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APPLIED MATHEMATICSAdvanced Higher

Second edition – published December 1999

NOTE OF CHANGES TO ARRANGEMENTSSECOND EDITION PUBLISHED WINTER 1999

COURSE TITLE: Applied Mathematics (Advanced Higher)

COURSE NUMBER: C057 13

National Course Specification

Detailed Content

The content of Statistics 1 (AH) and 2 (AH), as detailed on pages 5 to 12, has been amended. Most ofthe changes involve rewording of the sentences in the comments column but there are also a fewchanges in the content column. Under the section Assessment: Details of the instruments for externalassessment, the percentage of opportunities at grade C is specified.

National Unit Specification:

D321 13 Mathematics 1 (AH)

Both PCs in Outcome 1 have been reworded. PC (a) of Outcome 2 has been split into three PCs withconsequent changes to the wording of the outcome. In Outcome 3 PC (c) has been deleted and PC (b)split into two PCs. PC (a) and the outcome itself have been reworded. PC (b) in Outcome 4 has beendeleted and PC (a) split into three PCs. Outcome 5 and PC (a) have been reworded.

The guidance on approaches to assessment for the unit has been reworded to clarify that the thresholdof attainment for the unit assessment has been removed, and that therefore the only way to achievesuccess on the unit assessment is through the achievement of the threshold scores for all the outcomesin the unit.

D326 13 Statistics 1 (AH)

PC (d) of Outcome 2 and PC (b) of Outcome 4 have been deleted. There has also been some minorrewording to several PCs and Outcomes.


D327 13 Mechanics 1 (AH)

All of the PCs have been amended or reworded.


National Unit Specification (cont)

D328 13 Numerical Analysis 1 (AH)

Most of the PCs in Outcomes 1, 2 and 3 have been reworded.


D330 13 Statistics 2 (AH)

The statement of Outcome 1 and PC (b) have been reworded. In Outcome 3 PC (b) has beenreworded. PC (a) of Outcome 4 has been reworded.


D329 13 Numerical Analysis 2 (AH)

PC (b) of Outcome 1 has been deleted and PC (a) and (d) reworded. The statement of Outcome 2 andboth PCs of Outcome 4 have been reworded.


D331 13 Mechanics 2 ( AH )

PC (c) of Outcome 1, PC (b) of Outcome 2 and PC (b) of Outcome 5 have been deleted. PC (a) and(b) of Outcome 1, PC (a) of Outcome 2, PC (b) of Outcome 3 and PC (a) of Outcome 5 have beenreworded.


Administrative Information

Publication date: December 1999

Source: Scottish Qualifications Authority

Version: 02

© Scottish Qualifications Authority 1999

This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived fromreproduction and that, if reproduced in part, the source is acknowledged.

Additional copies of this course specification (including unit specifications) can be purchased from the Scottish QualificationsAuthority for £7.50. Note: Unit specifications can be purchased individually for £2.50 (minimum order £5).

1

National Course Specification

APPLIED MATHEMATICS (ADVANCED HIGHER)

COURSE NUMBER C057 13

COURSE STRUCTURE

This course has two mandatory and one optional units as follows:

Mandatory units

D326 13 Statistics 1 (AH) 1 credit (40 hours)D330 13or

Statistics 2 (AH) 1 credit (40 hours)

D327 13 Mechanics 1 (AH) 1 credit (40 hours)D331 13or

Mechanics 2 (AH) 1 credit (40 hours)

D328 13 Numerical Analysis 1 (AH) 1 credit (40 hours)D329 13 Numerical Analysis 2 (AH) 1 credit (40 hours)

Optional units – choose one from:

D326 13 Statistics 1 (AH) 1 credit (40 hours)D327 13 Mechanics 1(AH) 1 credit (40 hours)D328 13 Numerical Analysis 1 (AH) 1 credit (40 hours)D321 13 Mathematics 1 (AH) 1 credit (40 hours)

All courses include a further 40 hours for induction, extending the range of learning and teachingapproaches, additional support, consolidation, integration of learning and preparation for externalassessment. This time is an important element of the course and advice on the use of the overall 160hours is included in the course details. Candidates who are taking Advanced Higher Mathematics and Advanced Higher AppliedMathematics must complete a total of six different units at Advanced Higher level.

Applied Mathematics: Advanced Higher Course 2

National Course Specification: general information (cont)

COURSE Applied Mathematics (Advanced Higher)

RECOMMENDED ENTRY

While entry is at the discretion of the centre, candidates will normally be expected to have attainedone of the following: • Higher Mathematics course award or its component units• equivalent Statistics 1 (AH) assumes knowledge of Outcome 2 of Statistics (H), while experience of Outcomes 1and 4 of Statistics (H) would be advantageous.

CORE SKILLS

Core skills for Advanced Higher remain subject to confirmation and details will be available at a laterdate.

For information about the automatic certification of core skills for any individual unit in this course,please refer to the general information section at the beginning of the unit.

Additional information about core skills is published in Automatic Certification of Core Skills inNational Qualifications (SQA, 1999).


National Course Specification: course details (cont)


RATIONALE

As with all mathematics courses, Advanced Higher Applied Mathematics aims to build upon andextend candidates’ mathematical skills, knowledge and understanding in a way that recognisesproblem solving as an essential skill. Through the study of units chosen from Statistics, NumericalAnalysis and Mechanics, the focus within the course is placed firmly on applications of mathematicsto real-life contexts and the formulation and interpretation of mathematical models. Each appliedmathematics topic brings its own unique brand of application and extended activity to the course andprovides the opportunity to demonstrate the advantages to be gained from the power of calculatorsand computer software in real-life situations.

Because of the importance of these features, the grade descriptions for Advanced Higher AppliedMathematics emphasise the need for candidates to undertake extended thinking and decision makingso as to be able to model, solve problems and apply mathematical knowledge. The use of courseworktasks, therefore, to practise problem solving as set out in the grade descriptions and the involvementin recommended practical activities, are both strongly encouraged.

The choice of optional course component units available at this level is wider than at lower levels.This breadth of choice is provided in response to the variety of candidates’ aspirations in highereducation studies or areas of employment. The course satisfies the needs of a wide range of interests,regardless of whether or not a specialism in mathematics is the primary intention. The course offersdepth of applied mathematical experience and, thereby, achieves relevance to further study oremployment in the areas of mathematical and physical sciences, computer science, engineering,biological and social sciences, medicine, accounting, business and management. For these purposes,the course can be taken as an alternative to the Advanced Higher Mathematics course. However,when the Advanced Higher Applied Mathematics course is taken in addition to the Advanced HigherMathematics course, an opportunity is offered for the candidate to acquire exceptional breadth anddepth of mathematical experience.

The course offers candidates, in an interesting and enjoyable manner, an enhanced awareness of therange and power of mathematics and the importance of mathematical applications to society ingeneral. COURSE CONTENT

The syllabus is designed to allow candidates the opportunity to study at least one area of appliedmathematics in depth. The opportunity to specialise, introduced at Higher level through the optionalstatistics unit, is now extended to three areas of applied mathematics. Two units in each of Statistics,Numerical Analysis and Mechanics provide the opportunity to study one of these areas in depth.Whatever applied core is chosen, the two units have been designed in a progressive way, with thefirst unit of each core providing a rounded experience of the topic for those candidates who choose iteither as the optional unit in the Advanced Higher Mathematics or Advanced Higher AppliedMathematics courses or as a free-standing unit and do not wish to proceed to study the topic furtherin the second unit.


National Course Specification: course details


Additionally, the course makes demands over and above the requirements of individual units. Someof the 40 hours of flexibility time should be used to ensure that candidates satisfy the gradedescriptions for mathematics courses that involve solving problems/carrying out assignments andundertaking practical activities, thereby encouraging more extended thinking and decision making.Candidates should be exposed to coursework tasks which require them to interpret problems, selectappropriate strategies, come to conclusions and communicate the conclusions intelligibly.

In assessments candidates should be required to show their working in carrying out algorithms andprocesses.




Detailed content

The content listed below should be covered in teaching the course. All of this content will be subject to sampling in the external assessment. Where commentis offered, this is intended to help in the effective teaching of the course.

References in this style indicate the depth of treatment appropriate to grades A and B.

CONTENT COMMENTStatistics 1 (AH)

Conditional probability and expectationappreciate the existence of dependent events

calculate conditional probability

solve problems using conditional probability

Conditional probability can be calculated from first principles in every casebut the following formulae may prove useful in solving problems involvingdependent events.

P(E|F) =

P(E and F) = P(E|F)P(F) = P(F|E)P(E)

P(F) =

P(Ej |F) = for mutually exclusive and exhaustive

events Ei [A/B]

solve a problem involving Bayes’ theorem [A/B] Bayes’ theorem is used to ‘reverse the dependence’, so that if theprobability of F given that Ej has occurred is known then the probability ofEj given that F has occurred can be found.

0)F(P , P(F)

F) and P(E >

∑i

ii ))P(EE|P(F

∑i

ii

jj

) )P(EE|P(F

))P(EE|P(F




CONTENT COMMENTapply the laws of expectation and variance:

use the results E(aX + b) = aE(X) + b and E(X ± Y) = E(X) ± E(Y)When considering the laws of expectation and variance an intuitiveapproach may be taken, perhaps as follows:It should be clear from graphical considerations that if a random variablehas a scale factor applied followed by a translation then so must thecorresponding expectation (or mean) giving E(aX + b) = aE(X) + b.

use the results V(aX + b) = a2V(X)and V(X ± Y) = V(X) + V(Y) if X and Y are independent

It may be equally intuitive that variability (dispersion) is not affected bytranslation and a little numerical investigation should demonstrate thatV(aX + b) = a2V(X).It follows that V(−X) = V(X)

Probability distributionscalculate probabilities using the binomial, Poisson and normal distributions

use standard results for the mean and standard deviation of binomial,Poisson and normal distributions

use the Poisson or normal distribution to approximate the binomialdistribution

use a correction for continuity when required

The difference between discrete and continuous distributions should bediscussed. The former may be exemplified by the uniform, binomial andPoisson distributions and the latter by the uniform (a, b) and normaldistributions.

