MODULE 1 : BASIC ALGEBRA AND FUNCTIONS...........................7MODULE 2 : TRIGONOMETRY AND PLANE GEOMETRY .............18MODULE 3 : CALCULUS I ..............................................................23
IT 2: ANALYSIS, MATRICES AND COMPLEX NUMBERS
MODULE 1 : CALCULUS II .............................................................30MODULE 2 : SEQUENCES, SERIES AND APPROXIMATIONS........36MODULE 3 : COUNTING, MATRICES AND COMPLEX NUMBERS.43
TLINE OF ASSESSMENT..................................................................50
GULATIONS FOR PRIVATE CANDIDATES.......................................58
GULATIONS FOR RE-SIT CANDIDATES ..........................................58
e Caribbean Advanced Proficiency Examination (CAPE) is designed to provide certificationf the academic, vocational and technical achievement of students in the Caribbean who,aving completed a minimum of five years of secondary education, wish to further theirtudies. The examination addresses the skills and knowledge acquired by students under ae and articulated system where subjects are organised in 1-Unit or 2-Unit courses with eachntaining three Modules.
ts examined under CAPE may be studied concurrently or singly, or may be combined withts examined by other examination boards or institutions.
ribbean Examinations Council offers three types of certification. The first is the awardrtificate showing each CAPE Unit completed. The second is the CAPE diploma, awardeddidates who have satisfactorily completed at least six Units, including Caribbeans. The third is the CAPE Associate Degree, awarded for the satisfactory completion of aibed cluster of seven CAPE Units including Caribbean Studies and Communications. For the CAPE diploma and the CAPE Associate Degree, candidates must complete theof required Units within a maximum period of five years.
ematics is one of the oldest and most universal means of creating, communicating,ecting and applying structural and quantitative ideas. The discipline of Mathematics allowsrmulation and solution of real-world problems as well as the creation of new mathematical, both as an intellectual end in itself, but also as a means to increase the success andrality of mathematical applications. This success can be measured by the quantum leapoccurs in the progress made in other traditional disciplines once mathematics is introducedscribe and analyze the problems studied. It is, therefore essential that as many persons asble be taught not only to be able to use mathematics, but also to understand it.
ents doing this syllabus will have been already exposed to Mathematics in some formly through courses that emphasize skills in using mathematics as a tool, rather than givinght into the underlying concepts. To enable students to gain access to mathematics traininge tertiary level, to equip them with the ability to expand their mathematical knowledge andake proper use of it, it is, necessary that a mathematics course at this level should not onlyide them with more advanced mathematical ideas, skills and techniques, but encourage
to understand the concepts involved, why and how they "work" and how they areconnected. It is also to be hoped that, in this way, students will lose the fear associated withg to learn a multiplicity of seemingly unconnected facts, procedures and formulae. Inion, the course should show them that mathematical concepts lend themselves toralizations, and that there is enormous scope for applications to the solving of reallems.
ematics covers extremely wide areas. However, students can gain more from a study ofully selected, representative areas of Mathematics, for a "mathematical" understanding ofareas, rather than to provide them with only a superficial overview of a much wider field.
e proper exposure to a mathematical topic does not immediately make students intorts in it, that proper exposure will certainly give the students the kind of attitude which will
them to become experts in other mathematical areas to which they have not beenously exposed. The course is, therefore, intended to provide quality in selected areas ratherin a large number of topics.
timize the competing claims of spread of syllabus and the depth of treatment intended, alls in the proposed syllabus are required to achieve the aims of the Course. While both Units
2 can stand on their own, it is advisable that students should complete Unit 1 beforeUnit 2, since Unit1 contains basic knowledge for students of tertiary level courses
ding Applied Mathematics.
Mathematics Syllabus
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Through a development of understanding of these areas, it is expected that the course willenable students to:
(i) develop mathematical thinking, understanding and creativity;
(ii) develop skills in using mathematics as a tool for other disciplines;
(iii) develop the ability to communicate through the use of mathematics;
(iv) develop the ability to use mathematics to model and solve real-world problems;
(v) gain access to mathematics programmes at tertiary institutions.
