FAC Pack Jun14

The Actuarial Education Company IFE: 2014 Examinations

FAC P C 14

Combined Materials Pack

ActEd Study Materials: 2014 Examinations

Foundation ActEd Course (FAC)

Contents

Study Guide for the 2014 exams Course Notes

Question and Answer Bank Summary Test

If you think that any pages are missing from this pack, please contact ActEds admin team by email at [email protected] or by phone on 01235 550005.

How to use the Combined Materials Pack

Guidance on how and when to use the Combined Materials Pack is set out in the Study Guide for the 2014 exams.

Important: Copyright Agreement

This study material is copyright and is sold for the exclusive use of the purchaser. You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any

part of it. You must take care of your material to ensure that it is not used or copied by anybody else. By opening this pack you agree to these conditions.

IFE: 2014 Examinations The Actuarial Education Company

All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned

by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries.

You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material.

You must take care of your study material to ensure that it is not used or copied by anybody else.

Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the

profession or through your employer.

These conditions remain in force after you have finished using the course.

FAC: Study Guide Page 1


Foundation ActEd Course (FAC)

Study Guide

Objectives of the Study Guide The purpose of this Study Guide is to give you the information that you should have before studying FAC.

0 Introduction

This document has the following sections: Section 1 Syllabus Page 2 Section 2 The FAC course Page 10 Section 3 Study skills Page 12 Section 4 Contacts Page 13 Section 5 Course index Page 15

FAC Online Classroom

Please note that by purchasing this FAC you receive complimentary access to the FAC online classroom. This is a series of pre-recorded tutorials covering the main points from the course with examples as well as a dedicated forum for queries staffed by tutors. To access the online classroom please visit:

https://learn.bpp.com

You should have received an email with your access details. If you have lost this then enter your username (which is your email address used by ActEd) and click the Forgotten your password? to have a new password emailed to you. Should you have any problems with accessing the online classroom then please do email our admin team at [email protected].

ACET Mock Exam

A practice exam containing questions of the same standard as the ACET exam can be found in the reference resources section of the FAC online classroom.

FAC: Study Guide


1 Syllabus

1.1 Syllabus

This syllabus has been written by ActEd to help in designing introductory tuition material for new students. The topics are all required within the Core Technical subjects. From past experience we know that some students can be a little rusty on mathematical techniques so we have designed a course to help students brush up on their knowledge. Unlike other actuarial subjects, there is no official syllabus or Core Reading written by the profession. (a) Mathematical Notation, Terminology and Methods (a)(i) Be familiar with standard mathematical notation and terminology, so as to be

able to understand statements such as the following: 1. , , , , 3 : n n na b c n n a b c 2. 2( ,0], {0}x x 3. { : 1,2,3, }x x 4. x x 163 5. Zero is a non-negative integer; is a positive real number.

6. f x( ) tends to 0 as x tends to , is not defined when x 0 , but takes positive values for sufficiently large x .

(a)(ii) Know the representations and names of the letters of the Greek alphabet that are

commonly used in mathematical, statistical and actuarial work, including in particular, the following letters:

lower-case: upper-case:

(a)(iii) Understand the meaning of the following commonly used conventions:

round brackets used to denote negative currency amounts, K and m used as abbreviations for thousand and million used to denote the change in a quantity iff used as an abbreviation for if and only if.



(a)(iv) Understand the concept of a mathematical proof and the meaning of necessary, sufficient and necessary and sufficient as they are used in mathematical derivations.

(a)(v) Prove a result using the method of mathematical induction. (b) Numerical methods (b)(i) Evaluate numerical expressions using an electronic calculator with the

following features: arithmetic functions ( ), powers ( y x ) and roots ( yx ), exponential ( ex ) and natural log ( ln x ) functions. (The following features are also useful but not essential: factorial function ( n!), combinations ( n rC ), hyperbolic tangent function and its inverse ( tanh x and tanh

1 x ), fraction mode, at least one memory and an undo facility. Statistical and financial functions are not required.) Students should be able to make efficient use of memories and/or brackets.

(b)(ii) Estimate the numerical value of expressions without using a calculator and

apply reasonableness tests to check the result of a calculation. (b)(iii) Quote answers to a specified or appropriate number of decimal places or

significant figures (using the British convention for representing numbers), and be able to assess the likely accuracy of the result of a calculation that is based on rounded or approximated data values.

(b)(iv) Be able to carry out consistent calculations using a convenient multiple of a

standard unit (eg working in terms of 000s). (b)(v) Express answers, where appropriate, in the form of a percentage (%) or as an

amount per mil (). (b)(vi) Calculate the absolute change, the proportionate change or the percentage

change in a quantity (using the correct denominator and sign, where appropriate) and understand why changes in quantities that are naturally expressed as percentages, such as interest rates, are often specified in terms of basis points.

(b)(vii) Calculate the absolute error, the proportionate error or the percentage error in

comparisons involving actual versus expected values or approximate versus accurate values (using the correct denominator and sign, where appropriate).

FAC: Study Guide


(b)(viii)Determine the units of measurement (dimensions) of a quantity and understand the advantages of using dimensionless quantities in certain situations.

(b)(ix) Use linear interpolation to find an approximate value for a function or the

argument of a function when the value of the function is known at two neighbouring points.

(b)(x) Apply simple iterative methods, such as the bisection method or the Newton-

Raphson method, to solve non-linear equations. (b)(xi) Carry out simple calculations involving vectors, including the use of row/column

vectors and unit vectors, addition and subtraction of vectors, multiplication of a vector by a scalar, scalar multiplication (dot product) of two vectors, determining the magnitude and direction of a vector, finding the angle between two vectors and understanding the concept of orthogonality.

(b)(xii) Carry out calculations involving matrices, including transposition of a matrix,

addition and subtraction of matrices, multiplication of a matrix by a scalar, multiplication of two appropriately sized matrices, calculating the determinant of a matrix, calculating and understanding the geometrical interpretation of eigenvectors and eigenvalues, finding the inverse of a 2 2 matrix and using matrices to solve systems of simultaneous linear equations.

(c) Mathematical Constants and Standard Functions (c)(i) Be familiar with the mathematical constants and e . (c)(ii) Understand and apply the definitions and basic properties of the functions

x n (where n may be negative or fractional), c x (where c is a positive constant), exp( )x [ ex ], and ln x [ loge x or log x ].

(c)(iii) Sketch graphs of simple functions involving the basic functions in (c)(ii) by

identifying key points, identifying and classifying turning points, considering the sign and gradient, and analysing the behaviour near 0, 1, or other critical values.

(c)(iv) Simplify and evaluate expressions involving the functions x (absolute value),

[ ]x (integer part), max( ) and min( ) , and understand the concept of a bounded function. [The notation ( )x 100 will also be used as an abbreviation for max( , )x 100 0 .]



(c)(v) Simplify and evaluate expressions involving the factorial function n! for non-negative integer values of the argument and the gamma function ( )x for positive integer and half-integer values of the argument.

(c)(vi) Understand the concept of a complex number and be able to simplify

expressions involving i 1 , including calculating the complex conjugate. (c)(vii) Calculate the modulus and argument of a complex number, represent a complex

number on an Argand diagram or in polar form ( z rei ). (c)(viii) Apply Eulers formula e ii cos sin and use the basic properties of the

sine and cosine functions to simplify expressions involving complex numbers, including determining the real and imaginary parts of an expression.

(c)(ix) Understand the correspondence between the factors of a polynomial expression

and the roots of a polynomial equation and appreciate that a polynomial equation of degree n with real coefficients will, in general, have n roots consisting of conjugate pairs and/or real values.

(d) Algebra (d)(i) Manipulate algebraic expressions involving powers, logs, polynomials and

fractions. (d)(ii) Solve simple equations, including simultaneous equations (not necessarily

linear) by rearrangement, substitution, cancellation, expansion and factorisation.

(d)(iii) Solve an equation that can be expressed as a quadratic equation (with real

roots) by factorisation, by completing the square or by applying the quadratic formula, and identify which of the roots is appropriate in a particular context.