The process of standardising data to produce standard scores is an importantone and should be discussed.

Candidates should be introduced to the probability functions and theformulae for the mean and standard deviation for each of the discretedistributions.




CONTENT COMMENTapply an appropriate approximation (normal or Poisson) to a binomialdistribution and demonstrate the correct use of a continuity correctionif appropriate [A/B]

Candidates should be able to calculate probabilities with or without tables.It is often desirable to approximate binomial probabilities. The reasons forusing either the Poisson or normal should be exemplified, particularly theuse of a continuity correction in the latter case.One rule of thumb in current use is to use the normal approximationif np and nq are >5 and the Poisson otherwise, as long as p is small.

solve problems involving the sum or difference of two independentnormal variates [A/B]Sampling methodsdistinguish between different sampling methods:

appreciate the difference between sample and census, and describerandom, stratified, cluster, systematic and quota sampling

The difference between a sample and a census should be stressed. Theadvantages and disadvantages of each type of sampling should be discussed,noting quota is not an example of random sampling.

use the central limit theorem and distribution of the sample mean andsample proportion

The central limit theorem states that for sufficiently large n the distributionof the sample mean is normal, irrespective of the distribution of the samplevariates.Candidates should be able to show that sample mean has expectation µ and

standard error (standard deviation) and that the sample proportion has

mean p and standard error (deviation) for large n and q = 1 – p.

use the sample mean and standard deviation to estimate the population meanand standard deviation and the sample proportion to estimate the populationproportion

In elementary sampling theory the population mean/proportion is estimatedby the sample mean/proportion and the population variance by the samplevariance

given by

n

σ

n

pq

.)(1

1 22 xxn

s i −−= ∑




CONTENT COMMENTobtain a confidence interval for the population mean [A/B] An approximate 95% confidence interval for the population mean is given

by for large n.

We may interpret this as saying that if we take a large number of samplesand compute a confidence interval for each then 95% of these intervalswould be expected to contain the population mean.

It is also common practice to use a 99% confidence interval where 1.96 isreplaced by 2.58.

obtain a confidence interval for the population proportion [A/B] An approximate 95% confidence interval for the population proportion p is

given by for large n, where and is the sampleproportion.

Hypothesis Testingunderstand the terms null hypothesis, alternative hypothesis, one-tail test,two-tail test, distribution under H0, test statistic, critical region, level ofsignificance, p-value, reject H0 and accept H0

In a statistical investigation we often put forward a hypothesis – the nullhypothesis H0 – about a population parameter. In order to test thishypothesis we take a sample from the population and perform a statisticaltest. If we decide to reject H0, we do so in favour of an alternativehypothesis H1.

use a z-test on a statistical hypothesis:decide on a significance levelstate one-tail or two-tail testdetermine the p-valuedraw appropriate conclusions

A z-test should be used to test whether the population mean is equal to somespecific value, where the population variable is normally distributed withknown variance.eg Ho: µ=µo versus H1: µ≠µo for a two-tail test.

n

sx 96.1 ±

n

qpp

ˆˆ 96.1 ˆ ± pq ˆ1ˆ −= p̂




CONTENT COMMENTStatistics 2 (AH)

Simple control chartsconstruct a control chart for the sample mean or proportion

use and interpret such a chart

Two basic charts should be used as an introduction to the subject of control.

A 3-sigma chart has action limits when a sample mean or proportion goesbeyond three standard deviations from the expected value. The process shouldbe investigated for special causes of variation.

A 3-sigma Shewart chart has control limits drawn 3 standard deviations eitherside of the expected value.The control limits for the sample mean are

Similarly, a p-chart for the proportion has control limitswhere q = 1-p

Further hypothesis testingcarry out a chi-squared goodness-of-fit test

carry out a chi-squared test for association in a contingency table

The chi-squared statistic can be used to test goodness-of-fit ie to establishwhether a set of observed frequencies differ significantly from those of aspecified discrete probability distribution. It can also be used to test forassociation between two factors in a contingency table. Tables of the chi-squared distribution are required for this test. Candidates require to beintroduced to the notion of degrees of freedom and to the restriction of 1being the minimum allowable expected frequency with no more than 20% ofexpected frequencies being less than 5 as a general guideline for validapplication without loss of degrees of freedom. Candidates should be awarethat conclusions should be made with caution when any frequencies are low.

n

σµ 3±

n

pqp 3±




CONTENT COMMENTcarry out a sign test The sign test can be used as a non-parametric test to make inferences about

the median of a single population. It can also be used with paired data to testthat the population median difference is zero. The test relies on binomialprobabilities since a sign can be either positive or negative.

carry out a Mann-Whitney test

compare two different methods of applying the Mann-Whitney test[A/B]

The Mann-Whitney test is a non-parametric test used to compare the mediansof two populations using independent samples.Candidates should be able to find the p-value from first principles, fromtables or by using the normal approximation (formulae given) with continuitycorrection.

This test assumes that the two distributions have the same shape and hencethe same variance.

t-distributiondetermine a confidence interval for the population mean, given a randomsample from a normal population with unknown variance

use the appropriate number of degrees of freedom

carry out a one sample t-test for the population mean

The t-distribution was discovered by William S. Gosset (c.1900) during histime as a scientist in the Guinness breweries in Dublin. He published hisfindings under the pseudonym of ‘student’ which explains why thedistribution is often called the student’s t-distribution.

The t-distribution is used for small samples from a normal population withunknown variance which can be estimated by the sample variance. It can alsobe used with paired data to test that the population mean difference is zero.Without the underlying assumption of normality there is very little statisticalanalysis that can be done with a small sample.




CONTENT COMMENTAnalyse the relationship between two variablestest the significance of a product moment correlation coefficient Assuming that a plot of y against x shows that a linear model is appropriate,

we calculate the sample product moment correlation coefficient (r) and testthe null hypothesis that the population product moment correlation coefficient(p′m′c′c′) ρ = 0 using the test statistic

consider the linear model Yi = α + βxi +∈i

use a residual plot to check the model assumptions E (∈i) = 0 and V(∈i) = σ2

by calculating the residuals (yi – (a + bxi)) and plotting them against fittedvalues (a + bxi)

use a t-test to test the fit of this model

The model Yi = α + βxi +∈i assumes that ∈i are independent and that theE(∈i ) = 0 and V(∈i) = σ2 which is constant for all xi. Ideally the plot ofresiduals against fitted values should show a random scatter centred on zero.If this is not the case then the model may be inappropriate (perhaps non-linear) or the data may require to be transformed to restore constant variance.

A further assumption is that ∈i ~ N(0, σ2) permitting the construction of bothi) a prediction interval for an individual response andii) a confidence interval for a mean response.

It can be shown that the sum of squared residuals,

and that an estimate of σ2 is s2 =

We may also use a t-test to test whether the slope parameter β in the model iszero, using the test statistic

xxSs

bt =

2

1 2

−−

=

n

r

rt

xx

xyyy S

SSSSR

2)(−=

.2−n

SSR




CONTENT COMMENTconstruct a prediction interval for an individual response or confidenceinterval for mean response, as appropriate

In a prediction or confidence interval the reliability of the estimate dependson the sample size, the variability in the sample and the value of x.

A prediction interval for Yi is given by

A confidence interval for µYi

is given by

xx

ini S

xx

nstY

2

2,2/

)(11ˆ −++± −α

xx

ini S

xx

nstY

2

2,2/

)(1ˆ −+± −α




CONTENT COMMENTNumerical Analysis 1 (AH)

Taylor polynomialsunderstand and use the notations n! and f (n) for n ∈ N

know the Taylor polynomial of degree n for a given function in the form Simple functions involving square roots, rational functions, trigonometric,exponential and logarithmic functions will be approximated.

obtain the Taylor polynomial of a function about a given point

approximate a suitable function, f, near the point x = a, with A graph-plotting package should be used to display f and the polynomialapproximations.

understand the term truncation error in function approximation

determine the magnitude of the principal truncation error when a function isapproximated by a Taylor polynomial

When discussing truncation error, consideration should be given to thenumber of decimal places to be carried in the calculation. The idea of round-off error and guard figures in a calculation should be introduced.

approximate a function value near to a given point giving the principaltruncation error in the approximation

use first order Taylor polynomials to assess the sensitivity of functionvalues to small changes in the independent variable

Ability to discuss the significance of the sensitivity of function valuesto small changes in the independent variable [A/B].

n(n)

axn

(a)fax

afa)a)(x' ff(a) )(

!... )(

!2

)("( 2 −+−+−+

n(n)

hn

(a)fh

a" f(a)h' ff(a)h)f(a

!...

