IB Mathematics Standard Level

Revision guide

F.G. Groeneveld TopClassTutors.ORG

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Copyright © 2016 by F. Groeneveld

All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any

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Printed in the Netherlands

First Printing, 2016

ISBN 978-90-823459-0-2

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Contents Chapter 0 Prior Knowledge ..................................................................................................................... 1

0.1 Real numbers .......................................................................................................................... 1

0.2 Roots and surds ....................................................................................................................... 2

0.3 Exponents ................................................................................................................................ 3

0.4 Scientific notation (standard form) ......................................................................................... 5

0.5 Algebraic expressions.............................................................................................................. 6

0.6 Equations and formulae .......................................................................................................... 9

Rearranging formulae. .................................................................................................................... 9

Straight lines ................................................................................................................................. 10

Solving linear equations ................................................................................................................ 11

Solving simultaneous linear equations ......................................................................................... 12

Chapter 1 Algebra ................................................................................................................................. 14

1.1 Sequences and series ............................................................................................................ 14

Arithmetic sequences ................................................................................................................... 14

Geometric sequences ................................................................................................................... 16

1.2 The sum of a sequence and the Σ notation .......................................................................... 17

Arithmetic series ........................................................................................................................... 18

Geometric series ........................................................................................................................... 20

1.3 Converging series .................................................................................................................. 22

Applications and sequences and series. ....................................................................................... 23

1.4 Binomial expansions ............................................................................................................. 24

1.5 Exponents and logarithms .................................................................................................... 28

Solving exponential equations ...................................................................................................... 28

Exponential functions ................................................................................................................... 29

Logarithms .................................................................................................................................... 32

Logarithmic equations .................................................................................................................. 34

Chapter 2 Function and equations ........................................................................................................ 37

2.1 Functions: domain and range ............................................................................................... 37

2.2 Composite functions ............................................................................................................. 38

2.3 Inverse functions ................................................................................................................... 39

2.4 Transformations .................................................................................................................... 40

2.5 Quadratic functions .............................................................................................................. 42

Solving quadratic equations .......................................................................................................... 42

2.6 Rational functions ................................................................................................................. 45

Chapter 3 Circular functions and trigonometry .................................................................................... 47

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3.1 The circle ............................................................................................................................... 47

3.2 Trigonometric functions ........................................................................................................ 48

3.3 Graphs of trigonometric functions ....................................................................................... 50

3.4 Triangles ................................................................................................................................ 52

Chapter 4 Vectors ................................................................................................................................. 54

4.1 Properties of vectors ............................................................................................................. 54

4.2 The scalar product ................................................................................................................ 57

4.3 Equations of lines .................................................................................................................. 59

Chapter 5 Statistics and Probability ...................................................................................................... 62

5.1 The center of data ................................................................................................................. 62

5.2 Dispersion ............................................................................................................................. 64

5.3 Cumulative frequency ........................................................................................................... 66

5.4 Bivariate analysis ......................................................................................................................... 67

5.5 Probability ............................................................................................................................. 70

The addition rule ........................................................................................................................... 75

The product rule ........................................................................................................................... 75

Conditional probability ................................................................................................................. 76

5.6 Random variables .................................................................................................................. 77

5.7 The binomial distribution ...................................................................................................... 79

5.8 The normal distribution ........................................................................................................ 80

Chapter 6 Calculus ................................................................................................................................ 83

6.1 Limits and rate of change ............................................................................................................ 83

6.2 Derivatives .................................................................................................................................. 84

More derivatives and more rules .................................................................................................. 87

The second and higher order derivatives. .................................................................................... 89

6.3 Local maximum and minimum points, points of inflexion.......................................................... 90

Practical applications .................................................................................................................... 91

6.4 Indefinite integration as anti-differentiation .............................................................................. 92

Other standard integrals ............................................................................................................... 94

6.5 Anti-differentiation with a boundary condition and definite integrals ...................................... 97

Boundary condition ....................................................................................................................... 97

Definite integrals ........................................................................................................................... 98

Area between 2 curves ............................................................................................................... 100

Volumes of revolution................................................................................................................. 101

6.6 Kinematics ................................................................................................................................. 102

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Chapter 1 Algebra

1.1 Sequences and series

A number sequence is a pattern of number arranged in order according to a rule.

