University Mathematics I.pdf University Mathematics I: Olaniyi Evans
- 1. VOLUME I UNIVERSITY MATHEMATICS REVISED AND EXPANDED EDITION OLANIYI EVANS Pan-Atlantic University www.unimath.5profz.com
- 2. Copyright © 2019, 2025 by Olaniyi Evans. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without prior written permission of the author, except for the inclusion of brief quotations in a review. Limits of Liability/Disclaimer of Warranty: Every effort has been made to make this book as complete and as accurate as it possibly can. However, there may be mistakes, both typographical and content. Therefore, this book should be harnessed as helpful information and not as the ultimate bringer of mathematical genius. The author and publisher shall have neither liability nor responsibility to any person or entity regarding any loss or damage caused, or alleged to have been caused, directly or indirectly, from the information contained in this book. First edition 2019 Second edition 2025 ISBN: 978-978-977-880-5 Published by 5-Profz International Km. 42 Lekki-Epe Expressway, Lekki, Lagos. +2348127455000
- 3. Dedicated to students, chasing knowledge with might, Whether undergraduate, postgraduate, or professional, This book, a beacon through your scholarly days and nights, Guiding, inspiring, as you reach academic heights. PREFACE Dear Student, Bravo! Welcome to this epic book, University Mathematics, where numbers weave the fabric of understanding, and equations unlock the mysteries of the universe. The book has been meticulously re-crafted to support you in navigating the multifaceted terrain of higher mathematics, as it provides you a solid foundation for both academic inquiry and real-world applications. With its broad structure, the book is tailored to offer clear, insightful explanations of key mathematical principles, along with practical exercises to deepen your understanding and sharpen your problem-solving skills. Each chapter is structured to provide a clear and concise overview of the key concepts, with numerous examples and exercises to cultivate mastery. In the early chapters, you will explore business mathematics topics, including set theory, logic, algebra, equations and series, building a solid foundation for the more advanced topics that follow. You will then delve deeper into operations research topics such as capital budgeting, inventory control, decision and network analysis as well as queuing theory. The latter half of the book covers a range of topics in statistics and data analysis, including probability, hypothesis testing, regression and time series. These chapters are meticulously designed to help you apply your statistical knowledge to real-world data, and to develop the critical thinking skills required for success in both academia and industry. I hope that this book will serve as a valuable resource throughout your university studies, and that it will inspire you to explore the many fascinating and rewarding applications of mathematics in the world around us. Whether you are a beginner or an advanced student, I believe that this book will equip you with the tools and knowledge you need to succeed in your academic journey and beyond. Thanks for choosing University Mathematics – and best of luck! Olaniyi Evans, Ph.D. ACKNOWLEDGEMENTS To the dedicated reviewers of this book, my immense gratitude for your meticulous scrutiny, constructive critiques and insightful feedback. To my colleagues and students, I am profoundly grateful for your unwavering support and continuous encouragement. To Ayomi Ini-K, Olami Oba-X and Iremi Teni- D, I love you to the moon & back.
- 4. CONTENTS 0. Mathematical Preliminaries 1 Number Systems 1 Integers 2 Multiplying/Dividing with Negatives 2 Order of Operations: PEMDAS 2 Fractions 3 Proportions, Percents, Rates, and Ratios 4 Percents to Decimals and Decimals to Percents 4 Scientific Notation 5 Algebraic Terms 5 Metric Conversions 5 Approximations 5 Percentage Error & Degree of Accuracy 6 Modular Arithmetic 6 Changing the Subject of a Formula 7 Distance Formula 7 Functions & Relations 7 Perimeter & Area 8 Volume & Surface Area 8 Polygons 8 Variation 9 1. Set Theory 10 Sets 10 Set Operations 11 Fundamental Laws of Set Algebra 12 Venn Diagrams 13 Word Problems on Sets and Venn Diagrams 14 2. Logic 19 Propositional Logic 19 Truth Table 20 Connectives and Operations between Propositions 20 Types of Statements 21 Inverse, Converse, and Contrapositive 22 Universal and existential quantifiers 22 Negation of Propositions 22 Methods of Proofs 23 Duality Principle 23 3. Algebra, Factorization & Surds 26 Algebra 26 Factorization 28 Surds 29 4. Indices, Logarithms & Growth 31 Indices 31 Logarithms 32 Exponential Equations 34 Logarithmic Equations 35 Mathematics of Growth Rates 36
