MATHEMATICS Advanced Higher

Transcription

1 MATHEMATICS Advanced Higher Fourth edition published March 2002

2 NOTE OF CHANGES TO ARRANGEMENT FOURTH EDITION PUBLISHED MARCH 2002 TITLE: NUMBERS AND TITLES FOR ENTRY TO S : C Mathematics: Maths 1, 2 and 3 C Mathematics: Maths 1, 2 and Stats 1 C Mathematics: Maths 1, 2 and Numerical Analysis 1 C Mathematics: Maths 1, 2 and Mechanics 1 National Course Specification Course Details: Course structure section has been updated to show the new codes and titles for entry to courses in Mathematics with optional routes. Core skills details updated. National Unit Specification: All Units No changes

3 National Course Specification MATHEMATICS (ADVANCED HIGHER) NUMBER C Mathematics: Maths 1, 2 and 3 C Mathematics: Maths 1, 2 and Stats 1 C Mathematics: Maths 1, 2 and Numerical Analysis 1 C Mathematics: Maths 1, 2 and Mechanics 1 STRUCTURE In order to ensure the accurate and complete transfer of data to and from centres, new codes and titles for entry to courses in Mathematics with optional routes have been introduced to reflect the options chosen by candidates. The course code C for will no longer be acceptable for entry for the summer or winter diets. The codes detailed below must be used. Unit codes and titles remain unchanged. There will be no change to the titles of the Mathematics courses as they appear on the certificate. C Mathematics: Maths 1, 2 and 3 This course consists of three mandatory units as follows: D Mathematics 1 (AH) 1 credit (40 hours) D Mathematics 2 (AH) 1 credit (40 hours) D Mathematics 3 (AH) 1 credit (40 hours) C Mathematics: Maths 1, 2 and Stats 1 This course consists of three mandatory units as follows: D Mathematics 1 (AH) 1 credit (40 hours) D Mathematics 2 (AH) 1 credit (40 hours) D Statistics 1(AH) 1 credit (40 hours) Administrative Information Publication date: March 2002 Source: Scottish Qualifications Authority Version: 04 Scottish Qualifications Authority 2002 This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived from reproduction 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 Qualifications Authority for Note: Unit specifications can be purchased individually for 2.50 (minimum order 5). 1

4 National Course Specification: general information (cont) C Mathematics: Maths 1, 2 and Numerical Analysis 1 This course consists of three mandatory units as follows: D Mathematics 1 (AH) 1 credit (40 hours) D Mathematics 2 (AH) 1 credit (40 hours) D Numerical Analysis 1 (AH) 1 credit (40 hours) C Mathematics: Maths 1, 2 and Mechanics 1 This course consists of three mandatory units as follows: D Mathematics 1 (AH) 1 credit (40 hours) D Mathematics 2 (AH) 1 credit (40 hours) D Mechanics 1 (AH) 1 credit (40 hours) In common with all courses, this course includes 40 hours for induction, extending the range of learning and teaching approaches, additional support, consolidation, integration of learning and preparation for external assessment. This time is an important element of the course and advice on the use of the overall 160 hours is included in the course details. RECOMMENDED ENTRY While entry is at the discretion of the centre, candidates would normally be expected to have attained a Higher Mathematics course award or its component units or equivalent. Statistics 1 (AH) assumes knowledge of outcome 2 of Statistics (H), while experience of outcomes 1 and 4 of Statistics (H) would be advantageous. Mathematics 1 (AH) and Numerical Analysis 1 (AH) assume knowledge of outcomes 2 and 3 of Mathematics 3 (H). CORE SKILLS The course give automatic certification of the following: Complete core skills for the course Numeracy H Additional core skills components for the course Critical Thinking H 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 in National Qualifications (SQA, 1999). Mathematics: Advanced Higher Course 2

5 National Course Specification: course details RATIONALE As with all mathematics courses, Advanced Higher Mathematics aims to build upon and extend candidates mathematical skills, knowledge and understanding in a way that recognises problem solving as an essential skill and enables them to integrate their knowledge of different aspects of the subject. The aim of developing mathematical skills and applying mathematical techniques in context will be furthered by exploiting the power of calculators and computer software where appropriate. In addition, when an applied mathematics unit is chosen as the optional unit, it will bring its own unique brand of application and extended activity to the course. Because of the importance of these features, the grade descriptions for Advanced Higher Mathematics emphasise the need for candidates to undertake extended thinking and decision making to solve problems and integrate mathematical knowledge. The use of coursework tasks, therefore, to practise problem solving as set out in the grade descriptions, is 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 higher education studies or areas of employment, and should satisfy most needs regardless of whether or not a specialism in mathematics is the primary intention. The course offers breadth and depth of mathematical experience and thereby achieves relevance to further study or employment in the areas of mathematical and physical sciences, computer science, engineering, biological and social sciences, medicine, accounting, business and management. The course offers candidates, in an interesting and enjoyable manner, an enhanced awareness of the range and power of mathematics and the importance of mathematical applications to society in general. CONTENT The syllabus is designed to build upon and extend candidates mathematical learning in the areas of algebra, geometry, calculus and, where appropriate, statistics. The two mandatory units, Mathematics 1 (AH) and Mathematics 2 (AH), are progressive and continue the development of algebra and calculus from Higher level. Further progression and extension of such mathematics is offered through the optional unit Mathematics 3 (AH). Progression in statistics from Higher level is provided through the optional unit Statistics 1(AH) whilst the two remaining optional units, Mechanics 1(AH) and Numerical Analysis 1(AH), offer the opportunity to study additional areas of applied mathematics in some depth. The first unit in each of the applied areas is also designed to provide a rounded experience of the topic for those candidates who do not wish to study the topic further in the Advanced Higher Applied Mathematics course. Mathematics: Advanced Higher Course 3