!2

)( 2 +++≈+




CONTENT COMMENTInterpolate dataunderstand the notation ∆n

rf for r and n ∈ N and be able to construct afinite difference table for a function tabulated at equal intervals

Errors in finite difference tables should be covered in examples.Ability to obtain relations involving differences, eg∆3

0 3 2 1 03 3f f f f f= − + − [A/B].

know that the nth differences of an nth degree polynomial are constant

understand the terms interpolate and extrapolate

know the notation , p ∈ R, r ∈ N

know, establish and be able to use the Newton forward differenceinterpolation formula

use the sigma notation

Ability to derive the Newton forward difference formula [A/B].

know the Lagrange interpolation formula ,

where

and use the formula for n ≤ 3

r

p

... 321 0

30

200 +∆

+∆

+∆

+= f

pf

pf

pff p

i

n

iin yxLxp )()(

0∑=

=

))...()()...()((

))...()()...()(()(

1110

1110

niiiiiii

niii xxxxxxxxxx

xxxxxxxxxxxL

−−−−−−−−−−=

+−

+−




CONTENT COMMENTNumerical integrationestablish the trapezium rule over a strip and the corresponding principaltruncation error

Obtain the truncation error both in terms of derivatives and in terms of finitedifferences.The derivation of the principal truncation error in the trapezium rule[A/B].

know and establish the composite trapezium rule

understand how the principal truncation error in the composite trapeziumrule can be obtained and estimate the error when the rule is used to estimatea given definite integral

Round-off errors in function values should be discussed; candidates shouldbe able to estimate their effect on the approximation to the integral. Thenumber of figures to be carried in the computation should be considered.

know and establish Simpson’s composite rule

understand how the principal truncation error in the composite Simpson’srule can be obtained and estimate the error when the rule is used to estimatea given definite integral

The principal truncation error need not be derived in this case. As with thetrapezium rule, round-off errors should be discussed.

know the meaning of the order of a truncation error, the notation O(hn) andthe orders of the truncation errors in both the trapezium rule and Simpson’srule

use the trapezium rule and Simpson’s rule to estimate irregular areas fromtabulated data and to approximate definite integrals

establish and perform one application of Richardson’s formula to improvethe accuracy of the trapezium rule and of Simpson’s rule by interval halving

Ability to establish Richardson’s formula [A/B].




CONTENT COMMENTNumerical Analysis 2 (AH)

Non-linear equationsunderstand the meaning of the terms sequence, nth term, limit as n tends toinfinity, recurrence relation

identify a linear recurrence relation and the order of any recurrence relation

illustrate a first order recurrence relation using a cobweb diagram

obtain the fixed point and know the condition for convergence of a first orderrecurrence relation

locate the roots of an equation f(x) = 0, using the interval bisection method,and calculate the number of applications of the method required to give aroot to a prescribed accuracy

obtain the condition for convergence of xn+1 = g(xn) Ability to derive the condition for convergence [A/B].

solve an equation f(x) = 0 using simple iteration

understand what is meant by and determine the order of convergence of aniterative process

know and establish the Newton-Raphson method, both graphically and usingTaylor polynomials, and know the order of convergence of the method

Ability to derive the Newton-Raphson method [A/B].

solve an equation f(x) = 0 using the Newton-Raphson method

demonstrate ill-conditioning in the case of an equation with nearly equalroots

Understanding of ill-conditioning [A/B].




CONTENT COMMENTGaussian eliminationknow the meaning of the terms matrix, square matrix, diagonal matrix,triangular matrix, identity and null matrix

determine the product of a matrix and a scalar and of two matrices

know what is meant by the inverse of a matrix

know the meaning of the terms singular matrix and ill-conditioned matrix Ability to identify ill-conditioning in a matrix and to show appreciationof its significance. Reference should be made to the entries in theupper triangular matrix produced in the row reduction. [A/B]

understand the need for pivoting when solving a system of equations usingGaussian elimination

Systems larger than 3 x 3 need not be considered.

solve systems of linear equations using Gaussian elimination with partialpivoting

extend the method of Gaussian elimination with partial pivoting to obtain theinverse of a matrix

Matrix iterative techniquessolve a system of linear equations using the Jacobi and Gauss-Seidelmethods

know the meaning of diagonal dominance and be able to state its relationshipto the convergence of the Jacobi and Gauss-Seidel methods




CONTENT COMMENTApproximate the solution of first order ordinary differential equationsknow and establish Euler’s method for solving a first order ordinarydifferential equation for which initial values are given

Candidates should have experience of computing Euler’s method both byhand and by using a simple program.

know and establish the order of the global truncation error in Euler’s method Ability to derive the order of the global truncation error in Euler’smethod. [A/B]

know and establish the predictor-corrector formulae for solving first orderordinary differential equations in which Euler’s method is used as predictorand the trapezium rule as corrector (ie, the modified Euler method)

use both Euler’s method and the modified Euler method to solve suitabledifferential equationsErrors in numerical computationknow the terms significant figures, measurement error, round-off error,absolute error and relative error

express the sum or product of two data values to an appropriate degree ofaccuracy

Ability to compare errors due to truncation inherent in a process andround-off in the computation. [A/B]

calculate error bounds for given expressions

know that there is a loss of precision when two nearly equal quantities aresubtracted

express a polynomial in nested form and understand why this form ispreferred in computation

use synthetic division to divide a polynomial by a linear factor




CONTENT COMMENTMechanics 1 (AH)

Motion in a straight lineknow the meaning of position, displacement, velocity, acceleration, uniformspeed, uniform acceleration, scalar quantity, vector quantity

Concepts of position, velocity and acceleration should be introduced usingvectors.

Candidates should be very aware of the distinction between scalar andvector quantities, particularly in the case of speed and velocity.

draw, interpret and use distance/time, velocity/time and acceleration/timegraphs

Candidates should be able to draw these graphs from numerical or graphicaldata.

know that the area under a velocity/time graph represents the distancetravelled

know the rates of change andCandidates should be familiar with the dot notation for differentiation withrespect to time.

derive, by calculus methods, and use the equations governing motion in astraight line with constant acceleration, namely:

and from these,

Candidates need to appreciate that these equations are for motion withconstant acceleration only. The general technique is to use calculus.

solve analytically problems involving motion in one dimension underconstant acceleration, including vertical motion under constant gravity

xdt

dxv ==

xvdt

xd

dt

dv

dt

xda =====

2

2

221 atuts +=,atuv +=

2

)( tvus

+=,222 asuv +=




CONTENT COMMENTsolve problems involving motion in one dimension where the acceleration is

dependent on time, ie

Position, velocity and acceleration vectors including relative motionknow the meaning of the terms relative position, relative velocity andrelative acceleration, air speed, ground speed and nearest approach

be familiar with the notation: for the position vector of P for the velocity vector of P for the acceleration vector of P

for the position vector of Q relative to P for the velocity of Q relative to P for the acceleration of Q relative to P

resolve vectors into components in two and three dimensions

differentiate and integrate vector functions of time

use position, velocity and acceleration vectors and their components in twoand three dimensions; these vectors may be functions of time

This requires emphasis.

apply position, velocity and acceleration vectors to solve practical problems,including problems on the navigation of ships and aircraft and on the effectof winds and currents

Candidates should be able to solve such problems both by usingtrigonometric calculations in triangles and by vector components.Solutions by scale drawing would not be accepted

solve problems involving collision courses and nearest approach

f(t)dt

dva ==

PrPP rv =

PPP rva ==

PQ rrPQ −=→

PQPQ rrvv −=−PQPQPQ rrvvaa −=−=−




CONTENT COMMENTMotion of projectiles in a vertical planeknow the meaning of the terms projectile, velocity and angle of projection,trajectory, time of flight, range and constant gravity

Candidates also require to know how to resolve velocity into its horizontaland vertical components.

solve the vector equation to obtain in terms of its horizontaland vertical components

The vector approach is particularly recommended.

obtain and solve the equations of motion obtainingexpressions for x and y in any particular case

find the time of flight, greatest height reached and range of a projectile Only range on the horizontal plane through the point of projection isrequired.

find the maximum range of a projectile on a horizontal plane and the angleof projection to achieve this

find, and use, the equation of the trajectory of a projectile Candidates should appreciate that this trajectory is a parabola.

solve problems in two-dimensional motion involving projectiles under aconstant gravitational force and neglecting air resistance

Applications from ballistics and sport may be included and vectorapproaches should be used where appropriate.

Force and Newton’s laws of motionunderstand the terms mass, force, weight, momentum, balanced andunbalanced forces, resultant force, equilibrium, resistive forces

know Newton’s first and third laws of motion

resolve forces in two dimensions to find their components Resolution of velocities, etc., has been covered in previous sections.

jgr −= r

, ,0 gyx −==, , yx




CONTENT COMMENTcombine forces to find resultant force

understand the concept of static and dynamic friction and limiting friction

understand the terms frictional force, normal reaction, coefficient of frictionµ, angle of friction λ, and know the equations F = µ R and µ = tan λ

Balanced, unbalanced forces and equilibrium could arise here.Candidates should understand that for stationary bodies, F ≤ µR.

solve problems involving a particle or body in equilibrium under the actionof certain forces

Forces could include weight, normal reaction, friction, tension in an elasticstring, etc.

know Newton’s second law of motion, that force is the rate of change ofmomentum, and derive the equation F = ma

use this equation to form equations of motion to model practical problemson motion in a straight line

solve such equations modelling motion in one dimension, including caseswhere the acceleration is dependent on time

solve problems involving friction and problems on inclined planes Both rough and smooth planes are required.




