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Mathematics for Business, Science, and Technology, Third Edition
Table of Contents
1 Elementary Algebra 1−1
1.1 Introduction..................................................................................................1−1
1.2 Algebraic Equations......................................................................................1−2
1.3 Laws of Exponents ........................................................................................1−5
1.4 Laws of Logarithms.....................................................................................1−10
1.5 Quadratic Equations...................................................................................1−13
1.6 Cubic and Higher Degree Equations..........................................................1−15
1.7 Measures of Central Tendency...................................................................1−15
1.8 Interpolation and Extrapolation.................................................................1−20
1.9 Infinite Sequences and Series.....................................................................1−23
1.10 Arithmetic Series........................................................................................1−23
1.11 Geometric Series.........................................................................................1−24
1.12 Harmonic Series .........................................................................................1−26
1.13 Proportions .................................................................................................1−27
1.14 Summary.....................................................................................................1−29
1.15 Exercises .....................................................................................................1−33
1.16 Solutions to End−of−Chapter Exercises .....................................................1−35
Excel Computations: Pages 1−7, 1−11, 1−16, 1−17, 1−33, 1−36
MATLAB Computations: Pages 1−7, 1−11
2 Intermediate Algebra 2−1
2.1 Systems of Two Equations ........................................................................................ 2−1
2.2 Systems of Three Equations...................................................................................... 2−6
2.3 Matrices and Simultaneous Solution of Equations................................................... 2−7
2.4 Summary ................................................................................................................. 2−26
2.5 Exercises.................................................................................................................. 2−30
2.6 Solutions to End−of−Chapter Exercises ..................................................................2−32
Excel Computations: Pages 2−23, 2−24, 2−30, 2−31
MATLAB Computations: Pages 2−15, 2−16, 2−24, 2−25, 2−32, 2−36
3 Fundamentals of Geometry 3−1
3.1 Introduction...............................................................................................................3−1
3.2 Plane Geometry Figures.............................................................................................3−1
3.3 Solid Geometry Figures............................................................................................3−16
3.4 Using Spreadsheets to Find Areas of Irregular Polygons.........................................3−20
3.5 Summary..................................................................................................................3−233.6 Exercises.................................................................................................................. 3−27
3.7 Solutions to End−of−Chapter Exercises.................................................................. 3−29
Excel Computations: Pages 3−20, 3−30
4 Fundamentals of Plane Geometry 4−1
4.1 Introduction............................................................................................................. 4−1
4.2 Trigonometric Functions......................................................................................... 4−2
4.3 Trigonometric Functions of an Acute Angle .......................................................... 4−2
4.4 Trigonometric Functions of an Any Angle ............................................................. 4−3
4.5 Fundamental Relations and Identities .................................................................... 4−6
4.6 Triangle Formulas.................................................................................................. 4−12
4.7 Inverse Trigonometric Functions .......................................................................... 4−13
4.8 Area of Polygons in Terms of Trigonometric Functions....................................... 4−14
4.9 Summary................................................................................................................ 4−16
4.10 Exercises ................................................................................................................ 4−18
4.11 Solutions to End−of−Chapter Exercises ................................................................ 4−19
Excel Computations: Pages 4−9 through 4−11, 4−13, 4−19 through 4−21
MATLAB Computations: Page 4−15
5 Fundamentals of Calculus 5−1
5.1 Introduction............................................................................................................ 5−1
5.2 Differential Calculus ............................................................................................... 5−1
5.3 The Derivative of a Function ................................................................................. 5−3
5.4 Maxima and Minima............................................................................................. 5−11
5.5 Integral Calculus ................................................................................................... 5−15
5.6 Indefinite Integrals................................................................................................ 5−16
5.7 Definite Integrals .................................................................................................. 5−16
5.8 Summary ............................................................................................................... 5−21
5.9 Exercises................................................................................................................ 5−23