6 The outcomes and the performance criteria for each unit are statements of basic competence. Additionally, the course makes demands over and above the requirements of individual units. Within the 40 hours of flexibility time, candidates should be able to integrate their knowledge across the component units Mathematics 1 (AH) and Mathematics 2 (AH) and, if chosen as the optional unit, across Mathematics 3 (AH). If one of the three available applied units is chosen as the option, candidates following the course should experience an extension of problem solving or practical activity in that area in order to meet the grade descriptions for the course award. The experience of extended thinking and decision making is important and will be enhanced when candidates are exposed to coursework tasks which require them to interpret problems, select appropriate strategies, come to conclusions and communicate these intelligibly. In assessments, candidates should be required to show their working in carrying out algorithms and processes. Mathematics: Advanced Higher Course 4

7 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 comment is 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 COMMENT TEACHING NOTES Mathematics 1 (AH) Algebra n n n + 1 know and use the notation n!, n n + = C r and r r 1 r r n n know the results = r n r and 8 eg Calculate 5 n eg Solve, for n N, 2 =15 Candidates should be aware of the size of n! for small values of n, and that calculator results consequently are often inaccurate (especially if the n n! formula C r = is used). These results r!( n r)! can be established numerically or from Pascal s triangle. This will be linked with elementary number theory in Mathematics 2 (AH), where they provide an opportunity to introduce the concept of direct 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 ( a + b) n = n r = 0 n a r n r b r, for r, n N eg Expand (x + 3) 4 eg Expand (2u - 3v) 5 [A/B] In the work on series and complex numbers (de Moivre s theorem) the binomial theorem will be extended to integer and rational indices. Mathematics: Advanced Higher Course 5

8 CONTENT COMMENT TEACHING NOTES evaluate specific terms in a binomial expansion eg Find the term in x 7 2 in x + x 9 express a proper rational function as a sum of partial fractions (denominator of degree at most 3 and easily factorised) include cases where an improper rational function is reduced to a polynomial and a proper rational function by division or otherwise [A/B] 5 10x eg Express 2 in partial fractions. 1 3x 4x x 3 + 2x 2 2x + 2 eg Express in partial ( x 1)( x + 3) fractions [A/B]. The denominator may include a repeated linear factor or an irreducible quadratic factor. This is also required for integration of rational functions and useful for graph sketching when asymptotes are present. When the degree of the numerator of the rational function exceeds that of the denominator by 1, non-vertical asymptotes occur. Mathematics: Advanced Higher Course 6

9 CONTENT COMMENT TEACHING NOTES Differentiation know the meaning of the terms limit, derivative, differentiable at a point, differentiable on an interval, derived function, second derivative use the notation: f dy d y ( x), f ( x),, dx 2 dx recall the derivatives of x α (α rational), sin x and cos x 2 know and use the rules for differentiating linear sums, products, quotients and composition of functions: ( f(x) + gx ( )) = f ( x) + g ( x) ( kf(x) ) = kf ( x), where k is a constant the 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) f(x) f (x)g(x) f(x)g (x) the quotient rule: = 2 g(x) (g(x)) Candidates should be exposed to formal proofs of differentiation, although proofs will not be required for assessment purposes. Once the rules for differentiation have been learned, computer algebra systems (CAS) may be used for consolidation/extension. However, when CAS are being used for difficult/real examples the emphasis should be on the understanding of concepts rather than routine computation. When software is used for differentiation in difficult cases, candidates should be able to say which rules were used. differentiate given functions which require more than one application of one or more of the chain rule, product rule and the quotient rule [A/B] Mathematics: Advanced Higher Course 7

10 CONTENT COMMENT TEACHING NOTES know the derivative of tan x the definitions and derivatives of sec x, cosec x Link with the graphs of these functions. The definitions of e x and ln x should be revised and examples given of their occurrence. and cot x the derivatives of e x (exp x) and ln x know the definition know the definition of higher derivatives f n ( x), d n dx y n apply differentiation to: f (x) = lim h 0 f(x + h) f(x) h a) rectilinear motion b) extrema of functions: the maximum and minimum values of a continuous function f defined on a closed interval [a,b] can occur at stationary points, end points or points where f ' is not defined [A/B] c) optimisation problems eg Find the acceleration of a particle whose displacement s metres from a certain point at time t seconds is given by s = 8-75t + t 3. eg Find the maximum value of the function f(x) = x 2, 0 x 1 [A/B] 2 x, 1 x 2 Candidates should be aware that not all functions are differentiable everywhere, eg f(x) = x at x = 0. The use of software allows further exploration here. Candidates should also know that higher derivatives can have discontinuities and be aware of the graphical effects of this, ie lack of smoothness. An example of this is: 2 x, x < 0 f(x) = 2 x, x 0 for which f and f are continuous but f is not. Optimisation problems should be linked with graph sketching. Mathematics: Advanced Higher Course 8