CONTENT COMMENTMechanics 2 (AH)

Motion in a horizontal circle with uniform angular velocityknow the meaning of the terms angular velocity and angular acceleration

Know that for motion in a circle of radius r, the radial and tangentialcomponents of velocity are 0 and respectively, and of acceleration are and respectively, where and are the unit vectors in the radial and tangentialdirections, respectively

Vectors should be used to establish these, starting from , where r is constant and θ is varying.

know the particular case where ω being constant, when theequations are

from which

and

apply these equations to motion in a horizontal circle with uniform angularvelocity including skidding and banking and other applications

Examples should include motion of cars round circular bends, with skiddingand banking, the ‘wall of death’, the conical pendulum, etc.

θθerr

2r eθ− θθer jie θ+θ= sin cosr

jie θ+θ−=θ cos sin

,t ω=θ

jt rit ra

jt rit rvjt rit rr

)( sin)( cos

)( cos)( sin)( sin)( cos

22 ωω−ωω−=ωω+ωω−=

ω+ω=

r

vrra

rrv2

2 =θ=ω=

θ=ω=2

ra 2ω−=

jrirr θθ sin cos +=




CONTENT COMMENTknow Newton’s inverse square law of gravitation, namely that the magnitudeof the gravitational force of attraction between two particles is inverselyproportional to the square of the distance between the two particles

apply this to simplified examples of motion of satellites and moons Circular orbits only.

find the time for one orbit, height above surface, etcSimple harmonic motionknow the definition of simple harmonic motion (SHM) and the meaning ofthe terms oscillation, centre of oscillation, period, amplitude, frequency

know that SHM can be modelled by the equation

know the solutions and the special casesand of the SHM equation

At this stage these solutions can be verified or established from rotating round a circle. Solution of secondorder differential equations is not required.

know and be able to verify that , where v x=

maximum speed is ωa, the magnitude of the maximum acceleration isω2a and when and where these arise

Proof using differential equations is not required here but will arise in thesection of work on motion in a straight line later in this unit.

know the meaning of the term tension in the context of elastic strings andsprings

xx 2 ω−=

)( sin α+ω= t ax )( sin t ax ω=),( cos t ax ω=

)( 2222 xav −=ω

ωπ2=T

jt ait ar )( sin) cos ω+ω=




CONTENT COMMENTknow Hooke’s law, the meaning of the terms natural length, l, modulus ofelasticity, λ, and stiffness constant, k, and the connection between them,λ = kl

know the equation of motion of an oscillating mass and the meaning of theterm position of equilibrium

apply the above to the solution of problems involving SHM These will include problems involving elastic strings and springs, and smallamplitude oscillations of a simple pendulum but not the compoundpendulum.

Principles of momentum and impulseknow that force is the rate of change of momentum This was introduced in Mechanics 1 (AH).

know that impulse is change in momentum

ie

understand the concept of conservation of linear momentum

solve problems on linear motion such as motion in lifts, recoil of a gun,pile-drivers, etc.

The equation F = ma is again involved here. Equations of motion withconstant acceleration could recur.

dtFumvmI ∫=−=




CONTENT COMMENTPrinciples of work, power and energyknow the meaning of the terms work, power, potential energy, kinetic energy

understand the concept of work Candidates should appreciate that work can be done by or against a force.

calculate the work done by a constant force in one and two dimensions, ie,W = Fd (one dimension); W = F.d (two dimensions)

calculate the work done in rectilinear motion by a variable force using

integration, ie W = ∫F.idx ; W = ∫F.vdt , where

understand the concept of power as the rate of doing work,

ie, (constant force), and apply this in practical examples

Examples can be taken from transport, sport, fairgrounds, etc.

understand the concept of energy and the difference between kinetic (EK) andpotential (EP) energy

know that EK

know that the potential energy associated with:a. a uniform gravitational field is EP = mghb. Hooke’s law is EP = (extension)2

c. Newton’s inverse square law is EP =Link with simple harmonic motion.Link with motion in a horizontal circle.

idt

dxv =

vFdt

dWP .==

221 mv=

k21

r

GMm




CONTENT COMMENTunderstand and apply the work–energy principle

understand the meaning of conservative forces like gravity, and non-conservative forces like friction

know and apply the energy equation EK + EP = constant, including to thesituation of motion in a vertical circle

Motion in a straight line, where the solution of first order differentialequations is required

know that as well as

use Newton’s law of motion, F = ma, to form first order differentialequations to model practical problems, where the acceleration is dependenton displacement or velocity,

ie

solve such differential equations by the method of separation of variables It may be necessary to teach this solution technique, depending on themathematical background of the candidates.Examples will be straightforward with integrals which are covered inMathematics 1, 2 (AH). If more complex, then the anti-derivative will begiven.

dx

dvva =

dt

dv

),(vfdt

dv = ),(xfdx

dvv = )(vf

dx

dvv =




CONTENT COMMENT

derive the equation v a x2 2 2 2= −ω ( ) by solving

know the meaning of the terms terminal velocity, escape velocity andresistance per unit mass and solve problems involving differential equations

and incorporating any of these terms or making use of

This section can involve knowledge and skills from other topics within thisunit.

x dx

dvv 2ω−=

v

pF =




CONTENT COMMENT TEACHING NOTESMathematics 1 (AH)

Algebra

know and use the notation n!, nrC and

know the results

and

eg Calculate

eg Solve, for n ∈ N,

Candidates should be aware of the size of n! forsmall values of n, and that calculator resultsconsequently are often inaccurate (especially if the

formula is used). These results

can be established numerically or from Pascal’striangle. This will be linked with elementarynumber theory in Mathematics 2 (AH), where theyprovide an opportunity to introduce the concept ofdirect proof.

know Pascal’s triangle Pascal’s triangle should be extended up to n = 7. For assessment purposes n ≤ 5.

know and use the binomial theorem

, for r, n ∈ N

eg Expand (x + 3)4

eg Expand (2u - 3v)5 [A/B]

In the work on series and complex numbers (deMoivre’s theorem) the binomial theorem will beextended to integer and rational indices.

)!(!

!

rnr

nCr

n

−=

rn

−=

rnn

rn

+

=

=

− rn

rn

rn 1

1

rrnn

r

n barnba −

=∑

=+

0

)(

58

152

=

n




CONTENT COMMENT TEACHING NOTES

evaluate specific terms in a binomial expansion eg Find the term in x7 in

express a proper rational function as a sum ofpartial fractions (denominator of degree at most3 and easily factorised)

eg Express in partial fractions.The denominator may include a repeated linearfactor or an irreducible quadratic factor. This isalso required for integration of rational functionsand useful for graph sketching when asymptotesare present.

include cases where an improper rationalfunction is reduced to a polynomial and aproper rational function by division orotherwise [A/B]

eg Express in partial

fractions [A/B].

When the degree of the numerator of the rationalfunction exceeds that of the denominator by 1,non-vertical asymptotes occur.

92

+

xx

2431

105

xx

x

−−−

3)1)((

222 23

+−+−+

xx

xxx




CONTENT COMMENT TEACHING NOTESDifferentiationknow the meaning of the terms limit, derivative,differentiable at a point, differentiable on aninterval, derived function, second derivative

use the notation: ′f x( ) , ′′f x( ) ,

recall the derivatives of xα (α rational), sin x andcos x

know and use the rules for differentiating linearsums, products, quotients and composition offunctions:( ( )) ( ) ( )f(x) + ′ = ′ + ′g x f x g x( kf(x) kf x′ = ′) ( ) , where k is a constantthe chain rule: (f(g(x)) f (g(x)).g (x)′ = ′ ′the product rule: (f(x)g(x)) f(x)g (x)′ = ′ + ′f (x)g(x)

the quotient rule:

Candidates should be exposed to formal proofs ofdifferentiation, although proofs will not berequired for assessment purposes. Once the rulesfor differentiation have been learned, computeralgebra systems (CAS) may be used forconsolidation/extension. However, when CAS arebeing used for difficult/real examples theemphasis should be on the understanding ofconcepts rather than routine computation. Whensoftware is used for differentiation in difficultcases, candidates should be able to say which ruleswere used.

differentiate given functions which requiremore than one application of one or more ofthe chain rule, product rule and the quotientrule [A/B]

2

2

,dx

yd

dx

dy

2(g(x))

(x)gf(x) (x)g(x)f

g(x)

f(x) ′−′=

′




CONTENT COMMENT TEACHING NOTESknow

the derivative of tan xthe definitions and derivatives of sec x, cosec xand cot xthe derivatives of ex (exp x) and ln x

Link with the graphs of these functions.The definitions of ex and ln x should be revisedand examples given of their occurrence.

know the definitionCandidates should be aware that not all functionsare differentiable everywhere, eg f(x) = |x| at x = 0.The use of software allows further explorationhere.

know the definition of higher derivatives

f xn ( ) ,

Candidates should also know that higher derivativescan have discontinuities and be aware of the graphicaleffects of this, ie lack of smoothness. An example ofthis is:

for which f and ′f are continuous but ′′f is not.apply differentiation to:

a) rectilinear motionb) extrema of functions: the maximum and

minimum values of a continuous function fdefined on a closed interval [a,b] can occur atstationary points, end pointsor points where f ' is not defined [A/B]

c) optimisation problems

eg Find the acceleration of a particle whosedisplacement s metres from a certain point at timet seconds is given by s = 8 - 75t + t3.

eg Find the maximum value of the function

f(x) = [A/B]

Optimisation problems should be linked withgraph sketching.

h

f(x)h)f(x(x)f

h

−+=′→0

lim

n

n

dx

yd

≤≤−≤≤2x1 x,21x0 ,x2

≥<−

=0 x,x

0 x,xf(x)

2

2




CONTENT COMMENT TEACHING NOTESIntegrationknow the meaning of the terms integrate,integrable, integral, indefinite integral, definiteintegral and constant of integration