AIMS
The syllabus aims to:
1. provide understanding of mathematical concepts and structures, their development andthe relationships between them;
2. enable the development of skills in the use of mathematical tools;
3. develop an appreciation of the idea of mathematical proof, the internal logical coherenceof Mathematics, and its consequent universal applicability;
4. develop the ability to make connections between distinct concepts in Mathematics, andbetween mathematical ideas and those pertaining to other disciplines;
5. develop a spirit of mathematical curiosity and creativity, as well as a sense of enjoyment;
6. enable the analysis, abstraction and generalization of mathematical ideas;
7. develop in students the skills of recognizing essential aspects of concrete real-worldproblems, formulating these problems into relevant and solvable mathematical problemsand mathematical modelling;
8. develop the ability of students to carry out independent or group work on tasks involvingmathematical modelling;
9. provide students with access to more advanced courses in Mathematics and itsapplications at tertiary institutions.
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SKILLS AND ABILITIES TO BE ASSESSED
The assessment will test candidates’ skills and abilities in relation to three cognitive levels.
(i) Conceptual knowledge is the ability to recall, select and use appropriate facts,concepts and principles in a variety of contexts.
(ii) Algorithmic knowledge is the ability to manipulate mathematical expressions andprocedures using appropriate symbols and language, logical deduction and inferences.
(iii) Reasoning is the ability to select appropriate strategy or select, use and evaluatemathematical models and interpret the results of a mathematical solution in terms of agiven real-world problem and engage in problem-solving.
PRE-REQUISITES OF THE SYLLABUS
Any person with a good grasp of the contents of the syllabus of the Caribbean SecondaryEducation Certificate (CSEC) General Proficiency course in Mathematics, or equivalent, shouldbe able to undertake the course. However, successful participation in the course will alsodepend on the possession of good verbal and written communication skills.
STRUCTURE OF THE SYLLABUS
The syllabus is arranged into two (2) Units, Unit 1 which will lay foundations, and Unit 2 whichexpands on, and applies, the concepts formulated in Unit 1.
It is, therefore, recommended that Unit 2 be taken after satisfactory completion of Unit 1 or asimilar course. Completion of each Unit will be separately certified.
Each Unit consists of three Modules.
Unit 1: Algebra, Geometry and Calculus, contains three Modules, each requiringapproximately 50 hours. The total teaching time, therefore, isapproximately 150 hours.
Module 1 - Basic Algebra and FunctionsModule 2 - Trigonometry and Plane GeometryModule 3 - Calculus I
Unit 2: Analysis, Matrices and Complex Numbers, contains three Modules, eachrequiring approximately 50 hours. The total teaching time, therefore, isapproximately 150 hours.
Module 1 - Calculus IIModule 2 - Sequences, Series and ApproximationsModule 3 - Counting, Matrices and Complex Numbers
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RECOMMENDED 2-UNIT OPTIONS
(a) Pure Mathematics Unit 1 AND Pure Mathematics Unit 2.
(b) Applied Mathematics Unit 1 AND Applied Mathematics Unit 2.
(c) Pure Mathematics Unit 1 AND Applied Mathematics Unit 2.
MATHEMATICAL MODELLING
Mathematical Modelling should be used in both Units 1 and 2 to solve real-worldproblems.
A. The topic Mathematical Modelling involves the following steps:
1. identification of a real-world situation to which modelling is applicable;
2. carry out the modelling process for a chosen situation to which modelling isapplicable;
3. discuss and evaluate the findings of a mathematical model in a written report.
B. The Modelling process requires:
1. a clear statement posed in a real-world situation, and identification of itsessential features;
2. translation or abstraction of the problem, giving a representation of theessential features of the real-world;
3. solution of the mathematical problem (analytic, numerical, approximate);
4. testing the appropriateness and the accuracy of the solution against behaviourin the real-world;
5. refinement of the model as necessary.
C. Consider the two situations given below.
1. A weather forecaster needs to be able to calculate the possible effects ofatmospheric pressure changes on temperature.
2. An economic adviser to the Central Bank Governor needs to be able to calculatethe likely effect on the employment rate of altering the Central Bank’s interest
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rate.
In each case, people are expected to predict something that is likely to happen in thefuture. Furthermore, in each instance, these persons may save lives, time, money orchange their actions in some way as a result of their predictions.
One method of predicting is to set up a mathematical model of the situation. A mathematicalmodel is not usually a model in the sense of a scale model motor car. A mathematical model isa way of describing an underlying situation mathematically, perhaps with equations, withrules or with diagrams.
D. Some examples of mathematical models are:
1. Equations
i. Business
A recording studio invests $25 000 to produce a master CD of a singinggroup. It costs $50.00 to make each copy from the master and cover theoperating expenses. We can model this situation by the equation ormathematical model,
C = 50.00 x + 25 000
where C is the cost of producing x CDs. With this model, one can predictthe cost of producing 60 CDs or 6 000 CDs.
