Pocket Book of
Integrals and
Mathematical Formulas
4th Edition
Pocket Book of
Integrals and
Mathematical Formulas
4th Edition
Ronald J,Tallarida
Temple University
Philadelphia,Pennsylvania,U.S.A.
Pocket Book of
Integrals and
Mathematical Formulas
4th Edition
Ronald J,Tallarida
Temple University
Philadelphia,Pennsylvania,U.S.A.
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW,Suite 300
Boca Raton,FL 33487?2742
2008 by Taylor & Francis Group,LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group,an Informa business
No claim to original U.S,Government works
Printed in the United States of America on acid?free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number?13,978?1?4200?6304?2 (Softcover)
This book contains information obtained from authentic and highly regarded sources Reasonable
efforts have been made to publish reliable data and information,but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use,The Authors
and Publishers have attempted to trace the copyright holders of all material reproduced in this publi?
cation and apologize to copyright holders if permission to publish in this form has not been obtained,
If any copyright material has not been acknowledged please write and let us know so we may rectify
in any future reprint
Except as permitted under U.S,Copyright Law,no part of this book may be reprinted,reproduced,
transmitted,or utilized in any form by any electronic,mechanical,or other means,now known or
hereafter invented,including photocopying,microfilming,and recording,or in any information
storage or retrieval system,without written permission from the publishers.
For permission to photocopy or use material electronically from this work,please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center,Inc,(CCC)
222 Rosewood Drive,Danvers,MA 01923,978?750?8400,CCC is a not?for?profit organization that
provides licenses and registration for a variety of users,For organizations that have been granted a
photocopy license by the CCC,a separate system of payment has been arranged.
Trademark Notice,Product or corporate names may be trademarks or registered trademarks,and
are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging?in?Publication Data
Tallarida,Ronald J.
Pocket book of integrals and mathematical formulas / author,Ronald
J,Tallarida, 4th ed.
p,cm.
Includes bibliographical references and index.
ISBN 978?1?4200?6304?2 (alk,paper)
1,IntegralsTables,2,MathematicsTables,I,Title,
QA310.T35 2008
510.2’12dc22
2007045335
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Preface to the Fourth Edition
As in the previous works,this new edition preserves the
content,size,and convenience of this portable reference
source for students and workers who use mathematics,
while introducing much new material,New in this fourth
edition is an expanded chapter on series that now includes
many fascinating properties of the natural numbers that
follow from number theory,a field that has attracted
much new interest since the recent proof of Fermat’s last
theorem,While the proofs of many of these theorems
are deep,and in some cases still lacking,all the number
theory topics included here are easy to describe and form
a bridge between arithmetic and higher mathematics,The
fourth edition also includes new applications such as the
geostationary satellite orbit,drug kinetics (as an applica-
tion of differential equations),and an expanded statistics
section that now discusses the normal approximation of
the binomial distribution as well as a treatment of non-
linear regression,The widespread use of computers now
makes the latter topic amenable to all students,and thus
all users of the Pocket Book of Integrals can benefit from
the concise summary of this topic,The chapter on financial
mathematics,introduced in the third edition,has proved
successful and is retained without change in this edition,
whereas the Table of Integrals has been reformatted for
easier usage,This change in format also allowed the inclu-
sion of all the new topics without the necessity of increas-
ing the physical size of the book,thereby keeping its wide
appeal as a true,handy pocket book that students and pro-
fessionals will find useful in their mathematical pursuits.
R,J,T.
Philadelphia
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW,Suite 300
Boca Raton,FL 33487?2742
2008 by Taylor & Francis Group,LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group,an Informa business
No claim to original U.S,Government works
Printed in the United States of America on acid?free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number?13,978?1?4200?6304?2 (Softcover)
This book contains information obtained from authentic and highly regarded sources Reasonable
efforts have been made to publish reliable data and information,but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use,The Authors
and Publishers have attempted to trace the copyright holders of all material reproduced in this publi?
cation and apologize to copyright holders if permission to publish in this form has not been obtained,
If any copyright material has not been acknowledged please write and let us know so we may rectify
in any future reprint
Except as permitted under U.S,Copyright Law,no part of this book may be reprinted,reproduced,
transmitted,or utilized in any form by any electronic,mechanical,or other means,now known or
hereafter invented,including photocopying,microfilming,and recording,or in any information
storage or retrieval system,without written permission from the publishers.
For permission to photocopy or use material electronically from this work,please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center,Inc,(CCC)
222 Rosewood Drive,Danvers,MA 01923,978?750?8400,CCC is a not?for?profit organization that
provides licenses and registration for a variety of users,For organizations that have been granted a
photocopy license by the CCC,a separate system of payment has been arranged.
Trademark Notice,Product or corporate names may be trademarks or registered trademarks,and
are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging?in?Publication Data
Tallarida,Ronald J.
Pocket book of integrals and mathematical formulas / author,Ronald
J,Tallarida, 4th ed.
p,cm.
Includes bibliographical references and index.
ISBN 978?1?4200?6304?2 (alk,paper)
1,IntegralsTables,2,MathematicsTables,I,Title,
QA310.T35 2008
510.2’12dc22
2007045335
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Preface to the Third Edition
This new edition has been enlarged to contain all the mate-
rial in the second edition,an expanded chapter on statistics
that now includes sample size estimations for means and
proportions,and a totally new chapter on financial math-
ematics,In adding this new chapter we have also included a
number of tables that aid in performing the calculations on
annuities,true interest,amortization schedules,compound
interest,systematic withdrawals from interest accounts,etc,
The treatment and style of this material reflect the rest of the
book,i.e.,clear explanations of concepts,relevant formulas,
and worked examples,The new financial material includes
analyses not readily found in other sources,such as the effect
of lump sum payments on amortization schedules and a novel
“in-out formula” that calculates current regular deposits to
savings in order to allow the start of systematic withdrawals
of a specified amount at a later date,While many engineers,
mathematicians,and scientists have found much use for this
handy pocket book,this new edition extends its usage to
them and to the many business persons and individuals who
make financial calculations.
R,J,T.
Philadelphia
Preface to the Second Edition
This second edition has been enlarged by the addition of sev-
eral new topics while preserving its convenient pocket size,
New in this edition are the following topics,z-transforms,
orthogonal polynomials,Bessel functions,probability and
Bayes’ rule,a summary of the most common probability
distributions (binomial,Poisson,normal,t,Chi square,and
F),the error function,and several topics in multivariable
calculus that include surface area and volume,the ideal gas
laws,and a table of centroids of common plane shapes,A list
of physical constants has also been added to this edition.
I am grateful for many valuable suggestions from users of
the first edition,especially Lt,Col,W,E,Skeith and his
colleagues at the U.S,Air Force Academy.
R,J,T.
Philadelphia,1992
Preface to the First Edition
The material of this book has been compiled so that it may
serve the needs of students and teachers as well as profes-
sional workers who use mathematics,The contents and size
make it especially convenient and portable,The widespread
availability and low price of scientific calculators have
greatly reduced the need for many numerical tables (e.g.,
logarithms,trigonometric functions,powers,etc.) that make
most handbooks bulky,However,most calculators do not
give integrals,derivatives,series,and other mathematical
formulas and figures that are often needed,Accordingly,this
book contains that information in addition to a comprehen-
sive table of integrals,A section on statistics and the accom-
panying tables,also not readily provided by calculators,have
also been included.
The size of the book is comparable to that of many calcula-
tors,and it is really very much a companion to the calcula-
tor and the computer as a source of information for writing
one’s own programs,To facilitate such use,the author and
the publisher have worked together to make the format
attractive and clear,Yet,an important requirement in a book
of this kind is accuracy,Toward that end we have checked
each item against at least two independent sources.
Students and professionals alike will find this book a valu-
able supplement to standard textbooks,a source for review,
and a handy reference for many years.
Ronald J,Tallarida
Philadelphia
About the Author
Ronald J,Tallarida holds B.S,and M.S,degrees in physics/
mathematics and a Ph.D,in pharmacology,His primary
appointment is professor of pharmacology at Temple Univer-
sity School of Medicine,Philadelphia,For over 30 years he
also served as an adjunct professor of biomedical engineer-
ing at Drexel University in Philadelphia where he received
the Lindback Award for Distinguished Teaching of Math-
ematics,As an author and researcher,he has published over
250 works that include seven books,has been the recipient of
research grants from NIH,and has served as a consultant to
both industry and government agencies,His main research
interests are in the areas of mathematical modeling of bio-
logical systems,feedback control,and the action of drugs
and drug combinations.
GReek LeTTeRs
α Α Alpha
β Β Beta
γ Γ Gamma
δ Δ Delta
ε Ε Epsilon
ζ Ζ Zeta
η Η Eta
θ Θ Theta
ι Ι Iota
κ Κ Kappa
λ Λ Lambda
μ Μ Mu
ν Ν Nu
ξ Ξ Xi
ο Ο Omicron
π Π Pi
ρ Ρ Rho
σ Σ Sigma
τ Τ Tau
υ? Upsilon
Φ Phi
χ X Chi
ψ Ψ Psi
ω Ω Omega
The NumbeRs π aNd e
π = 3.14159 26535 89793
e = 2.71828 18284 59045
log
10
e = 0.43429 44819 03252
log
e
10 = 2.30258 50929 94046
PRime NumbeRs
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
… … …
important Numbers in science (Physical Constants)
Avogadro constant (N
A
) 6.02 × 10
26
kmole
–1
Boltzmann constant (k) 1.38 × 10
–23
J?°K
–1
Electron charge (e) 1.602 × 10
–19
C
Electron,charge/mass
(e/m
e
)
1.760 × 10
11
C?kg
–1
Electron rest mass (m
e
) 9.11 × 10
–31
kg (0.511 MeV)
Faraday constant (F) 9.65 × 10
4
C?mole
–1
Gas constant (R) 8.31 × 10
3
J?°K
–1
kmole
–1
Gas (ideal) normal volume
(V
o
)
22.4 m
3
kmole
–1
Gravitational constant (G) 6.67 × 10
–11
N?m
2
kg
–2
Hydrogen atom
(rest mass) (m
H
)
1.673 × 10
–27
kg (938.8 MeV)
Neutron (rest mass) (m
n
) 1.675 × 10
–27
kg (939.6 MeV)
Planck constant (h) 6.63 × 10
–34
J?s
Proton (rest mass) (m
p
) 1.673 × 10
–27
kg (938.3 MeV)
Speed of light (c) 3.00 × 10
8
m?s
–1
Contents
1 elementary algebra and Geometry
1,Fundamental Properties
(Real Numbers) 1
2,Exponents 2
3,Fractional Exponents 2
4,Irrational Exponents 2
5,Logarithms 2
6,Factorials 3
7,Binomial Theorem 3
8,Factors and Expansion 4
9,Progression 4
10,Complex Numbers 5
11,Polar Form 6
12,Permutations 7
13,Combinations 7
14,Algebraic Equations 8
15,Geometry 9
16,Pythagorean Theorem 9
2 determinants,matrices,and
Linear systems of equations
1,Determinants 15
2,Evaluation by Cofactors 16
3,Properties of Determinants 17
4,Matrices 18
5,Operations 18
6,Properties 19
7,Transpose 20
8,Identity Matrix 20
9,Adjoint 21
10,Inverse Matrix 21
11,Systems of Linear Equations 23
12,Matrix Solution 24
3 Trigonometry
1,Triangles 25
2,Trigonometric Functions of an Angle 26
3,Trigonometric Identities 28
4,Inverse Trigonometric Functions 31
4 analytic Geometry
1,Rectangular Coordinates 32
2,Distance between Two Points,Slope 33
3,Equations of Straight Lines 34
4,Distance from a Point to a Line 37
5,Circle 37
6,Parabola 37
7,Ellipse 39
8,Hyperbola (e > 1) 43
9,Change of Axes 44
10,General Equation of Degree 2 47
11,Polar Coordinates 47
12,Curves and Equations 50
13,Exponential Function (Half-Life) 56
5 series,Number Facts,and Theory
1,Bernoulli and Euler Numbers 57
2,Series of Functions 58
3,Error Function 63
4,Fermat’s Little Theorem 64
5,Fermat’s Last Theorem 64
6,Beatty’s Theorem 64
7,An Interesting Prime 66
8,Goldbach Conjecture 66
9,Twin Primes 67
10,Collatz Conjecture 67
6 differential Calculus
1,Notation 68
2,Slope of a Curve 68
3,Angle of Intersection of Two Curves 69
4,Radius of Curvature 69
5,Relative Maxima and Minima 70
6,Points of Inflection of a Curve 70
7,Taylor’s Formula 71
8,Indeterminant Forms 72
9,Numerical Methods 73
10,Functions of Two Variables 75
11,Partial Derivatives 76
7 integral Calculus
1,Indefinite Integral 77
2,Definite Integral 77
3,Properties 78
4,Common Applications of the
Definite Integral 78
5,Cylindrical and Spherical Coordinates 80
6,Double Integration 82
7,Surface Area and Volume by
Double Integration 83
8,Centroid 83
8 Vector analysis
1,Vectors 86
2,Vector Differentiation 88
3,Divergence Theorem (Gauss) 89
4,Stokes’ Theorem 89
5,Planar Motion in Polar Coordinates 90
6,Geostationary Satellite Orbit 90
9 special Functions
1,Hyperbolic Functions 92
2,Gamma Function (Generalized
Factorial Function) 93
3,Laplace Transforms 94
4,Z-Transform 97
5,Fourier Series 100
6,Functions with Period Other Than 2π 101
7,Bessel Functions 103
8,Legendre Polynomials 105
9,Laguerre Polynomials 107
10,Hermite Polynomials 108
11,Orthogonality 108
10 differential equations
1,First-Order,First-Degree Equations 110
2,Second-Order Linear Equations
(with Constant Coefficients) 112
3,Runge Kutta Method (of Order 4) 114
11 statistics
1,Arithmetic Mean 116
2,Median 116
3,Mode 116
4,Geometric Mean 116
5,Harmonic Mean 117
6,Variance 117
7,Standard Deviation 117
8,Coefficient of Variation 118
9,Probability 118
10,Binomial Distribution 120
11,Mean of Binomially Distributed
Variable 121
12,Normal Distribution 121
13,Poisson Distribution 122
14,Empirical Distributions 123
15,Estimation 123
16,Hypotheses Testing 124
17,t-Distribution 125
18,Hypothesis Testing with t- and
Normal Distributions 126
19,Chi-Square Distribution 129
20,Least Squares Regression 131
21,Nonlinear Regression Analysis 134
22,The F-Distribution (Analysis of
Variance) 138
23,Summary of Probability
Distributions 139
24,Sample Size Determinations 142
12 Financial mathematics
1,Simple Interest 146
2,True Interest Formula
(Loan Payments) 147
3,Loan Payment Schedules 148
4,Loan Balance Calculation 149
5,Accelerated Loan Payment 150
6,Lump Sum Payment 152
7,Compound Interest 153
8,Time to Double (Your Money) 155
9,Present Value of a Single
Future Payment 155
10,Regular Saving to Accumulate a
Specified Amount 156
11,Monthly Payments to Achieve a
Specified Amount 158
12,Periodic Withdrawals from an
Interest-Bearing Account 158
13,Periodic Withdrawals That
Maintain the Principal 161
14,Time to Deplete an Interest-
Bearing Account with Periodic
Withdrawals 162
15,Amounts to Withdraw for a
Specified Number of Withdrawals I,
Payments at the End of Each Year 163
16,Amounts to Withdraw for a Specified
Number of Withdrawals II,Payments
at the Beginning of Each Year 165
17,Present Value of Regular Payments 167
18,Annuities 168
19,The In-Out Formula 170
20,Stocks and Stock Quotations 172
21,Bonds 173
22,Tax-Free Yield 175
23,Stock Options (Puts and Calls) 176
24,Market Averages 177
25,Mutual Fund Quotations 177
26,Dollar Cost Averaging 179
27,Moving Average 180
Table of derivatives 182
Table of integrals,
indefinite and definite integrals 187
appendix 243
index 263
1
1
Elementary Algebra
and Geometry
1,Fundamental Properties (Real Numbers)
a b b a+ = + Commutative Law for
Addition
( ) ( )a b c a b c+ + = + + Associative Law for
Addition
a a+ = +0 0 Identity Law for
Addition
a a+? =? + =( ) ( ) 0a a Inverse Law for
Addition
a bc ab c( ) ( )= Associative Law for
Multiplication
a
a a
a a
1 1
1 0
=
= ≠,Inverse Law for
Multiplication
( )( ) ( )( )a a a1 1= = Identity Law for
Multiplication
ab ba= Commutative Law for
Multiplication
a b c ab ac( )+ = + Distributive Law
Division by zero is not defined.
2
2,Exponents
For integers m and n,
a a a
a a a
a a
ab a b
a b
n m n m
n m n m
n m n m
m m m
=
=
=
=
+
/
/
( )
( )
( )
m m m
a b= /
3,Fractional Exponents
a a
p q q p/ /
= ( )
1
where a
q1/
is the positive qth root of a if a > 0 and the
negative qth root of a if a is negative and q is odd,
Accordingly,the five rules of exponents given above
(for integers) are also valid if m and n are fractions,
provided a and b are positive.
4,Irrational Exponents
If an exponent is irrational,e.g.,2,the quantity,such as
a
2
,is the limit of the sequence a a a
1 4 1 41 1 414.,,
,,,.,,,
Operations with Zero
0 0 1
0m
a= =;
5,Logarithms
If x,y,and b are positive and b ≠1,
3
log log log
log log log
lo
b b b
b b b
xy x y
x y x y
( )
( / )
= +
=?
g log
log log
log
log
b
p
b
b b
b
b
x p x
x x
b
N
=
=?
=
=
( / )1
1
1 0 oote b x
x
:
log
.
b
=
Change of Base (a ≠ 1)
log log log
b a b
x x a=
6,Factorials
The factorial of a positive integer n is the product of
all the positive integers less than or equal to the inte-
ger n and is denoted n! Thus,
n n!,=…?1 2 3
Factorial 0 is defined 0! = 1.
Stirling’s Approximation
lim !
n
n
n e n n
→∞
=( / ) 2π
(See also 9.2.)
7,Binomial Theorem
For positive integer n,
4
( )
( )
!
( ( )
x y x nx y
n n
x y
n n n
n n n n
+ = + +
+
1 2 2
1
2
1 2)
3
3 3
1
!