CAS may be used for consolidation/extension.However, when CAS are being used fordifficult/real examples the emphasis should be onunderstanding of the concepts rather than routinecomputation. When software is being used forintegration in difficult cases, candidates should beable to say which rules were used.

recall standard integrals of xα (α ∈ Q, α ≠ −1),sin x and cos x

, a, b ∈ R

, a < c < b

b ≠ a

where

know the integrals of ex, x−1, sec2x

∫∫ ∫ +=+ dxxgbdxf(x)adxxbgf(x)a )( ))((

∫ −=b

a

aFbFdxf(x) ),()(

∫ ∫−=a

b

b

a

dxf(x)dxf(x) ,

∫ ∫ ∫+=b

a

c

a

b

c

dxf(x)dxf(x)dxf(x)

f(x)(x)F =′




CONTENT COMMENT TEACHING NOTES

integrate by substitution:expressions requiring a simple substitution

Candidates are expected to integrate simplefunctions on sight. eg

Where substitutions are given they will be of theform x = g(u) or u = g(x).

expressions where the substitution will be given eg u = cos x

the following special cases of substitution

eg

eg

use an elementary treatment of the integral as alimit using rectangles

apply integration to the evaluation of areasincluding integration with respect to y [A/B]. Other applications may include

(i) volumes of simple solids of revolution (disc/washer method)

(ii) speed/time graph [A/B].

dxxex∫2

dxx

x∫ + 3

22

∫ + b)dxf(ax

dxf(x)

xf∫ )('

∫ xdx, sinxcos3

∫ + dxx )23sin(




CONTENT COMMENT TEACHING NOTESProperties of functionsknow the meaning of the terms function, domain,range, inverse function, critical point, stationarypoint, point of inflexion, concavity, local maximaand minima, global maxima and minima,continuous, discontinuous, asymptote

Candidates are expected to recognise graphs ofsimple functions, be able to sketch the graphs byhand and know their key features, eg behaviour oftrigonometric and exponential function. Theassessment should be structured to ensure thatcandidates carry out and display the calculationsrequired to identify the important features on thegraph. Candidate learning can be enhancedthrough the use of calculators with a graphicfacility and CAS.

determine the domain and the range of a function

use the derivative tests for locating and identifyingstationary points

ie Concave up ⇔ ′′ >f x( ) 0 ;concave down ⇔ ′′ <f x( ) 0 ;a necessary and sufficient condition for a point ofinflexion is a change in concavity.

Care should be exercised when using the secondderivative test in preference to the first derivativetest. The second derivative may not exist, andeven when it does and can easily be computed, itmay not be helpful, eg the function f(x) = x4 at x =0. Here, ′′ =f ( )0 0 which is inconclusive. The firstderivative test, however, easily shows there is alocal minimum at x = 0.

sketch the graphs of sin x, cos x, tan x, ex, ln x andtheir inverse functions, simple polynomialfunctions




CONTENT COMMENT TEACHING NOTESknow and use the relationship between the graph ofy = f(x) and the graphs of

y = kf(x),y = f(x) + k,y = f(x + k),y = f(kx),

where k is a constant

know and use the relationship between the graph ofy = f(x) and the graphs of

y= |f(x)|y = f −1(x) ie Reflection in the line y = x

eg f(x) = ex, − ∞ < < ∞xf x x− =1 ( ) ln , x > 0

Care must be taken over the domain and rangewhen finding inverses.

given the graph of a function f, sketch the graph ofa related function

determine whether a function is even or odd orneither and symmetrical and use these propertiesin graph sketching

sketch graphs of real rational functions usingavailable information, derived from calculus and/oralgebraic arguments, on zeros, asymptotes (verticaland non-vertical), critical points, symmetry

For rational functions, the degree of both thenumerator and the denominator will be less than orequal to two.




CONTENT NOTES TEACHING NOTESSystems of linear equationsuse the introduction of matrix ideas to organise asystem of linear equations

know the meaning of the terms matrix, element,row, column, order of a matrix, augmented matrix

use elementary row operations (EROs)

reduce to upper triangular form using EROs

solve a 3 × 3 system of linear equations usingGaussian elimination on an augmented matrix

Only 3 × 3 cases are required for assessmentpurposes, and at grade C they are restricted to thosewith a unique solution. Larger systems can be tackledusing computer packages or advanced calculators.

find the solution of a system of linear equationsAx = b, where A is a square matrix, include casesof unique solution,no solution (inconsistency) and an infinitefamily of solutions [A/B].

know the meaning of the term ill-conditioned[A/B].

Ill-conditioning can be introduced by comparingthe solutions of the following systems:a. x + 0.99y = 1.99

0.99x + 0.98y = 1.97

b. x + 0.99y = 2.000.99x + 0.98y = 1.97 [A/B]

compare the solutions of related systems oftwo equations in two unknowns and recogniseill-conditioning [A/B]




ASSESSMENT

To gain the award of the course, the candidate must pass all unit assessments as well as the externalassessment. External assessment will provide the basis for grading attainment in the course award.

Where units are taken as component parts of a course, candidates will have the opportunity to achieveat levels beyond that required to attain each of the unit outcomes. This attainment may, whereappropriate, be recorded and used to contribute towards course estimates and to provide evidence forappeals. Additional details are provided, where appropriate, with the exemplar assessment materials.Further information on the key principles of assessment are provided in the paper Assessment,published by HSDU in May 1996.

DETAILS OF THE INSTRUMENTS FOR EXTERNAL ASSESSMENT The external assessment will take the form of an examination of up to three hours’ duration. Theexamination will assess the two chosen mandatory units and the optional unit of the course. Theexamination will contain a balance of short questions designed mainly to test knowledge andunderstanding and extended response questions which also test problem solving skills. These twostyles of questions will include ones which are set in more complex contexts to provide evidence forperformance at grades A and B. Where the two chosen mandatory units are Numerical Analysis 1(AH) and Numerical Analysis 2 (AH), an externally marked assignment will also contribute to courseassessment.

GRADE DESCRIPTIONS FOR ADVANCED HIGHER APPLIED MATHEMATICS

Advanced Higher Applied Mathematics courses should enable candidates to solve problems whichintegrate mathematical knowledge across performance criteria, outcomes and units, and whichrequire extended thinking and decision making. The award of grades A, B and C is determined by thecandidate’s demonstration of the ability to apply knowledge and understanding to problem solving.To achieve grades A and B in particular, this demonstration will involve more complex contextsincluding the depth of treatment indicated in the detailed content tables.

In solving these problems, candidates should be able to:

a) interpret the problem and consider what might be relevant;b) decide how to proceed by selecting an appropriate strategy;c) implement the strategy through applying mathematical knowledge and understanding and come

to a conclusion;d) decide on the most appropriate way of communicating the solution to the problem in an

intelligible form.

Familiarity and complexity affect the level of difficulty of problems/assignments. It is generallyeasier to interpret and communicate information in contexts where the relevant variables are obviousand where their inter-relationships are known. It is usually more straightforward to apply a knownstrategy than to modify one or devise a new one. Some concepts are harder to grasp and sometechniques more difficult to apply if they have to be used in combination.




Exemplification at grade C and grade A

a) Interpret the problem and consider what might be relevant

At grade C candidates should be able to interpret and model qualitative and quantitative informationas it arises within:

• the description of real-life situations• the context of other subjects• the context of familiar areas of mathematics

Grade A performance is demonstrated through coping with the interpretation of more complexcontexts requiring a higher degree of reasoning ability in the areas described above.

b) Decide how to proceed by selecting an appropriate strategy

At grade C candidates should be able to tackle problems by selecting algorithms drawn from relatedareas of mathematics or apply a heuristic strategy.

Grade A performance is demonstrated through an ability to decide on and apply a more extendedsequence of algorithms to more complex contexts.

c) Implement the strategy through applying mathematical knowledge and understanding, andcome to a conclusion

At grade C candidates should be able to use their knowledge and understanding to carry through theirchosen strategies and come to a conclusion. They should be able to process data in numerical andsymbolic form with appropriate regard for accuracy, marshal facts, sustain logical reasoning andappreciate the requirements of proof.

Grade A performance is demonstrated through an ability to cope with processing data in morecomplex situations and sustaining logical reasoning, where the situation is less readily identifiablewith a standard form.

d) Decide on the most appropriate way of communicating the solution to the problem in anintelligible form

At grade C candidates should be able to communicate qualitative and quantitative mathematicalinformation intelligibly and to express the solution in language appropriate to the situation.

Grade A performance is demonstrated through an ability to communicate intelligibly in morecomplex situations and unfamiliar contexts.




APPROACHES TO LEARNING AND TEACHING

The approaches to learning and teaching recommended for Higher level should be continued andreinforced, whenever possible. Exposition to a group or class remains an essential technique.However, candidates should be more actively involved in their own learning in preparation for futurestudy in higher education. Opportunities for discussion, problem solving, practical activities andinvestigation should abound in Advanced Higher Applied Mathematics. There also exists muchgreater scope to harness the power of technology in the form of mathematical and graphicalcalculators and computer software packages.

Independent learning is further encouraged in the grade descriptions for the course. Coursework tasksand projects/assignments are recommended as vehicles for the introduction of new topics, forillustration or reinforcement of mathematics in context and for the development of extended problemsolving, practical and investigative skills as well as adding interest to the course.