Is the equation x + 2 = 5 a mathematical model? Justify your answer.
ii. Banking
Suppose you invest $100.00 with a commercial bank which pays interestat 12% per annum. You may leave the interest in the account toaccumulate. The equation A = 100(1.12)n can be used to model theamount of money in your account after n years.
2. Table of Values
Traffic Management
The table below shows the safe stopping distances for cars recommended by theHighway Code.
We can predict our stopping distance when travelling at 50mph from thismodel.
3. Rules of Thumb
You might have used some mathematical models of your own without realizingit; perhaps you think of them as “rules of thumb”. For example, in the baking ofhams, most cooks used the rule of thumb that “bake ham fat side up in roastingpan in a moderate oven (160ºC) ensuring 25 to 40 minutes per ½kg”. The cookis able to predict how long it takes to bake his ham without burning it.
4. Graphs
Not all models are symbolic in nature; they may be graphical. For example, thegraph below shows the population at different years for a certain country.
25 x
x20
15 x
10 x
Po
pu
lati
on
(mil
lio
ns
)
5 x
1960 1970 1980 1990
Years
RESOURCE
Hartzler, J. S. and Swetz, F. Mathematical Modelling in the Secondary SchoolCurriculum, A Resource Guide of Classroom Exercises,Vancouver, United States of America: National Councilof Teachers of Mathematics, Incorporated, Reston,1991.
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UNIT 1 - ALGEBRA, GEOMETRY AND CALCULUSMODULE 1: BASIC ALGEBRA AND FUNCTIONS
GENERAL OBJECTIVES
On completion of this Module, students should:
1. understand the concept of number;
2. develop the ability to construct simple proofs of mathematical assertions;
3. understand the concept of a function;
4. be confident in the manipulation of algebraic expressions and the solutions of equationsand inequalities;
5. develop the ability to use concepts to model and solve real-world problems.
SPECIFIC OBJECTIVES
(a) The Real Number System – R
Students should be able to:
1. use subsets of R;
2. use the properties of the inclusion chain N⊂WZQ R, RQ ;
3. use the concepts of identity, closure, inverse, commutativity, associativity,distributivity of addition and multiplication of real numbers;
4. demonstrate that the real numbers are ordered;
5. perform operations involving surds;
6. construct simple proofs, specifically direct proofs, or proof by the use of counter
examples;
7. use the summation notation ( );
8. establish simple proofs by using the principle of mathematical induction.
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UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
CONTENT
(a) The Real Number System – R
(i) Axioms of the system - including commutative, associative and distributive laws;non-existence of the multiplicative inverse of zero.
(ii) The order properties.
(iii) Operations involving surds.
(iv) Methods of proof - direct, counter-examples.
(v) Simple applications of mathematical induction.
SPECIFIC OBJECTIVES
(b) Algebraic Operations
Students should be able to:
1. apply real number axioms to carry out operations of addition, subtraction,multiplication and division of polynomial and rational expressions;
2. factorize quadratic polynomial expressions leading to real linear factors (realcoefficients only);
3. use the Remainder Theorem;
4. use the Factor Theorem to find factors and to evaluate unknown coefficients;
5. extract all factors of an- bn for positive integers n≤ 6;
6. use the concept of identity of polynomial expressions.
CONTENT
(b) Algebraic Operations
(i) Addition, subtraction, multiplication, division and factorization of algebraicexpressions.
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UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
(ii) Factor Theorem.
(iii) Remainder Theorem.
SPECIFIC OBJECTIVES
(c) Indices and Logarithms
Students should be able to:
1. use the laws of indices to simplify expressions (including expressions involvingnegative and rational indices);
2. use the fact that logαb = c αc = b;
3. simplify expressions by using the laws of logarithms, such as:
(i) log (PQ) = log P + log Q,
(ii) log(P/Q) = log P – log Q,
(iii) log Pa = a log P;
4. use logarithms to solve equations of the form ax = b;
5. solve problems involving changing of the base of a logarithm.
CONTENT
(c) Indices and Logarithms
(i) Laws of indices, including negative and rational exponents.
(ii) Laws of logarithms applied to problems.
(iii) Solution of equations of the form ax = b.
(iv) Change of base.
