2009 A-level Maths Tutor All Rights Reserved

Size: px
Start display at page:

Download "2009 A-level Maths Tutor All Rights Reserved"

Transcription

1

2 2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents indices 3 laws of logarithms 7 surds 12 inequalities 18 quadratic equations 22 partial fractions 28 polynomials 33 The Binomial Theorem 37 iteration 41 Sets theory 46 functions 50

3 3 Indices The Laws of Indices Indices - Multiplication remembering that: Examples

4 4 Indices - Division remembering that: Examples:

5 5 Indices - Powers remembering that: Examples: Indices - Roots and Reciprocals remembering that: and

6 6 Examples:

7 7 The Laws of Logarithms The Laws of Logarithms

8 8 Proofs

9 9 Changing the base Remember that the change of base occurs in the term where the base is 'x' or some other variable. Example

10 10 Simultaneous equations 'Substitution' simultaneous equations are common problems. First find what x is in terms of y. Then substitute for x in the other equation. Solve for y. Example

11 11 Variable in the index Take logs on both sides. Move the indices infront of the logs. Expand the equation. Collect x-terms to the left. Sum the numbers to the right. These problems can be tricky with the amount of arithmetic involved. So make sure you write everything down to make checking your working easier. Example

12 12 Surds Rules Surds are mathematical expressions containing square roots. However, it must be emphasized that the square roots are 'irrational' i.e. they do not result in a whole number, a terminating decimal or a recurring decimal. The rules governing surds are taken from the Laws of Indices. rule #1 examples rule #2

13 13 examples Some Useful Expressions expression #1

14 14 expression #2 - (the difference of two squares) Rationalising Surds - This is a way of modifying surd expressions so that the square root is in the numerator of a fraction and not in the denominator. The method is to multiply the top and bottom of the fraction by the square root.

15 15 Rationalising expressions using 'difference of two squares' Remembering that :......from 'useful expressions' above. Example #1 - simplify multiplying top and bottom by

16 16 Example #2 rationalise multiply top and bottom by Reduction of Surds - This is a way of making the square root smaller by examining its squared factors and removing them.

17 17 Rational and Irrational Numbers - In the test for rational and irrational numbers, if a surd has a square root in the numerator, while the denominator is '1' or some other number, then the number represented by the expression is 'irrational'. examples of irrational surds:

18 18 Inequalities Symbols The rules of inequalities (sometimes called 'inequations') These are the same as for equations i.e that whatever you do to one side of the equation(add/subtract, multiply/divide by quantities) you must do to the other. However, their are two exceptions to these rules. When you multiply each side by a negative quantity '<' becomes '>', or '>' becomes '<'. That is, the inequality sign is reversed. Similarly, when you divide each side by a negative quantity < becomes >, or > becomes<. That is, the inequality sign is reversed.

19 19 Examples Inequalities with one variable Example #1 - Find all the integral values of x where, The values of x lie equal to and less than 6 but greater than -5, but not equal to it. The integral(whole numbers + or - or zero) values of x are therefore: 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4 Example #2 - What is the range of values of x where, Since the square root of 144 is +12 or -12(remember two negatives multiplied make a positive), x can have values between 12 and -12. In other words the value of x is less than or equal to 12 and more than or equal to -12. This is written:

20 20 Inequalities with two variables - Solution is by arranging the equation into the form Ax + By = C Then, above the line of the equation, Ax + By < C and below the line, Ax + By > C Consider the graph of -2x + y = -2 note - the first term A must be made positive by multiplying the whole equation by -1 The equation -2x + y = -2 becomes 2x - y =2 look at the points(red) and the value of 2x - y for each. The table below summarises the result. point(x,y) 2x - y value more than 2? above/below (1,1) 2(1)1(1) 1 no - less above (1,4) 2(1)-(4) -2 no - less above (2,3) 2(2)-(3) 1 no - less above (3,3) 2(3)-(3) 3 yes - more below (2,1) 2(2)-(1) 3 yes - more below (4,2) 2(4)-(2) 6 yes - more below

21 21 The Modulus The modulus is the numerical value of a number, irrespective of the sign it carries. hence l x l < 3 means -3 < x < 3 Example