(d)(iv) Solve inequalities (inequations) in simple cases and understand the concept of

a strict or weak inequality. (d)(v) State and apply the arithmetic-geometric mean inequality, and know the

conditions under which equality holds. (d)(vi) Understand and apply the and notation for sums and products, including

sums over sets (eg i0 ) and repeated sums.

FAC: Study Guide


(d)(vii) Calculate the sum of a series involving finite arithmetic or geometric progressions or infinite geometric progressions using the formulae:

AP a n d a ln n 2 22 1( ( ) ) ( ) or , GP a r

r

n

( )11

and

GPa

r1,

and be able to determine when an infinite geometric series converges.

(d)(viii) Apply the formulae: k n nk

n

1

12 1( ) and k n n n

k

n2

1

16 1 2 1

( )( ) . (d)(ix) Solve simple first or second order difference equations (recurrence relations),

including applying boundary conditions, by inspection or by means of an auxiliary equation.

(d)(x) Recognise and apply the binomial expansion of expressions of the form ( )a b n

where n is a positive integer, and ( )1 x p for any real value of p and, in the latter case, determine when the series converges.

(e) Calculus (e)(i) Understand the concept of a limit (including limits taken from one side) and

evaluate limits in simple cases using standard mathematical notation, including the use of order notation O x( ) and o x( ) , and the sup/ lub and inf / glb functions (considered as generalisations of max and min ).

(e)(ii) Understand the meaning of a derivative as the rate of change of a function when

its argument is varied (in particular, for functions dependent on t , the time measured from a specified reference point), including the interpretation of a derivative as the gradient of a graph.

(e)(iii) Differentiate the standard functions x n , c x , ex and ln x . (e)(iv) Evaluate derivatives of sums, products (using the product rule), quotients (using

the quotient rule) and functions of a function (using the chain rule). (e)(v) Understand the concept of a higher-order (repeated) derivative and be familiar

with the mathematical notation used to denote such quantities.



(e)(vi) Use differentiation to find the maximum or minimum value of a function over a specified range (including the application of a monotone function, such as the natural log function, to simplify the calculation) and determine the nature of stationary points.

(e)(vii) Understand the meaning of a partial derivative and how to express a partial

derivative in standard mathematical notation, and be able to evaluate partial derivatives in simple cases. Find extrema of functions of two variables.

(e)(viii)Use the method of Lagrangian multipliers. (e)(ix) Understand the meaning of an indefinite integral as the anti-derivative of a

function and the meaning of a definite integral as the limit of a sum of infinitesimal elements, including the interpretation of a definite integral as the area under a graph.

(e)(x) Integrate the standard functions x n , c x and ex . (e)(xi) Evaluate indefinite and definite integrals by inspection, by identifying and

applying an appropriate substitution, by integration by parts, by using simple partial fractions or by a combination of these methods.

(e)(xii) Determine when a definite integral converges. (e)(xiii)Understand the meaning of a multiple integral and how to express a multiple

integral in standard mathematical notation, and be able to evaluate a double integral as a repeated integral in simple cases, including determining the correct limits of integration. Swap the order of integration.

(e)(xiv) Apply the trapezium rule to find the approximate value of an integral. (e)(xv) State and apply Taylor series and Maclaurin series in their simplest form,

including using these to determine the approximate change in a function when the argument is varied by a small amount. (Knowledge of the error terms is not required.)

(e)(xvi) Recognise and apply the Taylor series expansions for ex and ln( )1 x and, in

the latter case, determine when the series converges. (e)(xvii)Solve simple ordinary first-order differential equations, including applying

boundary conditions, by direct integration (which may involve a function of the dependent variable), by separation of variables or by applying an integrating factor.

FAC: Study Guide


(e)(xvii)Differentiate expressions involving definite integrals with respect to a parameter, including cases where the limits of integration are functions of the parameter.

(g) General (g)(i) Be familiar with the currency systems of the United Kingdom (pounds and pence

sterling), the United States (dollars and cents), the European monetary system (Euro and cent) and other major economies, and be able to interpret and write down currency amounts using these systems.

(g)(ii) Be familiar with the Gregorian calendar, including determining when a

specified year is a leap year, the concepts of calendar years, quarters and tax years, and the abbreviations commonly used to represent dates in the United Kingdom, Europe and the United States.

(g)(iii) Understand the distinction between expression/equation/formula and

term/factor. (g)(iv) Understand the meaning of the words gross and net. (g)(v) Be able to spell the following words correctly: actuarial, appropriately,

basically, benefit, benefiting, bias(s)ed, calendar, cancelled, commission, consensus, correlation, cyclically, deferred, definitely, formatted, fulfil, gauge, hierarchy, immediately, independence, instalment (British spelling), lose, loose, millennium, necessary, occasion, occurred/occurring, offered, orthogonal, paid, particularly, pensioner, precede, proceed, receive, referred, relief, seize, separate, similarly, specifically, supersede, targeted, theorem, until, yield.

(g)(vi) Be able to determine the correct member of word pairs according to context: eg

affect/effect, principal/principle, dependant/dependent. (g)(vii) Be able to distinguish between the singular and plural forms of words of Latin

or Greek origin, including the following: criterion/criteria, formula/formulae, analysis/analyses. [The word data may be treated as singular or plural, according to the preferences of individual authors/speakers.]

(g)(viii)Be familiar with commonly used Latin expressions and abbreviations such as

per annum, vice versa, status quo, pro rata, ie, eg, cf, sic and stet.



1.2 The Professions Copyright

All of the course material is copyright. The copyright belongs to Institute and Faculty Education Ltd, a subsidiary of the Institute and Faculty of Actuaries. The material is sold to you for your own exclusive use. You may not hire out, lend, give, sell, transmit electronically, store electronically or photocopy any part of it. You must take care of your material to ensure it is not used or copied by anyone at any time. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer. These conditions remain in force after you have finished using the course.

FAC: Study Guide


2 The FAC course

2.1 Course Notes

The Foundation ActEd Course consists of a set of eight chapters of notes covering the following ideas: Chapter 1 Notation Chapter 2 Numerical Methods I Chapter 3 Mathematical constants and standard functions Chapter 4 Algebra Chapter 5 Numerical Methods II Chapter 6 Differentiation Chapter 7 Integration Chapter 8 Vectors and matrices We recommend that you work through the sections that you are unsure of, completing the questions that are given. If you need further practice, there is a Question and Answer Bank and a Summary Test. These both cover material from all the chapters. Section 5 of this study guide gives you an index of the topics that are covered in order to identify quickly which chapters you need to look at. When you are working through the Core Technical subjects you can continue to use this course as a reference document if you come across areas of mathematics that you are still unhappy about.



For further guidance, this grid shows which chapters from FAC are needed for Subjects CT1, CT3, CT4, CT5, CT6 and CT8: Chapter Section Course Notes

1 1-7 All This is just general information and notation. 2 1-4 All 3 1-3 All 4 1-9 All 5 1-6 All 5 7 CT6 Syllabus items (c)(vi), (c)(vii), (c)(ix) 5 8 CT6 Syllabus item (d)(ix) 6 1-8 All 6 9 CT8 Syllabus item e(viii) 7 1, 2 All 7 3.1-3.4 All 7 3.5 CT4, CT8 Syllabus item (e)(xviii) 7 4-7 All 7 8.1-8.3 CT4, CT8 Syllabus item (e)(xvii) 8 1, 2.1-2.4 CT4, CT6, CT8 Syllabus items (b)(xi), (b)(xii) 8 2.5 CT4 Syllabus item (b)(xii) (part)

2.2 Online Classroom

The Online Classroom is available to provide tuition on the material covered in FAC. It is a comprehensive, easily-searched collection of recorded tutorial units covering the same topics as the FAC Course Notes. These tutorial units are a mix of:

teaching, covering the relevant theory to help you get to grips with the course material, and

worked questions or examples, illustrating the various techniques you should be familiar with.

To find out more about the Online Classroom, and to watch example tutorial units, please visit the ActEd website at www.ActEd.co.uk.