In particular:

• in statistics, it is important that illustrative examples should, as far as possible, be drawn fromreal life, emphasising the relevance of the subject in the modern world and it is also highlydesirable that much of the experimental work, data analysis and simulation should beundertaken with suitable computer software

• in numerical analysis, practical work is important and has to be supported by a properunderstanding of the underlying theory enabling candidates to make judgements in the detaileddesign of a calculation and its execution using appropriate technology

• in mechanics, the demonstration to candidates of the application to real problems throughmodelling is fundamental. The tackling of realistic problems is essential to acquiring a betterunderstanding of the physical nature of the world. The teaching may be enhanced with the aidof computer software




SPECIAL NEEDS

This course specification is intended to ensure that there are no artificial barriers to learning orassessment. Special needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments or considering alternative outcomes for units.For information on these, please refer to the SQA document Guidance on Special Assessment andCertification Arrangements for Candidates with Special Needs/Candidates whose First Language isnot English (SQA, 1998).

SUBJECT GUIDES

A Subject Guide to accompany the Arrangements documents has been produced by the Higher StillDevelopment Unit (HSDU) in partnership with the Scottish Consultative Council on the Curriculum(SCCC) and Scottish Further Education Unit (SFEU). The Guide provides further advice andinformation about:

• support materials for each course• learning and teaching approaches in addition to the information provided in the Arrangements

document• assessment• ensuring appropriate access for candidates with special educational needs

The Subject Guide is intended to support the information contained in the Arrangements document.The SQA Arrangements documents contain the standards against which candidates are assessed.


Superclass: RB



Version: 02



Additional copies of this unit specification can be purchased from the Scottish Qualifications Authority. The cost for eachunit specification is £2.50 (minimum order £5).

42

National Unit Specification: general information

UNIT Statistics 1 (Advanced Higher)

NUMBER D326 13


SUMMARY

This unit is the first of two Advanced Higher units which, together with one optional unit, compriseone of the variants of the Advanced Higher Applied Mathematics course, and is an optional unit ofthe Advanced Higher Mathematics course. It is also an optional unit for the other variants of theAdvanced Higher Applied Mathematics course. It builds on the work of Statistics (H) and introducesspecial distributions, sampling, estimation and hypothesis testing. The unit provides a basis forprogression to Statistics 2 (AH).

OUTCOMES

1 Use conditional probability and the algebra of expectation and variance.2 Use probability distributions in simple situations.3 Identify sampling methods and estimate population parameters.4 Use a z-test on a statistical hypothesis where the significance level is given.5 Undertake a statistical assignment.

RECOMMENDED ENTRY

While entry is at the discretion of the centre, candidates will normally be expected to have attained:

• Higher Mathematics award including Statistics (H)

Applied Mathematics: Unit Specification – Statistics 1 (AH) 43

National Unit Specification: general information (cont)


CREDIT VALUE

1 credit at Advanced Higher.

CORE SKILLS




National Unit Specification: statement of standards


Acceptable performance in this unit will be the satisfactory achievement of the standards set out inthis part of the unit specification. All sections of the statement of standards are mandatory and cannotbe altered without reference to the Scottish Qualifications Authority.

OUTCOME 1

Use conditional probability and the algebra of expectation and variance.

Performance criteria

(a) Calculate a conditional probability.(b) Apply the laws of expectation and variance in simple cases.

OUTCOME 2

Use probability distributions in simple situations.


(a) Use the binomial distribution.(b) Use the Poisson distribution.(c) Use the normal distribution.

OUTCOME 3

Identify sampling methods and estimate population parameters.


(a) Identify a given sampling method.(b) Estimate a population parameter from a sample statistic.

OUTCOME 4

Use a z-test on a statistical hypothesis where the significance level is given.


(a) State null and alternative hypotheses.(b) State one-tail or two-tail test.(c) Determine the p-value.(d) State and justify an appropriate conclusion.


National Unit Specification: statement of standards (cont)


Evidence requirements

Although there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed book test under controlled conditions. Examples ofsuch tests are contained in the National Assessment Bank.

In assessment, candidates should be required to show their working in carrying out algorithms andprocesses.

OUTCOME 5

Undertake a statistical assignment.


(a) Pose the question that the assignment addresses.(b) Collect (generate) the relevant data.(c) Analyse the data.(d) Interpret and communicate the conclusions.


The assignment must satisfy the performance criteria, using the statistical content of the unit. A fullreport is to be written by the candidate individually. This report may include sets of data, graphs,computer printout, calculated statistics, consideration of probability and a conclusion.


National Unit Specification: support notes


This part of the unit specification is offered as guidance. The support notes are not mandatory.

While the time allocated to this unit is at the discretion of the centre, the notional design length is40 hours.

GUIDANCE ON CONTENT AND CONTEXT FOR THIS UNIT

Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed with the two-fold objective ofproviding a rounded experience of statistics for candidates who take the unit free-standing or as thethird unit of the Advanced Higher Mathematics course or Advanced Higher Applied Mathematicscourse and, at the same time, forming a sound basis for progression to Statistics 2 (AH) forcandidates specialising in statistics in the Advanced Higher Applied Mathematics course.

The first outcome of this unit extends the earlier work on probability to conditional probability andallows candidates the opportunity to study the algebra of expectation and variance.

Outcome 2 assumes knowledge of the Statistics (H) outcome on discrete probability distributions andextends the study to special distributions.

Outcome 3 requires demonstration of competence in sampling methods and estimation of populationparameters, for which experience of the Statistics (H) outcome on exploratory data analysis is ofsome advantage. Outcome 4 introduces the concept of hypothesis testing. Candidates are given theopportunity to apply statistical processes in Outcome 5 where competence will be demonstrated bycompleting an assignment.

The recommended content for this unit can be found in the course specification. The detailed contentsection provides illustrative examples to indicate the depth of treatment required to achieve a unitpass and advice on teaching approaches.


National Unit Specification: support notes (cont)


GUIDANCE ON LEARNING AND TEACHING APPROACHES FOR THIS UNIT

The investigative approaches to teaching and learning consistently recommended at earlier levels areequally beneficial at Advanced Higher level.

Where appropriate, statistical topics should be taught and skills in applying statistics developedthrough real-life contexts. Candidates should be encouraged throughout this unit to make efficient useof the arithmetical, mathematical, statistical and graphical features of calculators, to be aware of thelimitations of the technology and always to apply the strategy of checking.

Numerical checking or checking a result against the context in which it is set is an integral part ofevery mathematical process. In many instances, the checking can be done mentally, but on occasions,to stress its importance, attention should be drawn to relevant checking procedures throughout themathematical process. There are various checking procedures which could be used:

• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order

Further advice on learning and teaching approaches is contained within the subject guide for AppliedMathematics.

GUIDANCE ON APPROACHES TO ASSESSMENT FOR THIS UNIT

The assessment for this unit will normally be in the form of a closed book test. Such tests should becarried out under supervision and it is recommended that candidates attempt an assessment designedto assess all the outcomes within the unit. Successful achievement of the unit is demonstrated bycandidates achieving the threshold of attainment specified for all outcomes in the unit. Candidateswho fail to achieve the threshold(s) of attainment need only be retested on the outcome(s) where theoutcome threshold has not been attained. Further advice on assessment and retesting is containedwithin the National Assessment Bank.

The fifth outcome is assessed by means of a statistical assignment. Examples of such assignments,with marking schemes, are contained within the National Assessment Bank.

In assessments, candidates should be required to show their working in carrying out algorithms andprocesses.

SPECIAL NEEDS

This unit specification is intended to ensure that there are no artificial barriers to learning orassessment. Special needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments or considering alternative outcomes for units.For information on these, please refer to the SQA document Guidance on Special Assessment andCertification Arrangements for Candidates with Special Needs/Candidates whose First Language isnot English (SQA, 1998).


Superclass: RB



Version: 02




48



NUMBER D330 13


SUMMARY

This unit is the second of two Advanced Higher units which, together with one optional unit,comprise one of the variants of the Advanced Higher Applied Mathematics course. It builds on thework of Statistics 1 (AH) and introduces control charts and the t-distribution.

OUTCOMES

1 Use simple control charts.2 Test a statistical hypothesis.3 Use the t-distribution.4 Analyse the relationship between two variables.5 Undertake a statistical investigation.

RECOMMENDED ENTRY


• Statistics 1 (AH)

CREDIT VALUE





CORE SKILLS







OUTCOME 1

Use simple control charts.


(a) Construct a control chart.(b) Interpret a control chart.

OUTCOME 2

Test a statistical hypothesis.


(a) Carry out a chi-squared test.(b) Carry out a sign test.(c) Carry out a Mann-Whitney test.

OUTCOME 3

Use the t-distribution.


(a) Determine a confidence interval for a population mean.(b) Carry out a one sample t-test for the population mean.




OUTCOME 4

Analyse the relationship between two variables.


(a) Test the significance of the strength of a linear relationship.(b) Use a residual plot to check model assumptions.(c) Construct an interval estimate for a given response.


Although there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed-book test. Tests should be carried out undersupervision. Examples of such tests are contained in the National Assessment Bank.

In assessment, candidates should be required to show their working in carrying out algorithms andprocesses.

OUTCOME 5

Undertake a statistical investigation.


(a) Pose the question that the investigation addresses.(b) Collect (generate) the relevant data.(c) Analyse the data.(d) Interpret and communicate the conclusions.


The investigation must satisfy the performance criteria, using the statistical content of the unit. A fullreport on the investigation is to be written by the candidate individually. This report may include setsof data, graphs, computer printout, calculated statistics, consideration of probability and a conclusion.







Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed to build on the content ofStatistics 1(AH) and extend the specialism in this topic to a wider experience and to a more advancedlevel.