22 22 Quadratic Equations Introduction The general form of a quadratic equation is: ax 2 +bx + c where a, b & c are constants The expression b 2-4ac is called the discriminant and given the letter (delta). All quadratic equations have two roots/solutions. These roots are either REAL, EQUAL or COMPLEX *. * complex - involving the square root of 1

23 23 Solution by factorising - This is best understood with an example. solve: You must first ask yourself which two factors when multiplied will give 12? The factor pairs of 12 are : 1 x 12, 2 x 6 and 3 x 4 You must decide which of these factor pairs added or subtracted will give 7? 1 : 12...gives 13, 11 2 : 6...gives 8, 4 3 : 4...gives 7, 1 Which combination when multiplied makes +12 and which when added gives -7? these are the choices: (+3)(+4), (-3)(+4), (+3)(-4) (-3)(-4) Clearly, (-3)(-4) are the two factors we want. therefore Now to solve the equation. factorising, as above either

24 24 or for the equation to be true. So the roots of the equation are: Completing the square This can be fraught with difficulty, especially if you only half understand what you are doing. The method is to move the last term of the quadratic over to the right hand side of the equation and to add a number to both sides so that the left hand side can be factorised as the square of two terms. e.g. However, there is a much neater way of solving this type of problem, and that is by remembering to put the equation in the following form:

25 25 using the previous example, Using the Formula - the two solutions of quadratic equations in the form are given by the formula:

26 26 Proof The proof of the formula is by using 'completing the square'.

27 27 Example find the two values of x that satisfy the following quadratic equation:

28 28 Partial Fractions some definitions: Proper Fraction When the degree(index) of the function is higher in the denominator than the numerator. Improper Fraction When the degree(index) of the function is higher in the numerator than the denominator. Partial Fractions Factorising the denominator of a proper fraction means that the fraction can be expressed as the sum(or difference) of other proper fractions. Simple addition/subtraction of algebraic fractions As with simple fraction arithmetic, a common denominator is found from the denominators of either fraction and the numerators altered to be fractions of the new denominator. Equations & Identities Equations are satisfied by discrete values of the variable involved. Example: Identities are satisfied by any value of the variable used. Note the equals sign '=' is modified to reflect this.

29 29 Example: When we make partial fractions(below) we are creating an identity from the original expression. Denominator with only 'linear factors' By 'linear' we mean that x has a power no higher than '1'. In other words, this method does not work with x 2, x 3, x 4 etc. For each linear factor of the type: there is a partial fraction: Example: where x is a variable and A,B,a,b,c,d are constants, where 'a' is not equal to 'b'.

30 30 Example Denominator with 'repeated' linear factors For each 'repeated' linear factor of the type: there is a partial fraction: Example:

31 31 Example Denominator with a quadratic factor For each quadratic factor of the type: there is a partial fraction:

32 32 Example: Example

33 33 Polynomials Introduction A polynomial is an expression which: consists of a sum of a finite number of terms has terms of the form kx n (x a variable, k a constant, n a positive integer) Every polynomial in one variable (eg 'x') is equivalent to a polynomial with the form: Polynomials are often described by their degree of order. This is the highest index of the variable in the expression. eg: containing x 5 order 5, containing x 7 order 7 etc. These are NOT polynomials: 3x 2 +x 1/2 +x second term has an index which is not an integer(whole number) 5x -2 +2x -3 +x -5 indices of the variable contain integers which are not positive examples of polynomials: 5 2 x +5x +2x+3 7 (x +4x 2 )(3x-2) x+2x 2 3-5x +x 4-2x 5 +7x 6

34 34 Algebraic long division If f(x) the numerator and d(x) the denominator are polynomials and the degree of d(x) <= the degree of f(x) and d(x) does not =0 then two unique polynomials q(x) the quotient and r(x) the remainder exist, so that: Note - the degree of r(x) < the degree of d(x). We say that d(x) divides evenly into f(x) when r(x)=0. Example

35 35 The Remainder Theorem If a polynomial f(x) is divided by (x-a), the remainder is f(a). Example Find the remainder when (2x 3 +3x+x) is divided by (x+4). The reader may wish to verify this answer by using algebraic division.