11 CONTENT COMMENT TEACHING NOTES Integration know the meaning of the terms integrate, integrable, integral, indefinite integral, definite integral and constant of integration CAS may be used for consolidation/extension. However, when CAS are being used for difficult/real examples the emphasis should be on understanding of the concepts rather than routine computation. When software is being used for integration in difficult cases, candidates should be able to say which rules were used. recall standard integrals of x α sin x and cos x (α Q, α 1), + b a b b a ( a f(x) + bg( x)) dx = a f(x)dx b g( x) dx, a, b R f(x)dx = f(x)dx + a c a b f(x) dx = f(x)dx, b a a f(x)dx, a < c < b f(x) dx = F( b) F( a), where F (x) = know the integrals of e x, x 1, sec 2 x b c f(x) Mathematics: Advanced Higher Course 9

12 CONTENT COMMENT TEACHING NOTES integrate by substitution: expressions requiring a simple substitution Candidates are expected to integrate simple 2 functions on sight. eg xex dx Where substitutions are given they will be of the form x = g(u) or u = g(x). expressions where the substitution will be given the following special cases of substitution eg cos 3 x sin xdx, u = cos x f(ax + b)dx f '( x) dx f(x) use an elementary treatment of the integral as a limit using rectangles eg eg sin(3x + 2 ) dx 2x dx 2 x + 3 apply integration to the evaluation of areas including 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]. Mathematics: Advanced Higher Course 10

13 CONTENT COMMENT TEACHING NOTES Properties of functions know the meaning of the terms function, domain, range, inverse function, critical point, stationary point, point of inflexion, concavity, local maxima and minima, global maxima and minima, continuous, discontinuous, asymptote determine the domain and the range of a function Candidates are expected to recognise graphs of simple functions, be able to sketch the graphs by hand and know their key features, eg behaviour of trigonometric and exponential function. The assessment should be structured to ensure that candidates carry out and display the calculations required to identify the important features on the graph. Candidate learning can be enhanced through the use of calculators with a graphic facility and CAS. use the derivative tests for locating and identifying stationary points sketch the graphs of sin x, cos x, tan x, e x, ln x and their inverse functions, simple polynomial functions ie Concave up f ( x) > 0; concave down f ( x) < 0; a necessary and sufficient condition for a point of inflexion is a change in concavity. Care should be exercised when using the second derivative test in preference to the first derivative test. The second derivative may not exist, and even when it does and can easily be computed, it may not be helpful, eg the function f(x) = x 4 at x = 0. Here, f ( 0) = 0 which is inconclusive. The first derivative test, however, easily shows there is a local minimum at x = 0. Mathematics: Advanced Higher Course 11

14 CONTENT COMMENT TEACHING NOTES know and use the relationship between the graph of y = 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 of y = f(x) and the graphs of y= f(x) y = f 1 (x) given the graph of a function f, sketch the graph of a related function determine whether a function is even or odd or neither and symmetrical and use these properties in graph sketching sketch graphs of real rational functions using available information, derived from calculus and/or algebraic arguments, on zeros, asymptotes (vertical and non-vertical), critical points, symmetry ie Reflection in the line y = x eg f(x) = e x, < x < f 1 ( x) = ln x, x > 0 For rational functions, the degree of the numerator will be less than or equal to three and the denominator will be less than or equal to two. Care must be taken over the domain and range when finding inverses. Mathematics: Advanced Higher Course 12

15 CONTENT NOTES TEACHING NOTES Systems of linear equations use the introduction of matrix ideas to organise a system 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 using Gaussian elimination on an augmented matrix find the solution of a system of linear equations Ax = b, where A is a square matrix, include cases of unique solution, no solution (inconsistency) and an infinite family of solutions [A/B]. know the meaning of the term ill-conditioned [A/B]. compare the solutions of related systems of two equations in two unknowns and recognise ill-conditioning [A/B] Ill-conditioning can be introduced by comparing the solutions of the following systems: a. x y = x y = 1.97 b. x y = x y = 1.97 [A/B] Only 3 3 cases are required for assessment purposes, and at grade C they are restricted to those with a unique solution. Larger systems can be tackled using computer packages or advanced calculators. Mathematics: Advanced Higher Course 13

16 CONTENT NOTES TEACHING NOTES Mathematics 2 (AH) Further differentiation know the derivatives of sin 1 x, cos 1 x, tan 1 x Link with the graphs of these functions in Mathematics 1 (AH). differentiate any inverse function using the technique: y = f 1 1 (x) f(y) = x (f (x)) f (y) = 1, etc., and know the corresponding result dy 1 = dx dx dy understand how an equation f(x, y) = 0 defines y implicitly as one (or more) function(s) of x use implicit differentiation to find first and second derivatives [A/B] use logarithmic differentiation, recognising when it is appropriate in extended products and quotients and indices involving the variable [A/B]. understand how a function can be defined parametrically eg, x + y = 1 y = ± 1 x, ie two functions defined on [ 1, 1]. eg Find the derivatives of 2 x 7x 3 y =, y = 2 1+ x x [A/B]. This technique should be initially used on the inverse trigonometric functions. Calculators with graphic facility and computer packages may be used to investigate these derivatives. Link the last result with the chain rule applied to 1 1 f ( f ( x)) = f ( f ( x)) = x and with the logarithmic and exponential functions. Link with obtaining the derivatives of sin 1 x, cos 1 x, tan 1 x. Mathematics: Advanced Higher Course 14

