IB Mathematics Standard Level Revision guide

Transcription

1 IB Mathematics Standard Level Revision guide F.G. Groeneveld TopClassTutors.ORG

2 Copyright 2016 by F. Groeneveld All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews. Printed in the Netherlands First Printing, 2016 ISBN TopClassTutors.ORG Publishing i P a g e

3 Contents Chapter 0 Prior Knowledge Real numbers Roots and surds Exponents Scientific notation (standard form) Algebraic expressions Equations and formulae... 9 Rearranging formulae Straight lines Solving linear equations Solving simultaneous linear equations Chapter 1 Algebra Sequences and series Arithmetic sequences Geometric sequences The sum of a sequence and the Σ notation Arithmetic series Geometric series Converging series Applications and sequences and series Binomial expansions Exponents and logarithms Solving exponential equations Exponential functions Logarithms Logarithmic equations Chapter 2 Function and equations Functions: domain and range Composite functions Inverse functions Transformations Quadratic functions Solving quadratic equations Rational functions Chapter 3 Circular functions and trigonometry ii P a g e

4 3.1 The circle Trigonometric functions Graphs of trigonometric functions Triangles Chapter 4 Vectors Properties of vectors The scalar product Equations of lines Chapter 5 Statistics and Probability The center of data Dispersion Cumulative frequency Bivariate analysis Probability The addition rule The product rule Conditional probability Random variables The binomial distribution The normal distribution Chapter 6 Calculus Limits and rate of change Derivatives More derivatives and more rules The second and higher order derivatives Local maximum and minimum points, points of inflexion Practical applications Indefinite integration as anti-differentiation Other standard integrals Anti-differentiation with a boundary condition and definite integrals Boundary condition Definite integrals Area between 2 curves Volumes of revolution Kinematics iii P a g e

5 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 u n is commonly used to denote the nth term of a sequence. For the same sequence as above: u 1 = 2, u 2 = 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: u n+1 = u n + 2, with u 1 = 2. In the sequence 243, 81, 27, 9, the value of each term is one-third the value of the previous term. In formula: u 1 = 243 and u n+1 = 1 3 u n. 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 u n+1 = u n + 2, u 1 = 2 u n+1 = 1 3 u n, u 1 = 243 General formula u n = 2n u n = 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: u n = u 1 + (n 1)d Example 1: 14 P a g e

6 a. Find the 30 th term of the arithmetic sequence 9, 17, 25, 33,.. b. Find an expression for the nth term. Answers: a. u 1 = 9 and the common difference d = 17 9 = 8 u 30 = 9 + (30 1)8 = 241 b. u n = 9 + (n 1)8 = 9 + 8n 8 u n = 8n + 1 Example 2: Find the number of terms in the arithmetic sequence 34, 42, 50,.,274. Answer: u 1 = 34 and d = = 8 u n = 34 + (n 1)(8) = n 8 = n = 274 8n = 248 n = 31 There are 31 terms in the sequence. Exercise For each sequence i. Find the 18 th 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, 5,, 6 d. 348, 300, 252,.., e. 4x, 9x, 14x,., 99x f. y, 3y 4, 5y 8,,17y 32 Example 3: In arithmetic sequence, u 7 = 54 and u 15 = 150. Find the first term and the common difference. Answer: u 7 + 8d = u d = 150 8d = P a g e