36 36 The Factor Theorem ( a special case of the Remainder Theorem) (x a) is a factor of the polynomial f(x) if f(a) = 0 Example

37 37 The Binomial Theorem Introduction This section of work is to do with the expansion of (a+b) n and (1+x) n. Pascal's Triangle and the Binomial Theorem gives us a way of expressing the expansion as a sum of ordered terms. Pascal's Triangle This is a method of predicting the coefficients of the binomial series. Coefficients are the constants(1,2,3,4,5,6 etc.) that multiply each variable, or group of variables. Consider (a+b) n variables a, b. The first line represents the coefficients for n=0. (a+b) 0 = 1 The second line represents the coefficients for n=1. (a+b) 1 = a + b The third line represents the coefficients for n=2. (a+b) 2 = a 2 + 2ab + b 2 The sixth line represents the coefficients for n=5. (a+b) 5 = a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + b 5 The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out triangles of numbers has its limits.

38 38 The General Binomial Expansion ( n 1 ) This is a way of finding all the terms of the series, the coefficients and the powers of the variables. The coefficients, represented by n C r, are calculated using probability theory. For a deeper understanding you may wish to look at where n C r comes from; but for now you must accept that: where 'n' is the power/index of the original expression and 'r' is the number order of the term minus one If n is a positive integer, then: Example #1

39 39 Example #2 It is suggested that the reader try making similar questions, working through the calculations and checking the answer here (max. value of n=8)

40 40 The Particular Binomial Expansion This is for (1+x) n, where n can take any value positive or negative, and x is a fraction ( - 1<x<1 ). Example Find the first 4 terms of the expression (x+3) 1/2.

41 41 Iteration Introduction Repeatedly solving an equation to obtain a result using the result from the previous calculation, is called 'iteration'. The procedure is used in mathematics to give a more accurate answer when the original data is only approximate. Problems usually involve finding the root of an equation when only an approximate value is given for where the curve crosses an axis. Direct/Fixed Point Iteration method: 1. rearrange the given equation to make the highest power of x the subject 2. find the power root of each side, leaving x on its own on the left 3. the LHS x becomes x n+1 4. the RHS x becomes x n The equation is now in its iterative form. We start by working out x 2 from the given value x 1. x 3 is worked out using the value x 2 in the equation. x 4 is worked out using the value x 3 and so on.

42 42 Example Find correct to 3 d.p. a root of the equation f(x) = x 3-2x + 3 given that there is a solution near x = -2

43 43 Iteration by Bisection method: 1. reduce the interval where the root lies into two equal parts 2. decide in which part the solution resides 3. repeat the process until a consistent answer is achieved for the degree of accuracy required Example Find correct to 3 d.p. a root of the equation f(x) = 2x 2-2x + 7 given that there is a solution near x = -2

44 44 Newton-Raphson Method This uses a tangent to a curve near one of its roots and the fact that where the tangent meets the x-axis gives an approximation to the root. The iterative formula used is:

45 45 Example Find correct to 3 d.p. a root of the equation f(x) = 2x 2 + x - 6 given that there is a solution near x = 1.4

46 46 Set Theory Introduction A set is a collection of objects, numbers or characters. {abcdef...wxyz} {1,2,3,4,...45, 46, 47} etc. Note how the set is enclosed in brackets {...} A definite set is one in which all its members are known. Sets are given uppercase letters: A, B, C, etc. The elements of sets are given lowercase letters: a, b, c,..etc. An element x belonging to the set A is written: A constraint bar {......} is for setting a property that all members satisfy. A{x l x has the colour blue} - all elements of A are blue Common Sets

47 47 Venn Diagrams Venn diagrams are used to visualise sets and their relations to one another. Above is a diagramatic representation of set A. The set can be represented mathematically as: A{1,3,5,7,9}. Note that set A(the circle) is a subset of the Universal set(the rectangle). A' (A-dash)is called the complement of A. It contains all elements which are not members of A. A and A' together make up the Universal set.

48 48 The union of sets A and B contains all of the elements from both sets. The intersection of sets A and B contains a particular group of elements that exist in set A and in set B.

49 49 Subsets If B is a subset of A. Then all of the elements of B are also in A.