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UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
SPECIFIC OBJECTIVES
(d) Functions
Students should be able to:
1. use the terms: function, domain, range, open interval, half open interval, closedinterval, one-to-one function (injective function), onto function (surjectivefunction), one-to-one and onto function (bijective function), inverse andcomposition of functions;
2. show that there are functions which are defined as a set of ordered pairs and notby a single formula;
3. plot and sketch functions and their inverses (if they exist);
4. state the geometrical relationship between the function y= f(x) and its inverse[reflection in the line y = x];
5. interpret graphs of simple polynomial functions;
6. show that, if g is the inverse function of f, then f[g(x)] x, for all x, in the domainof g;
7. perform calculations involving given functions;
8. show graphical solutions of f(x) = g(x), f(x) g(x), f(x) g(x);
9. identify an increasing or decreasing function, using the sign ofba
bfaf
)()(when
a b;10. illustrate by means of graphs, the relationship between the function y = f(x) given
in graphical form and y a f(x); y f(x a); y = ƒ(x) ± a; y=a ƒ(x ± b); y =ƒ(ax), y f(x), where a, b are real numbers, and, where it is invertible, y f -1(x).
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UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
CONTENT
(d) Functions
(i) Domain, range, composition.
(ii) Injective, surjective, bijective functions, inverse function.
(iii) Graphical solutions of problems involving functions.
(iv) Simple transformations.
(v) Transformation of the graph y f(x) to y af(x); y f(x ± a); y f(x) ± a;y af(x ± b); y f(ax); y f(x) and, if appropriate, to y f -1(x).
SPECIFIC OBJECTIVES
(e) The Modulus Function
Students should be able to:
1. define the modulus function, for example, x = ;0xifx0xifx
5 = 5;
2. use the fact that x is the positive square root of x2 ;
3. use the fact that x < y if, and only if, x² < y²;
4. solve equations involving the modulus functions.
CONTENT
(e) The Modulus Function
(i) Definition and properties of the modulus function.
(ii) The triangle inequality.
CXC A6/U2/0712
UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
SPECIFIC OBJECTIVES
(f) Quadratic and Cubic Functions and Equations
Students should be able to:
1. express the quadratic function ax 2 + cbx in the form hxa 2 + k ;
2. sketch the graph of the quadratic function, including maximum or minimumpoints;
3. determine the nature of the roots of a quadratic equation;
4. find the roots of a cubic equation;
5. use the relationship between the sums and products of the roots and thecoefficients of:
(i) ax2 bx c = 0 ,
(ii) ax 3 bx2 + cx + d = 0 .
CONTENT
(f) Quadratic and Cubic Functions and Equations
(i) Quadratic equations in one unknown.
(ii) The nature of the roots of quadratic equations.
(iii) Sketching graphs of quadratic functions.
(iv) Roots of cubic equations.
(v) Sums and products, with applications, of the roots of quadratic and cubicequations.
CXC A6/U2/0713
UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
SPECIFIC OFBJECTIVES
(g) Inequalities
Students should be able to use algebraic and graphical methods to find the solution setsof:
1. linear inequalities;
2. quadratic inequalities;
3. inequalities of the form 0
bcx
bax;
4. inequalities of the form dcxbax .
CONTENT
(g) Inequalities
(i) Linear inequalities.
(ii) Quadratic inequalities.
(iii) Inequalities involving simple rational and modulus functions.
Suggested Teaching and Learning Activities
To facilitate students’ attainment of the objectives of this Module, teachers are advised to engagestudents in the teaching and learning activities listed below.
1. The Real Number System
The teacher should encourage students to practise different methods of proof byconstructing simple proofs of elementary assertions about real numbers, such as:
(i) (2) = 2;
(ii) For any real number a , 0a 0;
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UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
(iii) (1)(1) 1;
(iv) The statement “For all real x and y, x yxy” is false (by counter-example).
2. Proof by Mathematical Induction (MI)
Typical Question
Prove that some formula or /statement P is true for all positive integers n k, where k issome positive integer; usually k = 1.
Procedure
Step 1: Verify that when k = 1: P is true for n = k = 1. This establishes that P is true forn = 1.
Step 2: Assume P is true for n = k, where k is a positive integer > 1. At this point, thestatement k replaces n in the statement P and is taken as true.
Step 3: Show that P is true for n = k 1 using the true statement in step 2 with nreplaced by k.
Step 4: At the end of step 3, it is stated that statement P is true for all positive integersn k.
