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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 .............................................. 12 Binomial Expansion ................................................................. 13 Proof by Induction .................................................................... 15 Complex Numbers ................................................................... 17 The Complex Plane ................................................................. 18 De Moivre's Theorem ............................................................... 20 Systems of Equations .............................................................. 22 FUNCTIONS AND EQUATIONS ............................................................ 24 Basics of Functions .................................................................. 24 Graphs of Functions................................................................. 28 Reciprocal Functions ............................................................... 32 Quadratic Functions ................................................................. 33 Solving Quadratic Equations .................................................... 34 Sum and Product of Roots ....................................................... 35 Inequalities .............................................................................. 37 Polynomial Functions ............................................................... 38 Exponential and Logarithmic Functions.................................... 40 CIRCULAR FUNCTIONS AND TRIGONOMETRY ................................ 42 Definitions and Formulae ......................................................... 42 Trigonometric Formulae ........................................................... 46 Solving Trigonometric Equations ............................................. 47 The Solution of Triangles ......................................................... 49 Graphing Periodic Functions .................................................... 50 VECTORS ............................................................................................... 52 Basics of Vectors ..................................................................... 52 Scalar (Dot) Product ................................................................ 54 Vector (Cross) Product ............................................................ 55 Equations of Lines ................................................................... 56 Equations of Planes ................................................................. 57 Equations of Lines and Planes – Summary.............................. 58 Intersections ............................................................................ 59 Angles in Three Dimensions .................................................... 61 Miscellaneous Vector Questions .............................................. 62 STATISTICS AND PROBABILITY ......................................................... 63 Statistics .................................................................................. 63 Probability Notation and Formulae ........................................... 68 Lists and Tables of Outcomes.................................................. 69 Venn Diagrams ........................................................................ 70 Tree Diagrams ......................................................................... 71 Bayes’ Theorem....................................................................... 72 Discrete Probability Distributions ............................................. 73 Binomial Distribution ................................................................ 74 Poisson Distribution ................................................................. 77 Continuous Distributions .......................................................... 78 The Normal Distribution ........................................................... 80 CALCULUS ............................................................................................. 84 Differentiation – The Basics ..................................................... 84 Differentiation from First Principles .......................................... 85 The Chain Rule ........................................................................ 86 Product and Quotient Rules ..................................................... 87 IBDP Mathematics HL Page 1 Second Derivative ....................................................................88 Applications of Differentiation ...................................................89 Implicit Differentiation ...............................................................91 Graphical behaviour of functions ..............................................93 Sketching Graphs – Examples .................................................94 Indefinite Integration ................................................................95 Definite Integration ...................................................................97 Integration By Substitution .......................................................98 Integration by Parts ..................................................................99 General Methods for Integration.............................................100 Further Kinematics .................................................................102 Volumes of Revolution ...........................................................103 Calculus – Non-Calculator Techniques ..................................104 MAXIMISING YOUR MARKS ............................................................... 