50 50 Functions Introduction To thoroughly understand the terms and symbols used in this section it is advised that you visit 'sets of numbers' first. Mapping(or function) This a 'notation' for expressing a relation between two variables(say x and y). Individual values of these variables are called elements eg x 1 x 2 x 3... y 1 y 2 y 3... The first set of elements ( x) is called the domain. The second set of elements ( y) is called the range. A simple relation like y = x 2 can be more accurately expressed using the following format: The last part relates to the fact that x and y are elements of the set of real numbers R(any positive or negative number, whole or otherwise, including zero) One-One mapping Here one element of the domain is associated with one and only one element of the range. A property of one-one functions is that a on a graph a horizontal line will only cut the graph once.

51 51 Example R + the set of positive real numbers Many-One mapping Here more than one element of the domain can be associated with one particular element of the range. Example Z is the set of integers(positive & negative whole numbers not including zero)

52 52 Complete function notation is a variation on what has been used so far. It will be used from now on. Inverse Function f -1 The inverse function is obtained by interchanging x and y in the function equation and then rearranging to make y the subject. If f -1 exists then, ff -1 (x) = f -1 f(x) = x It is also a condition that the two functions be 'one to one'. That is that the domain of f is identical to the range of its inverse function f -1. When graphed, the function and its inverse are reflections either side of the line y = x. Example Find the inverse of the function(below) and graph the function and its inverse on the same axes.

53 53 Composite Functions A composite function is formed when two functions f, g are combined. However it must be emphasized that the order in which the composite function is determined is important. The method for finding composite functions is: find g(x) find f[g(x)]

54 54 Example For the two functions, find the composite functions (i fg (ii g f

55 55 Exponential & Logarithmic Functions Exponential functions have the general form: where 'a' is a positive constant However there is a specific value of 'a' at (0.1) when the gradient is 1. This value, or 'e' is called the exponential function. The function(above) has one-one mapping. It therefore possesses an inverse. This inverse is the logarithmic function.

56 56 Notes This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit.

Mathematics 1 Lecture Notes Chapter 1 Algebra Review

Mathematics 1 Lecture Notes Chapter 1 Algebra Review Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to

More information

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

More information

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010 Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or

More information

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010

Mathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010 Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,

More information

AS PURE MATHS REVISION NOTES

AS PURE MATHS REVISION NOTES AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are

More information

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:

More information

INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC

INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics

More information

Glossary. Glossary 981. Hawkes Learning Systems. All rights reserved.

Glossary. Glossary 981. Hawkes Learning Systems. All rights reserved. A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle

More information

Learning Objectives These show clearly the purpose and extent of coverage for each topic.

Learning Objectives These show clearly the purpose and extent of coverage for each topic. Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus

More information

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name: A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,

More information

Algebraic. techniques1

Algebraic. techniques1 techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

Maths Higher Prelim Content

Maths Higher Prelim Content Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of

More information

Chapter Five Notes N P U2C5

Chapter Five Notes N P U2C5 Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have

More information

Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document

Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.

More information

Core 1 Module Revision Sheet J MS. 1. Basic Algebra

Core 1 Module Revision Sheet J MS. 1. Basic Algebra Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12

More information

CHAPTER 1. Review of Algebra

CHAPTER 1. Review of Algebra CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you

More information

A Partial List of Topics: Math Spring 2009

A Partial List of Topics: Math Spring 2009 A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose

More information

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE ALGEBRA Module Topics Simplifying expressions and algebraic functions Rearranging formulae Indices 4 Rationalising a denominator

More information

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives

More information

C-1. Snezana Lawrence

C-1. Snezana Lawrence C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary

More information

Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition

Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important

More information

MATH Spring 2010 Topics per Section

MATH Spring 2010 Topics per Section MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

More information

Section 1.1 Notes. Real Numbers

Section 1.1 Notes. Real Numbers Section 1.1 Notes Real Numbers 1 Types of Real Numbers The Natural Numbers 1,,, 4, 5, 6,... These are also sometimes called counting numbers. Denoted by the symbol N Integers..., 6, 5, 4,,, 1, 0, 1,,,

More information

Algebra One Dictionary

Algebra One Dictionary Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.

