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Mathematics C Semesters 1 & 2 Syllabus 2011

01 2011 Maths C Syllabus

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Page 1: 01 2011 Maths C Syllabus

Mathematics C

Semesters 1 & 2

Syllabus

2011

Page 2: 01 2011 Maths C Syllabus

UNSW Foundation Studies UNSW Global Pty Limited UNSW Sydney NSW 2052 Copyright © 2011 All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, this publication may not be reproduced, in part or whole, without the permission of the copyright owner.

Page 3: 01 2011 Maths C Syllabus

TABLE OF CONTENTS

General Information for Students.............................. page 1

Summary of Syllabus Units........................................ page 4

Sections of the Syllabus Units.................................... page 5

Syllabus in Detail....................................................... page 8

Page 4: 01 2011 Maths C Syllabus

GENERAL INFORMATION FOR STUDENTS

1. Course Objectives

At the conclusion of the course, students should have developed an appreciation of the scope and usefulness of mathematics and the skills in applying mathematical techniques to the solution of practical problems. Students should have obtained and enhanced the ability to interpret and communicate mathematics and developed skills required for further studies in Commerce, Business, Economics and Social Sciences.

2. Semester Program At the beginning of each semester, students will be given a Lecture Program

for the Semester. The numbered references on the Semester Program refer to the Syllabus. The Semester Program should be used in close conjunction with the Syllabus. Teachers will keep closely to the Semester Program, but some minor

variations may be unavoidable. 3. Exercises The exercises listed in the syllabus are from the Tutorial Books for Semester 1

and Semester 2. Lecturers and tutors will advise students of the specific questions that should

be attempted from these references. Lecturers and tutors may give or set further exercises in order to practice, consolidate, extend or review the material.

4. Non Assessable Tests During each semester, there will be at least one non assessable test. These tests

will be designed to give students practice in examination technique and time allocation and also to provide feedback on student progress.

5. Assessment The grade for Mathematics will be based on the Semester 1 assessment and the

Final Examination. The Final Examination will be based on the whole course.

©UNSW Foundation Year 1 Mathematics C Syllabus

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6. Lectures Lecturers will review familiar areas, introduce new topics, cover relevant

theory and will give an overview of the main points and ideas as well as consolidating and extending basic concepts. Lecturers will also set exercises for students to attempt before the follow-up tutorial on the content.

Lectures are of 50 minutes duration. After each lecture, students are advised to read over their own lecture notes

and any material handed out during the lecture BEFORE attempting the exercises set for tutorial.

7. Tutorials In tutorials students will be in smaller groups where course material will be

discussed, consolidated, practised and extended. Before their tutorials, students should attempt the exercises set to ensure they

obtain maximum benefit from their tutorials. You should bring to each tutorial: (a) your lecture notes; (b) your Tutorial Book; (c) your attempted exercises. During tutorials, you should: (a) seek your tutor's help for any individual problems arising from lectures

or tutorial exercises attempted; (b) be prepared to consolidate and extend the material listed. Note: Rolls will be marked during each tutorial. After tutorials, if you need further help with the material you should attend

consultations at times appropriate to your timetable. 8. Consultations Consultations will be available to all students. These are additional times

during which students can seek help for individual problems. Although consultations are optional, students are encouraged to use them to clarify and reinforce material outside of lecture or tutorial time.

©UNSW Foundation Year 2 Mathematics C Syllabus

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9. Additional Material Sometimes additional material in the form of extra sheets, lecture notes and

booklets of notes and exercises may be given to all students. All additional material is examinable unless otherwise specified.

10. Calculators The Casio fx-82AU PLUS calculator is to be used in UNSW Foundation

Year examinations. 11. Reference Books The following books are suggested as references: Cambridge Mathematics 3 Unit Year 11 by Pender, Sadler, Shea, Ward. Cambridge Mathematics 3 Unit Year 12 by Pender, Sadler, Shea, Ward. Year 11 − 3 Unit (Extension 1) Maths by J. Coroneos. Year 12 − 3 Unit (Extension 1) Maths by J. Coroneos. 3 Unit Mathematics − Book 1 by S.B. Jones & K.E. Couchman. 3 Unit Mathematics − Book 2 by S.B. Jones & K.E. Couchman.

These books may provide alternative approaches to material in the syllabus. They also contain additional exercises that would be beneficial for reinforcement, consolidation, extension or revision.