.
x y
nxy y
n
n n
+…
+ +
8,Factors and Expansion
( )
)
( )
a b a ab b
a b a ab b
a b a
+ = + +
=? +
+ = +
2 2 2
2 2 2
3 3
2
2
3
(
a b ab b
a b a a b ab b
a b
2 2 3
3 3 2 2 3
2 2
3
3 3
+ +
=? +?
=
(
(
)
) (a b a b
a b a b a ab b
a b a
+
=? + +
+ =
) )
( ) ) ( )
(
(
(
) (
3 3 2 2
3 3
+? +b a ab b) ( )
2 2
9,Progression
An arithmetic progression is a sequence in which the
difference between any term and the preceding term
is a constant (d):
a a d a d a n d,,,...,.+ + +?2 1( )
If the last term is denoted l a n d[ ( )= +?1 ],then the
sum is
s
n
a l= +
2
( ).
5
A geometric progression is a sequence in which the
ratio of any term to the preceding terms is a constant
r,Thus,for n terms,
a ar ar ar
n
,,,...,
2 1?
The sum is
S
a ar
r
n
=
1
10,Complex Numbers
A complex number is an ordered pair of real num-
bers (a,b).
equality,(,) (,)a b c d= if and only if a = c and b = d
addition,( ) (,),a,b c d a c b d+ = + +( )
multiplication,( )(,) (a,b c d ac bd ad bc=? +,)
The first element of (a,b) is called the real part; the
second,the imaginary part,An alternate notation for
(a,b) is a + bi,where i
2
1 0=?( ),,and i (0,1) or 0 + 1i
is written for this complex number as a convenience,
With this understanding,i behaves as a number,
i.e.,( )( )2 3 4 8 12 2 3 11 10
2
+ =? +? =?i i i i i i,The
conjugate of a + bi is a bi?,and the product of a
complex number and its conjugate is a b
2 2
+,Thus,
quotients are computed by multiplying numerator
and denominator by the conjugate of the denomina-
tor,as illustrated below:
2 3
4 2
4 2 2 3
4 2 4 2
14 8
20
7+
+
=
+
+
=
+
=
i
i
i i
i i
i( ) ( )
( ) ( )
+4
10
i
6
11,Polar Form
The complex number x + iy may be represented by a
plane vector with components x and y:
x iy r i+ = +(cos sinθ θ)
(see Figure 1.1),Then,given two complex numbers
z r i
1 1 1 1
= +(cos )θ θsin and z r i
2 2
= +(cos sin ),
2 2
θ θ
the product and quotient are:
Product,z z r r i
1 2 1 2
= + + +[ ]cos( ) sin( )
1 2 1 2
θ θ θ θ
Quotient,z z r r
i
1 2 1 2
/ / [
]
=?
+?
( ) cos( )
sin( )
1 2
1 2
θ θ
θ θ
Powers,z r i
r n i n
n n
n
= +
= +
[ ( )]
[ ]
cos sin
cos sin
θ θ
θ θ
0
x
y
r
P (x,y)
θ
FiGuRe 1.1 Polar form of complex number.
7
Roots,z r i
r
k
n
n
1
1
36
/
/
(cos sin )
cos
.
= +
=
+
[ ]
1/n
θ θ
θ 0 360
n
i
k
n
n
+
+?
=?
sin
.θ
,
k 0,1,2,.,,,1
12,Permutations
A permutation is an ordered arrangement (sequence)
of all or part of a set of objects,The number of per-
mutations of n objects taken r at a time is
p n r n n n n r
n
n r
( ) ( )( ) )
( )
,.,,(
!
!
= +
=
1 2 1
A permutation of positive integers is even or odd if
the total number of inversions is an even integer or
an odd integer,respectively,Inversions are counted
relative to each integer j in the permutation by count-
ing the number of integers that follow j and are less
than j,These are summed to give the total number
of inversions,For example,the permutation 4132 has
four inversions,three relative to 4 and one relative
to 3,This permutation is therefore even.
13,Combinations
A combination is a selection of one or more objects
from among a set of objects regardless of order,The
number of combinations of n different objects taken
r at a time is
C n r
P n r
r
n
r n r
(,
,
!
!
! ( !
)
( )
)
= =
8
14,Algebraic Equations
Quadratic
If ax bx c
2
0+ + =,and a ≠ 0,then roots are
x
b b ac
a
=
±?
2
4
2
Cubic
To solve x bx cx d
3 2
0+ + + =,let x y b=? / 3,Then
the reduced cubic is obtained:
y py q
3
0+ + =
where p = c – (1/3)b
2
and q = d – (1/3)bc + (2/27)b
3
,
Solutions of the original cubic are then in terms of
the reduced cubic roots y
1
,y
2
,y
3
:
x y b x y b
x y b
1 1 2 2
3 3
1 3 1 3
1 3
=? =?
=?
( / ) ( / )
( / )
The three roots of the reduced cubic are
y A B
y W A W B
y W A
1
1 3 1 3
2
1 3 2 1 3
3
2
= +
= +
=
( ) ( )
( ) ( )
( )
/ /
/ /
1 3 1 3/ /
( )+W B
where
A q p q=? + +
1
2
1 27
1
4
3 2
( / ),
9
B q p q
W
i
W
i
= +
=
+
=
1
2
1 27
1
4
1 3
2
1 3
2
3 2
2
( / ),
,.
When ( )1 27 1 4
3 2
/ ( / )p p+ is negative,A is complex;
in this case,A should be expressed in trigonometric
form,A = r (cosθ + i sinθ),where θ is a first or second
quadrant angle,as q is negative or positive,The three
roots of the reduced cubic are
y r
y r
1
1 3
2
1 3
2 3
2
3
120
=
= + °
( ) ( / )
( )
/
/
cos
cos
θ
θ
= + °
y r2 240
1 3
( )
/
cos
3
θ
15,Geometry
Figures 1.2 to 1.12 are a collection of common geo-
metric figures,Area (A),volume (V),and other mea-
surable features are indicated.
16,Pythagorean Theorem
For any right triangle with perpendicular sides a and
b,the hypotenuse c is related by the formula
c
2
= a
2
+ b
2
This famous result is central to many geometric rela-
tions,e.g.,see Section 4.2.
10
h
b
FiGuRe 1.4 Triangle,
A bh=
1
2
.
b
h
FiGuRe 1.3 Parallelogram,A = bh.
b
h
FiGuRe 1.2 Rectangle,A = bh.
11
b
a
h
FiGuRe 1.5 Trapezoid,
A a b h= +
1
2
( ),
S
R
θ
FiGuRe 1.6 Circle,A π R=
2; circumference =
2πR; arc length S R= θ (θ in radians).
θ
R
FiGuRe 1.7 Sector of circle,
A R
sector;=
1
2
2
θ
A
segment
= A R
seg ent
( sin ).=?
1
2
2
θ θ
12
h
R
FiGuRe 1.9 Right circular cylinder,V R h=π
2;
lateral surface area = 2π Rh.
R
b
θ
FiGuRe 1.8 Regular polygon of n sides,
A
n
b
n
=
4
2
ctn ;
π
R
n
=
b
2
csc,
π
13
R
l
h
FiGuRe 1.11 Right circular cone,
V R h=
1
3
2
π ;
lateral surface area = = +π πRl R R h
2 2
.
h
A
FiGuRe 1.10 Cylinder (or prism) with parallel
bases,V = Ah.
14
R
FiGuRe 1.12 Sphere,
V R=
4
3
3
π ;
surface area =
4π R
2
.
15
2
Determinants,Matrices,
and Linear Systems
of Equations
1,Determinants
definition,The square array (matrix) A,with n rows
and n columns,has associated with it the determinant
det A
a a a
a a a
a a a
n
n
n n nn
=
…
…
… … … …
…
11 12 1
21 22 2
1 2
,
a number equal to
∑
± …( )a a a a
i j k nl1 2 3
where i,j,k,…,l is a permutation of the n integers
1,2,3,…,n in some order,The sign is plus if the
permutation is even and is minus if the permutation
is odd (see 1.12),The 2 × 2 determinant
a a
a a
11 12
21 22
has the value a a a a
11 22 12 21
since the permutation
(1,2) is even and (2,1) is odd,For 3 × 3 determinants,
permutations are as follows:
16
1,2,3 even
1,3,2 odd
2,1,3 odd
2,3,1 even
3,1,2 even
33,2,1 odd
Thus,
a a a
a a a
a a a
a a a
a
11 12 13
21 22 23
31 32 33
11 22 33
1
=
+
.,
1 23 32
12 21 33
12 23 31
13 21
.,
.,
.,
.,
a a
a a a
a a a
a a
+
+ a
a a a
32
13 22 31
.,
A determinant of order n is seen to be the sum of n!
signed products.
2,Evaluation by Cofactors
Each element a
ij
has a determinant of order (n – 1) called
a minor (M
ij
) obtained by suppressing all elements in
row i and column j,For example,the minor of element
a
22
in the 3 × 3 determinant above is
a a
a a
11 13
31 33
17
The cofactor of element a
ij
,denoted A
ij
,is defined as
±M
ij
,where the sign is determined from i and j:
A M
ij
i j
ij
=?
+
( )1,
The value of the n × n determinant equals the sum of
products of elements of any row (or column) and their
respective cofactors,Thus,for the 3 × 3 determinant,
det (first row)A a A a A a A= + +
11 11 12 12 13 13
or
= + +a A a A a A
11 11 21 21 31 31
(first column)
etc.
3,Properties of Determinants
a,If the corresponding columns and rows of A are
interchanged,det A is unchanged.
b,If any two rows (or columns) are interchanged,the
sign of det A changes.
c,If any two rows (or columns) are identical,det A = 0.
d,If A is triangular (all elements above the main
diagonal equal to zero),A = a
11
· a
22
· … · a
nn
:
a
a a
a a a a
n n n nn
11
21 22
1 2 3
0 0 0
0 0
…
…
… … … … …
…
18
e,If to each element of a row or column there is
added C times the corresponding element in
another row (or column),the value of the determi-
nant is unchanged.
4,Matrices
definition,A matrix is a rectangular array of numbers
and is represented by a symbol A or [a
ij
]:
A
a a a
a a a
a a a
n
n
m m m n
=
…
…
… … … …
…
11 12 1
21 22 2
1 2
= [ ]a
ij
The numbers a
ij
are termed elements of the matrix;
subscripts i and j identify the element,as the number
is row i and column j,The order of the matrix is m × n
(“m by n”),When m = n,the matrix is square and is
said to be of order n,For a square matrix of order n the
elements a
11
,a
22
,…,a
nn
constitute the main diagonal.
5,Operations
addition,Matrices A and B of the same order may
be added by adding corresponding elements,
i.e.,A + B = [(a
ij
+ b
ij
)].
scalar multiplication,If A = [a
ij
] and c is a constant
(scalar),then cA = [ca
ij
],that is,every element of
A is multiplied by c,In particular,(–1)A = –A =
[–a
ij
] and A + (–A) = 0,a matrix with all elements
equal to zero.
19
multiplication of matrices,Matrices A and B
may be multiplied only when they are conform-
able,which means that the number of columns
of A equals the number of rows of B,Thus,
if A is m × k and B is k × n,then the product
C = AB exists as an m × n matrix with elements c
ij
equal to the sum of products of elements in row i of
A and corresponding elements of column j of B:
c a b
ij il lj
l
k
=
=
∑
1
For example,if
a a a
a a a
a a
k
k
m m k
11 12 1
21 22 2
1
…
…
… … … …
… …
…
…
… … … …
…
b b b
b b b
b b b
n
n
k k k n
11 12 1
21 22 2
1 2
=
…
…
… … … …
c c c
c c c
c
n
n
m
11 12 1
21 22 2
1
c c
m m n2
…
then element c
21
is the sum of products a
21
b
11
+
a
22
b
21
+ … + a
2k
b
k1
.
6,Properties
A B B A
A B C A B C
+ = +
+ + = + +( ) ( )
20
( )
( )
( ) ( )
c c A c A c A
c A B cA cB
c c A c c A
1 2 1 2
1 2 1 2
+ = +
+ = +
=
(AB C A BC
A B C AC BC
AB BA
) ) ( )
( ) ( )
ingeneral
(
(
=
+ = +
≠ )
7,Transpose
If A is an n × m matrix,the matrix of order m × n
obtained by interchanging the rows and columns of A is
called the transpose and is denoted A
T
,The following
are properties of A,B,and their respective transposes:
( )
)
A A
A B A B
cA cA
AB B A
T T
T T T
T T
T T T
=
+ = +
=
=
(
( )
( )
A symmetric matrix is a square matrix A with the
property A = A
T
.
8,Identity Matrix
A square matrix in which each element of the main
diagonal is the same constant a and all other elements
zero is called a scalar matrix.
21
a
a
a
a
0 0 0
0 0 0
0 0 0
0
0 0 0
…
…
…
… … … …
…
When a scalar matrix multiplies a conformable second
matrix A,the product is aA,that is,the same as multi-
plying A by a scalar a,A scalar matrix with diagonal
elements 1 is called the identity,or unit matrix,and
is denoted I,Thus,for any nth-order matrix A,the
identity matrix of order n has the property
AI IA A= =
9,Adjoint
If A is an n-order square matrix and A
ij
the cofactor of
element a
ij
,the transpose of [A
ij
] is called the adjoint
of A:
adj A A
ij
T
= [ ]
10,Inverse Matrix
Given a square matrix A of order n,if there exists a
matrix B such that AB = BA = I,then B is called the
inverse of A,The inverse is denoted A
–1
,A neces-
sary and sufficient condition that the square matrix A
have an inverse is det A ≠ 0,Such a matrix is called
nonsingular; its inverse is unique and is given by
22
A
adj A
A
=
1
det
Thus,to form the inverse of the nonsingular matrix,
A,form the adjoint of A and divide each element of
the adjoint by det A,For example,
1 0 2
3 1 1
4 5 6
has matrix of cofactors
=
11 14 19
10 2 5
2 5 1
11 1
,
adjoint
0 2
14 2 5
19 5 1
27
and determinant,
Therefore,
A
=
1
11
27
10
27
2
27
14
27
2
27
5
27
19
27
5
27
1
27
.
23
11,Systems of Linear Equations
Given the system
a x a x a x b
a x a x a x b
n n
n n
11 1 12 2 1 1
21 1 22 2 2
+ +
…
+ =
+ +
…
+ =
2
1 1 2 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a x a x a x b
n n n n n n
+ +
…
+ =
a unique solution exists if det A ≠ 0,where A is the
n × n matrix of coefficients [a
ij
].
Solution by Determinants (Cramer’s Rule)
x
b a a
b a
b a a
A
n
n n nn
1
1 12 1
2 22
2
=
…
÷
.
.
.
.
.
.
.
.
.
det
x
a b a a
a b
a b a a
n
n n n nn
2
11 1 13 1
21 2
1 3
=
…
… …
÷
.
.
.
.
.
.
det A
x
A
A
k
k
det
det
,
.
.
.
=
where A
k
is the matrix obtained from A by replacing
the kth column of A by the column of b’s.
24
12,Matrix Solution
The linear system may be written in matrix form
AX = B where A is the matrix of coefficients [a
ij
] and
X and B are
X
x
x
x
B
b
b
b
n n
=
=
1
2
1
2
.
.
.
.
.
.
If a unique solution exists,det A ≠ 0; hence,A
–1
exists and
X A B=
1
.
25
3
Trigonometry
1,Triangles
In any triangle (in a plane) with sides a,b,and c and
corresponding opposite angles A,B,C,
a
A
b
B
c
C
a b c c
sin sin sin
,(Law of Sines)= =
= +?
2 2 2
2 b A
a b
a b
A B
cos,(Law of Cosines)
tan
tan
+
=
+
1
2
( )
1
2
( )A B?
,(Law of Tangents)
sin,where s a b c,
cos
1
2
1
2
1
2
A
s b s c
bc
=
= + +
( )( )
( )
A
s s a
bc
A
s b s c
s s a
Area
=
=
=
( )
( )( )
( )
.
tan
1
2
.
1
2
bbc A
s s a s b s c
sin
.=( )( )( )
If the vertices have coordinates (x
1
,y
1
),(x
2
,y
2
),(x
3
,y
3
),
the area is the absolute value of the expression
26
1
2
1
1
1
1 1
2 2
3 3
x y
x y
x y
2,Trigonometric Functions of an Angle
With reference to Figure 3.1,P(x,y) is a point in either
one of the four quadrants and A is an angle whose
initial side is coincident with the positive x-axis and
whose terminal side contains the point P(x,y),The
distance from the origin P(x,y) is denoted by r and
y
0
A
r
(II) (I)
(III) (IV)
x
P(x,y)
FiGuRe 3.1 The trigonometric point,Angle A
is taken to be positive when the rotation is counter-
clockwise and negative when the rotation is clock-
wise,The plane is divided into quadrants as shown.
27
is positive,The trigonometric functions of the angle
A are defined as
sin sine
cos cosine
tan tangent
A A y r
A A x r
A A
= =
= =
=
/
/
=
= =
= =
y x
A A x y
A A r x
/
/
/
ctn cotangent
sec secant
csc A A r y= =cosecant /
Angles are measured in degrees or radians,180° = π
radians; 1 radian = 180°/π degrees.
The trigonometric functions of 0°,30°,45°,and integer
multiples of these are directly computed:
0 30 45 60 90 120 135 150 180
2
2
3
2
1
3
2
° ° ° ° ° ° ° ° °
sin 0
1
2
2
2
1
2
0
3
2
2
2
1
2
0
1
2
2
2
3
2
1
0
3
3
1 3 3 1
3
cos 1
tan
∞
3
0
1
3
3
0
3
3
1 3
1
2 3
3
2 2 2 2
2 3
3
1
ctn 3
sec
csc
∞ ∞
∞
∞ ∞2 2
2 3
3
1
2 3
3
2 2
28
3,Trigonometric Identities
sin
csc
cos
sec
tan
ctn
sin
cos
csc
A
A
A
A
A
A
A
A
A
=
=
= =
1
1
1
=
=
= =
1
1
1
sin
sec
cos
ctn
tan
cos
sin
A
A
A
A
A
A
A
sin cos
tan sec
ctn csc
2
2
2 2
2
2
1
1
1
A A
A A
A A
+ =
+ =
+ =
sin( ) sin cos cos sin
cos( ) cos cos
A B A B A B
A B A
± = ±
± = B A B?sin sin
tan
tan tan
tan tan
sin sin cos
( )A B
A B
A B
A A A
± =
±
=
1
2 2
sin sin sin3 3 4
3
A A A=?
29
sin sin cos sin(
cos2 cos
nA n A A n A
A A
=
=
2 1 2
2
2
( ) )
=?