More information

Algebra. Mathematics Help Sheet. The University of Sydney Business School

Algebra. Mathematics Help Sheet. The University of Sydney Business School Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,

More information

( )( ) Algebra I / Technical Algebra. (This can be read: given n elements, choose r, 5! 5 4 3! ! ( 5 3 )! 3!(2) 2

( )( ) Algebra I / Technical Algebra. (This can be read: given n elements, choose r, 5! 5 4 3! ! ( 5 3 )! 3!(2) 2 470 Algebra I / Technical Algebra Absolute Value: A number s distance from zero on a number line. A number s absolute value is nonnegative. 4 = 4 = 4 Algebraic Expressions: A mathematical phrase that can

More information

Math Precalculus I University of Hawai i at Mānoa Spring

Math Precalculus I University of Hawai i at Mānoa Spring Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents

More information

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0)

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 2017 First certification from June 2018 2

More information

PLC Papers. Created For:

PLC Papers. Created For: PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use

More information

Common Core Standards Addressed in this Resource

Common Core Standards Addressed in this Resource Common Core Standards Addressed in this Resource 6.RP.3 - Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,

More information

2009 GCSE Maths Tutor All Rights Reserved

2009 GCSE Maths Tutor All Rights Reserved 2 This book is under copyright to GCSE Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents number types LCM HCF operators (+ - / x) powers & roots growth & decay

More information

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step

More information

MSM120 1M1 First year mathematics for civil engineers Revision notes 3

MSM120 1M1 First year mathematics for civil engineers Revision notes 3 MSM0 M First year mathematics for civil engineers Revision notes Professor Robert. Wilson utumn 00 Functions Definition of a function: it is a rule which, given a value of the independent variable (often

More information

More Polynomial Equations Section 6.4

More Polynomial Equations Section 6.4 MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

More information

Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Chapter 8. Exploring Polynomial Functions. Jennifer Huss Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

Newbattle Community High School Higher Mathematics. Key Facts Q&A

Newbattle Community High School Higher Mathematics. Key Facts Q&A Key Facts Q&A Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. ) Work with a friend who is also doing to take turns reading a random question

More information

CALCULUS ASSESSMENT REVIEW

CALCULUS ASSESSMENT REVIEW CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness

More information

Check boxes of Edited Copy of Sp Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and

Check boxes of Edited Copy of Sp Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Check boxes of Edited Copy of 10021 Sp 11 152 Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Additional Topics Appendix Course Readiness Multiplication

More information

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10). MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and

More information

Maths Scheme of Work. Class: Year 10. Term: autumn 1: 32 lessons (24 hours) Number of lessons

Maths Scheme of Work. Class: Year 10. Term: autumn 1: 32 lessons (24 hours) Number of lessons Maths Scheme of Work Class: Year 10 Term: autumn 1: 32 lessons (24 hours) Number of lessons Topic and Learning objectives Work to be covered Method of differentiation and SMSC 11 OCR 1 Number Operations

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

Mathematics High School Algebra

Mathematics High School Algebra Mathematics High School Algebra Expressions. An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels,

More information

Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A)

Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Updated 06/05/16 http://www.haesemathematics.com.au/ Note: Exercises in red text indicate material in the 10A textbook

More information

Core Mathematics 1 Quadratics

Core Mathematics 1 Quadratics Regent College Maths Department Core Mathematics 1 Quadratics Quadratics September 011 C1 Note Quadratic functions and their graphs. The graph of y ax bx c. (i) a 0 (ii) a 0 The turning point can be determined

More information

CHAPTER 2 POLYNOMIALS KEY POINTS

CHAPTER 2 POLYNOMIALS KEY POINTS CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x

More information

Mesaieed International School

Mesaieed International School Mesaieed International School SUBJECT: Mathematics Year: 10H Overview of the year: The contents below reflect the first half of the two-year IGCSE Higher course which provides students with the opportunity

More information

Semester Review Packet

Semester Review Packet MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

More information

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4 - SUBSTITUTION - 5 - REMOVING BRACKETS - 7 - FACTORISING - 8 - COMMON FACTORS - 8 - DIFFERENCE

More information

CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics

CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics SCHOLAR Study Guide CfE Higher Mathematics Course Materials Topic 4: Polynomials and quadratics Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy

More information

Math Prep for College Physics

Math Prep for College Physics Math Prep for College Physics This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (190 topics + 52 additional

More information

Algebra I. abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain.