IF YOU NEED HELP OR ADVICE . . . You should consult your lecturer or tutor about any problems you are experiencing and attend scheduled consultations. If you have any problems or questions regarding your mathematics course after you have consulted your lecturer or tutor and attended consultations, then you should come to the UNSW Foundation Year Office, level 1, L5 Building, UNSW. Dr Tony van Ravenstein Head of Department, Mathematics Foundation Studies UNSW [email protected]

©UNSW Foundation Year 3 Mathematics C Syllabus

Page 7: 01 2011 Maths C Syllabus

SUMMARY OF SYLLABUS UNITS

Semester 1

Prerequisite Knowledge and Terms in Mathematics

Unit 1 Basic Number and Algebra Review

Unit 2 The Number Plane, Functions and Graphs

Unit 3 Differential Calculus

Unit 4 Logarithmic and Exponential Functions

Unit 5 Sequences and Series

Semester 2

Unit 6 Further Differential Calculus and Applications

Unit 7 Mathematics of Finance

Unit 8 Integral Calculus

Unit 9 Probability

Unit 10 Data Description and Probability Distributions

©UNSW Foundation Year 4 Mathematics C Syllabus

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SECTIONS OF THE SYLLABUS UNITS

Semester 1 Prerequisite Knowledge and Terms in Mathematics Unit 1 Basic Number and Algebra Review

1-1 Calculators and Approximation 1-2 The Real Number System 1-3 Sets 1-4 Linear Equations and Inequalities in One Variable 1-5 Operations with Surds 1-6 Indices 1-7 Factorization of Algebraic Expressions 1-8 Algebraic Fractions 1-9 Simultaneous Equations 1-10 Absolute Value Equations and Inequalities 1-11 Quadratic Equations and Applications 1-12 Basic Operations with Polynomials

Unit 2 The Number Plane, Functions and Graphs

2-1 Gradient of a Straight Line 2-2 Equation of a Straight Line 2-3 Distance and Midpoint Formulae 2-4 Circles and Semicircles 2-5 Coordinate Geometry Problems 2-6 Relations and Functions 2-7 Basic Graphs and Transformations 2-8 Graphs of Linear Functions and Absolute Value Graphs 2-9 Graphs of Parabolas, Cubic Curves, Hyperbolas 2-10 More on Parabolas and Quadratic Inequalities 2-11 Polynomial Functions 2-12 More on Domain and Piecemeal Functions 2-13 Interpretation of Graphs

©UNSW Foundation Year 5 Mathematics C Syllabus

Page 9: 01 2011 Maths C Syllabus

Semester 1 (continued) Unit 3 Differential Calculus

3-1 Limits and Continuity 3-2 The Derivative 3-3 The Rules for Differentiation 3-4 Tangents and Normals 3-5 Rates of Change – Average and Instantaneous 3-6 Implicit Differentiation 3-7 Related Rates 3-8 The Sign of the First Derivative 3-9 The Sign of the Second Derivative 3-10 Curve Sketching Using Calculus

Unit 4 Logarithmic and Exponential Functions

4-1 The Definition of a Logarithm 4-2 Laws of Logarithms – Change of Base 4-3 Exponential Functions – Graphs and Applications 4-4 Inverse Functions 4-5 Graphs of Logarithmic Functions

Unit 5 Sequences and Series

5-1 Introduction to Sequences and Series – Sigma Notation 5-2 Arithmetic Sequences and Series 5-3 Geometric Sequences and Series 5-4 Infinite Geometric Series

©UNSW Foundation Year 6 Mathematics C Syllabus

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©UNSW Foundation Year 7 Mathematics C Syllabus

Semester 2

Unit 6 Further Differential Calculus and Applications

6-1 Review of Curve Sketching using Calculus 6-2 Sketching Rational Functions 6-3 Maximization and Minimization Problems 6-4 Differentiation of Logarithmic and Exponential Functions

Unit 7 Mathematics of Finance

7-1 Simple Interest 7-2 Compound Interest 7-3 Future Value of an Annuity; Sinking Funds 7-4 Present Value of an Annuity; Amortization 7-5 Mixed Problems in Finance Mathematics