17 CONTENT NOTES TEACHING NOTES understand simple applications of parametrically eg x 2 + y 2 = r 2, x = r cos θ, y = r sin θ defined functions use parametric differentiation to find first and second derivatives [A/B], and apply to motion in a plane apply differentiation to related rates in problems where the functional relationship is given explicitly or implicitly solve practical related rates by first establishing a functional relationship between appropriate variables [A/B] eg If the position is given by x = f(t), y = g(t) then the velocity components are given by dx dy and dt dt 2 2 dx dy and the speed by + dt dt The instantaneous direction of motion is given by dy dy/dt = dx dx/dt 1 2 dh dv eg Explicitly: V = πr h; given, find. 3 dt dt Implicitly: x + y = r where x, y are functions of dx dy dx dy t; given, find using 2 x + 2y = 0. dt dt dt dt Mathematics: Advanced Higher Course 15

18 CONTENT NOTES TEACHING NOTES Further integration know the integrals of 1 1 Link this with the derivatives of inverse, 2 trigonometric functions. 2 1 x 1+ x use the substitution x = at to integrate functions of the form 1 1, a x a + x integrate rational functions, both proper and improper, by means of partial fractions; the degree of the denominator being 3 the denominator may include: (i) two separate or repeated linear factors (ii) three linear factors [A/B] (iii) a linear factor and an irreducible quadratic factor [A/B] Candidates learning can be enhanced by completing the square but this will not be formally assessed. integrate by parts with one application integrate by parts involving repeated applications [A/B] eg xsin xdx 2 3x eg x e dx [A/B]. Mathematics: Advanced Higher Course 16

19 CONTENT NOTES TEACHING NOTES know the definition of differential equation and the meaning of the terms linear, order, general solution, arbitrary constants, particular solution, initial condition solve first order differential equations (variables separable) ie, equations that can be written in the form dy g( x) = dx h( y) Link with differentiation. Begin by verifying that a particular function satisfies a given differential equation. Candidates should know that differential equations arise in modelling of physical situations, such as electrical circuits and vibrating systems, and that the differential equation describes how the system will change with time so that initial conditions are required to determine the complete solution. formulate a simple statement involving rate of change as a simple separable first order differential equation, including the finding of a curve in the plane, given the equation of the tangent at (x, y), which passes through a given point know the laws of growth and decay: applications in practical contexts Link with motion in a straight line. Further similar contexts can be found in the Mechanics 1 (AH) unit. Scientific contexts such as chemical reactions, Newton s law of cooling, population growth and decay, bacterial growth and decay provide good motivating examples and can build on the knowledge and use of logarithms. Mathematics: Advanced Higher Course 17

20 CONTENT NOTES TEACHING NOTES Complex numbers know the definition of i as a solution of z = 0, so that i = 1 know the definition of the set of complex numbers as C = {a + ib : a, b R} know the definition of real and imaginary parts know the terms complex plane, Argand diagram A suggested approach is through the solution of quadratics. The introduction of i then allows the provision of two solutions for all quadratics. Thereafter, an essentially geometric approach is recommended. plot complex numbers as points in the complex plane perform algebraic operations on complex numbers: equality (equating real and imaginary parts), addition, subtraction, multiplication and division Plotting complex numbers as points can be made easier because some software (and calculators) write complex numbers as ordered pairs. Multiplication by i is equivalent to rotation by 90º and demonstrates a link between operations and transformations in the plane. This could be linked with Matrices in Mathematics 3 (AH). evaluate the modulus, argument and conjugate of complex numbers Mathematics: Advanced Higher Course 18

21 CONTENT NOTES TEACHING NOTES 2 2 y convert between Cartesian and polar form x = r cos θ, y = r sin θ, r = x + y and tan θ =. x know the fundamental theorem of algebra and the conjugate roots property The commonly used practice of writing 1 y θ = tan should be used with reference to x quadrants. The exponential form z = re iθ provides a link with power series in Further sequences and series in Mathematics 3 (AH). The facility in CAS to produce complex factors of polynomials allows candidates to conjecture that a polynomial of degree n has n roots, and that these occur in conjugate pairs when the coefficients are real. factorise polynomials with real coefficients eg Find the roots of a quartic when one complex root is given. solve simple equations involving a complex variable by equating real and imaginary parts interpret geometrically certain equations or inequalities in the complex plane eg z = 1 z a = b z 1 = z i z a > b The triangle inequality z + w z + w can be mentioned here. The proof of de Moivre s theorem for integer powers provides an opportunity to illustrate proof by induction. [Link with Further number theory in Mathematics 3(AH)]. For fractional powers the result can be stated without proof. Mathematics: Advanced Higher Course 19