=?
=?
1 1 2
3 4 3
2
2
sin
cos cos cos
cos cos
3
A
A A A
nA n( 1 2) ( )A A n Acos cos
sin sin sin
1
2
cos
1
2
sin sin
A B A B A B
A B
+ = +?
=
2
2
( ) ( )
cos
1
2
sin
1
2
cos cos cos c
( ) ( )
( )
A B A B
A B A B
+?
+ = +2
1
2
oos
cos cos sin sin
1
2
2
1
2
1
2
( )
( ) ( )
A B
A B A B A B
=? +?
tan tan
sin
sin cos
ctn ctn
sin
A B
A B
A B
A B
A
± =
±
± = ±
±
( )
( B
A B
)
sin sin
sin sin cos cos
cos cos co
A B A B A B
A B
= +
=
1
2
1
2
1
2
( ) ( )
s cos
sin cos sin sin
( ) ( )
( )
A B A B
A B A B
+ +
= + +
1
2
1
2
1
2
( )A B?
30
sin
2
cos
cos
2
cos
A A
A A
= ±
= ±
+
1
2
1
2
tan
2
cos
sin
sin
1 cos
cos
1 cos
s
A A
A
A
A
A
A
=
=
+
=±
+
1 1
in cos
cos cos
sin
2
2
3
1
2
1 2
1
2
1 2
1
4
A A
A A
A
=?
= +
=
( )
( )
(3 3
1
4
3
1
2
3
sin sin
cos cos3 cos
sin
A A
A A A
ix i
= +
=
)
( )
(
cos cosh
tan
e e i x
ix e e x
ix
x x
x x
=
= + =
) sinh
( )
1
2
=
+
=
= +
+
i e e
e e
i x
e e y i y
x x
x x
x iy x
(
(
)
tanh
cos sin )
)(cos sin ) cos sinx i x nx i nx
n
± = ±
31
4,Inverse Trigonometric Functions
The inverse trigonometric functions are multiple val-
ued,and this should be taken into account in the use
of the following formulas:
sin cos
tan ctn
sec
=?
=
=
=
1 1 2
1
2
1
2
1
1
1
x x
x
x
x
x
1
2
1
1
1
1
=
=
x
x
x
csc
sin
1
( )
cos sin
tan ctn
sec
=?
=
=
=
1 1 2
1
2
1
2
1
1
1
x x
x
x
x
x
1
2
1
1 1
1
x
x
π x
=
=
csc
cos
1
( )
tan ctn
sin cos
sec
=
=
+
=
=
1 1
1
2
1
2
1
1
1
1
1
1
x
x
x
x x
+ =
+
=
x
x
x
x
2
2
1
1
csc
tan
1
( )
32
4
Analytic Geometry
1,Rectangular Coordinates
The points in a plane may be placed in one-to-one
correspondence with pairs of real numbers,A
common method is to use perpendicular lines that are
horizontal and vertical and intersect at a point called
the origin,These two lines constitute the coordinate
axes; the horizontal line is the x-axis and the vertical
line is the y-axis,The positive direction of the x-axis
is to the right,whereas the positive direction of the
y-axis is up,If P is a point in the plane,one may draw
lines through it that are perpendicular to the x- and
y-axes (such as the broken lines of Figure 4.1),The
II
y
y
1
x
1
x
0
I
P(x
1
,y
1
)
IV III
FiGuRe 4.1 Rectangular coordinates.
33
lines intersect the x-axis at a point with coordinate x
1
and the y-axis at a point with coordinate y
1
,We call
x
1
the x-coordinate,or abscissa,and y
1
is termed the
y-coordinate,or ordinate,of the point P,Thus,point
P is associated with the pair of real numbers (x
1
,y
1
)
and is denoted P(x
1
,y
1
),The coordinate axes divide
the plane into quadrants I,II,III,and IV.
2,Distance between Two Points,Slope
The distance d between the two points P
1
(x
1
,y
1
) and
P
2
(x
2
,y
2
) is
d x x y y=? +?( ) ( )
2 1
2
2 1
2
In the special case when P
1
and P
2
are both on one of
the coordinate axes,for instance,the x-axis,
d x x x x=? =?( ) | |
2 1
2
2 1
,
or on the y-axis,
d y y y y=? =?( ) | |
2 1
2
2 1
.
The midpoint of the line segment P
1
P
2
is
x x y y
1 2 2
2 2
+ +?
,.
1
The slope of the line segment P
1
P
2
,provided it is not
vertical,is denoted by m and is given by
m
y y
x x
=
2 1
2 1
.
34
The slope is related to the angle of inclination α
(Figure 4.2) by
m = tan α
Two lines (or line segments) with slopes m
1
and m
2
are perpendicular if
m m
1 2
1=? /
and are parallel if
m m
1 2
=,
3,Equations of Straight Lines
A vertical line has an equation of the form
x = c
x
P
1
P
2
y
α
FiGuRe 4.2 The angle of inclination is the small-
est angle measured counterclockwise from the posi-
tive x-axis to the line that contains P
1
P
2
.
35
where (c,0) is its intersection with the x-axis,A line
of slope m through point (x
1
,y
1
) is given by
y y m x x? =?
1 1
( )
Thus,a horizontal line (slope = 0) through point
(x
1
,y
1
) is given by
y y=
1
.
A nonvertical line through the two points P
1
(x
1
,y
1
)
and P
2
(x
2
,y
2
) is given by either
y y
y y
x x
x x? =
1
2 1
2 1
1
( )
or
y y
y y
x x
x x? =
2
2 1
2 1
2
( ).
A line with x-intercept a and y-intercept b is given by
x
a
y
b
a b+ = ≠ ≠1 0 0(,).
The general equation of a line is
Ax By C+ + = 0
The normal form of the straight line equation is
x y pcos sinθ θ+ =
36
where p is the distance along the normal from the
origin and θ is the angle that the normal makes with
the x-axis (Figure 4.3).
The general equation of the line Ax + By + C = 0 may
be written in normal form by dividing by
± +A B
2 2
,
where the plus sign is used when C is negative and the
minus sign is used when C is positive:
Ax By C
A B
+ +
± +
=
2 2
0,
so that
cos,sinθ θ=
± +
=
± +
A
A B
B
A B
2 2 2 2
y
x
0
p
θ
FiGuRe 4.3 Construction for normal form of
straight line equation.
37
and
p
C
A B
=
+
| |
2 2
.
4,Distance from a Point to a Line
The perpendicular distance from a point P(x
1
,y
1
) to
the line Ax + By + C = 0 is given by d:
d
Ax By C
A B
=
+ +
± +
1 1
2 2
.
5,Circle
The general equation of a circle of radius r and center
at P(x
1
,y
1
) is
( ( )x x y y r? +? =
1 1
2 2
),
2
6,Parabola
A parabola is the set of all points (x,y) in the plane that
are equidistant from a given line called the directrix
and a given point called the focus,The parabola is
symmetric about a line that contains the focus and is
perpendicular to the directrix,The line of symmetry
intersects the parabola at its vertex (Figure 4.4),The
eccentricity e = 1.
The distance between the focus and the vertex,or
vertex and directrix,is denoted by p (> 0) and leads
to one of the following equations of a parabola with
vertex at the origin (Figures 4.5 and 4.6):
38
y
x
p
y
x
p
x
=
=?
=
2
2
4
4
(opens upward)
(opens downward)
y
p
x
y
p
2
2
4
4
(opens to right)
(opens to left)=?
For each of the four orientations shown in Figures 4.5
and 4.6,the corresponding parabola with vertex (h,k)
is obtained by replacing x by x – h and y by y – k,
Thus,the parabola in Figure 4.7 has the equation
x
y
0
F
x = h
(h,k)
FiGuRe 4.4 Parabola with vertex at (h,k),F iden-
tifies the focus.
39
x h
y k
p
=?
( )
.
2
4
7,Ellipse
An ellipse is the set of all points in the plane such
that the sum of their distances from two fixed points,
y
x
y = –p
y = p
P(x,y)
F
F
0
0
FiGuRe 4.5 Parabolas with y-axis as the axis of
symmetry and vertex at the origin,Upper,y
x
p
=
2
4;
lower,Y
x
p
=?
2
4
.
40
0
0
F F
y
x
x = –p
x = p
x
FiGuRe 4.6 Parabolas with x-axis as the
axis of symmetry and vertex at the origin,Left,
x
y
p
=
2
4; right,x
y
p
=?
2
4
.
y
y = k
(h,k)
x
0
FiGuRe 4.7 Parabola with vertex at (h,k) and
axis parallel to the x-axis.
41
called foci,is a given constant 2a,The distance
between the foci is denoted 2c; the length of the
major axis is 2a,whereas the length of the minor axis
is 2b (Figure 4.8),and
a b c= +
2 2
.
The eccentricity of an ellipse,e,is <1,An ellipse with
center at point (h,k) and major axis parallel to the
x-axis (Figure 4.9) is given by the equation
( ) ( )x h
a
y k
b
+
=
2
2
2
2
1.
An ellipse with center at (h,k) and major axis parallel
to the y-axis is given by the equation (Figure 4.10)
( )y k
a
x h
b
+
=
2
2
2
2
1
( )
.
y
x
b
aP
c
0
F
1
F
2
FiGuRe 4.8 Ellipse; since point P is equidistant
from foci F
1
and F
2
,the segments F
1
P
and F
2
P = a;
hence,a b c= +
2 2
.
42
x
y
x = h
F
1
F
2
0
y = k
FiGuRe 4.9 Ellipse with major axis parallel to
the x-axis,F
1
and F
2
are the foci,each a distance c
from center (h,k).
0
F
F
y
x
y = k
x = h
FiGuRe 4.10 Ellipse with major axis parallel to the
y-axis,Each focus is a distance c from center (h,k).
43
8,Hyperbola (e > 1)
A hyperbola is the set of all points in the plane such
that the difference of its distances from two fixed
points (foci) is a given positive constant denoted 2a,
The distance between the two foci is 2c,and that
between the two vertices is 2a,The quantity b is
defined by the equation
b c a=?
2 2
and is illustrated in Figure 4.11,which shows the
construction of a hyperbola given by the equation
x
a
y
b
2
2
2
2
1? =,
c
0
c
a
b
F
1
F
2
V
2
V
1
y
x
FiGuRe 4.11 Hyperbola,V
1
,V
2
= vertices; F
1
,F
2
= foci,A circle at center 0 with radius c contains the
vertices and illustrates the relations among a,b,and c,
Asymptotes have slopes b/a and –b/a for the orienta-
tion shown.
44
When the focal axis is parallel to the y-axis,the equa-
tion of the hyperbola with center (h,k) (Figures 4.12
and 4.13) is
( )y k
a
x h
b
=
2
2
2
2
1
( )
.
If the focal axis is parallel to the x-axis and center
(h,k),then
( ) (
2
x h
a
y k
b
+
=
2
2
2
1
)
9,Change of Axes
A change in the position of the coordinate axes will
generally change the coordinates of the points in the
y
x = h
x
y = k
0
a
b
FiGuRe 4.12 Hyperbola with center at (h,k),
( ) ( )x h
a
y k
b
=
2
2
2
2
1; slope of asymptotes,± b a/,
45
plane,The equation of a particular curve will also
generally change.
Translation
When the new axes remain parallel to the original,the
transformation is called a translation (Figure 4.14),
The new axes,denoted x′ and y′,have origin 0′ at
(h,k) with reference to the x- and y-axes.
A point P with coordinates (x,y) with respect to the
original has coordinates (x′,y′) with respect to the
new axes,These are related by
x x h
x y k
= ′ +
= ′ +
a
b
x = h
y = k
y
0
x
FiGuRe 4.13 Hyperbola with center at (h,k),
( ) ( )y k
a
x h
b
=
2
2
2
2
1; slopes of asymptotes,± a b/,
46
For example,the ellipse of Figure 4.10 has the
following simpler equation with respect to axes x′
and y′ with the center at (h,k):
′
+
′
=
y
a
x
b
2
2
2
2
1.
Rotation
When the new axes are drawn through the same ori-
gin,remaining mutually perpendicular,but tilted with
respect to the original,the transformation is one of
rotation,For angle of rotation φ (Figure 4.15),the coor-
dinates (x,y) and (x′,y′) of a point P are related by
x x y= ′? ′cos sinφ φ
y x y= ′ + ′sin cosφ φ
0
y
x
x?
y?
P
0? (h,k)
FiGuRe 4.14 Translation of axes.
47
10,General Equation of Degree 2
Ax Bxy Cy Dx Ey F
2 2
0+ + + + + =
Every equation of the above form defines a conic sec-
tion or one of the limiting forms of a conic,By rotat-
ing the axes through a particular angle φ,the xy term
vanishes,yielding
′ + ′ ′ + ′ ′+ ′ ′+ ′=A x C y D x E y F
2 2
0
with respect to the axes x′ and y′,The required angle
φ (see Figure 4.15) is calculated from
tan,( ).2 90φ φ=
< °
B
A C
11,Polar Coordinates (Figure 4.16)
The fixed point O is the origin or pole,and a line
OA drawn through it is the polar axis,A point P in
the plane is determined from its distance r,measured
from O,and the angle θ between OP and OA,Dis-
tances measured on the terminal line of θ from the
y?
y
x
φ
x?
FiGuRe 4.15 Rotation of axes.
48
pole are positive,whereas those measured in the
opposite direction are negative.
Rectangular coordinates (x,y) and polar coordinates
(r,θ) are related according to
x r y r
r x y y x
= =
= + =
cos,sin
,tan,
θ θ
θ
2 2 2
/
Several well-known polar curves are shown in
Figures 4.17 to 4.21.
The polar equation of a conic section with focus at the
pole and distance 2p from directrix to focus is either
P(r,θ)
+
–
A
y
x
O
θ
FiGuRe 4.16 Polar coordinates.
O
FiGuRe 4.17 Polar curve
r e
a
=
θ
.
49
O
FiGuRe 4.19 Polar curve r a b= +2 cos,θ
O
FiGuRe 4.18 Polar curve r a= cos,2θ
O
FiGuRe 4.20 Polar curve r a= sin,3θ
O
FiGuRe 4.21 Polar curve r a=?( )1 cos,θ
50
r
ep
e
=
2
1 cos
(directrix to left of pole)
θ
or
r
ep
e
=
+
2
1 cos
(directrix to right of pole)
θ
The corresponding equations for the directrix below
or above the pole are as above,except that sin θ
appears instead of cos θ.
12,Curves and Equations
y
a
b
x
a
1
2
0
FiGuRe 4.22 y
ax
x b.
=
+
51
x
y
e
1
0
FiGuRe 4.23 y = log x.
0
y
x
(0,1)
FiGuRe 4.24 y e
x
=,
52
y
x
0
(0,a)
FiGuRe 4.25 y ae
x
=
.
x
y
1
0
FiGuRe 4.26 y x x= log,
x
y
0.4
0
1 2 3 4
FiGuRe 4.27 y xe
x
=
.
53
x
y
0.5
0
x* x** 100
FiGuRe 4.28 y = e
–ax
– e
–bx
,0 < a < b (drawn
for a = 0.02,b = 0.1,and showing maximum and
inflection).
–3π –2π –π π 2π 3π
x
y
1
–1
FiGuRe 4.29 y = sin x.
–2π
–π π
2π 3π
x
y
1
–1
FiGuRe 4.30 y = cos x.
54
x
y
–
3pi
2
3pi
2
–
pi
2
pi
2
FiGuRe 4.31 y = tan x.
1
x
y
1
–1
pi
2
pi
2
–
FiGuRe 4.32 y = arcsin x.
55
1
0
x
y
–1
pi
2
pi
2
–
FiGuRe 4.34 y = arctan x.
1
x
y
1
–1
pi
pi
2
FiGuRe 4.33 y = arccos x.
56
13,Exponential Function (Half-Life)
The function given by y = e
x
is the well-known expo-
nential function (e = base of natural logarithms; see
Figures 4.24 and 4.25),In many applications,e.g.,
radioactive decay,pharmacokinetics,growth models,
etc.,one encounters this function with time (t) as the
independent variable,i.e.,y = Ae
kt
,for constants
A and k,For positive k,the function increases and
doubles in time ln(2)/k,When k is negative,the
function decreases and is often characterized by
the half-life,which is the time to decrease to A/2,
Half-life is therefore –ln(2)/k.
y
x
0
a
1
2a
1
FiGuRe 4.35 y e a e x
bx bx
= + ≥/ (1 0),(logistic
equation).
57
5
Series,Number
Facts,and Theory
1,Bernoulli and Euler Numbers
A set of numbers,B B B
n1
,,…,
3 2 1?
(Bernoulli num-
bers) and B B B
n2
,,…,
4 2
(Euler numbers),appears in
the series expansions of many functions,A partial
listing follows; these are computed from the follow-
ing equations:
B
n n
B
n n n n
B
n n2 2 2
2 2 1
2
2 2 3
4
+
( )
!
(2 1)(2 )(2 )
!
2 4
1 0
n
n
…
+? =( ),
and
2 2 1
2
2
2 2
2 1 2 2
n n
n n
n
B n B
n n
( )
(2 1)
(2 1)(2 )(
=?
2 )
!
( ),
n
B
n
n
+
…
+?
3
3
1
2 4
1
B
1
= 1/6 B
2
= 1
B
3
= 1/30 B
4
= 5
B
5
= 1/42 B
6
= 61
B
7
= 1/30 B
8
= 1385
B
9
= 5/66 B
10
= 50521
58
B
11
= 691/2730 B
12
= 2702765
B
13
= 7/6 B
14
= 199360981
2,Series of Functions
In the following,the interval of convergence is indi-
cated; otherwise,it is all x,Logarithms are to the
base e,Bernoulli and Euler numbers (B
2n–1
and B
2n
)
appear in certain expressions.
( )
(
a x a na x
n n
a x
n n n
n n n n
+ = + +
+
1 2 2
1
2
1 2
( )
!
) ( )
3
3 3
2 2
!
!
( ! !
[ ]
(
a x
n
n j j
a x x a
a bx
n
n j j
+
…
+
+
…
<
)
) [ ]
= + + + +
…
<
1
2 2
2
3 3
3
2 2 2
1
1
a
bx
a
b x
a
b x
a
b x a
(
( )
!
( )( )
!
[
1 1
1
2
1 2
3
2
3
± = ± +
±
+
…
x nx
n n
x
n n n x
n
)
x
x nx
n n
x
n n n
n
2
2
1
1 1
1
2
1 2
3
<
± = +
+
+ +
]
(
( )
!
( )( )
!