Unit 8 Integral Calculus

8-1 Basic Antiderivatives and Indefinite Integrals 8-2 Definite Integrals and The Fundamental Theorem of Calculus 8-3 Area Under a Curve 8-4 The Area Between Two Curves 8-5 Volumes of Solids of Revolution About the x-axis 8-6 Integration Involving Exponential Functions 8-7 Integration Involving Logarithmic Functions 8-8 Exponential Growth and Decay 8-9 Differential Equations, Integration and Rates of Change 8-10 Integration by Substitution

Unit 9 Probability

9-1 Basic Counting Principles 9-2 Basic Probability, Sample Spaces and Events 9-3 Permutations and Combinations 9-4 Union, Intersection, and Complement of Events; Odds 9-5 Successive Outcomes, Conditional Probability, and Independence 9-6 Random Variable, Probability Distribution and Expectation

Unit 10 Data Description and Probability Distributions

10-1 Graphing Data 10-2 Graphing Quantitative Data 10-3 Measures of Central Tendency 10-4 Measures of Dispersion 10-5 Normal Distributions 10-6 The Binomial Theorem 10-7 Bernouilli Trials and Binomial Distributions.

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SYLLABUS IN DETAIL Semester 1 PREREQUISITE KNOWLEDGE AND TERMS IN MATHEMATICS UNIT 1 BASIC NUMBER AND ALGEBRA REVIEW 1-1 Calculators and Approximation EXERCISE

Use of the calculator, exact values and approximations. Scientific notation.

Exercise 1-1

1-2 The Real Number System (a) Terms in Mathematics; the language of

Mathematics.

Exercise 1-2A

(b) The real number system. The real number line. Ordering on the real number line. Basic real number properties.

Exercise 1-2B

1-3 Sets (a) Symbols, union and intersection, the

Universal set, complementary sets, set properties, Venn diagrams.

Exercise 1-3A

(b) Further sets, Venn diagrams and word problems.

Exercise 1-3B

1-4 Linear Equations and Inequalities in One Variable

(a) Linear equations, change of subject Exercise 1-4A

(b) Linear inequalities; word problems and applications.

Exercise 1-4B

1-5 Operations with Surds

Properties of surds. Simplification; addition and subtraction, multiplication and division; rationalization of the denominator.

Exercise 1-5

1-6 Indices

Definition of , index laws, zero and negative exponents; exponent properties. Rational exponents, nth roots of real numbers; rational exponents and surds.

na

Exponential equations.

Exercise 1-6

©UNSW Foundation Year 8 Mathematics C Syllabus

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UNIT 1 BASIC NUMBER AND ALGEBRA REVIEW (CONTINUED) 1-7 Factoring of Algebraic Expressions EXERCISE Common factor, factoring by grouping

difference of two squares, factoring second-degree polynomials, sum and difference of two cubes, combined factoring techniques.

Exercise 1-7

1-8 Operations with Algebraic Fractions

Reducing to lowest terms, multiplication and division, addition and subtraction, compound fractions.

Exercise 1-8

1-9 Simultaneous Equations

Algebraic solution of systems of linear equations in two variables (elimination and substitution); applications.

Exercise 1-9

1-10 Absolute Value Equations and Inequalities.

(a) Absolute value – definition and equations.

Exercise 1-10A

(b) Absolute value - inequalities.

Exercise 1-10B

1-11 Quadratic Equations

(a) Solution by square root, by factoring and the quadratic formula; simultaneous equations - one linear and one non-linear leading to a quadratic; word problems and applications. Equations reducible to quadratic equations.

Exercise 1-11A

(b) Completing the square of a quadratic expression, maximum or minimum value of a quadratic expression. Apllications.

Exercise 1-11B

1-12 Basic Operations with Polynomials

(a) Definition, degree, leading term, monic polynomial, operations on polynomials, division of polynomials

Exercise 1-12A

(b) The remainder and factor theorems, zeros of a polynomial.

Exercise 1-12B

©UNSW Foundation Year 9 Mathematics C Syllabus

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©UNSW Foundation Year 10 Mathematics C Syllabus

UNIT 2 THE NUMBER PLANE, FUNCTIONS AND GRAPHS 2-1 Gradient of a Straight Line EXERCISE Gradient or slope of lines

Cartesian coordinate system; intercepts; gradients of lines, gradient of a straight line passing through two points. Relationship between gradients of parallel and perpendicular lines.