22 CONTENT NOTES TEACHING NOTES know and use de Moivre s theorem with positive eg Expand (cos θ + i sin θ) 3 integer indices and fractional indices [A/B] eg Expand (cosθ + i sin θ) 1/2 [A/B] apply de Moivre s theorem to multiple angle trigonometric formulae [A/B] apply de Moivre s theorem to find nth roots of unity [A/B] Sequences and series know the meaning of the terms infinite sequence, infinite series, nth term, sum to n terms (partial sum), limit, sum to infinity (limit to infinity of the sequence of partial sums), common difference, arithmetic sequence, common ratio, geometric sequence, recurrence relation know and use the formulae u n = a + (n 1)d and Sn = 1 n[ 2a + ( n 1) d] 2 for the nth term and the sum to n terms of an arithmetic series, respectively eg Express sin 5θ in terms of sin θ only [A/B] Express cos 3 θ in terms of cos θ and cos 3θ [A/B] For assessment purposes only multiples and powers less than or equal to 5 are required. Link with the Argand diagram. An investigative approach to solving z n = 1, n 2, may be adopted, leading to discussion of the geometrical significance. Higher work on first order linear recurrence relations should be revised. Develop arithmetic and geometric sequences as special cases by considering x n + 1 = ax n + b with a = 1, b 0 for arithmetic sequences and then a 0, b = 0 for geometric sequences. These results should be explored numerically, conjectures made and proved. Mathematics: Advanced Higher Course 20

23 CONTENT NOTES TEACHING NOTES know and use the formulae u n = ar n 1 and n a(1 r ) Sn =, r 1, for the nth term and the sum 1 r to n terms of a geometric series, respectively know and use the condition on r for the sum to a infinity to exist and the formula S = for the 1 r sum to infinity of a geometric series where r < 1 expand 1 as a geometric series 1 r 1 and extend to [A/B] a + b This result should be linked to the binomial expansion (1 r) 1. In addition, link to Maclaurin expansions in Mathematics 3 (AH). 1 know the sequence and its limit n 1 + know and use the Σ notation n 1 The definition should be explored n numerically. n e = lim 1 + n n 1 know the formula r = n( n + 1) and apply it to simple sums r = 1 2 eg n r = 1 ( ar + b) = a n r = 1 r + n r = 1 b Link with proof by mathematical induction in the next section. Mathematics: Advanced Higher Course 21

24 CONTENT NOTES TEACHING NOTES Elementary number theory and methods of proof understand the nature of mathematical proof understand and make use of the notations, and know the corresponding terminology implies, implied by, equivalence know the terms natural number, prime number, rational number, irrational number know and use the fundamental theorem of arithmetic disprove a conjecture by providing a counterexample Simple examples involving the real number line, inequalities and modulus. eg x + y x + y Candidates should appreciate the need for proof in mathematics. A general strategy which could be used would be to explore a situation, make conjectures, verify (perhaps using software) and then finally prove.the triangle inequality can be linked with complex numbers. use proof by contradiction in simple examples eg The irrationality of 2. eg The infinity of primes. It should be emphasised that examples do not prove a conjecture except in proof by exhaustion. Mathematics: Advanced Higher Course 22

25 CONTENT NOTES TEACHING NOTES use proof by mathematical induction in simple examples The concept of mathematical induction may be introduced in a familiar context such as demonstrating that 2p coins and 5p coins are sufficient to pay any sum of money greater than 3p. prove the following results n r = 1 1 r = n( n + 1) 2 Candidates should use the application of: n 1 r = n( n + 1) to prove results about other sums, r = 1 2 such as the sum of the first n odd numbers is a perfect square. the binomial theorem for positive integers de Moivre s theorem for positive integers Straightforward proofs could be asked for in assessments without guidance. eg n = 2 n + 1 1, for all n N eg 8 n is a factor of (4n)! for all n N eg n < 2 n for all n N Mathematics: Advanced Higher Course 23

26 CONTENT NOTES TEACHING NOTES Mathematics 3 (AH) Vectors know the meaning of the terms position vector, unit vector, scalar triple product, vector product, components, direction ratios/cosines Higher work on vectors should be revised. calculate scalar and vector products in three dimensions The triangle inequality can now be completed with the vector version: a + b a + b know that a b = b a find a b and a. b c in component form This should be linked with determinants of matrices. know the equation of a line in vector form, parametric and symmetric form know the equation of a plane in vector form, parametric and Cartesian form find the equations of lines and planes given suitable defining information eg Find the equation of a plane passing through a given point and perpendicular to a given direction (the normal). find the angles between two lines, two planes [A/B], and between a line and a plane Link with systems of equations. find the intersection of two lines, a line and a plane, and two or three planes Mathematics: Advanced Higher Course 24

27 CONTENT NOTES TEACHING NOTES Matrix algebra know the meaning of the terms matrix, element, Link determinant with vector product. row, column, order, identity matrix, inverse, determinant, singular, non-singular, transpose perform matrix operations: addition, subtraction, multiplication by a scalar, multiplication, establish equality of matrices Link with Systems of linear equations in Mathematics 1 (AH). know the properties of the operations: A + B = B + A AB BA in general (AB)C = A(BC) A(B + C) = AB + AC (A')' = A (A + B)' = A' + B' (AB)' = B'A' (AB) 1 = B 1 A 1 det (AB) = det A det B calculate the determinant of 2 2 and 3 3 matrices know the relationship of the determinant to invertability find the inverse of a 2 2 matrix 1 a b 1 d - b The result = should be c d ad bc - c a known. Mathematics: Advanced Higher Course 25