)
x x
3 2
1+
…
<[ ]
59
( )1 1
1
2
1
2 4
1 3
2 4 6
1 3 5
2 4 6
1
2
2 3
± = ±?
±
x x x x
8
1
1 1
1
2
1 3
2 4
1 3 5
2
4 2
1
2
2
x x
x x x
±
…
<
± = +
[ ]
( )
4 6
1 3 5 7
2 4 6 8
1
1 1
1
3
4 2
2
1
2
+
…
<
± = ±
x
x x
x
[ ]
( )
2 2 4
1 3
2 4 6
1 3 5
2 4 6 8
1
2
4
6
8 2
x
x
x
x x
±
±
…
<[ ]
( [ ]
(
1 1 1
1 1
1 2 3 4 5 2
2
± = + + +
…
<
± =
x x x x x x x
x
)
)
2 3 4 5 1
2 3 4 5 2
x x x x x x+ + +
…
<[ ]
e x
x x x
e x
x x
x
x
= + + + + +
…
=? +? +
1
2 3 4
1
2 3
2 3 4
2
4 6
2
! ! !
! !
x
8
4!
…
a x a
x a x a
x
= + + + +
…
1
2
3
log
( log )
!
( log )
3!
2
log ( ) ( )
2
x x x x x= +
…
< <1
1
2
1
1
3
1 0 2
3
( ) [ ]
60
log x
x
x
x
x
x
x
x=
+
+
+
…
>
1 1
2
1 1
3
1 1
2 3
2
log x
x
x
x
x
x
x
=
+
+
+
+
+
2
1
1
1
3
1
1
1
5
1
1
3
+
…
>
5
0[ ]x
log( ) [ ]1
1
2
1
3
1
4
1
2 3 4 2
+ =? +? +
…
<x x x x x x
log 2
1
1
1
3
1
5
1
7
3 5 7
+
= + + + +
…
x
x
x x x x x[
2
1< ]
log 2
x
x x x x
+
= +
+
1
1
1 1
3
1 1
5
1
3 5
+
…
<[ ]x
2
1
sin
! ! !
cos
! !
x x
x x x
x
x x x
=? +? +
…
=? +?
3 5 7
2 4 6
3 5 7
1
2 4 6 !
+
…
tan
!
(2 1)
2
x x
x x x
B
n n
n
= + +?
+
…
+
3 5 7
2
2 1
3
2
15
17
315
2 x
n
x
2 1
2
2
4
n?
<
(2 )!
π
61
ctn
( )
( )!
2
x
x
x x x
B x
n x
n
n
=
…
1
3 45
2
945
2
2
3 5
2 1
…
<[ ]x
2 2
π
sec
! ! !
( ) !
x
x x x
B x
n
x
n
n
= + + + +
…
+ +
…
1
2
5
4
61
6
2
2 4 6
2
2
2
<
π
2
4
csc
! ! !
( )
(
x
x
x x x
n
= + +
+
+
…
+
+
1
3
7
3 5
31
3 7
2 2 1
2
3 5
2 1
n
B x x
n
n
+
+
…
<
+
+
2
2 1
2 1 2 2
) !
[ ]π
sin
(1 3)
( )5
( )
(2 4 6)
5 7
= + +
+
1
3
6 2 4
1 3 5
x x
x x x
7
[ ]+
…
<x
2
1
tan [ 1]
1?
=? +? +
…
<x x x x x x
1
3
1
5
1
7
3 5 7 2
sec
( )5 ( )7
1
2
=
x
x x
x
π
2
1 1
6
1 3
2 4
1 3 5
2 4 6
3
x
x
7
[ 1]?
…
>
2
62
sinh
! ! !
cosh
! !
x x
x x x
x
x x x
= + + + +
…
= + + +
3 5 7
2 4
3 5 7
1
2 4
6 8
1
4
3
3
6 8
1
2
! !
tanh (2 )2
!
(2 1)2
2 2 4
+ +
…
=
x
x
x
B
x
b
4
1 2
6 4
1
1
6
5
5
2
2
!
(2
!
ctnh
6
+
…
<
= +
) B
x
x
x
x
π
2
2
2
4
2
6
2
1
2 4
3
4 6
2
6
2
B x B x B x
x
! ! !
[ ]
2
+?
…
<π
sech
! ! !
x
B x B x B x
x=? +? +
…
<
1
2 4 6 4
2
2
4
4
6
6
2
2
π
csch (2 1)2
!
(
!
[
1
3
x
x
B
x
B
x
x
=
+
…
<
1
2
2 1 2
4
3
3
2 2
) π ]
sinh [
=? +
+
…
1
3 5 7
1
2 3
1 3
2 4 5
1 3 5
2 4 6 7
x x
x x x
x
2
<1]
tanh [ 1]
= + + + +
…
<
1
3 5 7
2
3 5 7
x x
x x x
x
ctnh [ 1]
= + + +
…
>
1
3 5
2
1 1
3
1
5
x
x x x
x
63
csch
=?
+
1
3 5
7
1 1
2 3
1 3
2 4 5
1 3 5
2 4 6 7
x
x x x
x
+
…
>[ 1]x
2
0
3
5 7
2 1
3 5 2 7 3
x
t
e dt x x
x x
∫
=? +
+
…
! !
3,Error Function
The following function,known as the error function,
erf x,arises frequently in applications:
erf x
π
e dt
x
t
=
∫
2
0
2
The integral cannot be represented in terms of a finite
number of elementary functions; therefore,values of
erf x have been compiled in tables,The following is
the series for erf x:
erf
! !
x
π
x
x x x
=? +
+
…
2
3 5 2 7 3
3 5 7
There is a close relation between this function and
the area under the standard normal curve (Table A.1),
For evaluation,it is convenient to use z instead of x;
then erf z may be evaluated from the area F(z) given
in Table A.1 by use of the relation
erf ( )z F z= 2 2
64
example
erf (,) 2 [(1.414)(0.5)] 2 (,0 5 0 707= =F F )
By interpolation from Table A.1,F (0.707) =
0.260; thus,erf (0.5) = 0.520.
4,Fermat’s Little Theorem
This theorem provides a condition that a prime num-
ber must satisfy.
Theorem,If p is a prime,then for any integer a,
(a
P
– a) is divisible by p.
examples
2
8
– 2 = 254 is not divisible by 8; thus,8 cannot
be prime.
3
7
– 3 = 2184 is divisible by 7,because 7 is prime.
5,Fermat’s Last Theorem
If n is an integer greater than 2,then a
n
+ b
n
= c
n
has no solutions in nonzero integers a,b,and c,For
example,there are no integers a,b,and c such that
a
3
+ b
3
= c
3
,This author has generated,near misses,”
i.e.,a
3
+ b
3
= c
3
± 1,as shown below,and shown
further that if (a + b) is odd,c is even,whereas if
(a + b) is even,then c is odd.
6,Beatty’s Theorem
If a and b are positive and irrational with the prop-
erty that
1 1
1
a b
+ =
,then for positive integers n,the
65
integer parts of na and nb constitute a partition of the
set of positive integers,i.e.,the two sequences
a a a?
,,,2 3 …
b b b?
,,,2 3 …
“Near misses” in the Cubic Form of
Fermat’s Last Theorem
a
Near misses for integers a and b between 2 and 1,000 … and beyond
a b c a
3
+ b
3
c
3
6 8 9 728 729
9 10 12 1729 1728
64 94 103 1092728 1092727
71 138 144 2985983 2985984
73 144 150 3375001 3375000
135 138 172 5088447 5088448
135 235 249 15438250 15438249
242 720 729 387420488 387420489
244 729 738 401947273 401947272
334 438 495 121287376 121287375
372 426 505 128787624 128787625
426 486 577 192100032 192100033
566 823 904 738763263 738763264
791 812 1010 1030300999 1030301000
… … … … …
2304 577 2316 12422690497 12422690496
… … … … …
11161 11468 14258 2898516861513 2898516861512
… … … … …
a
Table derived from a computer program written by this author.
66
where
x?
is the greatest integer function,containing
all positive integers but having no common terms.
An interesting example occurs if a = 2,which
yields the two sequences
{S
1
} 1,2,4,5,7,8,9,11,12,…
{S
2
} 3,6,10,13,17,20,23,27,30 …
which partition the integers and also have the prop-
erty that the difference between successive terms
{S
2i
– S
1i
} is the sequence
2,4,6,8,10,12,14,…
7,An Interesting Prime
73939133 is a prime number as is each number
obtained by deleting the right-most digit; each of the
following is a prime number:
7393913,739391,73939,7393,739,73,7
8,Goldbach Conjecture
Every even number greater than or equal to 4 can be
expressed as the sum of two prime numbers.
examples
6 = 3 + 3
12 = 5 + 7
18 = 5 + 13
20 = 3 + 17 = 7 + 13
67
9,Twin Primes
Twin primes are pairs of primes that differ by 2,e.g.,
{3,5},{5,7},{11,13},{17,19},{29,31},…,{137,
139},etc,It is believed,but not proved,that there are
infinitely many twin primes.
10,Collatz Conjecture
Consider a sequence that begins with any positive
integer and applies the following rule for successive
terms,if it is odd,multiply by 3 and add 1; if it is
even,divide it by 2,All such sequences terminate
with 4,2,1,(This conjecture is still unproven.)
example
Start with 23 to give 23,70,35,106,53,160,
80,40,20,10,5,16,8,4,2,1.
68
6
Differential Calculus
1,Notation
For the following equations,the symbols f(x),g(x),
etc.,represent functions of x,The value of a function
f(x) at x = a is denoted f(a),For the function y = f(x)
the derivative of y with respect to x is denoted by one
of the following:
dy
dx
f x D y y
x
,,,.′ ′( )
Higher derivatives are as follows:
d y
dx
d
dx
dy
dx
d
dx
f x f x
d y
dx
2
2
3
3
=
= ′ = ′′
=
( ) ( )
d
dx
d y
dx
d
dx
f x f x
2
2
= ′′ = ′′′( ) ( ),etc.
and values of these at x = a are denoted ′′ ′′′f a f a( ) ( ),,
etc,(see Table of Derivatives).
2,Slope of a Curve
The tangent line at a point P(x,y) of the curve y = f(x)
has a slope f′(x) provided that f′(x) exists at P,The
slope at P is defined to be that of the tangent line at P,
The tangent line at P(x
1
,y
1
) is given by
y y f x x x? = ′?
1 1
( ) ( ).
1
69
The normal line to the curve at P(x
1
,y
1
) has slope
–1/ f′(x
1
) and thus obeys the equation
y y f x x x? =? ′?
1 1 1
1[ / ( )] ( )
(The slope of a vertical line is not defined.)
3,Angle of Intersection of Two Curves
Two curves y = f
1
(x) and y = f
2
(x),that intersect at a
point P(X,Y) where derivatives ′ ′f X f X
1 2
( ),( ) exist,
have an angle (α) of intersection given by
tan
( )
1 ( )
.α =
′? ′
+ ′? ′
f X f X
f X f X
2 1
2 1
( )
( )
If tan α > 0,then α is the acute angle; if tan α < 0,
then α is the obtuse angle.
4,Radius of Curvature
The radius of curvature R of the curve y = f(x) at point
P(x,y) is
R
f x
f x
=
+ ′
′′
{ [ ( )] }
( )
/
1
2 3 2
In polar coordinates (θ,r) the corresponding formula is
R
r
dr
d
r
dr
d
=
+
+
2
2
3 2
2
2
2
θ
θ
/
r
d r
d
2
2
θ
The curvature K is 1/R.
70
5,Relative Maxima and Minima
The function f has a relative maximum at x = a if
f(a) ≥ f(a + c) for all values of c (positive or negative)
that are sufficiently near zero,The function f has a
relative minimum at x = b if f(b) ≤ f(b + c) for all
values of c that are sufficiently close to zero,If the
function f is defined on the closed interval x
1
≤ x ≤ x
2
,
and has a relative maximum or minimum at x = a,
where x
1
< a < x
2
,and if the derivative f′(x) exists at
x = a,then f′(a) = 0,It is noteworthy that a relative
maximum or minimum may occur at a point where
the derivative does not exist,Further,the derivative
may vanish at a point that is neither a maximum nor
a minimum for the function,Values of x for which
f′(x) = 0 are called critical values,To determine
whether a critical value of x,say,x
c
,is a relative max-
imum or minimum for the function at x
c
,one may use
the second derivative test:
1,If f″(x
c
) is positive,f (x
c
) is a minimum.
2,If f″(x
c
) is negative,f (x
c
) is a maximum.
3,If f″(x
c
) is zero,no conclusion may be made.
The sign of the derivatives as x advances through x
c
may also be used as a test,If f′(x) changes from posi-
tive to zero to negative,then a maximum occurs at
x
c
,whereas a change in f′(x) from negative to zero
to positive indicates a minimum,If f′(x) does not
change sign as x advances through x
c
,then the point
is neither a maximum nor a minimum.
6,Points of Inflection of a Curve
The sign of the second derivative of f indicates
whether the graph of y = f(x) is concave upward or
concave downward:
71
′′ >
′′ <
f x
f x
( ) 0,concave upward
( ) 0,concave dowwnward
A point of the curve at which the direction of concav-
ity changes is called a point of inflection (Figure 6.1),
Such a point may occur where f″(x) = 0 or where
f″(x) becomes infinite,More precisely,if the function
y = f(x) and its first derivative y′ = f′(x) are continu-
ous in the interval a ≤ x ≤ b,and if y″ = f″(x) exists in
a < x < b,then the graph of y = f(x) for a < x < b is con-
cave upward if f″(x) is positive and concave downward
if f″(x) is negative.
7,Taylor’s Formula
If f is a function that is continuous on an interval that
contains a and x,and if its first (n + 1) derivatives are
continuous on this interval,then
P
FiGuRe 6.1 Point of inflection.
72
f x f a f a x a
f a
x a
f a
( ) ( ) ( )( )
( )
!
( )
= + ′? +
′′
+
′′′
2
2
( )
3
3
!
( ) …
( )
!
( ),
x a
f a
n
x a R
n
n
+
+? +
( )
where R is called the remainder,There are various
common forms of the remainder.
Lagrange’s Form
R f
x a
n
a x
n
n
=?
+
+
+
( )1
1
( )
( )
( 1)!; between and,β β
Cauchy’s Form
R f
x x a
n
a x
n
n
=?
+( );
1
( )
( ) ( )
!
between and,β
β
β
Integral Form
R
x t
n
f t dt
a
x
n
n
=
∫
+
( )
!
( ),
( )1
8,Indeterminant Forms
If f(x) and g(x) are continuous in an interval that
includes x = a,and if f/(a) = 0 and g(a) = 0,the limit
lim
x→a
(f(x)/g(x)) takes the form,0/0”,called an
indeterminant form,L’H?pital’s rule is
73
lim
( )
( )
lim
( )
( )
.
x a x a
f x
g x
f x
g x
→ →
=
′
′
Similarly,it may be shown that if f(x) → ∞ and
g(x) → ∞ as x → a,then
lim
( )
( )
lim
( )
( )
.
x a x a
f x
g x
f x
g x
→ →
=
′
′
(The above holds for x → ∞.)
Examples
lim
sin
lim
cos
lim lim
x x a
x
x
x
x
x
x
x
e
→ →
→∞ →∞
= =
=
0
2
1
1
2x
e e
x
x
x
= =
→∞
lim
2
0
9,Numerical Methods
a,Newton’s method for approximating roots of the
equation f(x) = 0,A first estimate x
1
of the root
is made; then provided that f′(x
1
) ≠ 0,a better
approximation is x
2
:
x x
f x
f x
.
2 1
1
1
=?
′
( )
( )
The process may be repeated to yield a third
approximation,x
3
,to the root:
x x
f x
f x
3 2
2
2
=?
′
( )
( )
provided f′(x
2
) exists,The process may be
repeated,(In certain rare cases the process will
not converge.)
74
b,Trapezoidal rule for areas (Figure 6.2),For the
function y = f(x) defined on the interval (a,b) and
positive there,take n equal subintervals of width
x = (b – a)/n,The area bounded by the curve
between x = a and x = b (or definite integral of f (x))
is approximately the sum of trapezoidal areas,or
A y y y y y x
n n
~ ( )
1
2
1
2
0 1 2 1
+ + +
…
+ +
Estimation of the error (E) is possible if the second
derivative can be obtained:
E
b a
f c x=
′′
12
2
( ) (,? )
where c is some number between a and b.
y
0
a
y
0
y
n
x
b
|?x|
FiGuRe 6.2 Trapezoidal rule for area.
75
10,Functions of Two Variables
For the function of two variables,denoted z = f(x,y),
if y is held constant,say,at y = y
1
,then the resulting
function is a function of x only,Similarly,x may
be held constant at x
1
,to give the resulting function
of y.
The Gas Laws
A familiar example is afforded by the ideal gas law
that relates the pressure p,the volume V,and the
absolute temperature T of an ideal gas:
pV nRT=
where n is the number of moles and R is the gas con-
stant per mole,8.31 (J?°K
–1
mole
–1
),By rearrangement,
any one of the three variables may be expressed as a
function of the other two,Further,either one of these
two may be held constant,If T is held constant,then
we get the form known as Boyle’s law:
p kV=
1
(Boyle’s law)
where we have denoted nRT by the constant k and,of
course,V > 0,If the pressure remains constant,we
have Charles’ law:
V bT= (Charles’ law)
where the constant b denotes nR/p,Similarly,volume
may be kept constant:
p aT=
where now the constant,denoted a,is nR/V.
76
11,Partial Derivatives
The physical example afforded by the ideal gas law
permits clear interpretations of processes in which one
of the variables is held constant,More generally,we
may consider a function z = f(x,y) defined over some
region of the x-y-plane in which we hold one of the
two coordinates,say,y,constant,If the resulting func-
tion of x is differentiable at a point (x,y),we denote
this derivative by one of the following notations:
f f x z x
x
,/ /,δ δ δ δ
called the partial derivative with respect to x,Simi-
larly,if x is held constant and the resulting function
of y is differentiable,we get the partial derivative
with respect to y,denoted by one of the following:
f f y z y
y
,/,/δ δ δ δ
Example
Given z x y y x y=? +
4 3
sin 4,then
δ δ
δ δ
z x xy y x
z y x y x
/
/
=?
