corresponding prices then the demand equations are
P1 = 50 – Q1 + Q2
P2 = 30 + 2Q1 – Q2.
Find the maximum profit if the firm is to produce a total of 15 goods of either type.
Solution Technique
An objective function is set up and then the technique of Lagrange Multipliers is used.
This involves partial derivatives and bringing in an additional dummy parameter.
3
Example 9 – ECONOMICS
The output levels of machinery, electricity and oil of a small country are 3000, 5000, and
2000 respectively.
Each unit of machinery requires inputs of 0.3 units of electricity and 0.3 units of oil.
Each unit of electricity requires inputs of 0.1 units of machinery and 0.2 units of oil.
Each unit of oil requires inputs of 0.2 units of machinery and 0.1 units of electricity.
Determine the machinery, electricity and oil available for export.
Solution Technique
⎛ 0 0.1 0.2 ⎞
With the matrix A given by A = ⎜⎜ 0.3 0 0.1 ⎟⎟ . The internal demand for each is given
⎜ 0.3 0.2 0 ⎟
⎝
⎠
⎛ 3000 ⎞
by the vector A ⎜⎜ 5000 ⎟⎟ . Therefore the amounts for export are the components in
⎜ 2000 ⎟
⎝
⎠
⎛ 3000 ⎞ ⎛ 3000 ⎞
A ⎜⎜ 5000 ⎟⎟ – ⎜⎜ 5000 ⎟⎟ .
⎜ 2000 ⎟ ⎜ 2000 ⎟
⎝
⎠ ⎝
⎠
Example 10 – GEOGRAPHY
Flooding
Suppose that we are interested in the frequency of flooding along a creek that runs
through a residential area. It would be useful to know how likely floods were whether we
were purchasing a house in the area, setting flood insurance premiums, or designing a
flood control project.
Floods can of course be of different magnitudes. The magnitude of an n-year flood is
such that it is exceeded with probability 1/n in any given year. Thus the probability of a
50-year flood in any given year is 1/50. A 100-year flood is larger, and occurs less
frequently; a 100-year flood occurs in any given year with probability 1/100.
What is the probability that there will be exactly one 50-year flood during the next 50
year period?
Solution Technique
The Poisson probability is again found by first recognising that the expected number of
floods during this period is equal to λ = 1 (if you have trouble deciding upon the correct
value of λ , it may be useful to realise that, because it is a mean, you can think in terms of
the binomial equivalent of np; in this case we have n = 50 years, and the probability of a
flood in a given year is 1/50, so that np = λ = 1). Then probability of observing exactly
4
one such flood is P(X=1) =
e −111
= 0.368 . The binomial approximation,
1!
⎛ 50 ⎞
⎜ ⎟ (1/ 50 )( 49 / 50 ) = 0.3716 ; this is actually the probability that there is precisely one
⎝ 1 ⎠
year in which (at least) one 50-year flood occurs.
Example 11 – GEOGRAPHY
Demographics
The world’s population grows at the rate of approximately 2% per year. If it is assumed
that the population growth is exponential, then the population t years from now will be
given by a function of the form P (t ) = P0 e0.02t , where P0 is the current population.
Assuming that this model of growth is correct, how long will it take for the world’s
population to double?
Solution Technique
Use of techniques to solve exponential growth.
Example 12 – LIFE SCIENCES
Marine Biology
When a fish swims up-stream at a speed v against a constant current vw, the energy it
expends in travelling to a point upstream is given by a function of the form
Cv k
E (v ) =
, where C > 0 and k > 2 is a number that depends on the species of fish
v − vw
involved.
a) Show that E(v) has exactly one critical number. Does it correspond to a relative
maximum or a relative minimum?
b) Note that the critical number in part (a) depends on k. Let F(k) be the critical
number. Sketch the graph of F(k). What can be said about F(k) if k is very large?
Example 13 – HUMAN SCIENCES
Concentration of Drug
The concentration of a drug in a patient’s blood t hours after an injection is decreasing at
the rate
−0.33t
C '(t ) =
mg/cm3 per hour
2
0.02t + 10
By how much does the concentration change over the first 4 hours after the injection?
5
Example 14 – SPORT SCIENCE
Pulse-rate on a run of long, slow distance
Most distance runners know that on a long, slow run, there is an immediate speed up of
the heart-rate, but then as the run progress, the heart rate slows down again, albeit not to
the resting level. As the run progresses, the heart-rate will increase as glycogen is used up
and the body turns to fat for calories. Converting fat to a useable energy source takes
more oxygen then burning glycogen, resulting in an increase respiration rate and heartrate.
We model this phenomenon with the function P (pulse-rate) below.