105 ASSESSMENT DETAILS ..................................................................... 107 PRACTICE QUESTIONS ...................................................................... 108 Answers to Practice Questions ..............................................114 Page 2 IBDP Mathematics HL THE NON-CALCULATOR PAPER The format of the two exam papers is the same – a section A consisting of short answer questions, and a section B involving extended response questions. However, calculators are only allowed to be used in the Paper 2. It is not intended that Paper 1 will test your ability to perform complicated calculations with the potential for careless errors. It is more to see if you can analyse problems and provide reasoned solutions without using your calculator as a prop. However, this doesn’t mean that there are no arithmetic calculations. You should, for example, be able to: Add and subtract using decimals and fractions: Examples: 18.43 + 12.87, 2 21 + 3 52 Multiply using decimals and fractions (brush up your multiplication tables): Examples: 0.5 0.1 2 0.5 432 × 14, 12.6 × 5, 21 × 52 + 32 × 41 , (2 × 106 ) × (5.1× 10−4 ), −0.1 0.2 1 −2 Carry out simple divisions using decimals and fractions Examples: 14 ÷ 0.02, 121 ÷ 53 , find x as a fraction is simplest form if 999x = 324 And don’t forget that divisions can be written as fractions, eg: 9 ÷ 15 = 159 = 53 = 0.6 Fraction simplification can help with more complex calculations: Convert 81km/h to m/s 81× 1000 81× 10 9 × 10 9 × 5 45 = = = = = 22.5m/s 3600 36 4 2 2 Percentage calculations: Examples: 15% of 600kg, Increase 2500 by 12%, what is 150 as a percentage of 500. Quadratic equations You will be called on to solve quadratic equations many times in the papers. Solving by factorisation is easier than using the formula when you are not using a calculator. Examples: 2 2 Solve x + 7x – 60 = 0; 3x – 19x + 20 = 0 NOTE: The Revision Guide contains many boxed questions which are either worked examples or practice questions. Any which would be hard to solve without a calculator will be shown with a double line (as in this box). For the remaining questions, calculator use is either irrelevant (for example, differentiating a function), or the question could be answered both with and without a calculator. In the latter case, it would be sensible for you to answer the question without a calculator, and then check your answer with a calculator. Page 4 IBDP Mathematics HL ALGEBRA Sequences and Series There are many different types of number sequence. You only need to know about two: the arithmetic sequence (AP) and the geometric sequence (GP). In an AP each number is the previous number plus a constant. In a GP each number is the previous number multiplied by a constant. A series is the same as a sequence except that the terms are added together: thus a series has a sum, whereas a sequence doesn’t. To answer most sequences and series questions, make sure you are familiar with the formulae below. First, the notation: u1 = the first term of the sequence (many formulae use a instead) n = the number of terms in the sequence l = the last term of the sequence d = the common difference (the number added on in an AP) r = the common ratio (the multiplier in a GP) un = the value of the nth term Sn = the sum of the first n terms S∞ = the sum to infinity Examples: Arithmetic sequences: 3, 5, 7, 9 ….. 1.1, 1.3, 1.5, 1.7 ….. 11, 7, 3, -1, -5 …… Geometric Sequences: 1, 3, 9, 27 ….. 4, 6, 9, 13.5 ….. 12, 6, 3, 1.5, 0.75 ….. 2, -6, 18, -54 ……. I use AP (Arithmetic Progression) and GP (Geometric Progression), but these are not terms used in IB. The formulae: For an AP: The value of the nth term: The sum of n terms: For a GP: The value of the nth term: un = u1 + (n – 1)d d = un + 1 - un n n Sn = (u1 + l ) = (2u1 + (n − 1)d ) 2 2 un = u1rn –1 u r = n +1 un u1( r n − 1) r −1 And for GPs only there is a formula for “the sum to infinity.” If the common ratio is has a value between -1 and 1 (ie -1 < r < 1) then the terms get ever smaller and approach zero. In this case, the sum of the series will converge on a particular value. To calculate this value: u1 S∞ = 1− r Series questions often involve algebra as well as numbers. Note that to find d given two consecutive terms in an AP, subtract the first from the second; and to find r in a GP, divide the second by the first. The sum of n terms: Sn = Sigma Notation: Sigma notation is just a shorthand for defining a series. The ∑ symbol means “the sum of” and will include a general formula for the terms of the series. For example, 4 ∑ (n 2 The sum formulae always start from the first term. If you wanted to sum, say, the 10th to the 20th terms, you would calculate s20 – s9. Think about it! Example: A GP has first two terms 2 and k. What range of values of k will ensure the series converges? The common ratio must be between –1 and 1. The k common ratio is , so: 2 k -1 < < 1 so -2 < k < 2 2 − 2) = (12 − 2) + (2 2 − 2) + (3 2 − 2) + (4 2 − 2) = 22 1 IBDP Mathematics HL Page 5