=? +
4
3 4
4 2
( ) cos ;
sin,
3
77
7
Integral Calculus
1,Indefinite Integral
If F(x) is differentiable for all values of x in the inter-
val (a,b) and satisfies the equation dy dx f x/ )= (,
then F(x) is an integral of f(x) with respect to x,The
notation is F x f x dx( ) ( )= ∫ or,in differential form,
dF x f x dx( ) ( ),=
For any function F(x) that is an integral of f(x) it follows
that F(x) + C is also an integral,We thus write
∫
= +f x dx F x C( ) ( ),
(See Table of Integrals.)
2,Definite Integral
Let f(x) be defined on the interval [a,b],which is par-
titioned by points x x x x
j n1 1
,,,,,
2
… …
between a = x
0
and b = x
n
,The jth interval has length?x x x
j j j
=?
1
,
which may vary with j,The sum Σ?
j
n
j j
f v x
=1
( ),
where v
j
is arbitrarily chosen in the jth subinterval,
depends on the numbers x x
n0
,,… and the choice of
the v as well as f,but if such sums approach a com-
mon value as all?x approach zero,then this value is
the definite integral of f over the interval (a,b) and
is denoted ∫
a
b
f x dx( ),The fundamental theorem of
integral calculus states that
78
a
b
f x dx F b F a
∫
=?( ) ( ) ( ),
where F is any continuous indefinite integral of f in
the interval (a,b).
3,Properties
a
b
j
a
b
a
b
f x f x f x dx f x dx
f
∫ ∫
∫
+ +
…
+ = +[ ]
1 2 1
2
( ) ( ) ( ) ( )
( ) ( ),
( ),i
x dx f x dx
c f x dx c f x dx
a
b
j
a
b
a
b
+
…
+
=
∫
∫ ∫
( ) f is a constant.
( ) ( ),
c
f x dx f x dx
a
b
b
a
a
b
∫ ∫
∫
=?
f x dx f x dx f (x dx
a
c
c
b
( ) ( ) ),= +
∫ ∫
4,Common Applications of the Definite Integral
Area (Rectangular Coordinates)
Given the function y = f(x) such that y > 0 for all x
between a and b,the area bounded by the curve
y = f(x),the x-axis,and the vertical lines x = a and
x = b is
A f x dx
a
b
=
∫
( ),
Length of Arc (Rectangular Coordinates)
Given the smooth curve f(x,y) = 0 from point (x
1
,y
1
)
to point (x
2
,y
2
),the length between these points is
79
L dy dx dx
L dx dy dy
x
x
y
y
= +
= +
∫
∫
1
2
1
2
1
1
2
2
( ),
( ),
/
/
Mean Value of a Function
The mean value of a function f(x) continuous on [a,b]
is
1
( )
.
b a
f x dx
a
b
∫
( )
Area (Polar Coordinates)
Given the curve r = f(θ),continuous and nonnegative
for θ θ θ
1 2
≤ ≤,the area enclosed by this curve and
the radial lines θ θ=
1
and θ θ=
2
is given by
A f d=
∫
θ
θ
θ θ
1
2 1
2
2
[ ( )],
Length of Arc (Polar Coordinates)
Given the curve r = f(θ) with continuous derivative
′f ( )θ on θ θ θ
1 2
≤ ≤,the length of arc from θ θ=
1
to θ θ=
2
is
L f f d= + ′
∫
θ
θ
θ θ θ
1
2
[ ( )] [ ( )],
2 2
Volume of Revolution
Given a function y = f(x) continuous and nonnegative
on the interval (a,b),when the region bounded by
80
f(x) between a and b is revolved about the x-axis,the
volume of revolution is
V π f x dx
a
b
=
∫
[ ( )],
2
Surface Area of Revolution (revolution
about the x-axis,between a and b)
If the portion of the curve y = f(x) between x = a and
x = b is revolved about the x-axis,the area A of the
surface generated is given by the following:
A f x f x dx
a
b
= + ′
∫
2
1 2
π ( ) {1 [ ( )]
2
}
/
Work
If a variable force f(x) is applied to an object in the
direction of motion along the x-axis between x = a and
x = b,the work done is
W f x dx
a
b
=
∫
( ),
5,Cylindrical and Spherical Coordinates
a,Cylindrical coordinates (Figure 7.1):
x r
y r
=
=
cos
sin
θ
θ
Element of volume,dV rdrd dz= θ,
b,Spherical coordinates (Figure 7.2):
81
P
z
y
Z
r
x
θ
FiGuRe 7.1 Cylindrical coordinates.
z
P
ρ
x
y
θ
FiGuRe 7.2 Spherical coordinates.
82
x
y
z
=
=
=
ρ
ρ
ρ
sin cos
sin sin
cos
φ θ
φ θ
φ
Element of volume,dV d d d=ρ φ ρ φ θ
2
sin,
6,Double Integration
The evaluation of a double integral of f(x,y) over a
plane region R,
R
f x y dA
∫∫
(,)
is practically accomplished by iterated (repeated)
integration,For example,suppose that a vertical
straight line meets the boundary of R in at most two
points so that there is an upper boundary,y = y
2
(x),
and a lower boundary,y = y
1
(x),Also,it is assumed
that these functions are continuous from a to b (see
Figure 7.3),Then
R a
b
y x
y x
f x y dA f x y dy dx
∫ ∫ ∫∫
=
(,) (,)
1
2
( )
( )
If R has a left-hand boundary,x = x
1
(y),and a right-
hand boundary,x = x
2
(y),which are continuous from
c to d (the extreme values of y in R),then
R c
d
x y
x y
f x y dA f x y dx dy
∫ ∫ ∫∫
=
(,) (,)
1
2
( )
( )
Such integrations are sometimes more convenient in
polar coordinates,x = r cos θ,y = r sin θ; dA = rdr dθ.
83
7,Surface Area and Volume by Double Integration
For the surface given z = f(x,y),which projects onto
the closed region R of the x-y-plane,one may calcu-
late the volume V bounded above by the surface and
below by R,and the surface area S by the following:
V zdA f x y dx dy
S z x z
R R
R
= =
= + +
∫ ∫ ∫∫
∫
(,)
[1 ( / ) ( /
2
δ δ δ δy dx dy) ]
2
∫
1 2/
[In polar coordinates,(r,θ),we replace dA by rdr dθ.]
8,Centroid
The centroid of a region R of the x-y-plane is a point
( )′ ′x y,where
y
y
2
(x)
y
1
(x)
a
x
b
FiGuRe 7.3 Region R bounded by y
2
(x) and y
1
(x).
84
′= ′=
∫∫ ∫∫
x
A
x dA y
A
y dA
R R
1 1;
and A is the area of the region.
example
For the circular sector of angle 2α and radius
R,the area A is αR
2; the integral needed for
x′,expressed in polar coordinates,is
∫∫ ∫∫
=
=
+
x dA r r drd
R
R
0
3
3
α
α
α
θ θ
θ
( cos )
sin
α
α=
2
3
3
R sin
and thus,
′= =x
R
R
R
2
3
2
3
3
2
sin
sin
.
α
α
α
α
Centroids of some common regions are shown in
Figure 7.4.
85
FiGuRe 7.4
Centroids of some common regions
are shown below:
Centroids
Area x′ y′
y
(rectangle)
h
x
b
bh b/2 h/2
b
x
(isos,triangle)*y
h
bh/2 b/2 h/3
y
R
x
(semicircle)
πR
2
/2 R 4R/3π
(quarter circle)y
R
x
πR
2
/4 4R/3π 4R/3π
(circular sector)y
x
A
R
R
2
A 2R sin A/3A 0
*
y′ = h/3 for any triangle of altitude h.
86
8
Vector Analysis
1,Vectors
Given the set of mutually perpendicular unit vectors
i,j,and k (Figure 8.1),any vector in the space may
be represented as F = ai + bj + ck,where a,b,and c
are components.
Magnitude of F
| | ( )F = + +a b c
2 2 2
1
2
Product by Scalar p
p pa pb pcF i j k= + +,
k
j
i
FiGuRe 8.1 The unit vectors i,j,and k.
87
Sum of F
1
and F
2
F F i j k
1 2
+ = + + + + +( ) ( ) (c
1 2 1
a a b b c
1 2 2
)
Scalar Product
F F
1 2
= + +a a b b c
1 2 21 2 1
c
(Thus,i i j j k k? =? =? =1 and i j j k k i? =? =?
= 0.)
Also
F F F F
F F F F F F F
1 2 2 1
1 2 3 1 3 2 3
=?
+? =? +?( )
Vector Product
F F
i j k
1 2
× = a b c
a b c
1 1 1
2 2 2
(Thus,i i j j k k 0 i j k j k i× = × = × = × = × =,,,
and,k i j)× =
Also,
F F F F
F F F F F F F
F F F
1 2 2 1
1 2 3 1 3 2 3
1 2 3
× =? ×
+ × = × + ×
× +
( )
( )
( ) ( ) ( )
= × + ×
× + =
F F F F
F F F F F F F F F
1 2 1 3
1 2 3 1 3 2 1 2 3
1 2 3 1 2 3
F F F F F F? × = ×?( ) ( )
88
2,Vector Differentiation
If V is a vector function of a scalar variable t,then
V i j k= + +a t b t c t( ) ( )( )
and
d
dt
da
dt
db
dt
dc
dt
V
i j k= + +,
For several vector functions V V V
1 2
,,…,
n
d
dt
d
dt
d
dt
d
dt
d
dt
n
(V V V
V V V
(V
1 2
1 2
1
+ +
…
+ = + +
…
+
n
),
V
V
V V
V
(V V
V
V
2
1
2 1
2
1 2
1
2
)
)
=? +?
× = × +
d
dt
d
dt
d
dt
d
dt
,
V
V
1
2
×
d
dt
.
For a scalar-valued function g(x,y,z),
(gradient)
grad g g
g
x
g
y
g
z
= = + +?
δ
δ
δ
δ
δ
δ
i j k.
For a vector-valued function V(a,b,c),where a,b,
and c are each a function of x,y,and z,respectively,
(divergence) div
y
c
z
V V=? = + +?
δ
δ
δ
δ
δ
δ
a
x
b
89
(curl) curl V V
i j k
= × =?
δ
δ
δ
δ
δ
δx y z
a b c
Also,
div grad
2 2
g g
g
x
g
y
g
z
=? = + +
2
2
2
2 2
δ
δ
δ
δ
δ
δ
and
curl grad ; div curl 0;
curl curl grad
g = =
=
0 V
V div,
2 2
V i j k? + +( )a b c
2
3,Divergence Theorem (Gauss)
Given a vector function F with continuous partial
derivatives in a region R bounded by a closed sur-
face S,
R S
div dV dS
∫∫∫ ∫∫
=?F n F,
where n is the (sectionally continuous) unit normal
to S.
4,Stokes’ Theorem
Given a vector function with continuous gradient
over a surface S that consists of portions that are
piecewise smooth and bounded by regular closed
curves such as C,
90
S C
dS d
∫ ∫∫
=?n F F rcurl
5,Planar Motion in Polar Coordinates
Motion in a plane may be expressed with regard to
polar coordinates (r,θ),Denoting the position vector by
r and its magnitude by r,we have r = rR(θ),where R is
the unit vector,Also,dR/dθ = P,a unit vector perpen-
dicular to R,The velocity and acceleration are then
v R P
a
= +
=?
dr
dt
r
d
dt
d r
dt
r
d
dt
θ
θ;
2
2
2
+ +
R Pr
d
dt
dr
dt
d
dt
.
2
2
2
θ θ
Note that the component of acceleration in the P direc-
tion (transverse component) may also be written
1
2
r
d
dt
r
d
dt
θ?
so that in purely radial motion it is zero and
r
d
dt
C
2
θ
= ( )constant
which means that the position vector sweeps out area
at a constant rate (see,Area (Polar Coordinates),”
Section 7.4).
6,Geostationary Satellite Orbit
A satellite in circular orbit with velocity v around the
equator at height h has a central acceleration,
v
R h
2
+
,
91
where R is the radius of the earth,From Newton’s
second law this acceleration equals
MG
R h( )+
2
,where
M is the mass of the earth and G is the gravita-
tional constant,thereby giving orbital velocity
MG
R h+
1 2/
and angular velocity ω =
( )
( )
/
/
MG
R h
1 2
3 2
+
.
Inserting constants M = 5.98 × 10
24
kg,R = 6.37 ×
10
6
m,G = 6.67 × 10
–11
N·m
2
/kg
2
,and earth’s angular
velocity ω = 7.27 × 10
–5
/s,one finds h ≈ 35,790 km,
Thus,a satellite orbiting around the equator at this
height above the earth’s surface appears stationary.
92
9
Special Functions
1,Hyperbolic Functions
sinh x
e e
x x
=
2
csch x
x
=
1
sinh
cosh x
e e
x x
=
+
2
sech x
x
=
1
cosh
tanh x
e e
e e
x x
x x
=
+
ctnh x
x
=
1
tanh
sinh( ) sinh? =?x x
ctnh ctnh( )? =?x x
cosh( ) cosh? =x x
sech sech( )? =x x
tanh( ) tanh? =?x x
csch csch( )? =?x x
tanh
sinh
cosh
x
x
x
=
ctnh x
x
x
=
cosh
sinh
cosh sinh
2 2
1x x? =
cosh cosh
2
1
2
2 1x x= +
( )
sinh cos
2
1
2
2 1x x=?
( )
ctnh csch
2 2
1x x? =
csch
2 2
2 2
x x
x x
=sech
csch sech
tanh
2 2
1x x+ =sech
93
sinh( ) sinh cosh cosh sinh
cosh( )
x y x y x y
x y
+ = +
+ = +
=
cosh cosh sinh sinh
sinh( ) sinh c
x y x y
x y x oosh cosh sinh
cosh( ) cosh cosh sin
y x y
x y x y
=? h sinh
tanh( )
tanh tanh
tanh tanh
x y
x y
x y
x
+ =
+
+1 y
x y
x y
x y
tanh( )
tanh tanh
tanh tanh
=
1
2,Gamma Function (Generalized Factorial Function)
The gamma function,denoted Γ(x),is defined by
Γ( ),0
0
x e t dt x
t x
= >
∞
∫
1
Properties
Γ Γ
Γ
Γ Γ
( ) ( )
( )
( ) !
x x x x
n n n n n
+ = >
=
+ = = =
1 0
1 1
1 1 2
,
( ) (,,
x x x
x
x
3
1
1
2
2
2 1
,…)
( ) /sin
(
Γ Γ
Γ
Γ
( )? =
=
π π
π
) ( )Γ Γx x+
=
1
2
2π
94
3,Laplace Transforms
The Laplace transform of the function f(t),denoted
by F(s) or L{f(t)},is defined
F s f t e dt
st
( ) =
∞
∫
0
( )
provided that the integration may be validly per-
formed,A sufficient condition for the existence of
F(s) is that f(t) be of exponential order as t →∞
and that it is sectionally continuous over every finite
interval in the range t ≥ 0,The Laplace transform of
g(t) is denoted by L g t{ ( )} or G(s).
Operations
f t( )
F s f t e dt
st
( ) ( )=
∞
∫
0
af t bg t( ) ( )+
aF s bG s( ) ( )+
′f t( )
sF s f( ) ( )? 0
′′f t( )
s F s sf f
2
0 0( ) ( ) ( ) ′
f t
n( )
( )
s F s s f
s f
f
n n
n
n
( ) ( )
( )
( )
( )
′
…
1
2
1
0
0
0
tf t( )
′F s( )
t f t
n
( )
( ) ( )
( )
1
n n
F s
95
e f t
at
( )
F s a( )?
f t g d
t
( ) ( )
∫
β β β
0
F s G s( ) ( )?
f t a( )?
e F s
as?
( )
f
t
a
aF as( )
g d
t
( )β β
0
∫
1
s
G s( )
f t c t c( ) ( )δ
e F s c
cs?
>( ),0
where
δ( )t c t c
t c
= ≤ <
= ≥
0 0
1
if
if
f t f t( ) ( )
(
= +ω
periodic)
e f d
e
s
s
∫
τ
ω
τ τ
ω
( )
0
1
Table of Laplace Transforms
f t( )
F s( )
1
1 / s
t
1
2
/ s
96
t
n
n?
1
1( )!
1 1 2 3/ (,,)s n
n
= …
t
1
2s s
π
1
t
π
s
e
at
1
s a?
te
at
1
2
( )s a?
t e
n
n at?
1
1( )!
1
1 2 3
( )
(,,)
s a
n
n
= …
t
x
x
Γ( )+1
1
1
1
s
x
x+
>?,
sinat
a
s a
2 2
+
cosat
s
s a
2 2
+
sinhat
a
s a
2 2
coshat
s
s a
2 2
97
e e
at bt
a b
s a s b
a b
≠
( )( )
,( )
ae be
at bt
s a b
s a s b
a b
( )
( )( )
,( )
≠
t atsin
2
2 2 2
as
s a( )+
t atcos
s a
s a
2 2
2 2 2
+( )
e bt
at
sin
b
s a b( )? +
2 2
e bt
at
cos
s a
s a b
+( )
2 2
sinat
t
Arc
a
s
tan
sinhat
t
1
2
log
e
s a
s a
+
4,Z-Transform
For the real-valued sequence {f(k)} and complex vari-
able z,the z-transform,F z Z f k( ) { }= ( ),is defined by
Z f k F f k z
k
k
{ ( )} (z) ( )= =
=
∞
∑
0
For example,the sequence f k k( ) 1,,,,…,= = 0 1 2
has the z-transform
98
F z z z z z
k
( ) 1= + + +
…
+ +
…
1 2 3
z-Transform and the Laplace Transform
For function U(t),the output of the ideal sampler U t*( )
is a set of values U(kT),k,,= 0 1 2,…,that is,
U t U t t kT
k
*( ) =?
=
∞
∑
0
( ) ( )δ
The Laplace transform of the output is
{ *( )} *( )
( ) ( )
U t e U t dt
e U t t kT d
st
st
=
=?
∞
∞
∫
∫
0
0
δ t
e U kT
k
sKT
k
=
=
∞
∞
∑
∑
=
0
0
( )
Defining z e
sT
= gives
{ *( )} ( )U t U kT z
k
k
=
=
∞
∑
0
which is the z-transform of the sampled signal U(kT).
Properties
Linearity,Z af k bf k
aZ f k bZ f k
aF z
{ ( ) ( )}
{ ( )} { ( )}
( )
1 2
1 2
1
+
= +
= +bF z
2
( )
99
Right-shifting property,Z f k n z F z
n
{ ( )} ( )? =
Left-shifting property,Z f k n z F z
f k z
n
n k
k
n
{ ( )} ( )
( )
+ =
=
∑
0
1
Time scaling,Z a f k F z a
k
{ ( )} ( / )=
Multiplication by k,Z kf k zdF z dz{ ( )} ( ) /=?