In this unit, Outcomes 1 and 3 introduce the statistical concepts of control charts and the t-distribution respectively.

Hypothesis testing, introduced in Statistics 1(AH), is now extended in Outcome 2 to include chi-squared, sign and Mann-Whitney tests.

Regression analysis studied in Statistics (H) is extended in Outcome 4 to include the significance ofthe product moment correlation coefficient and the consideration of interval estimates.

To reinforce the practical nature of the subject, candidates, in Outcome 5, are required to demonstratecompetence in all the stages of undertaking a statistical investigation.






The investigative approaches to teaching and learning consistently recommended at earlier levels areequally beneficial at Advanced Higher level.

Where appropriate, statistical topics should be taught and skills in applying statistics developedthrough real-life contexts. Candidates should be encouraged, throughout this unit, to make efficientuse of the arithmetical, mathematical, statistical and graphical features of calculators, to be aware ofthe limitations of the technology and always to apply the strategy of checking.



Further advice on learning and teaching approaches is contained within the Subject Guide forMathematics.



The fifth outcome is assessed by means of a statistical investigation. This investigation should be asubstantial piece of work, taking up to ten hours, in which candidates collect their own data by, forexample, carrying out an experiment. The analysis, interpretation and communication of theconclusions should be included in each candidate’s report. Examples of such investigations, arecontained within the National Assessment Bank.





SPECIAL NEEDS



Superclass: RB



Version: 02




55


UNIT Numerical Analysis 1 (Advanced Higher)

NUMBER D328 13


SUMMARY

This unit is the first of two Advanced Higher units which, together with one optional unit, compriseone of the variants of the Advanced Higher Applied Mathematics course, and is an optional unit ofthe Advanced Higher Mathematics course. It is also an optional unit for the other variants of theAdvanced Higher Applied Mathematics course. The unit uses Taylor polynomials together withforward difference and Lagrange interpolation to introduce the ideas of function approximation andinterpolation. These ideas are applied in the study of some methods of numerical integration. Thefinal outcome requires work to be carried out on an extended piece of work based on the content ofthe other outcomes. The unit builds on the work of Higher Mathematics and provides a progression toNumerical Analysis 2 (AH).

OUTCOMES

1 Use Taylor polynomials to approximate functions.2 Interpolate data.3 Use numerical integration.4 Use numerical methods to solve problems.

RECOMMENDED ENTRY


• Higher Mathematics award, including Mathematics 3 (H)

Applied Mathematics: Unit Specification – Numerical Analysis 1 (AH) 56



CREDIT VALUE


CORE SKILLS







OUTCOME 1

Use Taylor polynomials to approximate functions.


(a) Obtain the Taylor polynomial of up to degree 2 of a function about a given point.(b) Approximate a function value near to a given point giving the principal truncation error in the

approximation.

OUTCOME 2

Interpolate data.


(a) Construct a difference table for a given set of data consisting of up to 4 pairs of values.(b) Use the Newton forward difference interpolation formula of up to order 2.(c) Use the Lagrange interpolation formula to obtain an interpolating polynomial of up to degree 3.

OUTCOME 3

Use numerical integration.


(a) Use the composite trapezium rule to estimate a given definite integral.(b) Obtain an estimate of the magnitude of the principal truncation error in the composite

trapezium rule.(c) Use Richardson’s formula to reduce the error in the use of the composite trapezium rule.(d) Use the composite Simpson’s rule to estimate a given definite integral.


Although there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed book test. Tests should be carried out undersupervision. Examples of such tests are contained in the National Assessment Bank.




OUTCOME 4

Use numerical methods to solve problems.


(a) Demonstrate the use of technology in solving numerical problems.(b) Write a report on the work carried out on completion of a coursework task.


The coursework task must satisfy the performance criteria, using the content of the unit. A full reporton the coursework task is to be written by the candidate individually. Examples of such courseworktasks, with marking schemes, are contained in the National Assessment Bank.

In assessment candidates should be required to show their working in carrying out algorithms andprocesses.







Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed with the two-fold objective ofproviding a rounded experience of numerical analysis for candidates who take the unit free-standingor as the third unit of the Advanced Higher Mathematics course, or the Advanced Higher AppliedMathematics course and, at the same time, forming a sound basis for progression to NumericalAnalysis 2 (AH) for candidates specialising in numerical analysis in the Advanced Higher AppliedMathematics course.

In numerical analysis, practical work is particularly important and the fourth outcome is a courseworktask where the use of technology will be demonstrated in the solution of a numerical problem.

The use of technology is also strongly encouraged in the other outcomes. In Outcome 1, in the use ofTaylor polynomials, a graphical display should be used to illustrate function approximation usingpolynomials of increasing degree. Thus, the dependence of truncation error on the value of h in theapproximation of f(a + h) can be demonstrated. In Outcome 3, in numerical integration using thetrapezium and Simpson’s rules and Richardson’s formula, candidates should be encouraged to writeshort programs for programmable calculators to compute the integration formulae. In Outcome 2, ininterpolation of data using difference tables and Newton and Lagrange formulae and throughout thisunit, candidates should become aware of errors due to round-off and consideration should be given tothe number of figures to be carried in computation and quoted in answers.






The investigative approaches to teaching and learning consistently recommended at earlier levels areequally beneficial at Advanced Higher level Mathematics.

Where appropriate, mathematical topics should be taught and skills in applying Applied Mathematicsdeveloped through real-life contexts. Candidates should be encouraged throughout this unit to makeefficient use of the arithmetical, mathematical and graphical features of calculators, to be aware ofthe limitations of the technology and always to apply the strategy of checking.



Further advice on learning and teaching approaches is contained within the subject guide for AppliedMathematics.



The fourth outcome is assessed by means of a coursework task. Examples of such coursework tasks,with marking schemes, are contained within the National Assessment Bank.


SPECIAL NEEDS



Superclass: RB



Version: 02




61



NUMBER D329 13


SUMMARY

This unit is the second of two Advanced Higher units which, together with one optional unit,comprise one of the variants of the Advanced Higher Applied Mathematics course. Much of the unitis based on iterative methods applied to solving non-linear equations and linear systems of equations.Linear systems of equations are also solved by direct methods. Step by step methods are used to solvesimple first order differential equations. The two units Numerical Analysis 1 (AH) and 2 (AH)provide a good general foundation in numerical mathematics for candidates continuing to furtherstudy in disciplines with a mathematics component.

OUTCOMES

1 Solve non-linear equations.2 Use Gaussian elimination to invert a matrix and solve a system of up to three linear equations.3 Use matrix iterative techniques to solve systems of linear equations.4 Approximate the solution of first order ordinary differential equations.5 Understand the errors arising in numerical computation.

RECOMMENDED ENTRY


• Numerical Analysis 1 (AH)

Mathematics: Unit Specification – Numerical Analysis 2 (AH) 62



CREDIT VALUE


CORE SKILLS







OUTCOME 1

Solve non-linear equations.


(a) Use the interval bisection method to solve an equation.(b) Determine the order of convergence of an iterative process.(c) Solve an equation using the Newton-Raphson method.

OUTCOME 2

Use Gaussian elimination to invert a matrix and solve a system of up to three linear equations.


(a) Solve Ax = b using Gaussian elimination with partial pivoting.(b) Invert a non-singular square matrix.

OUTCOME 3

Use matrix iterative techniques to solve systems of linear equations.


(a) Solve Ax = b using the Jacobi method.(b) Solve Ax = b using the Gauss-Seidel method.

OUTCOME 4

Approximate the solution of first order ordinary differential equations.


(a) Solve an appropriate first order ordinary differential equation using Euler’s method.(b) Use the modified Euler method with the trapezium rule as corrector, to solve a suitable first

order ordinary differential equation.




OUTCOME 5

Understand the errors arising in numerical computation.


(a) Perform simple calculations using data values and give answers to the appropriate accuracy.(b) Calculate bounds on the error caused by data errors in polynomial evaluation.










Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed to build on the content ofNumerical Analysis 1(AH) and extend the specialism in this topic to a wider experience and to amore advanced level.

In this unit, in Outcome 1, candidates investigate the analysis of fixed point iteration. Programmablecalculators should be used to compute sequence values and the values displayed graphically. Themethods for solving non-linear equations should be compared and candidates should be aware of ill-conditioned problems. In Outcome 3, iterative methods are applied to the solution of systems oflinear equations. The matrix operations which are required for the numerical solution of linearsystems of equations are introduced in Outcome 2 and again, the existence of ill-conditionedproblems is noted. Competence in the solution of first order ordinary differential equations usingEuler’s method and the modified Euler method is required in Outcome 4. In the final outcome, errors,frequently referred to throughout the two numerical analysis units, are considered in more detail inthe contexts of numerical computation and polynomial evaluation. Candidates who proceed to theAdvanced Higher Applied Mathematics course will be required to complete an investigation based onthe content of the Numerical Analysis units. This will involve an extended piece of work beingcarried out by the candidate and a report on the work being written.






The investigative approaches to teaching and learning consistently recommended at earlier levels areequally beneficial at Advanced Higher level mathematics.

Where appropriate, mathematical topics should be taught and skills in applying mathematicsdeveloped through real-life contexts. Candidates should be encouraged throughout this unit to makeefficient use of the arithmetical, mathematical and graphical features of calculators, to be aware ofthe limitations of the technology and always to apply the strategy of checking.