FAC: Study Guide


3 Study skills

All the mathematical techniques covered in this course will be used in the context of one of the Core Technical subjects and will therefore not be directly tested on their own. For example, in the Core Technical exams you will not be set a question saying Integrate this function by parts but you will be asked to work out expectations in Subject CT3 which may involve integration by parts. It is therefore essential that you feel really comfortable with each method. You should study this course actively. In particular we recommend the following: 1. Annotate your Notes with your own ideas and questions. This will make your

study more active. 2. Attempt the questions in the Notes as you work through the course. Write down

your answer before you check against the solution. 3. Attempt the Question and Answer Bank and the Summary Test on a similar

basis, ie write down or work out your answer before looking at the solution provided.



4 Contacts on the course material

Queries From time to time, you may come across something in the study material that is not clear to you. If you cannot resolve the query through discussion with friends and colleagues, then you can use one of ActEds discussion forums: If you have access to the FAC Online Classroom, then you can post your query in

the forums within the Online Classroom itself.

Alternatively, you can post your query on our forum at www.ActEd.co.uk/forums (or use the link from our homepage at www.ActEd.co.uk). This forum includes a section for each actuarial exam theres one for FAC and StatsPack.

Our forums are dedicated to actuarial students so that you can get help from fellow students on any aspect of your studies from technical issues to general study advice. ActEd tutors monitor the forums to answer queries and ensure that you are not being led astray. If you are still stuck, then you can send queries by email to [email protected] (but we recommend you try a forum first). We will endeavour to contact you as soon as possible after receiving your query, but you should be aware that it may take some time to reply to queries, particularly when tutors are away from the office running tutorials. At the busiest teaching times of year, it may take us more than a week to get back to you. Corrections and feedback We are always happy to receive feedback from students, particularly concerning any errors, contradictions or unclear statements in the course. If you find an error, or have any comments on this course, please email them to [email protected].

FAC: Study Guide


Further reading If you feel that you would find it useful to obtain a different viewpoint on a particular topic, or to have access to more information and further examples, then the best place to look would be a mathematics textbook. The level of mathematics covered in FAC is broadly similar to that covered by those examinations taken immediately prior to going to university (A-Level or Higher exams in the UK). You may still have your old textbooks, or know which ones you used and be able to track them down. If not, one set of textbooks published to help students prepare for A-Level exams is: Edexcel AS and A Level Modular Mathematics Core Mathematics 1, 2, 3 & 4 These are available from internet retailers, including www.amazon.co.uk. Each textbook covers different topics, so you can choose which would be most suitable for you.



5 Course index

Topic Chapter Page Absolute change 5 5 Arithmetic-geometric inequality 4 20 Arithmetic progressions 4 24 Binomial expansion 4 35 Calculator, use of 2 4 Complex numbers 5 17 Convergence 7 16 Curve sketching 6 25 Determinants 8 12 Difference equations 5 24 Differential equations 7 31 Differentiating an integral 7 14 Differentiation, products and quotients 6 12 Differentiation, standard functions 6 11 Dimensions 5 9 Double integrals 7 18 Eigenvectors and eigenvalues 8 21 Errors 5 7 Estimation 2 9 Extrema 6 29 Factorial notation 3 12 Fractions, algebraic 4 5 Functions and graphs 3 2 Gamma function 3 13 Geometric progressions 4 26 Greek symbols 1 5 Indices 4 2 Induction 1 10 Inequalities 4 16 Infimum 6 7 Integer part 3 9 Integration, by parts 7 12 Integration, partial fractions 7 9 Integration, standard functions 7 5

FAC: Study Guide


Integration, substitution 7 11 Interpolation 5 12 Iteration 5 15 Lagrangian multipliers 6 30 Leibnizs formula 7 14 Limits 6 2 Logarithms 4 2 Maclaurin series 7 30 Mathematical notation 1 2 Matrices 8 8 Max and min notation 3 10 Modulus 3 8 Newton Raphson iteration 5 16 Order notation 6 3 Partial differentiation 6 27 Percentages 5 2 notation 4 21 Proof 1 7 Proportionate change 5 5 Quadratic equations 4 7 Rounding 2 2 Scalar product 8 5 Series 4 31 notation 4 21 Simultaneous equations 4 11 Stationary points 6 20 Supremum 6 7 Taylor series 7 27 Trapezium rule 7 25 Vectors 8 2

FAC-01: Notation Page 1


Chapter 1

Notation

You need to study this chapter to cover: standard mathematical notation and terminology the letters of the Greek alphabet conventions commonly used in financial and actuarial mathematics mathematical proof mathematical induction currencies, dates and ages.

0 Introduction

This chapter deals with the notation and terminology that you must be familiar with in order to study the actuarial exams. Much of this may be familiar to you already, in which case read the chapter quickly or use it as a reference guide.

FAC-01: Notation


1 Mathematical notation

Symbol Meaning Further explanation/examples Types of numbers: Integers (whole numbers) {...,3, 2, 1, 0, 1, 2, 3,...} Natural numbers {1, 2, 3, ...} (counting numbers) Rational numbers 32 , 651.2 = , 14990.141414... = etc (all can

(fractions) be written as pq ) Real numbers rational numbers plus irrational numbers

(such as 2 , p and e) ie no imaginary component

Complex numbers can be written in the form a ib+ , where 1i = -

Logic: " For all (values) 2 x x " $ : Such that (see next example) $ There exists : 1 5x x$ + = /$ There doesnt exist 2 : 4x x/$ = - Implies 22 4x x= - = Implies and is implied by 32 8x x= = iff If and only if equivalent to Set Theory: { }1,2,3,... A set etc ( ,0]- A set containing numbers from - to 0, not

including - (since the bracket is round next to - ), but including 0 (since the bracket is square next to 0)

{ } or Empty set the set of odd numbers divisible by 2 Is a member of 2 A B Union of two sets A or B or both, ie the things that are in one

or other or both



A B Intersection of two sets A and B, ie only the things that are in both A Complement of a set not A, ie the things that are not in A Miscellaneous: p Pi 3.14159...(the ratio of the circumference of

a circle to its diameter) e base of the natural logarithm 2.7182818... Infinity Tends to approaches eg x means that x is

becoming very large

Note: A superscript + or - on symbols such as refers to the positive or

negative numbers within the group ie + means 1, 2, 3, (excluding zero), ie the same as .

When a superscript + or - is used in situations such as 1x + , it means that x is approaching 1 from above, in other words x is taking values slightly bigger than 1.

is the set of complex numbers. These can be written in the form a ib+ , where ,a b and 2 1i = - , a being called the real part and b being called the imaginary part. You may have seen j used for 1- rather than i.

FAC-01: Notation


Example (i) Interpret the statement:

{ }: 4x x+ < (ii) For the function 1( ) xf x = , describe what happens as x tends towards 0 and . Solution (i) This is the set of whole numbers which are less than 4 but are positive ie the set

{ }1,2,3= (ii) ( )f x is not defined for 0x = , ( ) as 0f x x + , ( ) as 0f x x - - ,

( ) 0 as f x x .

Question 1.1

(i) If { } { } and P prime numbers Q even numbers= = what is M P Q= ? (ii) What values are in the set 2{ : 10}x x < ?



2 Greek symbols

You need to know the following Greek letters:

Letter Lower case Used for Upper case Used for

alpha a parameter beta b parameter B beta function

gamma g parameter G gamma function delta d small change D difference

epsilon e small quantity theta q parameter Q number of deaths kappa k parameter

lambda l parameter mu m mean and force of mortality

nu n force of mortality when sick pi p 3.14159= P product

rho r correlation coefficient and force of recovery

sigma s standard deviation and force of sickness S sum

tau t parameter (pronounced as in torn)

phi f probability density

function of standard normal distribution

F cumulative distribution

function of standard normal distribution

chi c 2c distribution

(pronounced as first syllable of Cairo)

psi y probability of ruin omega w limiting age in a life table

FAC-01: Notation


3 Conventions

There are many conventions and short-hand notations that are used in mathematical and financial work. The following are used in the course notes: Abbreviations are used for thousands and millions to save writing many zeros.

For example, 9K means 9,000 (the K comes from the kilo prefix seen in words such as kilometre, meaning 1,000 metres) and $6.2m or $6.2M means $6,200,000. This is used in preference to using standard form, where $6,200,000 would be written as 6$6.2 10 .

D is used to denote a change in a quantity, for example D profit = 534K means that the profit has risen by 534,000.