Exercise 2-1

2-2 Equation of a Straight Line

Equations of lines General form of a line ( )0=++ cbyax ; slope and y-intercept form, finding the slope from the equation of a line, point-slope form. Parallel and perpendicular lines Equations of lines parallel or perpendicular to a given line; testing whether a point lies on a given line. Applications.

Exercise 2-2

2-3 Distance and Midpoint Formula

Distance and Midpoint of an Interval Distance between two points and midpoint of an interval.

Exercise 2-3

2-4 Circles and Semicircles

(a) Circles Equation of a circle in the form of

, centre and ( ) ( )x a y b r− + − =2 2 2

radius of a circle, domain and range.

Exercise 2-4A

(b) Semicircles Equations of semicircles in the form of y a x y a x= − = − −2 2 2 and 2 , domain and range.

Exercise 2-4B

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UNIT 2 THE NUMBER PLANE, FUNCTIONS AND GRAPHS (CONTINUED) 2-5 Coordinate Geometry Problems EXERCISE

Coordinate Geometry Problems Multi-step problems using coordinate geometry.

Exercise 2-5

2-6 Relations and Functions

(a) Functions and Relations Relations; functions-dependent and independent variables; vertical line test, domain and range. Word problems.

Exercise 2-6A

(b) Function Notation Notation and applications.

Exercise 2-6B

(c) Even and Odd Functions Definitions of even and odd functions; graphs of even and odd functions.

Exercise 2-6C

2-7 Basic Graphs and Transformations

Elementary functions:

3

32

)( (vi))( )v()( (iv) )( (iii)

)( (ii) )( i)(

xxpxxnxxmxxhxxgxxf

======

used in the transformation of functions - for a given function ),(xfy = to graph for a constant c

.10 and 1for )( (v))( (iv))( (iii)

)( (ii))( i)(

<<>=

=−=

+=+=

ccxfcyxfyxfy

cxfycxfy

Exercise 2-7

2-8 Linear Functions and Straight Lines

(a) Linear functions and their graphs Graphs of equations in the form

cbyax =+ ; domain and range; Applications.

Exercise 2-8A

(b) Absolute value graphs involving straight lines, domain and range.

Exercise 2-8B

©UNSW Foundation Year 11 Mathematics C Syllabus

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UNIT 2 THE NUMBER PLANE, FUNCTIONS AND GRAPHS (CONTINUED) 2-9 Graphs of Parabolas, Cubic Curves and

Hyperbolas EXERCISE

(a) Basic parabolas in the form , where a and C are

constants and . Transformations. ( ) CBxay +−= 2

a ≠ 0

Exercise 2-9A

(b) Basic Cubic Curves Exercise 2-9B

(c) Hyperbolas Simple rectangular hyperbolas; domain and range, asymptotes.

Exercise 2-9C

2-10 Quadratic Functions, Parabolas and Quadratic Inequalities

(a) Quadratic functions and factored form and their graphs.

y ax bx c= + +2

Parabolas, vertex, axis of symmetry, domain and range; applications.

Exercise 2-10A

(b) Solving quadratic inequalities using a graph and applications.

Exercise 2-10B

2-11 Polynomial Functions Graphs of Polynomial Functions

Graphs of polynomial functions showing double and multiple zeros ; roots of a polynomial equation.

Exercise 2-11

2-12 More on Domain and Piecemeal Functions

(a) More on Domain Exclusion from the domain; largest possible domain; restriction on the domain.

Exercise 2-12A

(b) Piecemeal Functions Evaluation; sketching piecemeal graphs.

Exercise 2-12B

2-13 Interpretation of Graphs

Reading information from a graph. Using a graph to solve , ( ) 0=xf ( ) 0>xf etc

Exercise 2-13

©UNSW Foundation Year 12 Mathematics C Syllabus

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UNIT 3 DIFFERENTIAL CALCULUS 3-1 Limits and Continuity EXERCISE (a) Computation of limits

Limits as x approaches a constant c, existence of limits, properties of limits.

Exercise 3-1A

(b) Continuity, continuity of graphs (light, intuitive treatment only)

Exercise 3-1B

3-2 The Derivative (a) Tangent lines, slope of a secant, slope of a

tangent line, differentiation from first principles, the derivative.

Exercise 3-2A

(b)

Differentiability, Non-existence of the derivative.

Exercise 3-2B

(c) Basic differentiation rules. Derivative of , derivative of a constant, derivative of a constant times a function, derivatives of sums and differences.

nx

Exercise 3-2C

(d) The second derivative. Exercise 3-2D

3-3 The Rules for Differentiation (a) Derivatives of products and quotients.