- 5. 5. Linear Equations & Inequalities 38 Solving Linear Equations 38 Graphing Linear Equations 40 Word Problems with Linear equations 41 Linear Inequalities 43 Inequalities on a Number Line 44 Graphing Inequalities 44 Absolute Value Inequalities 44 Word Problems with Inequalities 45 6. Quadratic Equations 47 Standard Form 47 Factoring 48 Completing the Square 48 Quadratic Formula 49 Graphing Quadratic Functions 51 The Discriminant 52 Word Problems to Quadratic Equations 54 7. Simultaneous Equations 58 System of Equations in 2 Variables 58 Elimination Method 58 Substitution Method 59 Graphical Method 60 Linear-Quadratic System of Equations 62 System of Equations in 3 Variables 64 Word Problems on Simultaneous Equations 66 8. Functions & Applications 70 Functions 70 Identities 70 Types of Functions 70 Economic Functions 71 Supply & Demand Analysis 74 IS-LM Analysis 74 Income Determination Models 75 9. Sequences & Series 77 Sequences 77 Arithmetic Progression 77 Geometric Progression 79 Sum of an Infinite Geometric Series 81 Sigma Notation of a Series 82 10. Permutation & Combination 85 Factorial 85 Counting Principles 85 Permutations 86 I. Permutations when repetition is allowed 86 II. Permutations of n different objects 87 III. Permutation where some items are identical 87 Combinations 88 Combination with Repetition 89 Combinations without Repetition 89
- 6. 11. Index Numbers 92 Simple Price Index 92 Aggregate Price Indices 93 Weighted Aggregate Price Indices 94 Consumer Price Index (CPI) and Inflation 96 Stock Exchange Index 98 Indices of Development 99 Human Development Index (HDI) 100 Multidimensional Poverty Index 101 Base Shifting (or Rebasing) of Index Numbers 103 Deflating the Index Numbers 103 Criteria of a Good Index Number 103 12. Simple & Compound Interest: Annuities & Perpetuities Simple Interest 105 Compound Interest 106 Continuous Compounding 108 Annuities 110 Deferred Annuity 113 Perpetuity 113 13. Trade & Cash Discount: Markup & Markdown 116 Discount 116 Discount Series 117 Cash Discount 118 Ordinary Dating Method 118 End of Month Dating Method 118 Receipt of Goods Dating Method 119 Partial Payments 119 Markup 120 Rate of Markup 121 Markdown 121 14. Capital Budgeting Techniques 124 Discounted Cash Flow Technique 124 Classification of Investment Projects 124 Payback Period 125 Discounted Payback Period 126 Net Present Value 127 Internal Rate Of Return 128 Modified Internal Rate of Return (MIRR) 131 Profitability Index (PI) 133 15. Sensitivity Analysis 135 16. Project Selection under Capital Rationing 141 Capital Rationing 141 Single-period capital rationing 142 Single Period Capital Rationing with Divisible Projects 142 Single Period Capital Rationing with Indivisible Projects 143 Multi-Period Capital Rationing 144 17. Decision Analysis & Trees: Risk & Uncertainty 148 Decision Analysis 148 Decision-Making Criteria Under Uncertainty 149 Decision-Making Under Risk 150 Decision Trees 153
- 7. 18. Transportation & Assignment Models 159 Nature of a Transportation Model 159 Solution of Transportation Problems 159 Special Issues in Modeling 165 Assignment Model 166 19. Inventory Planning & Production Control 170 Inventory 170 Inventory Costs 170 Inventory Re-order Level 171 General Inventory Models 172 EOQ with Gradual Replenishment (Stockholder Manufacturer) 173 EOQ with Stock-outs 174 EOQ with Bulk Discounts 174 Probabilistic Inventory Models and Safety Stock 175 20. Network Analysis 178 Network Diagram 179 Critical Path Method 179 Project Scheduling 180 Program Evaluation and Review Technique (PERT) 183 Cost Scheduling 186 Gantt Chart 190 Linear Programming in Project Scheduling 191 21. Simulation Modeling 193 Modeling in Problem Solving 193 Basic Steps in Simulation 194 Types of Systems in Simulation Models 194 Simulation Methods 195 Monte Carlo Method 196 22. Queueing Theory & Models 201 Elements of a Queueing System 201 Types of Service Order/Queueing Disciplines 201 Important distributions in Queueing Theory 202 Types of Queueing Models 202 Kendall's Notation 202 Single Server (M/M/1) 203 Multiple Server (M/M/k) 204 23. Replacement Analysis 207 Replacement of Items that Deteriorate 207 Replacement of Items that Fail Suddenly 210 Individual Replacement Policy 210 Group Replacement Policy 211 24. Descriptive Statistics 215 Measures of Location 215 Mean, Median & Mode 216 Percentiles 216 Quartiles 217 Measures of Variability, or Dispersion 219 Range 219 Interquartile Range 219
- 8. Variance and Standard Deviation 220 Coefficient of Variation 220 Measures of Association 221 Covariance 221 Correlation Coefficient 221 Grouped Data 223 25. Probabilities 225 Basic Relationships of Probability 228 Complement 228 Union and Intersection 228 Mutually Exclusive Events 228 Addition Law 228 Conditional Probability 230 Multiplication Law 230 Tree Diagrams 233 Bayes’ Theorem 235 26. Discrete & Continuous Probability 240 Random Variables: Probability Distributions 240 Discrete Probability Distributions 240 Binomial Probability Distribution 242 Binomial Experiment 243 Binomial Probability Function 243 Poisson Probability Distribution 245 Continuous Probability Distributions 247 Normal Probability Distribution 247 Normal Curve 247 27. Hypothesis Testing 249 z-Tests: Large Sample Tests 250 t-Tests: Small Sample Tests 252 Chi-Square Tests 254 Chi-Square Test of Goodness of Fit 254 Chi-Square Test of Independence 256 F-Tests: One-Way ANOVA 257 28. Regression & Correlation 262 Regression 262 Simple Regression 262 Coefficient of Determination 264 Correlation Coefficient 265 Multiple Regression 267 29. Time Series & Forecasting 271 Forecasting 271 Time Series Patterns 271 Smoothing Methods 273 Moving Averages 274 Weighted Moving Average 275 Exponential Smoothing 275 Forecast Accuracy 276 Trend Projection 277 Linear Regression Analysis 277 Nonlinear Trend Regression 279 Seasonality and Trend 280