If interest rates were 6% in January, 8% in February and 347 % in March, this would often be described as an increase of 2 percentage points, followed by a reduction of 25 basis points (one basis point being one-hundredth of a percentage point). Basis points are sometimes abbreviated to bps.

In accounting, negative amounts of money are represented by placing them in brackets eg 5 (5)- = .

Example Here is an example of a very simple income statement for a company (sometimes called a profit and loss account), showing the negative cashflows in brackets. Dont worry if you dont understand what the individual items represent. Pre-tax profit 9.6m Tax (2.4m) Net profit 7.2m Dividends (1.7m) Retained profit 5.5m

Similarly you will see things like calculate the profit (loss) made last year, which means calculating the income minus the outgo and writing it as a positive number for a profit, or in brackets if it is negative, ie a loss.



4 Proof

To prove that something is true in mathematics, it is not sufficient to show that something works for a particular case if you are trying to prove it in general. However you can prove that something is not true in general by showing that it doesnt work for a particular case (this is called a counterexample). You need to be familiar with the terms sufficient, necessary and necessary and sufficient. If A is necessary for B, then B A (ie B implies A or B is true only if A is true). If A is sufficient for B, then A B (ie A implies B or B is true if A is true). If A is necessary and sufficient for B, then A B (ie A implies and is implied by B or A and B are equivalent statements or A is true if and only if B is true).

Example In a group of 50 people, there are 25 men, 11 people with beards (who are all male!) and 25 people who like football who are all men too. If we use the notation M for is a man, B for has a beard and F for likes football, then we have B M and M F . The first implication is true since if we know someone has a beard, they must be male. The second implication is true since if we know someone is male, they automatically like football and vice versa.

Example (i) If A is the statement the integer x ends in a 5, and B is the statement the

number x is divisible by 5, then A is sufficient for B, but not necessary (since a number ending in a 0 is also divisible by 5). So we can write A B but not B A .

(ii) If x is a solution of the equation 2 0ax bx c+ + = , then if P is the statement

2 4 0b ac- and Q is the statement x is a real number, then P is a necessary and sufficient condition for Q so that P Q . (More about quadratic equations later!)

FAC-01: Notation


5 Expressions, equations and formulae

While youre studying for the Core Technical exams youll use a lot of expressions, equations and formulas.

5.1 Expressions

A mathematical expression is any combination of mathematical symbols that can be evaluated to give an answer. Usually the expression involves more than one symbol, often it includes some letters ( , ,x y z etc), and usually the answer is numerical (but not necessarily). For example, the following are all mathematical expressions:

2 2+ , 251.09 , 1

nt

tv

= , 5(1 ) 1id+ - , ()G

In Subjects CT1 and CT5 youll meet some of the special symbols used in actuarial calculations, and your answers to assignment questions and exam questions will involve expressions containing actuarial symbols such as |20a and 1 |[30]:25A . If a question says Find an expression for , you should simplify your final expression as much as possible.

5.2 Equations

An equation is just a statement that two expressions are equal. For example, the following are all equations:

2 2 4+ = , 251.09 8.6231= , |1

nt

nt

v a=

= , 5 |5(1 ) 1i sd+ - = , () pG = The issue is slightly confused by the fact that a lot of word processing packages use the word equation to refer to anything that contains mathematical symbols. A word processing equation may really be an expression, a formula, an inequality(!) or just nonsense (eg * p " ).



5.3 Formulas (or formulae)

A formula is just an expression that can be used specifically for calculating the answer to a particular type of problem. A formula may be stated in the form of an expression or as an equation, eg:

The quadratic formula is 2 4

2b b ac

a- -

The formula for calculating the present value of a unit annuity-certain payable annually

in arrears is |1 n

nva

i-= . You will meet this in Subject CT1.

A formula usually involves standard letters for the variables (eg you know that the

, ,a b c in the quadratic formula are the coefficients of 2x , x and the constant term). Also the letters often stand for the quantities involved, as in F ma= (force = mass acceleration).

5.4 Terms and factors

A term is an element in an expression that is added or subtracted. A factor is an element in an expression that is multiplied or divided.

FAC-01: Notation


6 Induction

One method of proving a general mathematical result is the method of (mathematical) induction. To prove that a result is true for all positive integers, we prove that if the result is true for any particular integer k then it must also be true for the next integer

1k + . If we can also show that it is true when 1k = , then it must be true for all positive integers 1,2,k = .

Example Prove by induction that 121 2 3 ( 1)n n n+ + + + = + . Solution Assume the result is true for n k= , ie: 121 2 3 ( 1)k k k+ + + + = + Adding the next term on to both sides it follows that:

[ ]

12

12

12

1 2 3 ( 1) ( 1) ( 1)

( 1)( 2)

( 1) ( 1) 1

k k k k k

k k

k k

+ + + + + + = + + += + += + + +

Since this is what the original equation predicts when 1n k= + , we have shown that if the result is true for n k= then it is also true for 1n k= + . Consider whether the result is true when 1n = :

LHS = 1 RHS = 1 So the result is true for 1n = and by the above result it must also be true for

2,3,4,n = ie for all positive integer values of n.

Question 1.2

Prove by induction that 2 2 2 2 161 2 3 ( 1)(2 1)n n n n+ + + + = + + .



7 Some general knowledge

Currencies

Country Main currency unit Sub-division United Kingdom Pound () Pence (1 = 100p)

United States Dollar ($) Cent ($1 = 100) European Union Euro () Cent (1 = 100)

Japan Yen ()

Question 1.3

Todays exchange rates are shown as: / = 0.61 and /$ = 1.44 How much would 1,000 Euro be worth in US dollars?

FAC-01: Notation


Dates Gregorian calendar The calendar system used in Western Europe and America is officially called the Gregorian calendar (named after Pope Gregory XIII who introduced it). This system is recognised and understood worldwide, although a number of countries in other parts of the world have alternative calendar systems that they use as well. Leap years Calendar years usually have 365 days but, in order to prevent the seasons gradually drifting, an extra leap day is added at the end of February in some years. These leap years have 366 days, the extra day being 29 February. The general rule for determining whether a particular calendar year is a leap year is as follows:

LEAP YEAR OR NOT? A calendar year IS NOT a leap year unless it divides exactly by 4, in which case it IS a leap year unless it also divides exactly by 100, in which case it IS NOT a leap year unless it also divides exactly by 400, in which case it IS a leap year!

Question 1.4

In actuarial calculations involving weekly payments it is often assumed that there are 52.18 weeks in an average year. Where does this figure comes from?

In a lot of actuarial applications the exact number of days in each month makes very little difference to the numerical answers. In these cases you can assume that the months are of equal length ie each month is exactly 112 of a year long. This simplifies the calculations considerably.



Calendar years, quarters and tax years Many organisations divide each calendar year into four quarters for budgeting and accounting purposes. For example, the calendar year 2012 would be broken into the four quarters: 2012 Q1: 1 January 2012 31 March 2012 2012 Q2: 1 April 2012 30 June 2012 2012 Q3: 1 July 2012 30 September 2012 2012 Q4: 1 October 2012 31 December 2012 In actuarial calculations where payments are made quarterly it is normally sufficiently accurate to assume that each quarter is exactly 14 of a year long. In the UK the amount of tax payable by individuals is calculated based on the transactions during each tax year (sometimes also referred to as a fiscal year), which run from 6 April to 5 April. So, for example, the 2011/12 tax year is the period from 6 April 2011 to 5 April 2012 (both days inclusive). The actual dates will differ between countries, for example the New Zealand tax year runs from 1 April to 31 March. Fencepost errors

Question 1.5

If you need to erect a fence 10 metres long in an open field using 1 metre-long strips of wood, how many posts will you need to support it?

If you got the answer wrong, youll see that its very easy to make these fencepost errors. Its particularly easy to make a mistake in calculations involving dates. Almost everyone gets one wrong at some point.

FAC-01: Notation


Question 1.6

(i) Five payments are made at 9-month intervals with the first payment on 1 January 2012. On what date will the last payment be made?

(ii) A man was born on 9 September 1960. In New Zealand, how many complete

tax years are there between 1 May 1998 and his 60th birthday? (iii) How long is the period from 1 March 2005 to 28 February 2015?