5.10 Solutions to End−of−Chapter Exercises ............................................................... 5−24
Excel Computations: Pages 5−8, 5−11, 5−18
MATLAB Computations: Pages 5−24 through 5−26
6 Mathematics of Finance and Economics 6−1
6.1 Common Terms........................................................................................................ 6−1
6.1.1 Bond............................................................................................................... 6−1
6.1.2 Corporate Bond ............................................................................................. 6−1
6.1.3 Municipal Bond ............................................................................................. 6−1
Mathematics for Business, Science, and Technology, Third Edition iii
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6.1.4 Treasury Bond ............................................................................................... 6−1
6.1.5 Perpetuity ...................................................................................................... 6−1
6.1.6 Perpetual Bond.............................................................................................. 6−1
6.1.7 Convertible Bond .......................................................................................... 6−2
6.1.8 Treasury Note ............................................................................................... 6−2
6.1.9 Treasury Bill .................................................................................................. 6−2
6.1.10 Face Value ................................................................................................... 6−2
6.1.11 Par Value...................................................................................................... 6−2
6.1.12 Book Value................................................................................................... 6−3
6.1.13 Coupon Bond............................................................................................... 6−3
6.1.14 Zero Coupon Bond ...................................................................................... 6−3
6.1.15 Junk Bond.................................................................................................... 6−3
6.1.16 Bond Rating Systems................................................................................... 6−3
6.1.17 Promissory Note .......................................................................................... 6−4
6.1.18 Discount Rate.............................................................................................. 6−4
6.1.19 Prime Rate ................................................................................................... 6−4
6.1.20 Mortgage Loan ............................................................................................ 6−4
6.1.21 Predatory Lending Practices........................................................................ 6−5
6.1.22 Annuity ....................................................................................................... 6−5
6.1.23 Ordinary Annuity........................................................................................ 6−5
6.1.24 Sinking Fund ............................................................................................... 6−6
6.2 Interest .................................................................................................................... 6−6
6.2.1 Simple Interest............................................................................................. 6−6
6.2.2 Compound Interest...................................................................................... 6−8
6.2.3 Effective Interest Rate ............................................................................... 6−22
6.3 Sinking Funds........................................................................................................ 6−24
6.4 Annuities............................................................................................................... 6−29
6.5 Amortization ......................................................................................................... 6−34
6.6 Perpetuities............................................................................................................ 6−36
6.7 Valuation of Bonds................................................................................................ 6−39
6.7.1 Calculating the Purchase Price of a Bond................................................... 6−40
6.7.2 Total Periodic Bond Disbursement............................................................. 6−42
6.7.3 Calculation of Interest Rate of Bond .......................................................... 6−44
6.8 Spreadsheet Financial Functions ......................................................................... 6−45
6.8.1 PV Function.............................................................................................. 6−46
6.8.2 FV Function .............................................................................................. 6−48
6.8.3 PMT Function........................................................................................... 6−49
6.8.4 RATE Function......................................................................................... 6−50
6.8.5 NPER Function.......................................................................................... 6−51
6.8.6 NPV Function ........................................................................................... 6−52
6.8.7 IIR Function.............................................................................................. 6−54
iv Mathematics for Business, Science, and Technology, Third Edition
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6.8.8 MIIR Function........................................................................................... 6−56
6.8.9 IPMT Function.......................................................................................... 6−58
6.8.10 PPMT Function ........................................................................................ 6−59
6.8.11 ISPMT Function........................................................................................ 6−60
6.9 The MATLAB Financial Toolbox ....................................................................... 6−61