Algebra I. abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain. Algebra I abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain. absolute value the numerical [value] when direction or sign is not considered. (two words) additive inverse

More information

Parent Guide. Number System. Diocese of Cleveland

Parent Guide. Number System. Diocese of Cleveland Parent Guide Grade Eight Algebra Curriculum Diocese of Cleveland Below is a list of skills your child will be taught in Grade Eight Algebra. As parents, you are encouraged to support the work of your child

More information

2Algebraic. foundations

2Algebraic. foundations 2Algebraic foundations 2. Kick off with CAS 2.2 Algebraic skills 2.3 Pascal s triangle and binomial expansions 2.4 The Binomial theorem 2.5 Sets of real numbers 2.6 Surds 2.7 Review c02algebraicfoundations.indd

More information

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

Pre-calculus 12 Curriculum Outcomes Framework (110 hours)

Pre-calculus 12 Curriculum Outcomes Framework (110 hours) Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01

More information

Final Exam Study Guide Mathematical Thinking, Fall 2003

Final Exam Study Guide Mathematical Thinking, Fall 2003 Final Exam Study Guide Mathematical Thinking, Fall 2003 Chapter R Chapter R contains a lot of basic definitions and notations that are used throughout the rest of the book. Most of you are probably comfortable

More information

JUST THE MATHS UNIT NUMBER 1.6. ALGEBRA 6 (Formulae and algebraic equations) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.6. ALGEBRA 6 (Formulae and algebraic equations) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.6 ALGEBRA 6 (Formulae and algebraic equations) by A.J.Hobson 1.6.1 Transposition of formulae 1.6. of linear equations 1.6.3 of quadratic equations 1.6. Exercises 1.6.5 Answers

More information

Mathematics: Year 12 Transition Work

Mathematics: Year 12 Transition Work Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution

More information

MATHEMATICS. Higher 2 (Syllabus 9740)

MATHEMATICS. Higher 2 (Syllabus 9740) MATHEMATICS Higher (Syllabus 9740) CONTENTS Page AIMS ASSESSMENT OBJECTIVES (AO) USE OF GRAPHING CALCULATOR (GC) 3 LIST OF FORMULAE 3 INTEGRATION AND APPLICATION 3 SCHEME OF EXAMINATION PAPERS 3 CONTENT

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS ADDITIONAL MATHEMATICS GCE NORMAL ACADEMIC LEVEL (016) (Syllabus 4044) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE

More information

CCSS Math- Algebra. Domain: Algebra Seeing Structure in Expressions A-SSE. Pacing Guide. Standard: Interpret the structure of expressions.

CCSS Math- Algebra. Domain: Algebra Seeing Structure in Expressions A-SSE. Pacing Guide. Standard: Interpret the structure of expressions. 1 Domain: Algebra Seeing Structure in Expressions A-SSE Standard: Interpret the structure of expressions. H.S. A-SSE.1a. Interpret expressions that represent a quantity in terms of its context. Content:

More information

grasp of the subject while attaining their examination objectives.

grasp of the subject while attaining their examination objectives. PREFACE SUCCESS IN MATHEMATICS is designed with the purpose of assisting students in their preparation for important school and state examinations. Students requiring revision of the concepts covered in

More information

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

More information

Trigonometry Self-study: Reading: Red Bostock and Chandler p , p , p

Trigonometry Self-study: Reading: Red Bostock and Chandler p , p , p Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,

More information

8th Grade Math Definitions

8th Grade Math Definitions 8th Grade Math Definitions Absolute Value: 1. A number s distance from zero. 2. For any x, is defined as follows: x = x, if x < 0; x, if x 0. Acute Angle: An angle whose measure is greater than 0 and less

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

Glossary. Glossary Hawkes Learning Systems. All rights reserved.