Conventions for writing dates To save time, dates are often written in numbers, rather than in words. So make sure you know the numbers of the months (eg October = 10, November = 11). Also, just to make life difficult, there are two different conventions in use. In the UK and Europe we use the DD/MM/YY order, whereas Americans use MM/DD/YY. This can cause a lot of confusion since 01/11/12 would mean 1 November 2012 in the UK, but 11 January 2012 in the US. (The reason for this discrepancy is that in the UK we tend to say the first of November, whereas in the US they tend to say November one.) To decide which convention is being used, look which position contains numbers greater than 12. This must be the days bit. In actuarial symbols a fixed period of time is represented by using a right-angle symbol, so that 5 years, for example, is usually represented by 5 . Some of your actuarial colleagues may use this as a shorthand notation. For example, they might write: The pension incorporates a |5 guarantee or they might even use 312 as an abbreviation for 3 months.



Ages In life insurance work and pensions work, youll often have to work out peoples ages. This might sound easy, but there are actually three different ways commonly used to express ages: Age last birthday: This is the age one of your friends would tell you if you asked them how old they were. Its just the number of candles they had on their last birthday cake. Age nearest birthday: This is the persons age at their nearest birthday (which could be either the previous one or the following one). Pension fund calculations usually use this definition because, for a large group of people, age nearest birthday usually averages out at the true age. Youll get some people who are slightly older and some who are slightly younger, and these will normally balance out. However, age last birthday will always understate the true age. Age next birthday: This is the persons age at their next birthday. This definition is the one usually used by insurance companies. This will always overstate the age. These age definitions are often abbreviated to age last, age nearest and age next. Youll use these ways of defining ages in Subject CT4. You may hear people in life offices referring to their policyholders as a female aged 50 next (say). Some people are born on one of the leap days eg 29 February 2012. For calculation purposes they are normally treated as if they were born on the following day ie 1 March 2012 in this example.

Question 1.7

A man was born on 6 May 1954. What will his age be on 1 January 2015 using each of these three age definitions?

FAC-01: Notation


Chapter 1 Solutions Solution 1.1

(i) {2}M = . (Dont forget that a prime number is one that has no factors other than 1 and itself!)

(ii) { }10 10x- < < Solution 1.2

Assume the result is true for n k= , ie: 2 2 2 2 161 2 3 ( 1)(2 1)k k k k+ + + + = + + Adding the next term on to both sides:

( )

2 2 2 2 2 216

16

216

216

16

16

1 2 3 ( 1) ( 1)(2 1) ( 1)

( 1)[ (2 1) 6( 1)]

( 1)(2 6 6)

( 1)(2 7 6)

( 1)( 2) 2 3

( 1)[( 1) 1][2( 1) 1]

k k k k k k

k k k k

k k k k

k k k

k k k

k k k

+ + + + + + = + + + += + + + += + + + += + + += + + += + + + + +

So we have shown that if the result is true for n k= then it is also true for 1n k= + . Consider when 1n = : LHS = 1 RHS = 1 So the result is true for 1n = and by mathematical induction it is also true for n = 2, 3, 4, ie for all positive integer values of n.



Solution 1.3

1,000 Euro must be equivalent to 1,000 0.61 610 = . 610 must be equivalent to 610 1.44 $878.40 = . The precise rules that banks have to use for converting currencies when Euro are involved are actually quite complicated eg you have to work to 6 decimal places. Solution 1.4

Its 14365 7 (rounded to 2 DP). Solution 1.5

11 The obvious answer was to divide 10 by 1 and say 10. But, because you need a post at each end of the fence, you actually need an extra one. If you said 10, youve made a fencepost error. To avoid these, you need to pay careful attention to which, if either, of the endpoints is included. This problem comes up when you are trying to work out the number of payments in a stream of payments. Solution 1.6

(i) 1 January 2015 (There are 4 gaps of 9 months between these 5 payments. This makes a total period of 36 months, which equals 3 years.)

(ii) 21 (The period from 1 April 1999 to 31 March 2020 consists of 21 complete tax

years ie 1999/2000, 2000/01, , 2019/20.) (iii) 10 years (When youre dealing with a period of time, its a straight subtraction.) Solution 1.7

Age last = 60 (because his last birthday would be 6 May 2014 and 2014 1954 60- = ). Age next = 61 (add 1 to his age last). Age nearest = 61 (because his nearest birthday would be 6 May 2015).




Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or

photocopy any part of the study material.





FAC-02: Numerical methods I Page 1


Chapter 2

Numerical methods I

You need to study this chapter to cover: accurately rounding numbers the use of an electronic calculator estimation abbreviations.

0 Introduction

This chapter reminds you of the conventions about rounding and accuracy. Since the Core Technical subjects rely heavily on numerical accuracy, rounding is more important in your actuarial studies than perhaps it has been during your university studies.

FAC-02: Numerical methods I


1 Rounding

Numerical answers should be rounded to a suitable number of decimal places (DP) or significant figures (SF). Marks can be lost in examinations if a final answer is not rounded to the accuracy that the question requires. Use common sense when deciding what suitable means. For example, when giving an answer in monetary terms, round to two decimal places. If a question is asking for an interest rate it is unlikely that you will need to work to more than three decimal places eg 6.125%. Do be aware though that we are talking about final answers here: if you are going on to use the interest rate later you will need an accurate figure to work with. More of that later!

Example Round the following numbers to two decimal places: 3.784, 15.239, 6.028, 6, 2002,

0.399- . Solution The answers are 3.78, 15.24, 6.03, 6.00, 2002.00, 0.40. Notice that all answers have two figures (digits) after the decimal point.

Example Round these numbers to two significant (ie non-zero) figures: 3.784, 15.239, 6.028, 6, 2002, 0.399- . Solution The answers are 3.8, 15, 6.0, 6.0, 2000, 0.40. Notice that the first significant figure is the first non-zero figure.



Question 2.1

Round these numbers to three decimal places:

0.0678, 15.3489, 9.9999

Question 2.2

Round these numbers to three significant figures:

14.3678, 5.9879, 0.08006

Notes: When a number is written as 6.00, it implies that the author believes it is correct

to 2DP ie the exact value lies somewhere in the range [5.995,6.005) , using the convention that a digit greater than or equal to 5 causes a number to be rounded up.

In some countries (continental Europe in particular) commas and full stops in numbers are used with the opposite meaning from in the UK and the US ie the decimal point is written as a comma and full stops (or spaces) are used to separate a large number. For example 3,142p = and the population of the UK is approximately 55.000.000. You should use the UK notation in the exam.

Do not use accuracy that is not valid, for example quoting the price of something as 2.78643 is not appropriate.



2 Use of a calculator

For a description of the functions you will need to have on your calculator, refer to the syllabus objectives for this course. In the exams you are not allowed graphical calculators. You can get guidance from the Profession as to which types are permissible. You must be familiar with the calculator that you are going to use in the exam. Since there are so many calculators on the market, this section is not going to be about how to use your particular calculator but instead it will enable you to practise getting the correct answer with your calculator. For technical help see your manual (if you can find it!). It is really important for you to try each example below for yourself to get used to working with your calculator. Most numerical answers have been rounded to three decimal places. Always ensure that you copy numbers correctly when you are half way through a calculation or reading numbers from actuarial tables.

Example Calculate 1.4 3.66.23 5.8+ . Solution Using the power key ( or y xx y ), we get 573.1474.

Example

Calculate ( )5.141 1.723+ . Solution

Using the root key ( x or 1

yx ) and the power key, we get 49.091.

If you dont have this key, you can treat the expression as 14 5.1(1 1.723 )+ . Since

1xx y y= you can use your or y xx y key.



Example

Calculate 221

3 .

Solution

The answer is 729

or 2.778. If your calculator has a fractions facility, be aware of the

order in which your calculator performs fractions!

Example

Calculate ( )247 ln 5.2+ . Solution The answer is 4.929. Your calculator will have loge or ln for the natural logarithm key. (We will deal with this function later in the course.) Most calculators have a fraction key ( b ca ) which can be helpful. Notice also that most calculators have a

squared key ( 2x ) to avoid you having to use the power key.

Example

Calculate 3.124 3 e .