- 9. 30. Mathematical Induction 284 31. Coordinate Geometry 287 Distance Between Two Points 287 Mid-Point of a Line Segment 287 Gradient/Equation of a Straight Line 288 Parallel Lines 289 Perpendicular Lines 290 32. Complex Numbers 293 Arithmetic Operations with Complex Numbers 293 Distance in the Complex Plane 295 Midpoint in the Complex Plane 295 Argand Diagrams 296 Polar Form 296 Exponential Form 296 33. Trigonometric Functions 298 Triangle 298 Pythagorean Theorem 298 Trigonometric Identities 301 Scalene Triangles 303 Law of Sines/Cosines 304 Bearings 305 Elevation and Depression 307 34. Vectors 310 Vector Addition 310 Cartesian Components of a Vector 310 The Scalar Product of Two Vectors 311 Position Vectors & Direction Vectors 312 Collinearity (lying on a straight line) 313 Perpendicular Vectors 313 Vector Equation of a Line 314 Parallel & Skews Lines 314 Cross Product of Two Vectors 316 35. Data Analysis: Excel & SPSS 318 Types of Data 318 Excel 319 Excel Formulas 322 SPSS 323 SPSS Commands 325 36. Econometric Analysis: Eviews & Stata 327 Eviews 327 Eviews Commands 330 Stata 332 Stata Commands 335 Econometric Techniques for Time Series and Panel Data 338 Suggestions for Further Reading 343 Index 344 Tables 347
- 10. TOPICS IN VOLUME II 1. Calculus: Differentiation 2. Applications of Differential Calculus 3. Differentiation of Logarithmic & Trigonometric Functions 4. Multivariate Functions & Partial Derivatives 5. Further Applications of Multivariate Calculus 6. Limits, Continuity & Differentiability 7. Polynomials: Remainder & Factor Theorem 8. Partial Fractions & Binomial Series 9. Calculus: Integration 10. Calculus: Definite Integrals 11. Matrices & Determinants 12. Matrices & Simultaneous Equations 13. Quadratic Forms, Definite Matrices, Eigenvalues & Hessian 14. Input-Output Analysis 15. Partial & General Equilibrium Analysis 16. Linear Programming 17. Linear Programming: Simplex Method 18. Duality in Linear Programming 19. Game Theory 20. Non-linear Programming & Envelope Theorem 21. Differential Equations 22. Difference Equations 23. Differential & Difference Equations of Higher Order 24. Simultaneous Systems of Differential & Difference Equations 25. Optimal Control Theory
- 11. MATH STRATEGY AND STUDY TIPS 1. Practice Like a Pro! Consistent practice is your gateway to mathematical mastery. Solve new problems daily to keep your mind sharp and agile! "Ama solves 10 math problems every morning to stay sharp and focused for her exams." 2. Unlock Your Toolbox! Leverage every resource—textbooks, formula sheets, calculators, and online tools. Different perspectives unlock deeper understanding. "Oba often watches math tutorials on YouTube to gain fresh insights and better understand complex concepts." 3. Learn from Mistakes! Mistakes are your greatest teachers. Review your errors, identify gaps, and turn weaknesses into strengths. "Ola always reviews his math mistakes daily to pinpoint gaps and grow more confident in his problem-solving abilities." 4. Uncover the Why! Understanding the principles behind the formulas is key. Grasping the ‘why’ makes formulas more intuitive and powerful. "Ola always asks his teacher to explain the logic behind formulas, and it helps everything click into place." 5. Complete Homework! Homework is your training ground. Each problem solved strengthens your grasp on theoretical concepts, bringing them to life. "Every week, Sage completes his math homework with a study partner, and they quiz each other to reinforce what they've learned." 6. Break Down Problems! Conquer complex problems by breaking them into smaller, manageable steps. Tackling them one piece at a time makes them less intimidating. "Chi always breaks down challenging math problems into smaller steps, which makes solving them more straightforward." 7. Seek Help When Needed! Learning is a collective journey. Don’t hesitate to reach out to instructors, peers, or tutors for guidance. "Chi often joins study groups to clarify doubts and gain fresh perspectives from her classmates." 8. Prepare Strategically! Review thoroughly, practice relentlessly, and approach tests methodically. Build math confidence every day. "Ama always reviews her notes and tackles practice tests to ensure she’s fully prepared and confident for her exams." Remember, every problem you solve is a step toward mastery. Stay persistent, stay curious!
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