P (t ) =
130
t2
+ 2t + 25
t
−
2
150
+ 35e ,
t + 25
where P(t) is the pulse rate in beats per minute and t is the time in seconds.
Find the resting pulse-rate.
Solution Technique
Inserting t = 0 into the equation
P (0) =
130 25
+ 35 = 26 + 35 = 61 beats per minute
25
Example 15 – SPORT SCIENCE
Resting Energy Expenditure
The Resting Energy Expenditure (REE) for a person, in kilocalories/day, is calculated
from the formula:
REE = (40/7)*BSA*BMR
Where BSA is Body Surface Area and BMR is Nasal Metabolic Rate. BSA is given by:
BSA = 0.007184*M0.425*H0.725
Where M is the mass of the person (in kg) and H is the height (in cm). BMR is
determined by the individual’s age and sex in the table. The units of BMR are
kilocalories/hour.
6
Basal metabolic rate as a function of age and sex.
Age (years)
15-19
20-24
25-39
Female
163.2
152.4
151.5
Male
177.9
165.8
162
a) If Mike is 22 years old, 1.86 metre tall and has a mass of 78kg, what is his REE?
b) Georgina is 19 years old, her mass is 61kg and she is 1.62 metres tall. What is her
REE?
c) Howard is 23 years old, his mass is 68kg and his REE is 1670 kilocalories/day.
How tall is he, to the nearest cm?
d) Sanjay is 27 ears old, his REE is 1800 kilocalories/day and his height is 1.78
metre. What is Sanjay’s mass, to the nearest kg?
Answers
Mike’s REE is 1916 kilocalories/day
Georgina’s REE is 1537 kilocalories/ day
Howard is 167 cm tall
Sanjay’s mass is 77 kg.
Solution Technique
a) insert M = 78 and H = 186 into BSA equation to get BSA. Then insert BSA value
and BMR = 165.8 into REE equation.
b) insert M = 61 and H = 162 into BSA equation to get BSA. Then insert BSA value
and BMR = 163.2 into REE equation.
c) REE = 1670, BMR = 165.8, M = 68. Substitute BSA equation into REE equation
and solve for H.
1
⎛
⎞ 0.725
⎜ REE
⎟
× BMR × 0.007184 × M 0.425 ⎟
H=⎜
⎜ ⎛ 40 ⎞
⎟
⎜⎜ 7 ⎟
⎟
⎝⎝ ⎠
⎠
d) REE = 1670, BMR = 165.8, H = 178. Substitute BSA equation into REE equation
and solve for M.
1
⎛
⎞ 0.425
⎜ REE
⎟
× BMR × 0.007184 × H 0.725 ⎟
M=⎜
⎜ ⎛ 40 ⎞
⎟
⎜⎜ 7 ⎟
⎟
⎝⎝ ⎠
⎠
7
Example 16 – BUSINESS
Marginal Analysis
A manufacturer estimates that if x units of a particular commodity are produced, the total
cost will be C(x) ponds, where C ( x) = x3 − 24 x 2 + 350 x + 338
a) At what level of production will the marginal cost C’(x) be minimised?
C ( x)
b) At what level of production will the average cost A( x) =
be minimised?
x
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Core
Maths
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Geography
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Manufacturing and Materials Engineering M, U, AB, AC
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Engineering
AA
Electrical Engineering
D,E,F
Economics
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Design Technology
Y, Z
Civil Engineering
B
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Civil and Construction
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Computer Science
AD
Chemical Engineering
T, J
Aeronautical and Automotive
F Maths Complex Numbers
Proof
Coordinate Systems
Calculus
Matrices
Curve Sketching
Series
Hyper Trig Functions
Vectors
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Chemistry
K,P,Q
Actuarial Mathematics
Algebra
Trig
Exp and Logs
Coord Geometry
Parametric Equations
Vectors
Numerical Methods
Sequences and Series
Curve Sketching
Functions
Calculus
A
AE
Department
Business
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Motion Graphs
Const Accel. + SUVAT
Projectiles
Centre of Mass
Variable Acceleration
Uniform Motion in a Circle
Newton's Laws applied along a Line
Vector's and Newton's Laws in 2D
Collisions
Equilibrium of a Rigid Body
Energy, Work and Power
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Decision Graphs
Networks
Critical Path Analysis
Game Theory
Optimisation
Algorithms
Linear Programming
Simulation
Logic and Boolean Algebra
Stats
Correlation and Regression
The Binomial Distribution and probability
Exploring Data
Normal Distribution
Chi-Squared
Data Presentation
Discrete Random Variables
Probability
Hypothesis Testing
Poisson Distribution
Sampling/ Estimation
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Sport and Exercise Science
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Social Sciences
Sports Technology
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Operational Research
Physics
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Mechanical Engineering
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