Solution The answer is 1.961. On many calculators, xe and ln are on the same key. On recent calculators, to get 3.1e , you would need to type 3.1xe (followed by = or Ans), whereas on older models you would need to type these in reverse. This is the same as for the square root key.



Example Calculate 4! This is read as four factorial. Solution There should be a factorial key ( !n ) on your calculator. The answer is 24. Notice that !n means ( 1) ( 2) 1n n n - - . So here 4! 4 3 2 1= .

Example Calculate tanh 2 and 1tanh 0.4- . Solution These are hyperbolic tangents and inverse hyperbolic tangents respectively, and are needed for the Fisher transformation in Subject CT3. If your calculator is a recent model then you will need to press hyp then tan before the number; otherwise it will be the number then hyp then tan. The answers are 1tanh 2 0.964, tanh 0.4 0.424-= = .

Example

Calculate 215.2 3.74 4 1.68 2.49

2.7 19.86 3- - + .

Solution The answers are 2.493 and 1.477. Notice here that you can use brackets (or stack) for the first part of the expression and the memory function for the second, or alternatively the memory for both parts.



Question 2.3

Calculate 3.5 121 (1.04)

0.04 1.04

-- .

Question 2.4

Calculate 1 1 0.1ln2 1 0.1

.

Question 2.5

Calculate 2 4.5

32 (3.789 2.5)

5.5 2.1+ +

- .

Question 2.6

Calculate 23 ( 3) 4 2 42 2

- - - .

Question 2.7

Calculate 10

10

117 1100 1.035

ln1.035(1.035)

.

Question 2.8

Calculate 10!7!2!

.



When substituting a value for a variable that occurs several times within an expression, it can be helpful to use a shortcut called nested multiplication. The following example shows what this shortcut involves.

Example

Calculate 2 33 7v v v+ + , for 10.92591.08

v . Solution We can do this as a nested multiplication:

2 3 23 7 (3 7 )

[1 (3 7 )]

v v v v v v

v v v

+ + = + += + +

Start by multiplying v by 7, then add 3, then multiply by v, then add 1, and finally multiply by v. Using this version we get an answer of 9.055.

Question 2.9

Calculate 2 3 42 5 6v v v v+ + + , when 10.90911.1

v .

Question 2.10

For Question 2.9, work out the number of keystrokes you need to make on your calculator using nested multiplication and also for an alternative method that you can think of.



3 Estimation and accuracy

3.1 Estimation

We have looked at how to use calculators efficiently, but it is always possible to make a mistake. So it is important to have a rough idea of what the numerical answer to a calculation is likely to be. To do this, you can round the numbers involved to a convenient figure and then carry out the calculation without using a calculator (or you might make the same mistake again!).

Example

Estimate the value of 2 42.7 3.1

5.2 7.8 .

Solution

2 42.7 3.15.2 7.8

is roughly

2 43 3 9 81 305 8 3 .

The actual answer is 38.3.

Question 2.11

Find estimates for the answers to Question 2.5 and Question 2.6. Compare your estimates with the actual answers.

Note: You need to be very accurate with intermediate values in estimates and rounding when you are doing any of the following: subtracting two numbers of a similar size dividing by a small number raising a number to a high power.



3.2 Accuracy

When numbers have been rounded, and are then subsequently used in a calculation, the final answer is affected by the rounding. Therefore, when answering questions, it is essential to realise how accurately you need to quote your final answer. For example, if all figures in the question are given to three significant figures, do not give your final answer to five significant figures.

Example

In the formula 101 (1 )ip

i

, the accurate value of i is 0.034724. Compare the values of p obtained using: (i) the accurate value of i (ii) i rounded to three significant figures (iii) i rounded to one significant figure. Solution (i) 8.32819 (ii) 8.32920 (iii) 8.53020 Rounding to three significant figures gives a value of p that is still accurate to two significant figures. However rounding to one significant figure results in losing all accuracy.

Question 2.12

Using the same formula as in the example above (which is one that you will meet in Subject CT1) but with the accurate value of i being 0.04562378, how many significant figures can you round i to, in order to maintain three significant figures of accuracy in your final answer?



4 Convenient abbreviations

Rather than writing several zeros when dealing with large numbers, it is often more convenient to work in, say thousands or millions. It is common to write 000s to mean thousands of pounds, and m to mean millions of pounds.

Example

If 8

9 (1 )R vT Pvi , where 114,000, 700, , 0.05

1P R v i

i , what is T?

(Work in 000s). Solution Working in 000s, 14, 0.7P R . Then 13.549T ie 13,549.

Example

If 4,000A , calculate 2 2,000 1,000,000A A (i) directly (ii) working in units of 1,000 Solution (i) 25,000,000 5,000 (ii) In units of 1,000: 1,000,000 becomes 21 1 , 2,000 becomes 2 and A becomes 4 The calculation is then done as follows:

24 2 4 1 25 5 ie the real answer is 5,000.



This page has been left blank so that you can keep the chapter summaries together for revision purposes.



Chapter 2 Summary You need to round off your final answers to a sensible degree of accuracy, but intermediate figures should not be rounded if that will mean an inaccurate final answer. You should be able to use your calculator efficiently and correctly. Remember to make a rough estimate of what the answer should be, in order to pick up careless numerical errors.




The rounded numbers are:

0.068, 15.349, 10.000 Solution 2.2

The rounded numbers are:

14.4, 5.99, 0.0801 Solution 2.3

The numerical answer is 0.295729. Solution 2.4



The numerical answer is 2.351 or 0.851. Solution 2.7


The numerical answer is 360.



Solution 2.9

Using the nested multiplication method, we get:

2 3 4 2 3

2

2 5 6 2 (5 6 )

[2 (5 6 )]

(1 [2 (5 6 )])

v v v v v v v v

v v v v

v v v v

Now using 11.1

v , we get the answer to be 10.42. Solution 2.10

On the calculator Im using, nested multiplication took 30 key presses. Using another method (just working from left to right) I took 43 key presses, or using the memory I took 25 key presses. Nested multiplication can be more efficient. Solution 2.11

For Question 2.5 one estimate is 2 4 2

32 (4 3) 2 19 2 400 201

2 26 2 .

For Question 2.6 one estimate is 3 9 32 3 6 2.25 or 0.754 4

. In each case you can see that the estimate is fairly close to the accurate answer.



Solution 2.12

Using the accurate value of i we get p to be 7.888505805. Rounding i to 6SF we get p to be 7.888505. Rounding i to 5SF we get p to be 7.888497. Rounding i to 4SF we get p to be 7.888652. Rounding i to 3SF we get p to be 7.889427. Rounding i to 2SF we get p to be 7.873956. So i can be rounded off to 3SF to keep three significant figures of accuracy. Notice that your first guess might have been 4SF, in which case the preceding work would not be necessary.




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photocopy any part of the study material.





FAC-03: Mathematical constants and standard functions Page 1


Chapter 3

Mathematical constants and standard functions

You need to study this chapter to cover: the definitions and basic properties of the functions nx , xc , exp( )x , and ln x

the functions x , max( ) and min( ) the factorial function !n and the gamma function.

0 Introduction

This chapter covers some standard functions and notation that will be needed in the following chapters.

FAC-03: Mathematical constants and standard functions


1 Standard functions and graphs

1.1 Exponential function

The exponential function, xe , can be defined by:

lim 1n

xn

xen

or the series expansion 2 3

12! 3!

x x xe x . It is the inverse of the natural logarithmic function (which we will look at in Chapter 4), so that ln xe x . Do remember that e is just a number which you can find from your calculators:

1 2.718...e If the power is a long expression then a convenient alternative notation is exp( )xe x ,

for example 212 212expx xe

. This makes things clearer to read.

However dont mix up these two notations by writing something like expx , as this is meaningless. The graph of xy e looks like this:

y=exp(x)

0

2

4

6

8

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x



Notice that xe can never be negative. This is because you can never get a negative answer if you raise a positive number to any power. Notice also that: 0e is 1

as x gets large, so does xe

and as x gets large but negative, 0xe . Since xe is just a number to the power of x, then this graph is also the basic shape of the graph of xy c , where c is a positive constant greater than 1. When c is a positive constant less than 1, its graph is a reflection in the y-axis of the graph of xy c , with c replaced by 1c . The diagram below shows the graphs of 0.5

xy and 2xy :

0.5xy 2xy

Again notice that both graphs have an intercept of 1, ie they cross the y-axis at 1, but you must look at the value of c carefully to judge when the graph tends towards infinity or zero.