Exercise 3-3A

(b)

Chain rule: power form, general form. Combining rules of differentiation. Applications

Exercise 3-3B

3-4 Tangents and Normals

Gradients and equations of tangents and normals.

Exercise 3-4

3-5 Rates of Change – Average and Instantaneous

Average rate of change, instantaneous rate of change, rates of change involving time.

Exercise 3-5

©UNSW Foundation Year 13 Mathematics C Syllabus

Page 17: 01 2011 Maths C Syllabus

UNIT 3 DIFFERENTIAL CALCULUS (CONTINUED) 3-6 Implicit Differentiation EXERCISE

Explicit and implicit functions, implicit differentiation

Exercise 3-6

3-7 Related Rates Problems where two rates of change are

related. Rates of change involving time.

Exercise 3-7

3-8 The First Derivative and Graphs

Increasing and decreasing functions; stationary points. Use of the first derivative to determine the nature of stationary points (local maximum or local minimum, horizontal points of inflection); curve sketching.

Exercise 3-8

3-9 The Second Derivative and Graphs

(a) Second Derivative, concavity, concave up and concave down.

Exercise 3-9A

(b) Finding inflection points using the second derivative. Second derivative test for local maximum and minimum; curve sketching

Exercise 3-9B

3-10 Curve Sketching Using Calculus

Curve sketching, turning points, inflections, intercepts on the axes, restrictions on domain. Absolute maxima and minima for continuous functions.

Exercise 3-10

©UNSW Foundation Year 14 Mathematics C Syllabus

Page 18: 01 2011 Maths C Syllabus

UNIT 4 LOGARITHMIC AND EXPONENTIAL FUNCTIONS 4-1 The Definition of a Logarithm EXERCISE (a) Logarithms and the calculator

Use of the calculator keys and .

,10, xxe xlnxlog

Exercise 4-1A

(b) Definition of a logarithm; conversion from logarithmic form to exponential form and vice versa; the irrational number e; logarithms to base e; simple logarithmic equations.

Exercise 4-1B

4-2 Laws of Logarithms – Change of Base Law (a) Basic laws of logarithms. Equations and

problems involving logarithms; exponential and logarithmic equations

Exercise 4-2A

(b) Further laws of logarithms and logarithmic properties including the change of base formula. Inequalities with logarithms.

Exercise 4-2B

4-3 Exponential Functions – Graphs and Applications (a) Graphs of basic exponential

functions, y-intercept, equation of horizontal asymptote, domain and range.

Exercise 4-3A

(b) Applications of exponential functions The exponential function with base e .

Exercise 4-3B

4-4 Inverse Functions Notation; reflection property;

equations of inverse functions; domain and range of functions and their inverses.

Exercise 4-4

4-5 Logarithmic Functions – Graphs and Applications

Graphs of Basic Logarithmic Functions x-intercept, equation of vertical asymptote, domain and range.

Exercise 4-5

©UNSW Foundation Year 15 Mathematics C Syllabus

Page 19: 01 2011 Maths C Syllabus

©UNSW Foundation Year 16 Mathematics C Syllabus

UNIT 5 SEQUENCES AND SERIES 5-1 Introduction to Sequences and Series EXERCISE Sequence; general term of a sequence;

series; sigma notation.

Exercise 5-1

5-2 Arithmetic Sequences and Series (a) Arithmetic sequences (progressions) ; test for

an A.P. ; the formula for the nth term; applications and problems.

Exercise 5-2A

(b) Arithmetic series, Both formulae for the sum of the first n terms of an arithmetic series; applications and problems.

Exercise 5-2B

5-3 Geometric Sequences and Series

(a) Geometric sequences (progressions), Test for a G.P; the formula for the nth term of a geometric progressions; applications and problems.

Exercise 5-3A

(b) Geometric series; formulae for the sum of the first n terms of a geometric series; applications and problems.

Exercise 5-3B

5-4 Infinite Geometric Series

Infinite geometric series Limiting sum of a geometric progression for − < <1 1r ; applications and problems.

Exercise 5-4

Page 20: 01 2011 Maths C Syllabus

Semester 2 UNIT 6 FURTHER DIFFERENTIAL CALCULUS AND APPLICATIONS 6-1 Review of Curve Sketching using Calculus EXERCISE Review of curve sketching, turning points,

inflections, intercepts on the axes, restrictions on domain.