Summary
Proof by MI: For k > 1, verify Step 1 for k and proceed through to Step 4.
Observation
Most users of MI do not see how this proves that P is true. The reason for this is thatthere is a massive gap between Steps 3 and 4 which can only be filled by becoming awarethat Step 4 only follows because Steps 1 to 3 are repeated an infinity of times togenerate the set of all positive integers. The focal point is the few words “for all positiveintegers n k” which points to the determination of the set S of all positive
integers for which P is true.
CXC A6/U2/0715
UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
Step 1 says that 1 S for k = 1.
Step 3 says that k + 1 S whenever k S, so immediately 2 S since 1 S.
Iterating on Step 3 says that 3 S since 2 S and so on, so that S = {1, 2, 3 ...}, that is, S is theset of all positive integers when k = 1 which brings us to Step 4.
When k > 1, the procedure starts at a different positive integer, but the execution of steps is thesame. Thus, it is necessary to explain what happens between Steps 3 and 4 to obtain a fullappreciation of the method.
Example 1: Use Mathematical Induction to prove that n3 – n is divisible by 3,whenever n is a positive integer.
Solution: Let P (n) be the proposition that “n3 – n is divisible by 3”.
Basic Step: P(1) is true, since 13 - 1 = 0 which is divisible by 3.
Inductive Step: Assume P(n) is true: that is, n3 – n is divisible by 3.We must show that P(n + 1) is true, if P(n) is true. That is,(n + 1)3 – (n + 1) is divisible by 3.
Both terms are divisible by 3 since (n3 - n) is divisible by 3by the assumption and 3(n2 + n) is a multiple of 3.Hence, P (n+1) is true whenever P (n) is true.
Thus, n3 – n is divisible by 3 whenever n is a positiveinteger.
Example 2: Prove by Mathematical Induction that the sum Sn of the first n oddpositive integers is n2.
Solution: Let P (n) be the proposition that the sum Sn of the first n odd positiveinteger is n2.
Basic Step: For n=1 the first one odd positive integer is 1, so S1 = 1, thatis S1 = 1 = 12, hence P(1) is true.
UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
CXC A6/U2/0716
Inductive Step: Assume P(n) is true. That is, Sn = 1 + 3 + 5 + …. +
Since P(1) is true and P(n) P(n + 1), the propositionP(n) is true for all positive integers n.
3. Functions (Injective, surjective, bijective) – Inverse Function
Teacher and students should explore the mapping properties of quadratic functionswhich:
(i) will, or will not, be injective, depending on which subset of the real line is chosenas the domain;
(ii) will be surjective if its range is taken as the co-domain (completion of the squareis useful, here);
(iii) if both injective and surjective, will have an inverse function which can beconstructed by solving a quadratic equation.
Example: Use the function f :A B given by 56x23xf(x) , where the domain
A is alternatively the whole of the real line, or the set {xR x 1}, andthe co-domain B is R or the set { yR y 2}.
RESOURCES
Aub, M. R. The Real Number System, Barbados: CaribbeanExaminations Council, 1997.
Bostock, L. and Chandler, S. Core Mathematics for A-Levels, United Kingdom:Stanley Thornes Publishing Limited, 1997.
Cadogan, C. Proof by Mathematical Induction (MI), Barbados:Caribbean Examinations Council, 2004.
CXC A6/U2/0717
UNIT 1MODULE 1: BASIC ALGEBRA AND FUNCTIONS (cont’d)
Greaves, Y. Solution of Simultaneous Linear Equations by RowReduction, Barbados: Caribbean ExaminationsCouncil, 1998.
Hartzler, J. S. and Swetz, F. Mathematical Modelling in the Secondary SchoolCurriculum, A Resource Guide of Classroom Exercises,Vancouver, United States of America: National Councilof Teachers of Mathematics, Incorporated Reston,1991.
Hutchinson, C. Injective and Surjective Functions, Barbados:Caribbean Examinations Council, 1998.
Martin, A., Brown, K., Rigby, P.and Ridley, S.
Advanced Level Mathematics Tutorials PureMathematics CD-ROM sample (Trade Edition),Cheltenham, United Kingdom: Stanley Thornes(Publishers) Limited, Multi-user version and Single-user version, 2000.