Initial value,f z F z F
z
( ) lim( ) ( ) ( )0 1
1
=? = ∞
→∞
Final value,lim ( ) lim( ) ( )
k z
f k z F z
→∞ →
=?
1
1
1
Convolution,Z f k f k F z F z{ ( )* ( )} ( ) ( )
1 2 1 2
=
z-Transforms of Sampled Functions
f k( )
Z f kT F z{ ( )} ( )=
1 0at ; elsek
z
k?
1
z
z?1
kT
Tz
z( )?1
2
( )kT
2
T z z
z
2
3
1
1
( )
( )
+
sinωkT
z T
z z T
sin
cos
ω
ω
2
2 1? +
100
cosωT
z z T
z z T
( cos )
cos
+
ω
ω
2
2 1
e
akT?
z
z e
aT
kTe
akT?
zTe
z e
aT
aT
( )
2
( )kT e
akT2?
T e z z e
z e
aT aT
aT
2
3
+
( )
( )
e kT
akT?
sinω
ze T
z ze T e
aT
aT aT
+
sin
cos
ω
ω
2 2
2
e kT
akT?
cosω
z z e T
z ze T e
aT
aT aT
( cos )
cos
+
ω
ω
2 2
2
a kT
k
sinω
az T
z az T a
sin
cos
ω
ω
2 2
2? +
a kT
k
cosω
z z a T
z az T a
( cos )
cos
+
ω
ω
2 2
2
5,Fourier Series
The periodic function f(t),with period 2π,may be
represented by the trigonometric series
a a nt b nt
n n0
1
+ +
∞
∑
( cos sin )
where the coefficients are determined from
101
a f t dt
a f t nt dt
b
n
n
0
1
2
1
1
=
=
=
∫
∫
π
π
π
π
π
π
π
( )
)cos(
π
π
∫
=f t nt dt n( )sin (,,,…)1 2 3
Such a trigonometric series is called the Fourier
series corresponding to f(t),and the coefficients are
termed Fourier coefficients of f(t),If the function is
piecewise continuous in the interval? ≤ ≤π πt,and
has left- and right-hand derivatives at each point in
that interval,then the series is convergent with sum
f(t) except at points t
i
at which f(t) is discontinuous,
At such points of discontinuity,the sum of the series
is the arithmetic mean of the right- and left-hand
limits of f(t) at t
i
,The integrals in the formulas for the
Fourier coefficients can have limits of integration that
span a length of 2π,for example,0 to 2π (because of
the periodicity of the integrands).
6,Functions with Period Other Than 2π
If f(t) has period P,the Fourier series is
f t a a
n
P
t b
n
P
t
n n
( ) ~ cos
2
sin
2
,
0
+ +
∞
∑
1
π π
where
a
P
f t dt
a
P
f t
P
P
n
P
P
0
2
2
2
2
1
2
=
=
∫
∫
/
/
/
/
( )
( ) cos
2πn
P
t dt
b
P
f t
n
P
t dt
n
P
P
=
∫
2
2
2
/
/
( ) sin
2
.
π
102
f(t)
a
t
0_
1
2
P
_
1
4
P
1
4
P
1
2
P
FiGuRe 9.1 Square wave:
f t
a a t
P
t
P
t
P
( ) ~
2
cos
2
cos
6
cos
10
+? + +
…
2 1
3
1
5π
π π π
.
f(t)
t
a
0
P
1
2
FiGuRe 9.2 Sawtooth wave:
f t
a t
P
t
P
t
P
( ) ~ sin
2
sin
4
sin
62 1
2
1
3π
π π π
+?
…
.
103
Again,the interval of integration in these formulas
may be replaced by an interval of length P,for
example,0 to P.
7,Bessel Functions
Bessel functions,also called cylindrical functions,
arise in many physical problems as solutions of the
differential equation
x y xy x n y
2 2 2
0′′+ ′+? =( )
which is known as Bessel’s equation,Certain solu-
tions of the above,known as Bessel functions of the
first kind of order n,are given by
J x
k n k
x
n
k
k
n k
( )
( )
( )
0
=
+ +
=
+
∞
∑
1
1 2
2
! Γ
f(t)
t
A
0
ω
π
FiGuRe 9.3 Half-wave rectifier:
f t
A A
t
A
t
( ) ~ sin
( ) (3)
cos
( ) (5)
cos
π
ω
π
ω
+
+
2
2 1
1
2
1
3
4ωt +
…
.
104
J x
k n k
x
n
k
k
n k
=
∞
+
=
+ +
∑
( )
( )
( )
0
1
1 2
2
! Γ
In the above it is noteworthy that the gamma function
must be defined for the negative argument q,Γ(q) =
Γ(q 1) /,+ q provided that q is not a negative integer,
When q is a negative integer,1 / Γ(q) is defined to
be zero,The functions J
–n
(x) and J
n
(x) are solutions
of Bessel’s equation for all real n,It is seen,for
n =1 2 3,,,… that
J x J x
n
n
n?
=?( ) ( ) ( )1
and therefore,these are not independent; hence,a
linear combination of these is not a general solution,
When,however,n is not a positive integer,negative
integer,nor zero,the linear combination with arbi-
trary constants c
1
and c
2
y c J x c J x
n n
= +
1 2
( ) ( )
is the general solution of the Bessel differential
equation.
The zero-order function is especially important as
it arises in the solution of the heat equation (for a
“long” cylinder):
J x
x x x
0
2
2
4
2 2
6
2 2 2
2 2 4 2 4 6
( ) 1=? +? +
…
while the following relations show a connection to
the trigonometric functions:
105
J x
x
x
1
2
2
1 2
( ) sin=
π
/
J x
x
x
=
1
2
2
1 2
( ) cos
π
/
The following recursion formula gives J
n+1
(x) for any
order in terms of lower-order functions:
2
1
n
x
J x J x J x
n n n
( ) ( ) ( )
1
= +
+
8,Legendre Polynomials
If Laplace’s equation,?
2
V = 0,is expressed in spherical
coordinates,it is
r
V
r
r
V
r
V V
2
2
2
2
2
2sin sin sin cosθ
δ
δ
θ
δ
δ
θ
δ
δθ
θ
δ
δθ
+ + +
+ =
1
0
2
2
sin θ
δ
δφ
V
and any of its solutions,V r(,,),θ φ are known as
spherical harmonics,The solution as a product
V r R r(,,) ( ) ( )θ φ θ= Θ
which is independent of φ,leads to
sin [ ( )]
2 2
1 0θ θ θ θ′′+ ′+ + =Θ Θ Θsin cos sin ]n n
Rearrangement and substitution of x = cos θ leads to
(1 ) ( ) 0 + + =x
d
dx
x
d
dx
n n
2
2
2
2 1
Θ Θ
Θ
106
known as Legendre’s equation,Important special
cases are those in which n is zero or a positive integer,
and for such cases,Legendre’s equation is satisfied
by polynomials called Legendre polynomials,P
n
(x),
A short list of Legendre polynomials,expressed in
terms of x and cos θ,is given below,These are given
by the following general formula:
P x
n j
j n j n j
x
n
j
L j
n
n
( )
( ) (2 )!
!( )!( !
=
=
∑
0
1 2
2 2 )
2 j
where L = n/2 if n is even and L = (n – 1)/2 if n is odd,
Some are given below:
P x
P x x
P x x
P x x
0
1
2
2
3
3
2
2
( ) 1
( )
( )
1
(3 1)
( )
1
(5 3
=
=
=?
=? x
P x x x
P x x x
)
( )
1
(35 3)
( )
1
(63
4
4 2
5
5 3
8
30
8
70
=? +
=? +1 )
(cos )
(cos ) cos
(cos )
1
(3 co
5
1
4
0
1
2
x
P
P
P
θ
θ θ
θ
=
=
= s 2 1)
(cos )
1
(5cos3 cos )
θ
θ θ θ
+
= +P
3
8
3
107
P
4
1
64
4 20 2 9(cos ) (35cos cos )θ θ θ= + +
Additional Legendre polynomials may be determined
from the recursion formula
( ) ( ) ( ( )
( ) 0 ( 1,2,
n P x n ) x P x
nP x n
n n
n
+? +
+ = =
+
1 2 1
1
1
……)
or the Rodrigues formula
P x
n
d
dx
x
n n
n
n
n
( )
!
( )=?
1
2
1
2
9,Laguerre Polynomials
Laguerre polynomials,denoted L
n
(x),are solutions of
the differential equation
xy x y ny
n
+? ′+ =(1 ) 0
and are given by
L x
j
C x n,
n
j
n j
n,j
j
( )
( )
!
(,,…)=
=
=
∑
0
1
0 1 2
( )
Thus,
L x
L x x
L x x x
L x x
0
1
2
2
3
1
1
2
1
2
3
3
( )
( )
( ) 1
( ) 1
=
=?
=? +
=? +
2
1
6
2 3
x x?
108
Additional Laguerre polynomials may be obtained
from the recursion formula
( ) ( ) ( ) ( )
( ) 0
n L x n x L x
nL x
n n
n
+? +?
+ =
+
1 2 1
1
1
10,Hermite Polynomials
The Hermite polynomials,denoted H
n
(x),are given
by
H H x e
d e
dx
n,,
n
n x
n x
n0
1 1 2
2
2
= =? =
,( ) ( 1),( …)
and are solutions of the differential equation
y xy ny n
n
′+ = =2 2 0 ( 0,1,2,…)
The first few Hermite polynomials are
H H x x
H x x H x x x
H x
0 1
2
2
3
3
4
1 2
4 2 8 12
1
= =
=? =?
=
( )
( ) ( )
( ) 6 48 12
4 2
x x? +
Additional Hermite polynomials may be obtained
from the relation
H x xH x H x
n n n+
=? ′
1
2( ) ( ),( )
where prime denotes differentiation with respect to x.
11,Orthogonality
A set of functions { ( )},,…)f x n
n
( =1 2 is orthogonal
in an interval (a,b) with respect to a given weight
function w(x) if
109
a
b
m n
w x f x f x dx m n
∫
= ≠( ) ( ) ( ) 0 when
The following polynomials are orthogonal on the
given interval for the given w(x):
Legendre polynomials,( ) ( ) 1
Lagu
P x w x
a,b
n
=
=? =1 1
erre polynomials,( ) ( ) exp ( )
,
He
L x w x x
a b
n
=?
= =∞0
rmite polynomials,( ) ( ) exp ( )
,
H x w x x
a b
n
=?
=?∞ =
2
∞
The Bessel functions of order n,J x J ( x,,
n n
( ),…λ λ
1 2
)
are orthogonal with respect to w(x) = x over the
interval (0,c) provided that the λ
i
are the positive
roots of J c
n
( ) 0:λ =
0
0
c
n j n k
x J x J x dx j k
∫
= ≠( ) ( ) ( )λ λ
where n is fixed and n ≥ 0.
110
10
Differential Equations
1,First-Order,First-Degree Equations
M x y dx N x y dy(,) (,)+ = 0
a,If the equation can be put in the form A(x)dx +
B(y)dy = 0,it is separable and the solution follows
by integration,
∫ +∫ =A x dx B y dy C( ) ( ) ;
thus,
x y dx ydy( )1 0
2
+ + = is separable since it is
equivalent to x dx ydy y+ + =/ )(,1 0
2
and inte-
gration yields x C
2 1
2
2 0/ + + + =log (1 y ),
2
b,If M(x,y) and N(x,y) are homogeneous and of the
same degree in x and y,then substitution of vx
for y (thus,dy v dx x dv= + ) will yield a separa-
ble equation in the variables x and y,[A function
such as M(x,y) is homogeneous of degree n in
x and y if M cx cy c M x y
n
(,),.= ( ) ] For example,
( ( )y x dx y x dy? + +2 2) has M and N each homog-
enous and of degree 1 so that substitution of
y = vx yields the separable equation
2 2 1
1
0
2
x
dx
v
v v
dv,+
+
+?
=
c,If M x y dx N x y d(,),y+ ( ) is the differential of
some function F(x,y),then the given equation
is said to be exact,A necessary and sufficient
condition for exactness is =M y N x/ /,
When the equation is exact,F is found from the
relations =F x M/ and =F y N/,and the
solution is F (x,y) = C (constant),For example,
111
( ) ( )x y dy xy x dx
2 2
2 3+ +? is exact since
=M y x/ 2 and =N x x./ 2 F is found from
=?F x xy x/ 2 3
2
and = +F y x y./
2
From
the first of these,F x y x y=? +
2 3
φ( ); from the
second,F x y y x,= + +
2 2
2/ Ψ( ) It follows that
F x y x y=? +
2 3 2
2/,and F = C is the solution.
d,Linear,order 1 in y,Such an equation has the
form dy P x ydx Q x dx+ =( ),( ) Multiplication by
exp [P(x)dx] yields
d y P dx Q x Pdx dx.exp exp∫( )
= ∫
( )
( )
For example,dy x ydx x dx+ =(2
2
/ ) is linear
in y,P x x( ) 2/,= so ∫ = =Pdx x x5 ln ln,
2
and
exp ( ),
2
∫ =P dx x Multiplication by x
2
yields
d x y x dx( )
2 4
=,and integration gives the solution
x y x C
2 5
5= +/,
Application of Linear-Order 1 Differential Equations,
Drug Kinetics
A substance (e.g.,a drug) placed in one compartment
is eliminated from that compartment at a rate propor-
tional to the quantity it contains,and this elimination
moves it to a second compartment (such as blood) that
originally does not contain the substance,The sec-
ond compartment also eliminates the substance to an
external sink and does so at a rate proportional to the
quantity it contains,If D denotes the initial amount
in the first compartment,and the elimination rate
constants from each compartment are denoted k
1
and
k
2
,respectively,then the quantities in compartment 1
112
(denoted X) and compartment 2 (denoted Y) at any
time t are described by
dX
dt
k X=?
1
X(0) = D (compartment 1)
dY
dt
k X k Y=?
1 2
Y(0) = 0 (compartment 2)
from which
X De
k t
=
1
so that
dY
dt
k Y k De
k t
+ =
2 1
1
,a linear order 1 equation with
solution
Y
k D
k k
e e
k t k t
=
( )
1
2 1
1 2
This illustrates a model that is commonly used to
describe the movement of a drug from some entry
site into and out of the blood.
2,Second-Order Linear Equations
(with Constant Coefficients)
( ) ( )b D b D b y f x D
d
dx
0
2
1 2
+ + = =,.
113
a,Right-hand side = 0 (homogeneous case)
( ),b D b D b y
0
2
1 2
0+ + =
The auxiliary equation associated with the above
is
b m b m b
0
2
1 2
0+ + =,
If the roots of the auxiliary equation are real and
distinct,say,m
1
and m
2
,then the solution is
y C e C e
m x m x
= +
1 2
1 2
where the C’s are arbitrary constants.
If the roots of the auxiliary equation are real and
repeated,say,m
1
= m
2
= p,then the solution is
y C e C xe
px px
= +
1 2
.
If the roots of the auxiliary equation are complex
a + ib and a – ib,then the solution is
y C e bx C e bx
ax ax
= +
1 2
cos sin,
b,Right-hand side ≠ 0 (nonhomogeneous case)
( ) ( )b D b D b y f x
0
2
1 2
+ + =
The general solution is y = C
1
y
1
(x) + C
2
y
2
(x) + y
p
(x),
where y
1
and y
2
are solutions of the corresponding
homogeneous equation and y
p
is a solution of the
given nonhomogeneous differential equation,y
p
has the form y x A x y x B x y
p
( ) ( ) ( ) ( ) (x)
2
= +
1
,and
A and B are found from simultaneous solution of
114
A′y
1
+ B′y
2
= 0 and A′y′
1
+ B′y′
2
= f(x)/b
0
,A solution
exists if the determinant
y y
y y
1 2
1 2
′ ′
does not equal zero,The simultaneous equations
yield A′ and B′ from which A and B follow by inte-
gration,For example,
D D y e
x2 3
2+?
( )
=
.
The auxiliary equation has the distinct roots 1 and
–2; hence,y
1
= e
x
and y
2
= e
–2x
,so that y
p
= Ae
x
+
Be
–2x
,The simultaneous equations are
′? ′ =
′ + ′ =
A e B e e
A e B e
x x x
x x
2
0
2 3
2
and give A′ = (1/3)e
–4x
and B′ = (–1/3)e
–x
,Thus,
A = (–1/12)e
–4x
and B = (1/3)e
–x
,so that
y e e
e
y C e C e
p
x x
x
x
=? +
=
∴ = +
( 1/12) (1/3)
.
1
3 3
1
4
3
2
+
2 1
4
3x x
e,
3,Runge Kutta Method (of Order 4)
The solution of differential equations may be approxi-
mated by numerical methods as described here for the
differential equation dy/dx = f(x,y),with y = y
0
at x = x
0
,
Step size h is chosen and the solution is approximated
115
over the interval [x
0
,x
n
],where x
n
= nh,The approxi-
mation follows from the recursion formula
y
n+1
= y
n
+ (1/6) (K
1
+ 2K
2
+ 2K
3
+ K
4
)
where
K
1
= hf(x
n
,y
n
)
K
2
= hf(x
n
+ h/2,y
n
+ K
1
/2)
K
3
= hf(x
n
+ h/2,y
n
+ K
2
/2)
K
4
= hf(x
n
+ h,y
n
+ K
3
)
116
11
Statistics
1,Arithmetic Mean
μ=
ΣX
N
i
,
where X
i
is a measurement in the population and N is
the total number of X
i
in the population,For a sample
of size n,the sample mean,denoted X,is
X
X
n
i
=
∑
.
2,Median
The median is the middle measurement when an
odd number (n) of measurement are arranged in
order; if n is even,it is the midpoint between the
two middle measurements.
3,Mode
It is the most frequently occurring measurement in
a set.
4,Geometric Mean
geometric mean …= X X X
i n
n
2
117
5,Harmonic Mean
The harmonic mean H of n numbers X,X X
n1 2
,…,,
is
H
n
Xi
=
∑ ( )1 /
6,Variance
The mean of the sum of squares of deviations from
the means (μ) is the population variance,denoted σ
2
:
σ μ
2 2
=∑?( )X N.
i
/
The sample variance,s
2
,for sample size n is
s X X n
i
2
1=∑( ) ( ).
2
/
A simpler computational form is
s
X
n
n
i
2
2
1
=
∑?
∑
( X )
i
2
7,Standard Deviation
The positive square root of the population variance is
the standard deviation,For a population,
σ=
∑?