Exercise 6-1

6-2 Sketching Rational Functions (a) Limits at infinity.

Exercise 6-2A

(b) Asymptotes Horizontal, vertical and oblique asymptotes.

Exercise 6-2B

(c) Graphs of Rational Functions Asymptotes, intercepts; turning points.

Exercise 6-2C

(d) More on Graphs of Rational Functions Applications and problems.

Exercise 6-2D

6-3 Maximization and Minimization Problems

(a) Application of calculus to maximum and minimum problems

Exercise 6-3A

(b) Further applications of calculus to maximum and minimum.

Exercise 6-3B

6-4 Derivatives of Logarithmic and Exponential Functions

(a) Derivative of e , , x baxe + ( )xfe ; applications

Exercise 6-4A

(b) Derivative of ln x, ( )bax +ln , ( )[ ]xfln ; logarithmic properties; applications.

Exercise 6-4B

(c) Further applications of the derivatives of exponential and logarithmic functions.

Exercise 6-4C

©UNSW Foundation Year 17 Mathematics C Syllabus

Page 21: 01 2011 Maths C Syllabus

UNIT 7 MATHEMATICS OF FINANCE 7-1 Simple Interest EXERCISE

Simple interest formula tPrI = , The principal or present value P and the amount or future value A; applications. Use of the formula . tPrPA +=

Exercise 7-1

7-2 Compound Interest

(a) Compound interest formula n)i(PA += 1 ;depreciation by reducing balance method using the formula n)i(PA −= 1 .

Exercise 7-2A

(b) Effective rate of interest , and nominal rate; er

effective rate formula 11 −+= me )

mr(r

Exercise 7-2B

7-3 Future Value of an Annuity; Sinking Funds Derivation of formula for future value

of an annuity ( )i

iPMTFVn 11 −+

= and

applications of this formula; sinking funds.

Exercise 7-3

7-4 Present Value of an Annuity; Amortization

Present value of an annuity, use of the formula ( )

iiPMTPV

n−+−=

11 ; amortization.

Exercise 7-4

7-5 Mixed Problems in Finance Mathematics

A general problem-solving strategy to delineate between present value and future value problems.

Exercise 7-5

©UNSW Foundation Year 18 Mathematics C Syllabus

Page 22: 01 2011 Maths C Syllabus

UNIT 8 INTEGRAL CALCULUS 8-1 Antiderivatives and Indefinite Integrals EXERCISE (a) Basic antiderivatives; the primitive function;

indefinite integral of , nx 1−≠n

Notation cnxdxx

nn +

+=

+

∫ 1

1;

Exercise 8-1A

(b) Further indefinite integrals; Use of the

formulae cna

baxdxbaxn

n ++

+=+

+

∫ )1()()(

1 and

[ ] cnxfdxxfxf

nn+

+=′

+

∫ 1)]([)()(

1

Exercise 8-1B

8-2 Definite Integrals and The Fundamental theorem of Calculus

Proof of the fundamental theorem of calculus and the link between areas and the definite integral; properties of definite integrals; evaluating definite integrals; using definite integrals to find simple areas above the x-axis.

Exercise 8-2

8-3 Area Under a Curve Areas above and below the x-axis only.

Exercise 8-3

8-4 The Area Between Two Curves

Area between two curves; addition and subtraction of areas. Using the x-axis only.

Exercise 8-4

8-5 Volumes of Solids of Revolution

Volumes of solids of revolution about the x-axis only.

Exercise 8-5

©UNSW Foundation Year 19 Mathematics C Syllabus

Page 23: 01 2011 Maths C Syllabus

©UNSW Foundation Year 20 Mathematics C Syllabus

UNIT 8 INTEGRAL CALCULUS (CONTINUED) 8-6 Integration Involving Exponential Functions EXERCISE (a)

Indefinite integrals and ⎮⌡⌠ += Cedxe xx

⎮⌡⌠ ≠+ 0a,C

= +e Cf x( )

= ++ 1 ea

dxe baxbax and

and ⌠⌡⎮

′f x e dxf x( ) ( )

⌠⌡⎮

= +a dxa

a C ax x1 0ln

, > but a ≠ 1.