Below some examples of number sequences:

2, 4, 6, 8,……

400, 300, 200,……..

243, 81, 27, 9,…….

1, 4, 9, 16,……….

Each number of a sequence is called a term. In the first sequence, the first term is 2, the second term

4, and so on.

The notation 𝑢𝑛is commonly used to denote the nth term of a sequence.

For the same sequence as above: 𝑢1 = 2, 𝑢2 = 4 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛.

For this sequence the pattern is simple, every next term is two larger as the previous term. Its

recursive formula would be: 𝑢𝑛+1 = 𝑢𝑛 + 2,𝑤𝑖𝑡ℎ 𝑢1 = 2.

In the sequence 243, 81, 27, 9,… the value of each term is one-third the value of the previous term.

In formula: 𝑢1 = 243 𝑎𝑛𝑑 𝑢𝑛+1 =1

3𝑢𝑛.

For each of the two recursive formulas it is also possible to write the general formula for the nth

term. With a general formula you can find the value of a term without knowing the value of the

previous term.

Recursive formula General formula

𝒖𝒏+𝟏 = 𝒖𝒏 + 𝟐, 𝒖𝟏 = 𝟐 𝑢𝑛 = 2𝑛

𝒖𝒏+𝟏 =𝟏

𝟑𝒖𝒏, 𝒖𝟏 = 𝟐𝟒𝟑 𝑢𝑛 = 243 ∙ (

1

3)𝑛−1

Arithmetic sequences

In an arithmetic sequence the term increase or decrease by a constant value, called the common

difference d.

The nth term of an arithmetic sequence can be found using the formula:

𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑

Example 1:

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a. Find the 30th term of the arithmetic sequence 9, 17, 25, 33,………..

b. Find an expression for the nth term.

Answers:

a. 𝑢1 = 9 and the common difference 𝑑 = 17 − 9 = 8

𝑢30 = 9 + (30 − 1)8 = 241

b. 𝑢𝑛 = 9 + (𝑛 − 1)8 = 9 + 8𝑛 − 8 𝑢𝑛 = 8𝑛 + 1

Example 2:

Find the number of terms in the arithmetic sequence 34, 42, 50,…….,274.

Answer:

𝑢1 = 34 𝑎𝑛𝑑 𝑑 = 42 − 34 = 8

𝑢𝑛 = 34 + (𝑛 − 1)(8) = 274

34 + 8𝑛 − 8 = 26 + 8𝑛 = 274

8𝑛 = 248

𝑛 = 31

There are 31 terms in the sequence.

Exercise 1.1

1. For each sequence i. Find the 18th term

ii. Find an expression for the nth term

a. 4, 8, 12, …… b. 18, 34, 50,…

c. 93, 88, 83, ……… d. 112, 98, 84, ….

e. 4.7, 5.8, 6.9, ……… f. 𝑎, 𝑎 − 𝑏, 𝑎 − 2𝑏,….

2. Find the number of terms in each sequence.

a. 6, 12, 18, ….., 126 b. 829, 818, 807, ….,, 444

c. 𝟒

𝟑,𝟑

𝟐,𝟓

𝟑, … , 𝟔 d. 348, 300, 252,… . . , − 660

e. 𝟒𝒙, 𝟗𝒙, 𝟏𝟒𝒙,… . , 𝟗𝟗𝒙 f. 𝑦, 3𝑦 − 4, 5𝑦 − 8,…… ,17𝑦 − 32

Example 3:

In arithmetic sequence, 𝑢7 = 54 𝑎𝑛𝑑 𝑢15 = 150. Find the first term and the common difference.

Answer:

𝑢7 + 8𝑑 = 𝑢15

54 + 8𝑑 = 150

8𝑑 = 96