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Use of A level mathematics in University Degrees and

in the workplace

Richard Lissaman Stephen Lee

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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.

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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

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So…

• Which university courses use A-level Mathematics and Further Mathematics?

• How much do they use? Which topics?

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Review of University Courses

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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

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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 APPLIEDmodules, thus an AS in FM is very useful!

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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 Seriesrecommended by the Higher Education Academy

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Economics

• Recommended text by a Business department for first year study

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Geography

• Recommended text by a Geography department for first year study

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• Very descriptive• Not ‘mathematicallySet 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

– 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

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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

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Examples from Geography

• Example 10 – Flooding– Poisson Distribution

• Example 11 – Demographics – Exponential Growth

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Examples from Human and Life Sciences

• Example 12 – Marine Biology

• Example 13 – Concentration of Drug

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Examples from Sports Science

• Example 14 – Resting Energy Expenditure

• Example 15 – Pulse-Rate on a long slow run

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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

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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.

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Maths in the real world

• Google worksheet

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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

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Undergraduate Students’ Posters

• Examples found on FMN Site – Curves of constant width– Projective geometry– Markovian spam filtering

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New diplomasThe 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

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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

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Discussion

• What type of resource would be useful to you?

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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

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4Vr rσ σε

⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

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

when r = 1

62 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

( ) sin n xx Aaπ⎛ ⎞Ψ = ⎜ ⎟

⎝ ⎠

In this case the function is said to be normalised if 2

0

1a

dxΨ =∫ .

Show that this will be the case if 2Aa

= .

Solution Technique Requires integration of a relatively difficult function involving sin2 and unknown constants.

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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

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(ln ) eHd pdT RT

Δ=

In which eHΔ is the enthalpy of vapourisation and R is the gas constant. If the vapour pressure is p0 when the temperature is T0 show that

00

1 1exp eHp pR T T

⎛ ⎞⎛ ⎞Δ= −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

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

*1 1 1 1 1

1 1

1 1

0.8 2000.2

Y C I X MC Y

M Y

= + + −= +

=

*2 2 2 2 2

2 2

2 1

0.9 1000.1

Y C I X MC Y

M Y

= + + −= +

=

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.

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Example 6 – ECONOMICS An individual’s utility function is given by 2 2

1 2 1 2 1 2260 310 5 10U x x x x x x= + + − − 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: 0.25 0.75

1 2U x x=

Determine the values of the marginal utilities: 1

Ux

∂∂

and 2

Ux∂∂

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.

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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

With the matrix A given by 0 0.1 0.2

0.3 0 0.10.3 0.2 0

A⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

. The internal demand for each is given

by the vector A300050002000

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

. Therefore the amounts for export are the components in

A300050002000

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

– 300050002000

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

.

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 1λ = (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

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one such flood is P(X=1) = 1 11 0.368

1!e−

= . The binomial approximation,

( )( )50 1/ 50 49 / 50 0.37161

⎛ ⎞ =⎜ ⎟⎝ ⎠

; this is actually the probability that there is precisely one

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 0.02

0( ) tP t P e= , 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

( )k

w

CvE vv v

=−

, where C > 0 and k > 2 is a number that depends on the species of fish

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

2

0.33'( )0.02 10

tC tt

−=

+ mg/cm3 per hour

By how much does the concentration change over the first 4 hours after the injection?

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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 heart-rate. We model this phenomenon with the function P (pulse-rate) below.

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150130 2 25

2( ) 3525

tt t

P t et

−+ +

= ++

,

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

130 25(0) 35 26 35 6125

P = + = + = beats per minute

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.

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Basal metabolic rate as a function of age and sex.

Age (years) Female Male 15-19 163.2 177.9 20-24 152.4 165.8 25-39 151.5 162

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.

10.725

0.425REEH BMR 0.007184 M407

⎛ ⎞⎜ ⎟⎜ ⎟= × × ×⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

d) REE = 1670, BMR = 165.8, H = 178. Substitute BSA equation into REE equation and solve for M.

10.425

0.725REEM BMR 0.007184 H407

⎛ ⎞⎜ ⎟⎜ ⎟= × × ×⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

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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 3 2( ) 24 350 338C x x x x= − + + a) At what level of production will the marginal cost C’(x) be minimised?

b) At what level of production will the average cost ( )( ) C xA xx

= be minimised?

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