6.9.1 irr MATLAB Function............................................................................... 6−61
6.9.2 effrr MATLAB Function............................................................................ 6−62
6.9.3 pvfix MATLAB Function .......................................................................... 6−62
6.9.4 pvvar MATLAB Function ......................................................................... 6−63
6.9.5 fvfix MATLAB Function ........................................................................... 6−64
6.9.6 fvvar MATLAB Function .......................................................................... 6−65
6.9.7 annurate MATLAB Function.....................................................................6−66
6.9.8 amortize MATLAB Function .....................................................................6−67
6.10 Comparison of Alternate Proposals .......................................................................6−68
6.11 Kelvin’s Law...........................................................................................................6−71
6.12 Summary ................................................................................................................6−75
6.13 Exercises.................................................................................................................6−78
6.14 Solutions to End−of−Chapter Exercises ................................................................6−81
Excel Computations: Pages 6−10 through 6−14, 6−28, 6−29, 6−33, 6−35,
6−45 through 6−52, 6−54, 6−56, 6−58 through 6−61, 6−81
MATLAB Computations: Pages 6−61, 6−68
7 Depreciation, Impairment, and Depletion 7−1
7.1 Depreciation Defined................................................................................................ 7−1
7.1.1 Items that Can Be Depreciated ...................................................................... 7−2
7.1.2 Items that Cannot Be Depreciated ................................................................. 7−2
7.1.3 Depreciation Rules.......................................................................................... 7−2
7.1.4 When Depreciation Begins and Ends ............................................................. 7−3
7.1.5 Methods of Depreciation ................................................................................ 7−3
7.1.5.1 Straight−Line (SL) Depreciation Method.......................................... 7−4
7.1.5.2 Sum of the Years Digits (SYD) Method ............................................. 7−5
7.1.5.3 Fixed-Declining Balance (FDB) Method ........................................... 7−6
7.1.5.4 The 125%, 150%, and 200% General Declining Balance Methods... 7−8
7.1.5.5 The Variable Declining Balance Method........................................... 7−9
7.1.5.6 The Units of Production Method ..................................................... 7−10
7.1.5.7 Depreciation Methods for Income Tax Reporting ........................... 7−11
7.2 Impairments............................................................................................................ 7−18
7.3 Depletion ................................................................................................................ 7−19
7.4 Valuation of a Depleting Asset .............................................................................. 7−20
7.5 Summary................................................................................................................. 7−25
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7.6 Exercises ..................................................................................................................7−27
7.7 Solutions to End−of−Chapter Exercises..................................................................7−28
Excel Computations: Pages 7−4, 7−6, 7−7, 7−9, 7−10, 7−29 through 7−32
MATLAB Computations: Pages 7−30, 7−31, 7−33
8 Introduction to Probability and Statistics 8−1
8.1 Introduction ..............................................................................................................8−1
8.2 Probability and Random Experiments ......................................................................8−1
8.3 Relative Frequency ...................................................................................................8−2
8.4 Combinations and Permutations ..............................................................................8−4
8.5 Joint and Conditional Probabilities...........................................................................8−7
8.6 Bayes’ Rule ..............................................................................................................8−11
8.7 Summary .................................................................................................................8−13
8.8 Exercises..................................................................................................................8−15
8.9 Solutions to End−of−Chapter Exercises..................................................................8−16
Excel Computations: Pages 8−6, 8−16
9 Random Variables 9−1
9.1 Definition of Random Variables...............................................................................9−1
9.2 Probability Function .................................................................................................9−2
9.3 Cumulative Distribution Function...........................................................................9−2
9.4 Probability Density Function....................................................................................9−9
9.5 Two Random Variables ..........................................................................................9−11
9.6 Statistical Averages ................................................................................................9−12
9.7 Summary.................................................................................................................9−20
9.8 Exercises .................................................................................................................9−23
9.9 Solutions to End−of−Chapter Exercises .................................................................9−24
Excel Computations: Page 9−3
10 Common Probability Distributions and Tests of Significance 10−1
10.1 Properties of Binomial Coefficients ....................................................................10−1