SPECIAL NEEDS



Superclass: RB



Version: 02




67


UNIT Mechanics 1 (Advanced Higher)

NUMBER D327 13


SUMMARY

This unit is the first of two Advanced Higher units which, together with one optional unit, compriseone of the variants of the Advanced Higher Applied Mathematics course, and is an optional unit ofthe Advanced Higher Mathematics course. It is also an optional unit for the other variants of theAdvanced Higher Applied Mathematics course. This unit uses some of the skills of HigherMathematics applied to problems in mechanics. The unit provides a basis for progression toMechanics 2 (AH).

OUTCOMES

1 Interpret and solve problems on motion in a straight line.2 Solve problems involving position, velocity and acceleration vectors, including relative

motion.3 Solve problems on the motion of projectiles in a plane, under constant gravity and ignoring

resistances.4 Solve problems on forces in equilibrium and on linear motion involving forces.

RECOMMENDED ENTRY


• Higher Mathematics award

Mathematics: Unit Specification – Mechanics 1 (AH) 68



CREDIT VALUE


CORE SKILLS







OUTCOME 1

Interpret and solve problems on motion in a straight line.


(a) Draw a velocity-time graph and use it to calculate distance.(b) Use an equation of motion in a simple situation.(c) Given an expression for displacement, find corresponding expressions for velocity and

accerleration (where these are functions of time).

OUTCOME 2

Solve problems involving position, velocity and acceleration vectors, including relative motion.


(a) Find the relative velocity of one body with respect to another (a simple trigonometric approachis allowed).

(b) Solve a simple problem involving collision.

OUTCOME 3

Solve problems on the motion of projectiles in a plane, under constant gravity and ignoringresistances.

Performance criterion

(a) Given appropriate information about initial velocity, find the time of flight, the greatest heightreached and the range of a projectile.




OUTCOME 4

Solve problems on forces in equilibrium and on linear motion involving forces.


(a) Resolve forces in two dimesions.(b) Find the resultant of two forces.(c) Given a set of three forces in equilibrium, derive appropriate results by resolving horizontally

and vertically(d) Solve a simple problem involving the coefficient of friction, µ.










Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed with the two-fold objective ofproviding a rounded experience of mechanics for candidates who take the unit free-standing or as thethird unit of the Advanced Higher Mathematics course or Advanced Higher Applied Mathematicscourse and, at the same time, forming a sound basis for progression to Mechanics 2 (AH) forcandidates specialising in mechanics in the Advanced Higher Applied Mathematics course.

The four outcomes of this unit provide the candidate with the opportunity to demonstrate competencein the solution of problems in a range of aspects of motion. Outcome 1 introduces motion in a straightline and Outcome 2 extends this concept to position, velocity and acceleration vectors includingrelative motion. Outcome 3 requires demonstration of ability to solve projectile problems andOutcome 4 the ability to deal with forces in equilibrium and linear motion involving forces.









• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order Further advice on learning and teaching approaches is contained within the subject guide forMathematics.




SPECIAL NEEDS



Superclass: RB



Version: 02




73



NUMBER D331 13


SUMMARY

This unit is the second of two Advanced Higher units which, together with one optional unit,comprise one of the variants of the Advanced Higher Applied Mathematics course. It builds on thework of Mechanics 1 (AH) and introduces circular motion and simple harmonic motion. It alsoapplies some further calculus from Mathematics 1 and 2 (AH) to problems in mechanics.

OUTCOMES

1 Interpret and solve problems involving motion in a horizontal circle with uniform angularvelocity.

2 Solve problems involving simple harmonic motion.3 Apply the principles of momentum and impulse.4 Apply the principles of work, power and energy.5 Interpret and solve problems involving first order differential equations.

RECOMMENDED ENTRY

While entry is at the discretion of the centre, candidates would normally be expected to have attained:

• Mechanics 1 (AH)

Candidates will also require to have knowledge of the calculus content of Mathematics 1 and 2 (AH).




CREDIT VALUE


CORE SKILLS







OUTCOME 1

Interpret and solve problems involving motion in a horizontal circle with uniform angular velocity.


(a) Solve a simple problem involving the inverse square law of gravitation.(b) Solve a simple problem involving motion in a horizontal circle.

OUTCOME 2

Solve problems involving simple harmonic motion.


(a) Solve a simple problem involving simple harmonic motion in a straight line, making use ofbasic equations, period, amplitude, maximum velocity or maximum acceleration as appropriate.

OUTCOME 3

Apply the principles of momentum and impulse.


(a) Use the concept of conservation of linear momentum.(b) Use impulse appropriately in a simple situation.

OUTCOME 4

Apply the principles of work, power and energy.


(a) Evaluate work done appropriately.(b) Use P = F.v appropriately.(c) Use the concept of conservation of energy.




OUTCOME 5

Interpret and solve problems involving first order differential equations.

Performance criterion

(a) Form and solve a differential equation of motion, modelling a simple practical problem wherethe acceleration is a function of the velocity.


Although there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be in the form of a closed book test under controlled conditions. Examples of such tests arecontained in the National Assessment Bank.








Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills with the emphasis on the application of mathematical ideas andtechniques to relevant and accessible problems. This unit is designed to build on the content ofMechanics 1(AH) and extend the specialism in this topic to a wider experience and to a moreadvanced level.

In this unit, the study of motion and vector methods in Mechanics 1(AH) and the emphasis onproblem solving throughout the unit is continued and extended to circular motion in outcome 1 andsimple harmonic motion in Outcome 2.

In Outcome 3, competence in solution of problems on impulse and conservation of linear momentumrequires to be demonstrated and likewise for problems on work, power and energy in Outcome 4.

Outcome 5 involves the modelling and solution of practical problems involving first order differentialequations. The calculus involved will be within the content of Mathematics 1 (AH) and 2 (AH).














SPECIAL NEEDS



Superclass: RB



Version: 02




79


UNIT Mathematics 1 (Advanced Higher)

NUMBER D321 13


SUMMARY

This unit is the first of the two mandatory units which together with one optional unit comprise theAdvanced Higher Mathematics course. This unit extends the calculus and graphicacy work fromHigher level and introduces matrices for solving systems of linear equations. It provides a basis forprogression to Mathematics 2 (AH). This unit is also an optional unit of the Advanced HigherApplied Mathematics course.

OUTCOMES

1 Use algebraic skills.2 Use the rules of differentiation on the elementary functions xn (n ∈ Q), sin x, cos x, ex and

ln x and their composites3 Integrate using standard results, and the substitution method.4 Use properties of functions.5 Use matrix methods to solve systems of linear equations.

RECOMMENDED ENTRY

While entry is at the discretion of the centre, candidates will normally be expected to have attained :

• Higher Mathematics award, including Mathematics 3 (H)

Applied Mathematics: Unit Specification – Mathematics 1 (AH) 80



CREDIT VALUE


CORE SKILLS







OUTCOME 1

Use algebraic skills.


(a) Expand an expression of the form (x + y)n, n ∈ N and n ≤ 5.(b) Express a proper rational function as a sum of partial fractions where the denominator is a

quadratic in factorised form.

OUTCOME 2

Use the rules of differentiation on the elementary functions xn, (n ∈ Q), sinx, cosx, ex and lnx andtheir composites.


(a) Differentiate a product.(b) Differentiate a quotatient.(c) Differentiate a simple composite function using the chain rule.

OUTCOME 3

Integrate using standard results and the substitution method.


(a) Integrate an expression requiring a standard result.(b) Integrate using a substitution method where the substitution is given.(c) Integrate an expression requiring a simple substitution.

OUTCOME 4

Use properties of functions.


(a) Find the vertical asymptote of a rational function.(b) Find the non-vertical asymptote of a rational function.(c) Sketch the graph of a rational function including appropriate analysis of stationary points.




OUTCOME 5

Use matrix techniques to solve systems of linear equations.


(a) Use Gaussian elimination to solve a 3 × 3 system of linear equations



In assessments candidates should be required to show their working in carrying out algorithms andprocesses.







Each mathematics unit at Advanced Higher level aims to build upon and extend candidates’mathematical knowledge and skills in a manner which reinforces the essential nature of problemsolving. New mathematical concepts and skills are within theoretical or practical applications, andthe importance of algebraic manipulative skills is emphasised throughout. At the same time, thebenefits of advanced technology in securing and consolidating understanding are acknowledged andthere are frequent references to the use of such technology throughout the course content. Equallyimportant is the need, where appropriate, for the limitations of the technology to be demonstrated andfor checking of accuracy and sensibility of answers to be ever present.

In this unit the algebraic skills learnt at Higher level are extended in Outcome 1 to binomialexpansions and partial fractions.

In Outcomes 2 and 3, the elementary calculus studied at Higher level is extended to differentiation ofsums, products, quotients and composites of elementary functions and to integration using standardresults and substitution methods respectively. In both of Outcomes 2 and 3, computer algebra systemscan be used extensively for consolidation and extension.

In Outcome 4 the work at Higher level on using calculus methods to sketch graphs of functions andgiven graphs of functions to sketch graphs of related functions is taken further, with enhancementthrough the use of graphic calculators recommended.

Outcome 5 is the only outcome which does not build upon Higher content. It provides an introductionto matrix techniques leading to the use of Gaussian elimination to solve a 3 × 3 system of linearequations.










Further advice on learning and teaching approaches is contained within the subject guide forMathematics. GUIDANCE ON APPROACHES TO ASSESSMENT FOR THIS UNIT


It is expected that candidates will be able to demonstrate attainment in the algebraic and calculuscontent of the unit without the use of computer software or sophisticated calculators.


SPECIAL NEEDS


applied mathematics advanced higher mathematics_advanced higher.pdf · core skills core skills for...

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