Exercise 8-6A

(b) Definite integrals of , and , areas, volumes and applications.

e x baxe + f x e f x' ( ). ( )

Exercise 8-6B

8-7 Integration Involving Logarithmic Functions (a)

Indefinite integrals ⎮⌡⌠ += Cxdx

xln1 and

⎮⌡⌠ ≠++=

+0ln11 a,Cbax

adx

bax and

′= +⌠

⌡⎮f xf x

dx f x C( )( )

ln ( ) .

Exercise 8-7A

(b) Definite integrals involving logarithmic functions, areas, volumes and applications

Exercise 8-7B

8-8 Exponential Growth and Decay (a) Exponential Growth and Decay

Proportionality; the equation dNdt

kN= ,

the graph of , growth and decay problems and applications.

N t Aekt( ) =

Exercise 8-8A

(b) Further exponential growth and decay problems and applications.

Exercise 8-8B

8-9 Differential Equations, Integration and Rates of Change Solving differential equations; differential

equations involving rates of change and applications.

Exercise 8-9

8-10 Integration by Substitution

Indefinite and definite integrals by change of variable, substitution techniques

Exercise 8-10

UNIT 9 PROBABILITY

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9-1 Basic Counting Principles EXERCISE (a) Addition principle for counting,

Venn diagrams, tables.

Exercise 9-1A

(b) Multiplication principle, successive outcomes, tree diagrams.

Exercise 9-1B

9-2 Basic Probability, Sample Spaces and Events

Experiments; outcomes, events, sample spaces, probability of an event, equally likely events.

Exercise 9-2

9-3 Permutations and Combinations

(a) Factorials Factorial notation, manipulation of factorials, use of the calculator.

Exercise 9-3A

(b) Permutations Definition; notation; arrangements with restrictions; arrangements in a row with repetitions; arrangements of n objects in a row when they are not all different.

Exercise 9-3B

(c) Combinations Definition; notation; selections with conditions or restrictions.

Exercise 9-3C

(d) Probability involving permutations and combinations.

Exercise 9-3D

9-4 Union, Intersection, and Complement of Events; Odds

Union and intersection, complement of an event, odds, applications to empirical probability.

Exercise 9-4

9-5 Successive Outcomes, Conditional Probability, and Independence

(a) Successive outcomes, product rule, probability trees, dependent and independent events.

Exercise 9-5A

(b) Conditional probability, testing for independence.

Exercise 9-5B

©UNSW Foundation Year 21 Mathematics C Syllabus

Page 25: 01 2011 Maths C Syllabus

UNIT 9 PROBABILITY ( CONTINUED ) 9-6 Discrete Random Variables, Probability Distribution and Expectation (a) Discrete random variables, probability

distribution; expected value of a random variable.

Exercise 9-6A

(b) Expected value of a random variable, decision making and applications

Exercise 9-6B

©UNSW Foundation Year 22 Mathematics C Syllabus

Page 26: 01 2011 Maths C Syllabus

UNIT 10 - DATA DESCRIPTION AND PROBABILITY DISTRIBUTIONS 10-1 Graphing Data EXERCISE Bar and line graphs; pie charts.

Exercise 10-1

10-2 Graphing Quantitative Data

Frequency distributions, cumulative and relative frequency distributions, histograms, frequency polygons.

Exercise 10-2

10-3 Measures of Central Tendency

Finding mean, median and mode for grouped and ungrouped data. Finding the mean on the calculator.

Exercise 10-3

10-4 Measures of Dispersion

(a) Measures of spread: range, variance and standard deviation. The formula for standard

deviation, ( )

1

2

1 −

−= ∑

− nxxi

nσ . Mean and

standard deviation and its significance for ungrouped and grouped data

Exercise 10-4A

(b) Applications of measures of central tendency and dispersion.

Exercise 10-4B

10-5 Normal Distributions

(a) Normal distributions Normal distributions, area under the normal curve, use of tables;

Exercise 10-5A

(b) Applications of the normal distribution.

Exercise 10-5B

10-6 The Binomial Theorem Binomial theorem, expansions; general

term, coefficients.

Exercise 10-6

10-7 Bernoulli Trials and Binomial Distributions Bernoulli trials, binomial distribution,

its mean and variance; applications.

Exercise 10-7

©UNSW Foundation Year 23 Mathematics C Syllabus

Page 27: 01 2011 Maths C Syllabus
Page 28: 01 2011 Maths C Syllabus

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