10.2 The Binomial (Bernoulli) Distribution...............................................................10−2
10.3 The Uniform Distribution ..................................................................................10−6
10.4 The Exponential Distribution ..........................................................................10−10
10.5 The Normal (Gaussian) Distribution ...............................................................10−12
10.6 Percentiles.........................................................................................................10−32
10.7 The Student’s t−Distribution ...........................................................................10−36
10.8 The Chi−Square Distribution...........................................................................10−40
vi Mathematics for Business, Science, and Technology, Third Edition
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10.9 The F Distribution ........................................................................................... 10−43
10.10 Chebyshev’s Inequality .................................................................................... 10−46
10.11 Law of Large Numbers ..................................................................................... 10−46
10.12 The Poisson Distribution ................................................................................. 10−47
10.13 The Multinomial Distribution ......................................................................... 10−52
10.14 The Hypergeometric Distribution ................................................................... 10−53
10.15 The Bivariate Normal Distribution ................................................................. 10−56
10.16 The Rayleigh Distribution ............................................................................... 10−56
10.17 Other Probability Distributions ....................................................................... 10−59
10.17.1 The Cauchy Distribution.................................................................... 10−59
10.17.2 The Geometric Distribution .............................................................. 10−59
10.17.3 The Pascal Distribution...................................................................... 10−59
10.17.4 The Weibull Distribution................................................................... 10−60
10.17.5 The Maxwell Distribution.................................................................. 10−61
10.17.6 The Lognormal Distribution.............................................................. 10−61
10.18 Critical Values of the Binomial Distribution ................................................... 10−62
10.19 Sampling Distribution of Means ...................................................................... 10−63
10.20 Z −Score ........................................................................................................... 10−64
10.21 Tests of Hypotheses and Levels of Significance............................................... 10−65
10.22 Tests of Significance......................................................................................... 10−72
10.22.1 The z−Test ......................................................................................... 10−72
10.22.2 The t−Test ......................................................................................... 10−73
10.22.3 The F−Test......................................................................................... 10−74
10.22.4 The Chi−SquareTest.......................................................................... 10−75
10.23 Summary.......................................................................................................... 10−80
10.24 Exercises.......................................................................................................... 10−89
10.24 Solutions to End−of−Chapter Exercises.......................................................... 10−91
Excel Computations: Pages 10−28, 10−29, 10−35, 10−36, 10−39, 10−40, 10−43,
10−45, 10−51, 10−55, 10−60, 10−62, 10−69 through 10−75,
10−77, 10−78, 10−93, 10−95
MATLAB Computations: Pages 10−21, 10−28, 10−31, 10−32, 10−37, 10−41
11 Curve Fitting, Regression, and Correlation 11−1
11.1 Curve Fitting ....................................................................................................... 11−1
11.2 Linear Regression ................................................................................................ 11−2
11.3 Parabolic Regression ........................................................................................... 11−7
11.4 Covariance ........................................................................................................ 11−10
11.5 Correlation Coefficient ..................................................................................... 11−12
11.6 Summary ........................................................................................................... 11−17
11.7 Exercises ............................................................................................................ 11−20
11.8 Solutions to End−of−Chapter Exercises ............................................................11−22
Excel Computations: Pages 11−3, 11−4, 11−6, 11−7, 11−8, 11−11 through 11−13,
11−15, 11−22 through 11−25
12 Analysis of Variance (ANOVA) 12−1
12.1 Introduction.........................................................................................................12−1
12.2 One−way ANOVA..............................................................................................12−1
12.3 Two−way ANOVA..............................................................................................12−8
12.3.1 Two−factor without Replication ANOVA............................................. 12−8
12.3.2 Two−factor with Replication ANOVA.................................................12−14
12.4 Summary............................................................................................................12−25
12.5 Exercises.............................................................................................................12−29
12.6 Solutions to End−of−Chapter Exercises ............................................................12−31
Excel Computations: Pages 12−6 through 12−8, 12−13, 12−14, 12−22 through 12−24,
12−31, 12−33, 12−35, 12−37
A Numbers and Arithmetic Operations A−1