∑
( )
X
X
N
N
i
i2
2
1 2/;
118
for a sample,
s
X
X
n
n
i
i
=
∑?
∑
( )
2
2
1 2
1
/
8,Coefficient of Variation
V s X.= /
9,Probability
For the sample space U,with subsets A of U (called
events),we consider the probability measure of an
event A to be a real-valued function p defined over all
subsets of U such that
0 1
2
≤ ≤
= =
p A
p U p
A A
( )
( ) 1and ( ) 0
If and are subs
1
Φ
ets of
( ) ( ) ( ) (
2 2 1
U
p A A p A p A p A A
1 1 2
∪ = +? ∩ )
Two events A
1
and A
2
are called mutually exclu-
sive if and only if A A
1 2
∩ =φ (null set),These
events are said to be independent if and only if
p A A p A p A( ) ( ) ( ).
1 2 1 2
∩ =
Conditional Probability and Bayes’ Rule
The probability of an event A,given that an event B
has occurred,is called the conditional probability
and is denoted p(A/B),Further,
119
p A B
p A B
p B
( )
( )
( )
/ =
∩
Bayes’ rule permits a calculation of an a posteriori
probability from given a priori probabilities and is
stated below:
If A A A
n1 2
,,…,are n mutually exclusive
events,and p A p A p A
n
( ) (,
1 2
1) ( )+ +
…
+ = and
B is any event such that p(B) is not 0,then the
conditional probability p B
i
(A )/ for any one
of the events A
i
,given that B has occurred,is
p A B
p A p B A
p A p B A p A p B
i
i i
( )
( ) ( )
( ) ( / ) ( ) (
/
/
=
+
1 1 2
/ /A p A p B A
n n2
) ( ) ( )+
…
+
example
Among five different laboratory tests for
detecting a certain disease,one is effective
with probability 0.75,whereas each of the
others is effective with probability 0.40,A
medical student,unfamiliar with the advan-
tage of the best test,selects one of them and is
successful in detecting the disease in a patient,
What is the probability that the most effective
test was used?
Let B denote (the event) of detecting the
disease,A
1
the selection of the best test,and A
2
the selection of one of the other four tests; thus,
p A p A,p B / A(,( ) ( ) 0.75
21 1
1 5 4 5) / /= = =
and p B / A( ) 0.40.
2
= Therefore,
p A B
.
.( )
1
5
(0.75)
1
5
(,)
4
5
( )
1
0 75 0 40
0 319/ =
+
=
120
Note,the a priori probability is 0.20; the out-
come raises this probability to 0.319.
Expected Value
For the random variable X that assumes n finite values
x
1
,x
2
,…,x
n
,with corresponding probabilities P(x
i
) such
that P x
i
n
( ) =
∑
1
1
,the expected value (also called the
mean) is given by E x x P x
i i
( ) ( )=
∑
,For a continuous
random variable with a x b≤ ≤,
E x xP x
a
b
( ) ( )=
∫
.
10,Binomial Distribution
In an experiment consisting of n independent trials in
which an event has probability p in a single trial,the
probability p
X
of obtaining X successes is given by
P C p q
X n X
X n X
=
( ),
( )
where
q p C
n
X n X
n,X
=? =
(1 )and
!
! ( )!
)(
.
The probability of between a and b successes (both a
and b included) is P P P
a a b
+ +
…
+
+1
,so if a = 0 and
b = n,this sum is
X
n
n X
X n X n
n
n
n
n
C p q q C q p
C q
=
∑
= +
+
0
1
1
2
( ) ( )
)
,
( )
,
(,
+
…
+ = + =
2 2
1p p q p
n n
( ),
121
11,Mean of Binomially Distributed Variable
The mean number of successes in n independent tri-
als is m = np with standard deviation σ= npq.
12,Normal Distribution
In the binomial distribution,as n increases the histo-
gram of heights is approximated by the bell-shaped
curve (normal curve),
Y e
x m
=
1
2
2 2
2
σ π
σ( ) /
where m = the mean of the binomial distribution =
np,and σ= npq is the standard deviation,For any
normally distributed random variable X with mean
m and standard deviation σ,the probability function
(density) is given by the above.
The standard normal probability curve is given by
y e
z
=
1
2
2
2
π
/
and has mean = 0 and standard deviation = 1,The
total area under the standard normal curve is 1,Any
normal variable X can be put into standard form by
defining Z X m=?( ) ;/ σ thus,the probability of X
between a given X
1
and X
2
is the area under the stan-
dard normal curve between the corresponding Z
1
and
Z
2
(Table A.1).
122
Normal Approximation to the Binomial Distribution
The standard normal curve is often used instead of
the binomial distribution in experiments with discrete
outcomes,For example,to determine the probability
of obtaining 60 to 70 heads in a toss of 100 coins,we
take X = 59.5 to X = 70.5 and compute correspond-
ing values of Z from mean np = 100
1
2
50=,and the
standard deviation σ= =( ) (1/2)(1/2),100 5 Thus,
Z = (59.5 – 50)/5 = 1.9 and Z = (70.5 – 50)/5 = 4.1,
From Table A.1,the area between Z = 0 and Z = 4.1
is 0.5000,and between Z = 0 and Z = 1.9 is 0.4713;
hence,the desired probability is 0.0287,The binomial
distribution requires a more lengthy computation:
C C
(100,60)
60 40 61
(1/2) (1/2) (1/2) (1+
(,)100 61
/2)
(1/2) (1/2)
39
70 30
+
…
+C,
(,)100 70
Note that the normal curve is symmetric,whereas
the histogram of the binomial distribution is sym-
metric only if p = q = 1/2,Accordingly,when p
(hence q) differs appreciably from 1/2,the difference
between probabilities computed by each increases,It
is usually recommended that the normal approxima-
tion not be used if p (or q) is so small that np (or nq)
is less than 5.
13,Poisson Distribution
P
e m
r
m r
=
!
is an approximation to the binomial probability for r
successes in n trials when m = np is small (<5) and
123
the normal curve is not recommended to approxi-
mate binomial probabilities (Table A.2),The vari-
ance σ
2
in the Poisson distribution is np,the same
value as the mean.
example
A school’s expulsion rate is 5 students per
1,000,If class size is 400,what is the prob-
ability that 3 or more will be expelled? Since
p = 0.005 and n = 400,m = np = 2 and r = 3,
From Table A.2 we obtain for m = 2 and r (= x)
= 3 the probability p = 0.323.
14,Empirical Distributions
A distribution that is skewed to the right (positive
skewness) has a median to the right of the mode and
a mean to the right of the median,One that is nega-
tively skewed has a median to the left of the mode
and a mean to the left of the median,An approximate
relationship among the three parameters is given by
Median (mean) /3(mode)= +
2 3 1/
Skewness may be measured by either of the follow-
ing formulas:
Skewness (mean mode)/s
Skewness (mean media
=?
=?3 n /s)
15,Estimation
Conclusions about a population parameter such as
mean μ may be expressed in an interval estimation
124
containing the sample estimate in such a way that
the interval includes the unknown μ with probability
(1 – α),A value Z
α
is obtained from the table for
the normal distribution,For example,Z
α
= 1.96 for
α = 0.05,Sample values X X X
n1 2
,,…,permit compu-
tation of the variance s
2
,which is an estimate of σ
2
,
A confidence interval for μ is
( )X Z s n X Z s n? +
α α
/,/
For a = 0.05 this interval is
(,,)X s n X s n? +1 96 1 96/,/
The ratio s n/ is the standard error of the mean
(see Section 17).
16,Hypotheses Testing
Two groups may have different sample means and it
is desired to know if the apparent difference arises
from random or significant deviation in the items of
the samples,The null hypothesis (H
0
) is that both
samples belong to the same population,i.e.,the dif-
ferences are random,The alternate hypothesis (H
1
)
is that these are two different populations,Test pro-
cedures are designed so one may accept or reject
the null hypothesis,The decision to accept is made
with probability α of error,The value of α is usually
0.05,0.01,or 0.001,If the null hypothesis is rejected,
though correct,the error is called an error of the first
kind,The error of acceptance of the null hypothesis,
when false,is an error of the second kind.
125
17,t-Distribution
In many situations,μ and σ are unknown and must be
estimated from X and s in a sample of small size n,
so use of the normal distribution is not recommended,
In such situations the Student’s t-distribution is used
and is given by the probability density function
y A t f
f
= +
+
(1 )
2 1 2
/
( )/
where f stands for degrees of freedom and A is a
constant
= +Γ Γ( )f f f π/ / / ( / )2 1 2 2
so that the total area (probability) under the curve of
y vs,t is 1,In a normally distributed population with
mean μ,if all possible samples of size n and mean
X are taken,the quantity ( ) nX s?μ / satisfies the
t-distribution with
f = n – 1
or
t
X
s n
=
μ
/
.
Thus,confidence limits for μ are
(,)X X +?t s n t s n/ /
where t is obtained from Table A.3 for (n – 1) degrees
of freedom and confidence level (1 – α).
126
18,Hypothesis Testing with t- and Normal Distributions
When two normal,independent populations with
means μ
X
and μ
Y
and standard deviations σ
X
and σ
Y
are
considered,and all possible pairs of samples are taken,
the distribution of the difference between sample
means X Y? is also normally distributed,This distri-
bution has mean μ
X
– μ
Y
and standard deviation
σ σ
X Y
n n
2
1
2
2
+
where n
1
is the sample size of X
i
variates and n
2
is
the sample size of Y
i
variates,The quantity Z com-
puted as
Z
X Y
n n
Y
X Y
=
+
( ) ( )
X
μ μ
σ σ
2
1
2
2
satisfies a standard normal probability curve (Sec-
tion 12).
Accordingly,to test whether two sample means differ
significantly,i.e.,whether they are drawn from the
same or different populations,the null hypothesis
(H
0
) is μ
X
– μ
Y
=0,and
Z
X Y
n n
X Y
=
+
σ σ
2
1
2
2
127
is computed,For sufficiently large samples (n
1
> 30
and n
2
> 30),sample standard deviations s
X
and s
Y
are used as estimates of σ
X
and σ
Y
,respectively,The
difference is significant if the value of Z indicates a
small probability,say,<0.05 (or |Z| > 1.96; Table A.1).
For small samples where the standard deviation of
the population is unknown and estimated from the
sample,the t-distribution is used instead of the stan-
dard normal curve.
t
X Y
s
n
s
n
X Y
=
+
( ) ( )
,
μ μ
2
1
2
2
where s is the,pooled estimate of the standard devia-
tion” computed from
s
n s n s
n n
X Y2 1
2
2
2
1 2
1
2
=
+?
+?
( ) ( 1)
The computed t is compared to the tabular value
(Table A.3) for degrees of freedom f = n
1
+ n
2
– 2 at
the appropriate confidence level (such as α = 0.05 or
0.01),When the computed t exceeds in magnitude the
value from the table,the null hypothesis is rejected
and the difference is said to be significant,In cases
that involve pairing of the variates,such as heart rate
before and after exercise,the difference D = X – Y is
analyzed,The mean (sample) difference D is com-
puted and the null hypothesis is tested from
128
t
D
s n
,
D
=
/
where s
D
is the standard deviation of the set of
differences:
s D D n
D
= ∑
( ) )
2
1 2
1/ (
/
In this case,f = n – 1.
Example,Mean exam scores for two groups of students
on a standard exam were 75 and 68,with other per-
tinent values:
X Y
s s
n n
x y
= =
= =
= =
75 68
4 3
20 18
1 2
Thus,
s
2
2 2
19 4 17 3
36
12 7=
+
=
( )( ) ( )( )
.,
and
t =
+
=
75 68
12 7
20
12 7
18
6 05
.,
.,
From Table A.3,t
0.01
,for 36 degrees of freedom,is
between 2.756 and 2.576; hence,these means are
significantly different at the 0.01 level.
129
19,Chi-Square Distribution
In an experiment with two outcomes (e.g.,heads or
tails),the observed frequencies can be compared
to the expected frequencies by applying the normal
distribution,For more than two outcomes,say,n,the
observed frequencies O,O,O
n1 2
…,and the expected
frequencies,e,e,e,
n1 2
…,are compared with the
chi-square statistic (χ
2
):
χ
2
1
=
=
∑
i
n
i i
i
O e
e
.
( )
2
The X
2
is well approximated by a theoretical distribu-
tion expressed in Table A.4,The probability that X
2
is
between two numbers χ
1
2
and χ
2
2
is the area under
the curve between χ
1
2
and χ
2
2
for degrees of free-
dom f,The probability density function is
y
f
e,
f
f
= ≤ ≤∞
1
2 2
2
2 2
1
2
2
/
)/
/Γ( )
( ) ( 0 ).
2 ( 2
χ
χ χ
In a contingency table of j rows and k columns,
f = (j – 1) (k – 1),In such a matrix arrangement the
observed and expected frequencies are determined
for each of the j × k = n cells or positions and entered
in the above equation.
When f = 1,the adjusted χ
2
formula (Yates’ correc-
tion) is recommended:
χ
adj
2
(| | )
.
2
1
1 2
=
=
∑
i
n
i i
i
O e
e
/
130
X
2
is frequently used to determine whether a popula-
tion satisfies a normal distribution,A large sample
of the population is taken and divided into C classes,
in each of which the observed frequency is noted
and the expected frequency calculated,The latter
is calculated from the assumption of a normal dis-
tribution,The class intervals should contain an
expected frequency of 5 or more,Thus,for the inter-
val ( )X X
i i
,
+1
,calculations of Z X X s
i i
=?( /) and
Z X X s
i i+ +
=?
1 1
( /) are made and the probability is
determined from the area under the standard normal
Example—Contingency Table,Men and women were
sampled for preference of three different brands of
breakfast cereal,The number of each gender that
liked the brand is shown in the contingency table,
The expected number for each cell is given in paren-
theses and is calculated as row total × column total/
grand total,Degrees of freedom = (2 – 1) × (3 – 1) = 2
and X
2
is calculated as:
χ
2
2 2
50 59 7
59 7
60 75 7
75 7
11 4=
+
…
+
=
(,)
.
(,)
.
.
Brands
A B C Totals
Men 50 (59.7) 40 (45.9) 80 (64.3) 170
Women 80 (70.3) 60 (54.1) 60 (75.7) 200
Totals 130 100 140 370
Since the tabular value at the 5% level for f = 2 is 5.99,
the result is significant for a relationship between
gender and brand preference.
131
curve,This probability,(P
i
) × N,gives the expected
frequency for the class interval,Degrees of freedom
= C – 3 in this application of the X
2
test.
20,Least Squares Regression
A set of n values (X
i
,Y
i
) that display a linear trend is
described by the linear equation
Y X
i i
= +α β,Vari-
ables α and β are constants (population parameters)
and are the intercept and slope,respectively,The rule
for determining the line is minimizing the sum of the
squared deviations,
i
n
i i
Y Y
=
∑
1
( )
2
and with this criterion the parameters α and β are
best estimated from a and b,calculated as
b
X Y
X Y
n
X
n
i i
i i
i
=
∑?
∑ ∑
∑?
∑
( )( )
( X )
i
2
2
and
a Y bX=?,
where X and Y are mean values,assuming that for
any value of X the distribution of Y values is normal
with variances that are equal for all X,and the latter
(X) are obtained with negligible error,The null hypoth-
esis,H
0
:β = 0,is tested with analysis of variance:
132
Source SS DF MS
Total ( )Y Y
i
Σ( )Y Y
i
2
n?1
Regression (
)Y Y
i
Σ(
)Y Y
i
2
1
Residual (
)Y Y
i i
Σ(
)Y Y
i i
2
n?2
SS
n
S
Y X
resid
( )?
=
2
2
Computing forms for SS terms are
SS Y Y Y Y n
SS Y
i i i
i
total
2
regr.
( ) ( )
(
=∑? = ∑? ∑
=∑
2 2
/
=
∑? ∑ ∑?
∑? ∑
Y
X Y X Y n
X X n
i i i i
i i
)
( ) ( )
( )
2
2
2
2
/
/
F MS MS
.
=
regr resid.
/ is calculated and compared with
the critical value of F for the desired confidence level
for degrees of freedom 1 and n – 2 (Table A.5),The
coefficient of determination,denoted r
2
,is
r SS SS
2
=
regr,total
/
Example,Given points,(0,1),(2,3),(4,9),(5,16),Analysis
proceeds with the following calculations:
SS DF MS
Total 136.7 3 F = =
121
7 85
15 4
.
.
(significant)
a
Regression 121 1 121
Residual 15.7 2 7 85
2
,S
Y X
=
r
s
b
2
0 885
0 73
=
=
.
.
a
See F-distribution,Section 21.
133
r is the correlation coefficient,The standard error of
estimate is s
Y X?
2
and is used to calculate confidence
intervals for α and β,For the confidence limits of β
and α,
b ts
X X
a ts
n
X
X X
Y X
i
Y X
i
±
∑?
± +
∑?
1
1
2
( )
( )
2
2
where t has n – 2 degrees of freedom and is obtained
from Table A.3 for the required probability.
The null hypothesis,H
0
:β = 0,can also be tested with
the t-statistic:
t
b
s
b
=
where s
b
is the standard error of b:
s
s
X X
b
Y X
i
=
∑
( )
2
1 2/
Standard Error of
Y
An estimate of the mean value of Y for a given value
of X,say,X
0
,is given by the regression equation
Y a bX
0 0
= +,
134
The standard error of this predicted value is given by
S S
n
X X
X X
Y Y X
0
1
2
1
= +
∑?
( )
( )
0
2
i
2
and is a minimum when X X
0
= and increases as
X
0
moves away from X in either direction.
21,Nonlinear Regression Analysis
Given a data set (x
i
,y
i
),i = 1,…,N,it is desired to fit
these to a nonlinear equation.
The basis of nonlinear curve fitting is as follows,A
function Y of x contains,say,two parameters denoted
here by α and β,that is,Y = f(x,α,β),We seek here
a representation in which α and β are estimated by
a and b,These estimates are initially a
0
and b
0
,A
Taylor series representation is made about these ini-
tial estimates a
0
and b
0
:
Y f a b x f a f b≈ + +(,,) ( / )( ) ( / )( )
0 0 0 0
α α β β
Y f a b x f a f b? ≈ +(,,) ( / )( ) ( / )( )
0 0 0 0
α α β β
For this choice of a
0
for α and b
0
for β,each value
x
i
gives the left-hand side of the above equation,
Y
i
– f(a
0
,b
0
,z
i
),denoted here by
Y,The partial deriv-
ativef a/ uses the a
0
and b
0
values and also has a
value for each x
i
value,denoted here by X
1i
,Similarly,
the partial derivativef / β has a value at this x
i
,
which we denote by X
2i
,Thus,we get a set of values
of a dependent variable
Y = cX
1
+ dX
2
that is linearly
135
related to the independent variables X
1
and X
2
,A
multiple linear regression (described below) yields
the two regression coefficients c and d.