28 CONTENT NOTES TEACHING NOTES find the inverse, where it exists, of a 3 3 matrix by elementary row operations Extension to larger matrices may be considered as a use of technology. know the role of the inverse matrix in solving linear systems use 2 2 matrices to represent geometrical transformations in the (x, y) plane Reference can also be made to the problem of finding the intersection of three lines or three planes in vector work and link its solubility to the existence of the matrix of coefficients. The transformations should include rotations, reflections, dilatations and the role of the transpose in orthogonal cases. Link with complex numbers in Mathematics 2 (AH). Further sequences and series know the term power series understand and use the Maclaurin series: r x ( r) f(x) = f (0) r = 0 r! find the Maclaurin series of simple functions: e x, sin x, cos x, tan 1 x, (1 + x) α, ln(1 + x), knowing their range of validity The approach taken should be through polynomials and their derivatives, linking coefficients to values at zero. Graphics software provides a powerful illustration of the concept of convergence. The difficulties of differentiating or integrating power series term by term should be stressed, pointing out that these processes are only valid within the interval of convergence Mathematics: Advanced Higher Course 26

29 CONTENT NOTES TEACHING NOTES Link (1 + x) α to (1 r) 1 from series work and to the extension of the binomial theorem. find the Maclaurin expansions for simple composites, such as e 2x use the Maclaurin series expansion to find power series for simple functions to a stated number of terms use iterative schemes of the form x n + 1 = g(x n ), n = 0, 1, 2, to solve equations where x = g(x) is a rearrangement of the original equation use graphical techniques to locate approximate solution x 0 know the condition for convergence of the sequence {x n } given by x n + 1 = g(x n ), n = 0, 1, 2, x x eg xe = 1 x = e to give the scheme xn e + 1 = with x 0 = 0.5 xn Candidates should be made aware that the equation f(x) = 0 can be rearranged in a variety of ways and that some of these may yield suitable iterative schemes while others may not. Cobweb and staircase diagrams help to demonstrate the test for convergence of an iterative scheme for finding a fixed point α ( α = g (α)) in a neighbourhood of α, namely g (x) < 1. Mathematics: Advanced Higher Course 27

30 CONTENT NOTES TEACHING NOTES Further ordinary differential equations solve first order linear differential equations using eg Write the linear equation the integrating factor method dy a ( x) + b( x) y = g( x) in the standard form dx dy + P( x) y = f ( x) and hence as dx d ( ) ( ) e P x dx P x dx y = e f ( x) dx find general solutions and solve initial value problems know the meaning of the terms: second order linear differential equation with constant coefficients, homogeneous, non-homogeneous, auxiliary equation, complementary function and particular integral eg Mixing problems, such as salt water entering a tank of clear water which is then draining at a given rate. eg Growth and decay problems, an alternative method of solution to separation of variables. eg Simple electrical circuits. eg The general solution of the homogeneous equation 2 d y dy a + b + cy = 0. 2 dx dx If f(x) and g(x) are solutions and f(x) Cg(x), ie linearly independent solutions, then the general solution is C 1 f(x) + C 2 g(x). Mathematics: Advanced Higher Course 28

31 CONTENT NOTES TEACHING NOTES solve second order homogeneous ordinary differential equations with constant coefficients Search for a trial solution using y = e mx and hence 2 derive the auxiliary equation am + bm + c = 0. 2 d y dy a + b + cy = 0 2 dx dx find the general solution in the three cases where the roots of the auxiliary equation: (i) are real and distinct (ii) coincide (are equal) [A/B] (iii) are complex conjugates [A/B] Link with Complex numbers in Mathematics 2 (AH). Context applications could include the motion of a spring, both with and without a damping term. solve initial value problems 2 d y dy eg y = 0 2 with y = 1 and dx dx when x = 0. dy dx = 2 solve second order non-homogeneous ordinary differential equations with constant coefficients: 2 d y dy a + b + cy = f(x) 2 dx dx using the auxiliary equation and particular integral method [A/B] The general solution is the sum of the general solution of the corresponding homogeneous equation (complementary function) and a particular solution. For assessment purposes, only cases where the particular solution can easily be found by inspection will be required, with the right-hand side being a low order polynomial or a constant multiple of sin x, cos x or e kx. Mathematics: Advanced Higher Course 29

32 CONTENT NOTES TEACHING NOTES Further number theory and further methods of proof know the terms necessary condition, sufficient condition, if and only if, converse, negation, contrapositive use further methods of mathematical proof: some simple examples involving the natural numbers Number theory can be drawn upon for likely but unproved conjectures, and also for erroneous conjectures, such as: (i) Are there an infinite number of twin primes, n and n + 2? (unproved) 2 (ii) Is 2 n + 1 prime for all n N? (erroneous, holds for n = 1, 2, 3, 4 but not for all higher values). direct methods of proof: sums of certain series and other straightforward results further proof by contradiction 2 eg x> 1 x > 1, the triangle inequality, the sum to n terms of an arithmetic or geometric series. eg If x, y R such that x + y is irrational then at least one of x, y is irrational. eg If m,n are integers and mn = 100, then either m 10 or n 10. further proof by mathematical induction prove the following result n r = 1 r 2 = 1 6 n( n + 1)(2n + 1); n N eg Proof of the result is a useful extension. n r = 1 r 2 n = r r = 1 n ( n 1) = 4 2 Mathematics: Advanced Higher Course 30