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

x



1.2 Log function

A logarithm is the inverse of a power. The graph of the natural logarithm function lny x is:

-6-5-4-3-2-10123

0 1 2 3 4 5 6 7

We will meet the idea of a base in the next chapter, but it is worth noting that this is the shape of the logarithmic graph whatever base is used. We will look at logarithms and bases in a later chapter. Notice that log1 0 , and that you cant take the log of a negative number (or zero). The limits are log as x x , and log as 0x x . These will be important in statistical work in later subjects. You will see the notation log x and ln x used for the natural logarithm function.



1.3 Powers of x

The graph of ny x , where n is an even positive integer has the same general shape as 4y x which is shown below.

-202468

1012141618

-2 -1 0 1 2x

y

Notice that the values of y for these functions can never be less than zero. The graph of ny x , where n is an odd positive integer has the same general shape as

3y x which is shown below:

-10-8-6-4-202468

10

-3 -2 -1 0 1 2 3

x

y

You may notice that out of the last two graphs, one was symmetrical about the y-axis and the other one wasnt. In general a function ( )f x which has the property

( ) ( )f x f x is called an even function. If ( ) ( )f x f x then ( )f x is called an odd function.



Remembering that 1n nx x , the graph of ny x will have a discontinuity (where the

function is not defined) at 0x . For example, look at the graph of 3y x :

-300

-200

-100

0

100

200

300

-2 -1 0 1 2

x

y

The graph of y x is (remembering that x represents the positive square root):

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5

x

y

This is the inverse function of 2y x , so the graph of y x is the reflection of

2y x in the line y x . The same relationship applies to the graphs of log x and xe which we have already looked at. Notice that this graph does not exist for negative values of x since a real value of the square root of a negative number cannot be found.



When you see an expression involving a power, such as zt , you need to think carefully whether: it is like nx , ie t is a variable and z is a fixed number, or

it is like xc , ie t is a fixed number and z is a variable.

1.4 Transformations

These standard graphs ( )y f x= can be transformed as follows: ( )y f x d= + causes a vertical translation (ie slide) of d ( )y f x c= + causes a horizontal translation left of c ( )y af x= causes a vertical stretch by a factor of a ( )y f bx= causes a horizontal stretch by a factor of 1 b (ie the graph is

squashed by a factor of b ).

Note how constants outside the function affect the function vertically, whereas constants inside the function affect it horizontally. The following question illustrates these points.

Question 3.1

Sketch the graphs of the following functions: (i) 1y x (ii) 53y x (iii) 4 1xy e (iv) ln(1 )y x (v) 4( 3)y x

Question 3.2

Describe what the graphs of 2 ( )y f x and (2 )y f x look like in relation to the graph of ( )y f x .



2 Other functions

2.1 Modulus function

x is the absolute value of x or the modulus of x. In practice, if x is a number, it means

ignore any negative sign, for example, 4 | 4 | 4 . Note that | |x c is the same as saying c x c because x must either be a positive number in the range [0, )c or a negative number in the range ( ,0)c .

Example Write the expression 2 1 3x without the use of the modulus sign. Solution 2 1 3 3 2 1 3x x

For those of you that can remember dealing with inequalities, we will further simplify this expression. For those that cant, we will pick up on this again in the next chapter. Simplifying the inequality:

3 2 1 3 2 2 4

1 2

x x

x

Question 3.3

Write the expression 3 2x without the modulus sign.



2.2 Integer part

x means the integer part of x, for example 2.89 is 2. This could be used, amongst other contexts, to give the complete number of years that someone has lived. It is also used a lot in computer calculations.

When you divide m by n, where both are positive integers, you get mn

as the quotient

with a remainder of mm nn

. When dealing with negative numbers you need to check whether x really means the integer part, so that [ ] 3 , or whether it means the greatest integer not exceeding x, so that [ ] 4 . The second definition is more common.

Example A baby boy was born on 26 November 1979. If x is defined to be his exact age, what is x on 11 February 2012? Solution He is 32 on 26 November 2011, so x is 32.



2.3 Max and min

The notation max( ) or min( ) is used to denote the largest or smallest of a set of values. If the quantities involved are variables, the value of these functions may depend on the ranges of values that you are considering.

Example What is max( 2,10)x for the region 0 20x ? Solution We could write max( 2,10)x as:

10 if 0 82 if 8 20

xx x

So max( 2,10) 10 for 0 8x x , and max( 2,10) 2 for 8 20x x x . The abbreviation ( 100)x is often used for max( 100,0)x .

Question 3.4

What is 2min( ,15)x for 0 6x ?



Question 3.5

In the UK the calculation of an employees National Insurance contributions is based on their Upper Band Earnings (UBE), which is calculated from the formula: min( , ) min( , )UBE S UEL S LEL In this formula, S is annual salary, and UEL and LEL are two published figures called the Upper and Lower Earnings Limits (which might for example be 30K and 4K). Describe in words what UBE represents and give an alternative mathematical formula in the form:

if

if

if

UBE



3 Factorial and gamma functions

3.1 Factorial notation

We have already looked at how to evaluate !n on a calculator. The definition of !n is as follows: ! ( 1) ( 2) 1n n n n where n is a non-negative integer. The factorial function satisfies the relationship ! ( 1)!n n n . If we put 1n , this tells us that 1! 1 0! 0! 1 .

Question 3.6

Evaluate 5!

Example

Simplify the expression !(2 1)!(2 )!( 2)!n nn n

.

Solution This expression can be written as:

! (2 1)!( 2)! (2 )!

n nn n

The first factor equals ( 1)n n , because !n contains an extra n and 1n in its expansion, which are not contained in the expansion of ( 2)!n , and the second factor is 12n

.

So we get 12!(2 1)! ( 1) ( 1)

(2 )!( 2)! 2n n n n nn n n

.



Question 3.7

Simplify the expression (3 )!(2 1)!(3 2)!(2 1)!

n nn n

.

3.2 Gamma function

( )x , where 0x , is defined by the integral 10

x tt e dt . This is the gamma

function. This function is used in statistics in connection with the gamma distribution, and there are several properties of the function that you are going to need to know. In this chapter we will quote the results without proof. We will look at their proofs later in the course after we have covered the necessary integration techniques. Result 1: ( ) ( 1) ( 1)x x x where 1x Result 2: ( ) ( 1)!n n where 1, 2, 3,...n Result 3: 12( ) These results can be found in the Tables page 5. The graph of the gamma function looks like this:

1 2 3 4

1234567



Example Evaluate (6.5) . Solution Using Result 1:

(6.5) 5.5 (5.5) By repeating this process:

(6.5) 5.5 4.5 3.5 2.5 1.5 0.5 (0.5)

162.42

287.89

Question 3.8

Evaluate (5) .

Question 3.9

Evaluate (4.5) .

Question 3.10

Simplify ( 1)(2 1)!(2 )( 1)!n n

n n where n is a natural number.

Question 3.11

Show that, if n is a nonnegative whole number, 2(2 )!( )2 !n

nnn

.



Stirlings approximation Working out a factorial by multiplying all the numbers together would be very tedious for large values of n . The following approximations (called Stirlings approximation) are sometimes useful in derivations involving large values of n : ! 2n nn n e and ( ) 2n nn n e (Dont worry that these appear to be inconsistent with the relationship ( ) ( 1)!n n Remember that for very large values of n , the ratio 1n

n is very close to 1.)



Chapter 3 Summary Standard functions

You should know the properties and be able to sketch the graphs of , , , and logn x xx c e x .

Transforming functions

( )y af bx c d= - + causes the function ( )y f x= to be: Stretched vertically by a factor a Squashed horizontally by a factor b Translated horizontally to the right by c Translated vertically up by d Modulus

The modulus of x, written as x , is the absolute value of x ie negative signs are ignored. Integer part

The integer part of x is written as x . Max and min

max( ) or min( ) is used to denote the largest or smallest of a set of values. Factorial

The definition of n factorial is ! ( 1) ( 2) 1n n n n . Note that 0! 1 . The gamma function

The gamma function is defined by 10

( ) x tx t e dt , when 0x .