CXC A6/U2/0718
CXC A6/U2/0719
UNIT 1MODULE 2: TRIGONOMETRY AND PLANE GEOMETRY
GENERAL OBJECTIVES
On completion of this Module, students should:
1. develop the ability to represent and deal with objects in the plane through the use ofcoordinate geometry, vectors;
2. understand that the alternative descriptions of objects are equivalent;
3. develop the ability to manipulate and describe the behaviour of trigonometric functions;
4. develop the ability to establish trigonometric identities;
5. develop skills to solve trigonometric equations;
6. develop the ability to use concepts to model and solve real-world problems.
SPECIFIC OBJECTIVES
(a) Trigonometric Functions, Identities and Equations (all angles will beassumed to be in radians unless otherwise stated)
Students should be able to:
1. graph the functions sin kx, cos kx, tan kx, k R;
2. relate the periodicity, symmetries and amplitudes of the functions in SpecificObjective 1
above to their graphs;
3. use the fact that xx cos2
sin
;
4. use the formulae for sin(A B), cos(A B) and tan (A B);
5. derive the multiple angle identities for sin kA, cos kA, tan kA, for kQ;
6. derive the identity cos2 sin2 1;
7. use the reciprocal functions sec x, cosec x and cot x;
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UNIT 1MODULE 2: TRIGONOMETRY AND PLANE GEOMETRY (cont’d)
8. derive the corresponding identities for tan2x, cot2x, sec2x and cosec2x;
9. develop and use the expressions for sin A ± sin B, cos A ± cos B;
10. use Specific Objectives 3, 4, 5, 6, 7, 8 and 9 above to prove simple identities;
11. express θθ sin b+ cos a in the form )( cos r αθ and )sin( r αθ where r is positive
0 α 2
;
12. find the general solution of equations of the form
(i) sin k c,
(ii) cos k c,
(iii) tan k =c,
(iv) a sin +b sin = c,
for a, b, c, k, R;
13. find the solutions of the equations in 12 above for a given range;
14. obtain maximum or minimum values of f() for 0 2 .
CONTENT
(a) Trigonometric Functions, Identities and Equations (all angles will beassumed to be radians)
(i) The circle, radian measure, length of an arc and area of a sector.
(ii) Sine rule, cosine rule.
(iii) Area of a triangle, using Area 12 ab sin C.
(iv) The functions sin x, cos x, tan x, cot x, sec x, cosec x.
(v) Compound-angle formulae for sin (A±B), cos (A±B), tan (A±B).
CXC A6/U2/0721
UNIT 1MODULE 2: TRIGONOMETRY AND PLANE GEOMETRY (cont’d)
(vi) Multiple-angle formulae.
(vii) Formulae for sin A ± sin B, cos A ± cos B.
(viii) Use of appropriate formulae to prove identities.
(ix) Expression of a sin + b cos in the forms r sin (±) and r cos (±), where r is
positive, 0 ≤ α <2
.
(x) General solution of simple trigonometric equations, including graphicalinterpretation.
(xii) Maximum and minimum values of functions of sin and cos .
SPECIFIC OBJECTIVES
(b) Co-ordinate Geometry
Students should be able to:
1. use the gradient of the line segment;
2. use the relationships between the gradients of parallel and mutuallyperpendicular lines;
3. find the point of intersection of two lines;
4. write the equation of a circle with given centre and radius;
5. find the centre and radius of a circle from its general equation;
6. find equations of tangents and normals to circles;
7. find the points of intersection of a curve with a straight line;
8. find the points of intersection of two curves;
9. obtain the Cartesian equation of a curve given its parametric representation.
CXC A6/U2/0722
CXC A6/U2/0723
UNIT 1MODULE 2: TRIGONOMETRY AND PLANE GEOMETRY (cont’d)
CONTENT
(b) Co-ordinate Geometry
(i) Properties of the circle.
(ii) Tangents and normals.
(iii) Intersections between lines and curves.
(iv) Cartesian equations and parametric representations of curves.
SPECIFIC OBJECTIVES
(c) Vectors
Students should be able to:
1. express a vector in the form
y
xor xi+yj;
2. define equality of two vectors;
3. add and subtract vectors;
4. multiply a vector by a scalar quantity;
5. derive and use unit vectors;
6. find displacement vectors;
7. find the magnitude and direction of a vector;
8. apply properties of parallel vectors and perpendicular vectors;
9. define the scalar product of two vectors:
(i) in terms of their components,
(ii) in terms of their magnitudes and the angle between them;
10. find the angle between two given vectors.