33 CONTENT NOTES TEACHING NOTES n n ( n + 1) know the result r = r = 1 4 n n 1 apply the above results and the one for r to eg r( r + 1) = n( n + 1)( n + 2). The generalisation of these results provides an r = 1 r = 1 3 interesting extension exercise. prove by direct methods results concerning other sums know the division algorithm and proof eg n r = 1 1 r( r + 1)( r + 2) = n( n + 1)( n + 2)( n + 3). 4 use Euclid s algorithm to find the greatest common divisor (g.c.d.) of two positive integers know how to express the g.c.d. as a linear combination of the two integers [A/B] use the division algorithm to write integers in terms of bases other than 10 [A/B] ie Use the division algorithm repeatedly. The special cases of finding the g.c.d. of two Fibonacci numbers are possible extensions. A possible context for this is binary arithmetic. Mathematics: Advanced Higher Course 31

34 Statistics 1 (AH) CONTENT Conditional probability and expectation appreciate the existence of dependent events calculate conditional probability solve problems using conditional probability solve a problem involving Bayes theorem [A/B] COMMENT Conditional probability can be calculated from first principles in every case but the following formulae may prove useful in solving problems involving dependent events. P(E and F) P(E F) =, P(F) > 0 P(F) P(E and F) = P(E F)P(F) = P(F E)P(E) i P(F) = P(F Ei )P(Ei ) P(F E j )P(Ej ) P(E j F) = for mutually exclusive and exhaustive P(F E )P(E ) i events E i [A/B] i i Bayes theorem is used to reverse the dependence, so that if the probability of F given that E j has occurred is known then the probability of E j given that F has occurred can be found. Mathematics: Advanced Higher Course 32

35 CONTENT apply the laws of expectation and variance: use the results E(aX + b) = ae(x) + b and E(X ± Y) = E(X) ± E(Y) use the results V(aX + b) = a 2 V(X) and V(X ± Y) = V(X) + V(Y) if X and Y are independent Probability distributions calculate probabilities using the binomial, Poisson and normal distributions use standard results for the mean and standard deviation of binomial, Poisson and normal distributions COMMENT When considering the laws of expectation and variance an intuitive approach may be taken, perhaps as follows: It should be clear from graphical considerations that if a random variable has a scale factor applied followed by a translation then so must the corresponding expectation (or mean) giving E(aX + b) = ae(x) + b. It may be equally intuitive that variability (dispersion) is not affected by translation and a little numerical investigation should demonstrate that V(aX + b) = a 2 V(X). It follows that V( X) = V(X) The difference between discrete and continuous distributions should be discussed. The former may be exemplified by the uniform, binomial and Poisson distributions and the latter by the uniform (a, b) and normal distributions. Candidates should be able to calculate probabilities with or without tables. The process of standardising data to produce standard scores is an important one and should be discussed. Candidates should be introduced to the probability functions and the formulae for the mean and standard deviation for each of the discrete distributions. Mathematics: Advanced Higher Course 33

36 CONTENT apply an appropriate approximation (normal or Poisson) to a binomial distribution and demonstrate the correct use of a continuity correction if appropriate [A/B] COMMENT It is often desirable to approximate binomial probabilities. The reasons for using either the Poisson or normal should be exemplified, particularly the use of a continuity correction in the latter case. One rule of thumb in current use is to use the normal approximation if np and nq are >5 and the Poisson otherwise, as long as p is small. solve problems involving the sum or difference of two independent normal variates [A/B] Sampling methods distinguish between different sampling methods: appreciate the difference between sample and census, and describe random, stratified, cluster, systematic and quota sampling use the central limit theorem and distribution of the sample mean and sample proportion use the sample mean and standard deviation to estimate the population mean and standard deviation and the sample proportion to estimate the population proportion The difference between a sample and a census should be stressed. The advantages and disadvantages of each type of sampling should be discussed, noting quota is not an example of random sampling. The central limit theorem states that for sufficiently large n the distribution of the sample mean is normal, irrespective of the distribution of the sample variates. Candidates should be able to show that sample mean has expectation µ and σ standard error (standard deviation) and that the sample proportion has n mean p and standard error (deviation) pq for large n and q = 1 p. n In elementary sampling theory the population mean/proportion is estimated by the sample mean/proportion and the population variance by the sample variance 2 given by s = 1 ( x x) 2 i. n 1 Mathematics: Advanced Higher Course 34

37 CONTENT obtain a confidence interval for the population mean [A/B] COMMENT An approximate 95% confidence interval for the population mean is given s by x ±1.96 for large n. n We may interpret this as saying that if we take a large number of samples and compute a confidence interval for each then 95% of these intervals would be expected to contain the population mean. It is also common practice to use a 99% confidence interval where 1.96 is replaced by obtain a confidence interval for the population proportion [A/B] Hypothesis Testing understand the terms null hypothesis, alternative hypothesis, one-tail test, two-tail test, distribution under H 0, test statistic, critical region, level of significance, p-value, reject H 0 and accept H 0 use a z-test on a statistical hypothesis: decide on a significance level state one-tail or two-tail test determine the p-value draw appropriate conclusions An approximate 95% confidence interval for the population proportion p is pq ˆ ˆ given by pˆ ± 1.96 for large n, where qˆ = 1 pˆ and pˆ is the sample proportion. n In a statistical investigation we often put forward a hypothesis the null hypothesis H 0 about a population parameter. In order to test this hypothesis we take a sample from the population and perform a statistical test. If we decide to reject H 0, we do so in favour of an alternative hypothesis H 1. A z-test should be used to test whether the population mean is equal to some specific value, where the population variable is normally distributed with known variance. eg H o : µ=µ o versus H 1 : µ µ o for a two-tail test. Mathematics: Advanced Higher Course 35