It has the following properties:

( ) ( 1) ( 1)x x xG = - G - where 1x > ( ) ( 1)!n nG = - where 1,2,3,n = () pG = .




(i) The graph of 1y x is as follows:

-10

-5

0

5

10

-1.5 -1 -0.5 0 0.5 1 1.5

x

y

(ii) The graph of 53y x is as follows:

-100

-50

0

50

100

-2 -1 0 1 2

x

y



(iii) The graph of 4xe is like that of xe , only steeper. In fact the whole graph is squeezed horizontally to a quarter of the original width. Adding 1 then shifts everything up by 1 unit.

0123456789

-1.5 -1 -0.5 0 0.5

x

y

(iv) This is like the graph of ln x shifted 1 unit to the left

-2.5-2

-1.5-1

-0.50

0.51

1.52

-1 -0.5 0 0.5 1 1.5

x

y



(v) This is like the graph of 4x , but shifted 3 units to the right.

02468

10121416

0 1 2 3 4 5

x

y

Solution 3.2

The graph of 2 ( )f x will have all vertical distances doubled. The graph of (2 )f x will have all horizontal distances halved. Solution 3.3

3 2x or 3 2x If you wish to simplify these expressions further, proceed as follows:

22 33

x x

or 22 33

x x Here there are two separate ranges of values of x where the inequality holds.



Solution 3.4

2 2min( ,15)x x for 0 3.87x

2min( ,15) 15x for 3.87 6x Solution 3.5

UBE is the portion of the employees salary that falls in the band ( , )LEL UEL , which would be (4,000,30,000) using the illustrative figures given. The graphs in the diagram represent the two min functions. UBE is the difference between them, which corresponds to the height of the gap between the upper and lower lines.

LEL UEL

LEL

UEL

The alternative mathematical formula is:

0 if

if

if

S LEL

UBE S LEL LEL S UEL

UEL LEL S UEL

Solution 3.6

5! 5 4 3 2 1 120



Solution 3.7

(3 )!(2 1)! (3 )! (2 1)! 1 2 (2 1)2 (2 1)(3 2)!(2 1)! (3 2)! (2 1)! (3 1)(3 2) (3 1)(3 2)

n n n n n nn nn n n n n n n n

Solution 3.8

(5) 4! 24 Solution 3.9

(4.5) 3.5 (3.5) 3.5 2.5 1.5 0.5 (0.5) 6.5625 11.63 Solution 3.10

( 1)(2 1)! ( 2)!(2 1)! 1(2 )( 1)! (2 1)!( 1)! ( 1)( 1)n n n n

n n n n n n n or 3

1n n



Solution 3.11

We can prove this using mathematical induction.

When 0n , the equation says 00!()

2 0! , which is correct.

If we assume its true for a typical value of n , say n k , then we know that:

2(2 )!( )2 !k

kkk

This would imply that:

2(2 )!( 1) ( ) ( ) ( )2 !k

kk k k kk

The last expression can be written as:

2 2 2 1(2 )! 2 1 (2 )! (2 1)!( )

22 ! 2 ! 2 !k k kk k k kkk k k

This doesnt quite match the formula we were hoping for. But, if we include an extra

factor of 2 22( 1)

kk (which wont affect the answer), we get:

2 1 2 22 2 (2 1)! (2 2)!

2( 1) 2 ! 2 ( 1)!k kk k kk k k

This now matches the formula given when 1n k . So, if it is true for n k , its also true for 1n k , and by the principle of mathematical induction it must be true for all values of n .

FAC-04: Algebra Page 1


Chapter 4

Algebra

You need to study this chapter to cover: manipulating algebraic expressions involving powers, logs, polynomials and

fractions

solving quadratic equations solving simultaneous equations solving inequalities the arithmetic-geometric mean inequality using the S and P notation for sums and products arithmetic and geometric progressions and other series

the binomial expansion of expressions of the form ( )na b+ where n is a positive integer, and (1 ) px+ for any real value of p .

0 Introduction

This chapter reminds you of the algebraic results that you need to be able to handle very easily in order to cope with the Core Technical subjects.

FAC-04: Algebra


1 Algebraic expressions

1.1 Indices

There are three laws governing indices (or powers):

( )

a b a b

a b a b

a b ab

x x x

x x x

x x

+

- =

==

Question 4.1

Simplify the following: (i) 3 55 2x x (ii) 2 716 6y y (iii) 3 2(5 )b

1.2 Logarithms

A logarithm is the inverse of a power:

21010 100 log 100 2= = (read as log base 10 of 100)

So if we want to find a log, we need to answer the question: To what power must we raise the base in order to get the number whose log we are trying to find? They have bases (the 10 here), the most commonly used one being e. Logarithms to these bases can be calculated directly from a scientific calculator, eg log 15 2.708e = . An alternative abbreviation for log to base e is ln. On your calculator, the ln button gives you loge and the log button gives you 10log .



There are three laws of logarithms that can be derived from the laws of indices:

log log log

log log log

log log

a a a

a a a

na a

x y xy

xx yy

x n x

+ =

- =

=

In words, the log of a product is the sum of the logs, and you can bring down the power.

Question 4.2

Starting from the laws of indices, prove the second of these three laws.

Notice that the base of the logarithm has been missed out in the following examples. Here we are looking at the properties of logs in general. So the logarithm used can be to any base, but if you need to use a calculator then you will have to use base 10 or base e.

Example (i) Simplify log 2 log 4 log5x x x+ - . (ii) Write 101 2log x in the form log ( )f x . Solution

(i) This simplifies to 2 4 8log log5 5

x x xx = .

(ii) 2 210 10 10log log 10 log 10x x .

Question 4.3

Simplify ln e.

FAC-04: Algebra


Logarithms are also used to solve equations where the unknown forms part of a power.

Example Solve the equation 2 11.1 2n+ = . Solution Take logs:

2 1log1.1 log 2

(2 1)log1.1 log 2

log 2(2 1)log1.1

1 log 2 1 3.1362 log1.1

n

n

n

n

Question 4.4

Solve the equation 2 2 12 1.05 1.1025t t- - = . Note:

The value of loglog

b

b

xy

is independent of the base b used.



1.3 Fractions

It is possible to manipulate algebraic fractions using the same rules that are used for numerical fractions. If you have forgotten how to factorise a quadratic you might want to look at the next section now!

Example Simplify the following:

(i) 2 3 3 21 6 1

x xx x+ +-+ -

(ii) 2

23 4

2 7 4x xx x- -- -

(iii) 2 2

2 22 3 2

2 5 12 2 5 3x x x xx x x x

- - + +- - + + Solution (i) Putting these over a common denominator:

2 2

2

2 3 3 2 (2 3)(6 1) (3 2)( 1)1 6 1 ( 1)(6 1)

(12 16 3) (3 5 2)( 1)(6 1)

9 11 5( 1)(6 1)

x x x x x xx x x x

x x x xx x

x xx x

+ + + - - + +- =+ - + -+ - - + += + -

+ -= + -

(ii) Factorising the numerator and denominator:

2

23 4 ( 1)( 4)

(2 1)( 4)2 7 4

12 1

x x x xx xx x

xx

- - + -= + -- -+= +

FAC-04: Algebra


(iii) Remembering that when you divide by a fraction you invert and multiply:

2 2

2 22 3 2 ( 2)( 1) ( 2)( 1)

( 4)(2 3) (2 3)( 1)2 5 12 2 5 3

( 2)( 1) (2 3)( 1)( 4)(2 3) ( 2)( 1)

( 2)( 1)( 4)( 2)

x x x x x x x xx x x xx x x x

x x x xx x x x

x xx x

- - + + - + + + = - + + +- - + +- + + += - + + +- += - +

Example

Simplify

1 1a ba bb a

++

.

Solution Multiplying the numerator and denominator of a fraction by the same quantity leaves the value of the fraction unaffected. We can pick a suitable quantity to simplify the fraction:

2 2

1 1ab b aa b

a b ab a bb

FAC Pack Jun14

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