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UNIT 1MODULE 2: TRIGONOMETRY AND PLANE GEOMETRY (cont’d)
CONTENT
(c) Vectors
(i) Expression of a given vector in the form
y
xor xi + yj.
(ii) Equality, addition and subtraction of vectors; multiplication by a scalar.
(iii) Position vectors, unit vectors, displacement vectors.
(iv) Length (magnitude/modulus) and direction of a vector.
(v) Scalar (Dot) Product.
Suggested Teaching and Learning Activities
To facilitate students’ attainment of the objectives of this Module, teachers are advised to engagestudents in the teaching and learning activities listed below.
1. Trigonometric Identities
Much practice is required to master proofs of Trigonometric Identities using identitiessuch as the formulae for:
sin (A ± B), cos (A ± B), tan (A ± B), sin 2A, cos 2A, tan 2A
Example: The identity
2tan
4sin
4cos1
can be established by realizing that
cos 4 1 – 2 sin2 2 and sin 4 2 sin 2 cos 2.
Derive the trigonometric functions sin x and cos x for angles x of any value (includingnegative values), using the coordinates of points on the unit circle.
RESOURCE
Bostock, L. and Chandler, S. Mathematics - The Core Course for A-Level, United Kingdom:Stanley Thornes (Publishers) Limited, 1997.
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UNIT 1MODULE 3: CALCULUS I
GENERAL OBJECTIVES
On completion of this Module, students should:
1. understand the concept of continuity of a function and its graph;
2. appreciate that functions need not be continuous;
3. develop the ability to find the limits (when they exist) of functions in simple cases;
4. know the relationships between the derivative of a function at a point and the behaviourof the function and its tangent at that point;
5. be confident in differentiating given functions;
6. know the relationship between integration and differentiation;
7. know the relationship between integration and the area under the graph of the function;
8. know the properties of the integral and the differential;
9. develop the ability to use concepts to model and solve real-world problems.
SPECIFIC OBJECTIVES
(a) Limits
Students should be able to:
1. use graphs to determine the continuity and continuity of functions;
2. describe the behaviour of a function f(x) as x gets arbitrarily close to some givenfixed number, using a descriptive approach;
3. use the limit notation axasLf(x)lim →→,)(→
Lxfax
;
CXC A6/U2/0726
UNIT 1MODULE 3: CALCULUS I (cont’d)
4. use the simple limit theorems:
If ,)( Fxfax
lim Gxgax
)(lim and k is a constant,
then kF,kf(x)ax
lim
FG,f(x)g(x)ax
lim
G,Fg(x)f(x)ax
lim
and, provided G 0,G
F
g(x)
f(x)
axlim
;
5. use limit theorems in simple problems, including cases in which the limit of f(x)
at a is not f(a), for example,
2x
42
x
xlim
2(the use of L’Hopital’s Rule is not
allowed);
6. use the fact that 1sin
lim x
xx 0
, demonstrated by a geometric approach (the use of
L’Hopital’s Rule is not allowed);
7. solve simple problems involving limits and requiring algebraic manipulation (theuse of L’Hopital’s Rule may be allowed);
8. identify the region over which a function is continuous;
9. identify the points where a function is discontinuous and describe the nature ofits discontinuity;
10. use the concept of left handed or right handed continuity, and continuity on aclosed interval.
CONTENT
(a) Limits
(i) Concept of limit of a function.
(ii) Limit Theorems.
(iii) Continuity and Discontinuity.
CXC A6/U2/0727
UNIT 1MODULE 3: CALCULUS I (cont’d)
SPECIFIC OBJECTIVES
(b) Differentiation I
Students should be able to:
1. demonstrate understanding of the concept of the derivative at a point x = c as thegradient of the tangent to the graph at x c;
2. define the derivative at a point as a limit;
3. use the f (x) notation for the first derivative at x;
4. differentiate, from first principles, such functions as:
(i) f kx )( where k R,
(ii) f( x ) = xn, where n {-3, -2, -1, - ½, ½, 1, 2, 3},
(iii) f )(x = sin x ;
5. demonstrate an understanding of how to obtain the derivative of xn, where n isany number;
6. demonstrate understanding of simple theorems about derivatives of y c f(x),y f(x) g(x); where c is a constant;
7. use 5 and 6 above repeatedly to calculate the derivatives of:
(i) polynomials,
(ii) trigonometric functions;
8. use the product and quotient rules for differentiation;
UNIT 1MODULE 3: CALCULUS I (cont’d)
CXC A6/U2/0728
9. differentiate products and quotients of:
(i) polynomials,
(ii) trigonometric functions;
10. apply the chain rule in the differentiation of composite functions (substitution);
11. demonstrate an understanding of the concept of the derivative as a rate ofchange;
12. use the sign of the derivative to investigate where a function is increasing ordecreasing;
13. demonstrate the concept of stationary (critical) points;
14. determine the nature of stationary points;
15. locate stationary points, maxima and minima by considering sign changes of thederivative;
16. calculate second derivatives;
17. interpret the significance of the sign of the second derivative;
18. use the sign of the second derivative to determine the nature of stationary points;
19. sketch graphs of polynomials, rational functions and trigonometric functionsusing the features of the function and its first and second derivatives;
20. describe the behaviour of such graphs for large values of the independentvariable;
21. obtain equations of tangents and normals to curves.