Glossary. Glossary Hawkes Learning Systems. All rights reserved. A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Acute triangle A triangle in which all three angles are acute Addends The

More information

Polynomial Functions

Polynomial Functions Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

ax 2 + bx + c = 0 where

ax 2 + bx + c = 0 where Chapter P Prerequisites Section P.1 Real Numbers Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Real number line A line used to graphically

More information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

More information

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET INTRODUCTION TO A LEVEL MATHS AT MGGS Thank you for choosing to study Mathematics in the sixth

More information

CONTENTS. IBDP Mathematics HL Page 1

CONTENTS. IBDP Mathematics HL Page 1 CONTENTS ABOUT THIS BOOK... 3 THE NON-CALCULATOR PAPER... 4 ALGEBRA... 5 Sequences and Series... 5 Sequences and Series Applications... 7 Exponents and Logarithms... 8 Permutations and Combinations...

More information

Check boxes of Edited Copy of Sp Topics (was 261-pilot)

Check boxes of Edited Copy of Sp Topics (was 261-pilot) Check boxes of Edited Copy of 10023 Sp 11 253 Topics (was 261-pilot) Intermediate Algebra (2011), 3rd Ed. [open all close all] R-Review of Basic Algebraic Concepts Section R.2 Ordering integers Plotting

More information

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS MAIDSTONE GRAMMAR SCHOOL FOR GIRLS King Edward VI High School DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET CONTENTS Reading List... 3 Section 1: FRACTIONS... 4 Section : EXPANDING...

More information

Ron Paul Curriculum Mathematics 8 Lesson List

Ron Paul Curriculum Mathematics 8 Lesson List Ron Paul Curriculum Mathematics 8 Lesson List 1 Introduction 2 Algebraic Addition 3 Algebraic Subtraction 4 Algebraic Multiplication 5 Week 1 Review 6 Algebraic Division 7 Powers and Exponents 8 Order

More information

Algebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher

Algebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work

More information

Mathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010

Mathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010 Link to past paper on AQA website: http://store.aqa.org.uk/qual/gce/pdf/aqa-mpc1-w-qp-jan10.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are

More information

Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

More information

The Grade Descriptors below are used to assess work and student progress in Mathematics from Year 7 to

The Grade Descriptors below are used to assess work and student progress in Mathematics from Year 7 to Jersey College for Girls Assessment criteria for KS3 and KS4 Mathematics In Mathematics, students are required to become familiar, confident and competent with a range of content and procedures described

More information

S4 (4.3) Quadratic Functions.notebook February 06, 2018

S4 (4.3) Quadratic Functions.notebook February 06, 2018 Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State

More information

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

EXPONENTIAL AND LOGARITHMIC FUNCTIONS Mathematics Revision Guides Exponential and Logarithmic Functions Page 1 of 14 M.K. HOME TUITION Mathematics Revision Guides Level: A-Level Year 1 / AS EXPONENTIAL AND LOGARITHMIC FUNCTIONS Version : 4.2

More information

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers

More information

1 Solving Algebraic Equations

1 Solving Algebraic Equations Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 1 Solving Algebraic Equations This section illustrates the processes of solving linear and quadratic equations. The Geometry of Real

More information

College Algebra Notes

College Algebra Notes Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

More information

MA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra

MA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra 0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required. Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;

More information

MODULE 1: FOUNDATIONS OF MATHEMATICS

MODULE 1: FOUNDATIONS OF MATHEMATICS MODULE 1: FOUNDATIONS OF MATHEMATICS GENERAL OBJECTIVES On completion of this Module, students should: 1. acquire competency in the application of algebraic techniques; 2. appreciate the role of exponential

More information

Grade 11 or 12 Pre-Calculus

Grade 11 or 12 Pre-Calculus Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard

More information

Variables and Expressions

Variables and Expressions Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic

More information

crashmaths Schemes of Work New A Level Maths (2017)

crashmaths Schemes of Work New A Level Maths (2017) crashmaths Schemes of Work New A Level Maths (2017) This scheme of work is for a class: with one teacher with 5 contact hours each week sitting the AS exams Textbook references are for our Pure/Applied

More information

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4 2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using

More information

Pure Mathematics P1

Pure Mathematics P1 1 Pure Mathematics P1 Rules of Indices x m * x n = x m+n eg. 2 3 * 2 2 = 2*2*2*2*2 = 2 5 x m / x n = x m-n eg. 2 3 / 2 2 = 2*2*2 = 2 1 = 2 2*2 (x m ) n =x mn eg. (2 3 ) 2 = (2*2*2)*(2*2*2) = 2 6 x 0 =

More information