38 Numerical Analysis 1 (AH) CONTENT Taylor polynomials understand 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 (n) f "( a) 2 f (a) n f(a) + f ' ( a)(x a) + ( x a) +... ( x a) 2! n! obtain the Taylor polynomial of a function about a given point approximate a suitable function, f, near the point x = a, with (n) f " ( a) f (a) n f(a + h) f(a) + f ' (a)h + h h 2! n! understand the term truncation error in function approximation determine the magnitude of the principal truncation error when a function is approximated by a Taylor polynomial COMMENT Simple functions involving square roots, rational functions, trigonometric, exponential and logarithmic functions will be approximated. A graph-plotting package should be used to display f and the polynomial approximations. When discussing truncation error, consideration should be given to the number of decimal places to be carried in the calculation. The idea of roundoff error and guard figures in a calculation should be introduced. approximate a function value near to a given point giving the principal truncation error in the approximation use first order Taylor polynomials to assess the sensitivity of function values to small changes in the independent variable Ability to discuss the significance of the sensitivity of function values to small changes in the independent variable [A/B]. Mathematics: Advanced Higher Course 36

39 CONTENT Interpolate data understand the notation n f r for r and n N and be able to construct a finite difference table for a function tabulated at equal intervals COMMENT Errors in finite difference tables should be covered in examples. Ability to obtain relations involving differences, eg f = f 3f + 3f f [A/B] know that the nth differences of an nth degree polynomial are constant understand the terms interpolate and extrapolate p know the notation, p R, r N r know, establish and be able to use the Newton forward difference interpolation formula p p 2 p 3 f p = f + f0 + f0 + f use the sigma notation Ability to derive the Newton forward difference formula [A/B]. know the Lagrange interpolation formula p ( x) = L ( x) y, where ( x x0)( x x1)...( x xi Li ( x) = ( x x )( x x )...( x x and use the formula for n 3 i 0 i 1 i 1 i 1 n )( x xi+ )( x x i n i= 0 1 i )...( x xn ) )...( x x ) i+ 1 i i n Mathematics: Advanced Higher Course 37

40 CONTENT Numerical integration establish the trapezium rule over a strip and the corresponding principal truncation error COMMENT Obtain the truncation error both in terms of derivatives and in terms of finite differences. 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 trapezium rule can be obtained and estimate the error when the rule is used to estimate a given definite integral Round-off errors in function values should be discussed; candidates should be able to estimate their effect on the approximation to the integral. The number 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 s rule can be obtained and estimate the error when the rule is used to estimate a given definite integral The principal truncation error need not be derived in this case. As with the trapezium rule, round-off errors should be discussed. know the meaning of the order of a truncation error, the notation O(h n ) and the orders of the truncation errors in both the trapezium rule and Simpson s rule use the trapezium rule and Simpson s rule to estimate irregular areas from tabulated data and to approximate definite integrals establish and perform one application of Richardson s formula to improve the accuracy of the trapezium rule and of Simpson s rule by interval halving Ability to establish Richardson s formula [A/B]. Mathematics: Advanced Higher Course 38

41 Mechanics 1 (AH) CONTENT Motion in a straight line know the meaning of position, displacement, velocity, acceleration, uniform speed, uniform acceleration, scalar quantity, vector quantity COMMENT Concepts of position, velocity and acceleration should be introduced using vectors. Candidates should be very aware of the distinction between scalar and vector quantities, particularly in the case of speed and velocity. draw, interpret and use distance/time, velocity/time and acceleration/time graphs Candidates should be able to draw these graphs from numerical or graphical data. know that the area under a velocity/time graph represents the distance travelled dx know the rates of change v = = x! and 2 dt d x dv dx! a = = = = v! =! x 2 dt dt dt derive, by calculus methods, and use the equations governing motion in a straight line with constant acceleration, namely: 2 v = u + at, s = ut + 1 at 2 and from these, 2 2 ( u + v) t v = u + 2as, s = 2 solve analytically problems involving motion in one dimension under constant acceleration, including vertical motion under constant gravity Candidates should be familiar with the dot notation for differentiation with respect to time. Candidates need to appreciate that these equations are for motion with constant acceleration only. The general technique is to use calculus. Mathematics: Advanced Higher Course 39

42 CONTENT solve problems involving motion in one dimension where the acceleration is dv dependent on time, ie a = = f(t) dt Position, velocity and acceleration vectors including relative motion know the meaning of the terms relative position, relative velocity and relative acceleration, air speed, ground speed and nearest approach COMMENT be familiar with the notation: r P for the position vector of P v P = r! P for the velocity vector of P a = v! =! r for the acceleration vector of P P P P PQ = r Q r P for the position vector of Q relative to P v Q v P = r! Q r! P for the velocity of Q relative to P a a = v! v! = r!!! r for the acceleration of Q relative to P Q P Q P Q P resolve vectors into components in two and three dimensions This requires emphasis. differentiate and integrate vector functions of time use position, velocity and acceleration vectors and their components in two and three dimensions; these vectors may be functions of time apply position, velocity and acceleration vectors to solve practical problems, including problems on the navigation of ships and aircraft and on the effect of winds and currents Candidates should be able to solve such problems both by using trigonometric calculations in triangles and by vector components. Solutions by scale drawing would not be accepted solve problems involving collision courses and nearest approach Mathematics: Advanced Higher Course 40