CONTENT
(b) Differentiation I
(i) The Gradient.
(ii) The Derivative as a limit.
UNIT 1MODULE 3: CALCULUS I (cont’d)
CXC A6/U2/0729
(iii) Rates of change.
(iv) Differentiation from first principles.
(v) Differentiation of simple functions, product, quotients.
(vi) Stationary points and chain rule.
(vii) Second derivatives of functions.
(viii) Curve sketching.
(ix) Tangents and Normals to curves.
SPECIFIC OBJECTIVES
(c) Integration I
Students should be able to:
1. define integration as the inverse of differentiation;
2. demonstrate an understanding of the indefinite integral and the use of the
integration notation dxf(x) ;
3. show that the indefinite integral represents a family of functions which differ byconstants;
4. demonstrate use of the following integration theorems:
(i) dxf(x)cdxcf(x) , where c is a constant,
(ii) dx;g(x)dxf(x)dxg(x)}{f(x)
5. find:
(i) indefinite integrals using integration theorems,
(ii) integrals of polynomial functions,
(iii) integrals of simple trigonometric functions;
UNIT 1MODULE 3: CALCULUS I (cont’d)
6. define and calculate ba dxxf )( F(b) F(a), where F(x) is an indefinite integral
CXC A6/U2/0730
of f(x) and integrate, using substitution;
7. use the results:
(i) ba
ba dtf(t)dxf(x) ,
(ii) a0 dxf(x)
a0 dx,x)f(a for a ;
8. apply integration to:
(i) finding areas under the curve,
(ii) finding volumes of revolution by rotating regions about both the x and yaxes;
9. formulate and solve differential equations of the form y´ f(x) where f is apolynomial or a trigonometric function.
CONTENT
(c) Integration I
(i) Integration as the inverse of differentiation.
(ii) Linearity of integration.
(iii) Indefinite integrals (concept and use).
(iv) Definite integrals.
(v) Applications of integration – areas, volumes and solutions to elementarydifferential equations.
(vi) Integration of polynomials.
(vii) Integration of simple trigonometric functions.
(viii) Use of b
adxxf )( F(b) F(a), where F '(x) f(x).
(ix) First order differential equations.
CXC A6/U2/0731
UNIT 1MODULE 3: CALCULUS I (cont’d)
Suggested Teaching and Learning Activities
To facilitate students’ attainment of the objectives of this Module, teachers are advised to engagestudents in the teaching and learning activities listed below.
The Area under the Graph of a Continuous Function
Class discussion should play a major role in dealing with this topic. Activities such as that whichfollows may be performed to motivate the discussion.
Example of classroom activity:
Consider a triangle of area equal to21 units, bounded by the graphs of y = x, y = 0 and x = 1.
(i) Sketch the graphs and identify the triangular region enclosed.
(ii) Subdivide the interval [0, 1] into n equal subintervals.
(iii) Evaluate the sum, s(n), of the areas of the inscribed rectangles and S(n), of thecircumscribed rectangles, erected on each subinterval.
(iv) By using different values of n, for example, for n = 5, 10, 25, 50, 100, show that both s(n)and S(n) get closer to the required area of the given region.
RESOURCES
Aub, M. R. Differentiation from First Principles: The PowerFunction, Barbados: Caribbean Examinations Council,1998.
Bostock, L., and Chandler, S. Mathematics - The Core Course for A-Level, UnitedKingdom: Stanley Thornes Publishing Limited, (Chapters5, 8 and 9), 1991.
Ragnathsingh, S. Area under the Graph of a Continuous Function,Barbados: Caribbean Examinations Council, 1998.
