Learner’s Book Senior Six - REB

Advanced Mathematics

for Rwanda Schools Learner's Book Senior Six

Authors Emmanuel Ngezahayo Pacifique Icyingeneye

FOUNTAIN PUBLISHERS

fountainpublishers.co.ug

Fountain Publishers Rwanda Ltd

P.O. Box 6567

Kigali, Rwanda

E-mail: Website:

fountainpublishers.rwanda@ sales@fountainpublishers.co.ug; publishing@fountainpublishers.co.ug fountainpublishers.co.ug

? Emmanuel Ngezahayo, Pacifique Icyingeneye 2017 First Published 2017

All rights reserved; no part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without prior written permission from the authors and the publishers.

ISBN: 978-9970-19-419-3

Contents

Unit 1

Introduction........................................................................... viii

Complex Numbers........................................................ 1

Introduction..............................................................................1

1.1. Concepts of complex numbers......................................2

1.2. Algebraic form of a complex number.............................4 1.2.1. Definition and properties of "i"................................. 4 1.2.2. Geometric representation of complex numbers....... 5 1.2.3. Modulus of a complex number................................ 7 1.2.4. Operations on complex numbers........................... 12 1.2.5. Square root of a complex number......................... 21 1.2.6. Equations in the set of complex numbers.............. 24 1.2.7. Polynomials in set of complex numbers................ 28

1.3. Polar form of a complex number.................................31 1.3.1. Argument of a complex number............................ 31 1.3.2. Polar form of a complex number........................... 38 1.3.3. Multiplication and division in polar form................. 41 1.3.4. Powers in polar form.............................................. 45 1.3.5. nth roots of a complex number............................... 46

1.4. Exponential form of a complex number.......................56

1.5. Applications ................................................................58

1.5.1. Trigonometric numbers of a multiple of an angle... 58

1.5.2. Linearisation of trigonometric expressions (product to sum)................................ 61

1.5.3. 1.5.4.

Solving equation

a cos x + b sin x = c a,b, c (a,b 0) .......... 63

Alternating current problems ................................ 65

Unit Summary........................................................................70

End of unit assessment.........................................................75

iii

Advanced Mathematics for Rwanda Schools Learner's Book Senior Six

Unit 2 Unit 3

Logarithmic and Exponential Functions................... 81

Introduction............................................................................81

2.1. Logarithmic functions...................................................82 2.1.1. Natural logarithm................................................... 82 2.1.2. Limit and asymptotes for natural logarithmic functions.............................................. 84 2.1.3. Logarithmic function with any base....................... 91

2.2. Exponential functions................................................102 2.2.1. Exponential function with base "e" ................... 102 2.2.2. Exponential function with any base..................... 110

2.3. Applications ..............................................................120 2.3.1. Compound interest problems.............................. 120 2.3.2. Mortgage amount problems................................. 121 2.3.3. Population growth problems................................ 124 2.3.4. Depreciation value problems............................... 126 2.3.5. Earthquake problems.......................................... 127 2.3.6. Carbon-14 dating problems................................. 130

Unit Summary......................................................................134 End of unit assessment.......................................................138

Taylor and Maclaurin's Expansions........................ 143

Introduction..........................................................................143

3.1. Generalities on series................................................144 3.1.1. Finite series......................................................... 144 3.1.2. Infinite series....................................................... 148

3.2. Power series .............................................................157 3.2.1. Taylor and Maclaurin series ................................ 159

3.3. Applications ..............................................................166 3.3.1. Calculation of limits.............................................. 166 3.3.2. Estimation of the number e...................................... 168

3.3.3. Estimation of the number ............................... 170

3.3.4. Estimation of trigonometric number of an angle.. 171 3.3.5. Estimation of an irrational number....................... 173 3.3.6. Estimation of natural logarithm of a number........ 176 3.3.7. Estimation of roots of equations.......................... 178 Unit summary.......................................................................180 End of unit assessment.......................................................184

iv

Contents

Unit 4 Unit 5

Integration.................................................................. 187 Introduction..........................................................................187

4.1. Differentials................................................................188

4.2. Indefinite integrals.....................................................191 4.2.1. Definition.............................................................. 191 4.2.2. Properties of integrals.......................................... 193

4.3. Techniques of integration...........................................195 4.3.1. Integration by substitution.................................... 195 4.3.2. Integration of rational functions........................... 197 4.3.3. Integration of trigonometric functions.................. 210 4.3.4. Integration of irrational functions......................... 223 4.3.5. Integration by parts.............................................. 232 4.3.6. Integration by reduction formulae........................ 236 4.3.7. Integration by Maclaurin series............................ 238

4.4. Definite integrals........................................................239 4.4.1. Definition.............................................................. 239 4.4.2. Properties of definite integrals............................. 243 4.4.3. Improper integrals................................................ 245

4.5. Applications...............................................................249 4.5.1. Calculation of area of a plane surface................. 249 4.5.2. Calculation of volume of a solid of revolution...... 255 4.5.3. Calculation of the arc length of a curved line....... 269

Unit summary.......................................................................276 End of unit assessment.......................................................285

Differential Equations............................................... 289 Introduction..........................................................................289

5.1. Definition and classification.......................................290

5.2. First order differential equations................................293 5.2.1. Differential equations with separable variables... 293 5.2.2. Simple homogeneous equations......................... 294 5.2.3. Linear equations.................................................. 298 5.2.4. Particular solution................................................ 300

5.3. Second order differential equations...........................301 5.3.1. Homogeneous linear equations with constant coefficients............................................ 302 5.3.2. Non-homogeneous linear equations with constant coefficients............................................ 307

v

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download