A.1 Number Systems..................................................................................................A−1
A.2 Positive and Negative Numbers..........................................................................A−1
A.3 Addition and Subtraction ...................................................................................A−2
A.4 Multiplication and Division ................................................................................A−7
A.5 Integer, Fractional, and Mixed Numbers..........................................................A−10
A.6 Reciprocals of Numbers ....................................................................................A−11
A.7 Arithmetic Operations with Fractional Numbers.............................................A−13
A.8 Exponents..........................................................................................................A−22
A.9 Scientific Notation ............................................................................................A−25
A.10 Operations with Numbers in Scientific Notation.............................................A−27
A.11 Square and Cubic Roots....................................................................................A−30
A.12 Common and Natural Logarithms ....................................................................A−32
A.13 Decibel ..............................................................................................................A−33
A.14 Percentages........................................................................................................A−34
A.15 International System of Units (SI)....................................................................A−35
A.16 Graphs ...............................................................................................................A−40
A.17 Summary............................................................................................................A−44
A.18 Exercises ............................................................................................................A−50
A.19 Solutions to End−of−Appendix Exercises .........................................................A−51
Excel Computations: Pages A−13 through A−18, A−26, A−30 through A−35,
A−37, A−39, A−51
viii Mathematics for Business, Science, and Technology, Third Edition
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B Introduction to MATLAB® B−1
B.1 MATLAB® and Simulink®...............................................................................B−1
B.2 Command Window .............................................................................................B−1
B.3 Roots of Polynomials ...........................................................................................B−3
B.4 Polynomial Construction from Known Roots .....................................................B−4
B.5 Evaluation of a Polynomial at Specified Values..................................................B−5
B.6 Rational Polynomials...........................................................................................B−8
B.7 Using MATLAB to Make Plots ........................................................................B−10
B.8 Subplots .............................................................................................................B−18
B.9 Multiplication, Division, and Exponentiation ..................................................B−18
B.10 Script and Function Files ..................................................................................B−26
B.11 Display Formats .................................................................................................B−31
MATLAB Computations: Entire Appendix
C The Gamma and Beta Functions and Distributions C−1
C.1 The Gamma Function ..........................................................................................C−1
C.2 The Gamma Distribution ...................................................................................C−15
C.3 The Beta Function .............................................................................................C−17
C.4 The Beta Distribution ........................................................................................C−20
Excel Computations: Pages C−5, C−11, C−16, C−17, C−19
MATLAB Computations: Pages C−3 through C−5, C−10, C−16, C−19, C−20
D Introduction to Markov Chains D−1
D.1 Stochastic Processes............................................................................................ D−1
D.2 Stochastic Matrices............................................................................................. D−1
D.3 Transition Diagrams ........................................................................................... D−4
D.4 Regular Stochastic Matrices ............................................................................... D−5
D.5 Some Practical Examples .................................................................................... D−7
Excel Computations: Pages D−8, D−11, D−14
E The Lambda Index E−1
E.1 Introduction.......................................................................................................... E−1
E.2 The Lambda Index Defined.................................................................................. E−1
Mathematics for Business, Science, and Technology, Third Edition ix
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E.3 Spreadsheet Construction .....................................................................................E−2
Excel Computations: Page E−2
F The Black−Scholes Stock Options Model F−1
F.1 Stock Options ....................................................................................................... F−1
F.2 The Black-Scholes Model Equations .................................................................... F−2
F.3 Spreadsheet for the Black−Scholes Model............................................................ F−3
Excel Computations: Page F−3
G Forecasting Bankruptcy G−1
G.1 The Financial Ratios..............................................................................................G−1
G.2 Interpretation of The Altman Z−Score .................................................................G−2
G.3 Spreadsheet for The Altman Z−Score...................................................................G−2
Excel Computations: Page G−2
References R−1
Index IN−1