There are N data points (x
i
,y
i
),Using estimates (a
0
,b
0
)
of parameters,the data are transformed into three differ-
ent sets,denoted by
Y,X
1
,and X
2
,defined as follows:
(,,)Y y f a b x
i i i
=?
0 0
X
1i
= ( / )f α
X
2i
= ( / )f β
where the partial derivatives are evaluated with a
0
,b
0
at each x
i
value.
Thus,the original data set gives rise to three data
columns of length N:
Y
X
1
X
2
… … …
… … …
The values of
Y,X
1
,and X
2
in the table are entered
into a linear multiple regression procedure to yield
Y = c
X
1
+ d X
2
The coefficients c and d are determined (with stan-
dard errors) from equations given below; these allow
improved estimates of parameters a and b by taking a
new set of estimates a
1
= c + a
0
and b
1
= d + b
0
.
136
The new set of estimates,a
1
and b
1
,are then used to
calculate
Y,X
1
,and X
2
,and the process is repeated
to yield new parameters,a
2
and b
2
(with standard
errors),A stopping criterion is applied,e.g.,if the dif-
ference between two iterates is less than some speci-
fied value,This last set is retained and the last set’s
standard errors are retained as the standard errors of
the final estimate.
Multiple Regression (Equations)
In the discussion on nonlinear curve fitting above,we
saw the need for iterative use of the two-parameter
linear regression given by
Y cX dX= +
1 2
.
At every step of the iterative process a set of X
1
,X
2
and corresponding
Y values is calculated,and at that
step we wish to calculate the coefficients c and d,The
procedure for doing this is a special case of the gen-
eral multiple regression algorithm based on Y = b
0
+
b
1
X
1
+ b
2
X
2
+ … + b
N
X
N
,which estimates all the
coefficients,In our application (two-parameter non-
linear analysis) there is no b
0
term and N = 2,The
data array is that shown above,Our model equation
is given by
Y cX dX= +
1 2
Using a least squares procedure we calculate the fol-
lowing by first getting the determinant D:
D
X X X
X X X
=
∑ ∑
∑ ∑
1
2
1 2
1 2 2
2
=
( )( )
( )∑ ∑ ∑
X X X X
1
2
2
2
1 2
2
The coefficients c and d are calculated:
137
c
YX X X
YX X
D= ÷
∑ ∑
∑ ∑
1 1 2
2 2
2
d
X YX
X X YX
D= ÷
∑ ∑
∑ ∑
1
2
1
1 2 2
The following Gaussian coefficients are needed in
the error estimates and these are given by
c
X
D
11
2
2
=
∑
c
X
D
22
1
2
=
∑
c
X X
D
12
1 2
=
∑
The squared differences between the observed
and estimated
Y values are summed to give
SS Y Y
res obs est
=?
∑
(
)
2
,From SS
res
we get the variance
s
SS
N
res2
2
=
which is used to obtain the needed variances and
standard errors from the following:
V c c s( ) =
11
2
V d c s( ) =
22
2
SE c V c( ) ( )=
SE d V d( ) ( )=
It is seen that the procedure for nonlinear curve fitting
requires extensive computation that is almost always
done on a computer,The iteration stops when the
changes in coefficients c and d become sufficiently
138
small,At that point in the process the standard errors
are those given above at this last turn of the cycle.
22,The F-Distribution (Analysis of Variance)
Given a normally distributed population from which
two independent samples are drawn,these provide
estimates,s
1
2
and s
2
2
,of the variance,σ
2
,Quotient
F s s=
1
2
2
2
/ has this probability density function for f
1
and f
2
degrees of freedom of s
1
and s
2
:
y
f f
f f
f f
f
=
+?
Γ
Γ Γ
1 2
1 2
1 2
2
2 2
1
2
f
f
f f
F
f f F
f
2
2
1
2
2
1 2
2
2 1
+
≤ <∞
+
( )
,(0 )
In testing among k groups (with sample size n) and
sample means A A A
k1 2
,,,…,the F-distribution tests
the null hypothesis,μ μ μ
1 2
= = =…
k
,for the means
of populations from which the sample is drawn,Indi-
vidual values from the jth sample (j = 1 to k) are
denoted A i n
ij
( to ).=1 The,between means” sums
of squares (S.S.T.) is computed
S.S T ( ) ( ) ( ),
2 2 2
,n A A n A A n A A
k
=? +? +
…
+?
1 2
where A is the means of all group means,as well as
the,within samples” sum of squares (S.S.E.),where
S.S.E,( ) (A )
2 2
=? +? +
…
+
= = =
∑ ∑
i
n
i
i
n
i
i
n
A A A
1
1 1
1
2 2
1
∑
( )
2
A A
ik k
139
Then
s
k
1
2
=
S.S.T.
1
and
s
k n
2
2
=
S.S.E
( 1)
are calculated and the ratio F is obtained:
F
s
s
=
1
2
2
2
,
with numerator degrees of freedom k – 1 and denomi-
nator degrees of freedom k(n – 1),If the calculated F
exceeds the tabular value of F at the desired probabil-
ity (say,0.05),we reject the null hypothesis that the
samples came from populations with equal means
(see Table A.5 and gamma function,Section 9.2).
23,Summary of Probability Distributions
Continuous Distributions
Normal
y x m=
1
2
2
σ π
σexp [ ( /2 ]
2
)
Mean = m
Variance
=σ
2
140
Standard Normal
y z=?
1
2
2
π
exp ( /2)
Mean = 0
Variance = 1
F-Distribution
y A
F
f f F
f
f f
=
+
+
1
2
2
1 2
2 1
( );
2
where A
f f
f f
f
f
=
+?
Γ
Γ Γ
1 2
1 2
1
2
2 2
1
2
2
2
2
f
f
Mean =
f
f
2
2
2
Variance
2 (
( 2) ( )
1
2
2
=
+?
f f f
f f f
2
2
2
1 2
2
4
)
Chi-Square
y
f
x x
f
f
=?
1
2 2
1
2
2
2 2
2
/
/Γ( )
exp ( )
2
141
Mean = f
Variance = 2f
Students t
y A t f A
f
f f
f
= + =
+
+
( where
( )
(
1
2 1 2
2 1 2
/ )
/ /
( )/
Γ
Γπ / 2)
Mean = 0
Variance
2
(for )=
>
f
f
f 2
Discrete Distributions
Binomial Distribution
y C p p
n x
x n x
=?
( ),
(1 )
Mean = np
Variance = np (1 – p)
Poisson Distribution
y
e m
x
m x
=
!
Mean = m
Variance = m
142
24,Sample Size Determinations
Single Proportion
The sample size required to detect a difference
between a test proportion,p
1
,and a standard propor-
tion value,p
0
,is calculated from
n
z p p z p p
p p
=
α β0 0 1 1
1 0
2
(1 ) (1 )
where z
α
is the two-tailed z-value from the standard
normal curve for the desired level of significance
and z
β
is the lower one-tailed z-value selected for
the power required (probability of rejecting the null
hypothesis when it should be rejected),For α < 0.05,
z
α
is 1.96,while z
β
is one of the following,–1.28 (90%
power),–0.84 (80% power),or –0.525 (70% power).
example
It is well established that 30% of the residents
of a certain community experience allergy
symptoms each year,It is desired to show that
newly developed preventive inoculations can
reduce this proportion to 10%,We have p
0
=
0.30 and p
1
= 0.10,and thus,at the 5% level of
significance and power 80%,n is given by
n,,= +
{ }
1 96 0 3 0 84 0 3
2
,(,)(0 7) (0.1)(0.9) (0.10,0
33 07
)
.
2
=
meaning that 34 patients should be tested.
143
Two Proportions
When control and treatment groups are sampled,and
the respective proportions expected are p
c
and p
t
,the
needed sample size of each group to show a differ-
ence between these is calculated from
n
z p p z p p p p
p p
c c t t c c
c t
=
+?
α β
2 1(1 ) ) (1 )(
2
example
Suppose shock is known to occur in 15% of
the patients who get a certain infection and
we wish to show that a new preventive treat-
ment can reduce this proportion to 5%; thus,
p
c
= 0.15 and p
t
= 0.05,Using z
α
= 1.96 and
z
β
= –0.84 (for 80% power),the sample size
needed in each group is calculated from
n
.,,,
=
+ +1 96 2 0 84 0 0 5 0(0.15)(0.85),( 05)( 9 ) ( 15)( 8 )
(0.15 )
.
2
0 5
0 05
179 9
2
.
.?
=
Thus,180 patients are needed in each group.
Sample Mean
When the mean of a sample (μ
1
) is to be compared to a
standard value (μ
0
),the number to be sampled in order
to show a significant difference is calculated from
144
n
z z
=
( )
α β
σ
μ μ
1 0
2
where σ is an estimate of the population standard
deviation.
example
A certain kind of light bulb is known to have
a mean lifetime of 1,000 hours,with standard
deviation = 100 hours,A new manufacturing
process is installed by the manufacturer and it
is desired to know whether the mean lifetime
changes by,say,25 hours; thus,μ μ
1 0
25? =,
The sample size required for testing the new
bulbs,based on the 0.05 level of significance
and 90% power,is calculated from
n = +
{ }
=(1 96 1 28 167
2
.,)(100)/25,96
so that 168 bulbs should be tested.
Two Means
When two groups are sampled with the aim of detect-
ing a difference in their means,μ
1
– μ
2
,the sample
size of each group is calculated from
n
z z
=
( )
2
1 2
2
α β
σ
μ μ
145
example
Examination scores of students from two dif-
ferent school districts are being compared
in certain standardized examinations (scale,
200–800,where the standard deviation is
100),A difference in mean scores of 20 would
be regarded as important,Using the 5% level
of significance and 80% power,the number of
student scores from each school district that
should be included is
n = + =2 1 96 0 84 392
2
{( ) }.,(100)/20
146
12
Financial Mathematics
1,Simple Interest
An item or service costs an amount C and is to be
paid off over time in equal installment payments,The
difference between the cost C and the total amount
paid in installments is the interest I,The interest rate
r is the amount of interest divided by the cost and the
time of the loan T (usually expressed in years):
r = I/CT
example
An item purchased and costing $4,000 is to be
paid off in 18 equal monthly payments of $249.
The total amount paid is 18 × $249 = $4,482,
so that I = $482,The time of the loan is 1.5
years; hence,the rate is r = 482/(4000 × 1.5) =
0.0803 or 8.03%.
Note,While the above computation is correct,
the computed rate,8.03%,is misleading,This
would be the true rate only if the $4,482 were
repaid in one payment at the end of 18 months,
But since you are reducing the unpaid balance
with each payment,you are paying a rate higher
than 8.03%,True interest rates are figured on
the unpaid balance,The monthly payment
based on the true rate is discussed below.
147
2,True Interest Formula (Loan Payments)
The interest rate is usually expressed per year; thus,
the monthly rate r is 1/12th of the annual interest
rate,The monthly payment P is computed from the
amount borrowed,A,and the number of monthly
payments,n,according to the formula
P Ar
r
r
r
n
n
=
+
+?
>
( )
( )
( )
1
1 1
0
example
A mortgage of $80,000 (A) is to be paid over
20 years (240 months) at a rate of 9% per year,
The monthly payment is computed from the
above formula with n = 240 months and r =
0.09/12 = 0.0075 per month.
It is necessary to calculate (1 + 0.0075)
240
for use in the formula,This is accomplished
with the calculator key [y
x
]; that is,enter
1.0075,press the [y
x
] key,then 240 = to give
6.00915,The above formula yields
P = × ×?
=
80000 0 0075 6 00915 6 00915 1
719 78
.,/ (,)
$,
example
An automobile costing $20,000 is to be
financed at the annual rate of 8% and paid in
equal monthly payments over 60 months,Thus,
n = 60,A = 20000,and r = 0.08/12 = 0.006667.
First compute (1 + 0.006667)
60
(by entering
1.006667 then pressing the key [y
x
],followed by
60) = 1.48987,Thus,the monthly payment is
148
P = × ×?
=
20000 006667 1 48987 1 48987 1
405 53
.,/ (,)
$,
Table A.6 gives the monthly payment for
each $1,000 of the loan at several different
interest rates.
example
Use Table A.6 to get the monthly payment for
the previous example.
Note that the table entry for 8% and 5 years is
$20.28 per thousand,Since the loan is $20,000,
you must multiply $20.28 by 20,which gives
$405.60,(This differs by a few cents from the
above due to rounding in the tables.)
3,Loan Payment Schedules
Once the monthly loan payment is determined,it usu-
ally remains constant throughout the duration of the
loan,The amount that goes to interest and principal
changes with each payment as illustrated below.
example
Show the payment schedule for a loan of
$10,000 at the annual interest rate of 12%,
which is to be paid in equal monthly payments
over 5 months.
The monthly payment P is computed using
the monthly interest rate r = 0.12/12 = 0.01 and
the formula in Section 2:
149
P = ×
×
10000
0 01 1 01
1 01 1
5
5
(,) (,)
(,)
The value (1.01)
5
is calculated by entering
1.01 then pressing [y
x
] followed by 5 to give
1.0510101,so that the above becomes
P = ×
×
=10000
0 01 1 0510101
1 0510101 1
2060 40
.,
.
.
Thus,monthly payments are $2,060.40,
The first month’s interest is 1% of $10,000,or
$100,Since the monthly payment is constant,
the following table shows the application of
the monthly payment to both principal and
interest as well as the balance.
Payment schedule
Payment To Interest To Principal Balance
1 100 1960.40 8039.60
2 80.40 1980.00 6059.60
3 60.60 1999.80 4059.80
4 40.60 2019.80 2040
5 20.40 2040.00 —
4,Loan Balance Calculation
The balance after some number of payments,illus-
trated in Section 3 above,may be calculated directly
from a formula that is given below,In this calculation
it is assumed that the monthly payments in amount
P are made every month,The amount of these pay-
ments was determined from the original amount of
the loan,denoted A,the number of months of the loan
150
(e.g.,120 months for a 10-year loan),and the monthly
interest rate r as given in Section 3,We now wish to
determine what the balance is after a specific number
of payments,denoted by k,have been made,The bal-
ance is given by
Bal r A
P
r
P
r
r
k
k
= +?
+ >( ) ( )1 0
example
A 15-year loan of $100,000 at 7% annual
interest rate was made and requires a monthly
payment of $899,This monthly payment was
determined from the formula in Section 3,It
is desired to know what the balance is after
5 years (60 payments).
The calculation requires the use of r at the
monthly rate; thus,r = 0.07/12 = 0.0058333,
and substitution yields
Bal
60
60
1 0 0058333 100000
899
0 0058333
= +?
(,)
.
+
=?
899
0 0058333
1 41762 100000 154115 1
.
(,)[,7 154115 17
77 400 43
],
$,.
+
=
5,Accelerated Loan Payment
The monthly payment P on a loan depends on the
amount borrowed,A,the monthly interest rate,r,and
the number of payments,n (the length of the loan),If
the monthly payment is increased to a new amount,
P,′ then the number of monthly payments will be
reduced to some lesser number,n′,which is calcu-
lated as follows:
151
First,calculate term 1 from the formula
term
P
P Ar
1=
′
′?
and term 2:
term r2 1= +( )
From term 1 and term 2 the number of months n′ is
calculated as
′=n
term
term
log( )
log( )
1
2
example
A mortgage of $50,000 for 30 years (360
months) at an annual rate of 8% requires
monthly payments of $7.34 per thousand;
thus,50 thousand requires a monthly payment
of 50 × $7.34 = $367 (see Table A.6),If the
borrower decides to pay more than the required
monthly payment,say $450,how long would it
take to pay off the loan?
The monthly interest rate is 0.08/12 and is
used in the calculations of term 1 and term 2:
term
term
1
450
450 50000 0 08 12
3 8571
2
=
=
=
( )(,/ )
.
(1 0 08 12 1 00667+ =,/ ),
Thus,
′= = =n
log(,)
log(,)
.
.
3 8571
1 00667
0 5863
0 002887
2003 1,months
152
The loan time is reduced to 203.1 months
(16.9 years).
6,Lump Sum Payment
A way to reduce the length of a loan is to make a lump
payment that immediately reduces the amount owed
to some lower amount,which we denote by Bal,The
original monthly payment remains at the amount P,
which was previously determined from the original
terms of the loan,but now the number of future pay-
ments M will be fewer because of the reduction in
the amount owed,This number M is calculated from
quantities X and Y,defined as follows:
X
P
P Bal r
Y r r
=
= + >
( )( )
( )1 0
and
M
X
Y
=
log( )
log( )
example
In a previous example (Section 4) we con-
sidered a situation at the end of 5 years of a
loan of $100,000 for 15 years at the annual
interest rate of 7% (0.0058333/month),The
balance after 5 years was $77,400.43 and the
monthly payment is $899.00 and scheduled to
remain at that amount for the remaining 120
months,Suppose a lump payment of $20,000
is made,thereby reducing the amount owed to
$57,400.43,denoted here by Bal,The monthly
payments remain at $899,The number of
153
future payments M is calculated from the
above formulas:
X
Y
=
=
= +
899
899 57400 43 0 0058333
1 59350
1
(,)(,)
.
( 0 0058333 1 0058333,),=
The quantity M is then calculated:
M = =
log(,)
log(,)
.
1 59350
1 0058333
80 1 months
7,Compound Interest
An amount of money (A) deposited in an interest-
bearing account will earn interest that is added to the
deposited amount at specified time intervals,Rates
are usually quoted on an annual basis,as a percent,
The interest is added at some fixed time interval or
interest period such as a year,a month,or a day,The
annual rate is divided by this interval for the purpose
of calculation; e.g.,if the annual rate is 9% and the
interest period is 1 month,then the periodic rate r is
0.09/12 = 0.0075; if the period is 3 months (quarter of
a year),then r = 0.09/4 = 0.0225,After n time inter-
vals (compounding periods) the money grows to an
amount S given by
S = A(1 + r)
n
where
A = original amount
n = number of interest periods
r = rate per period
154
example
$500 is deposited with an annual interest rate
of 10% compounded quarterly,What is the
amount after 2 years?
A = $500
r = 0.10/4 = 0.025 (the periodic rate =
12-month rate/4)
n = 2/(1/4) = 8 (no,of interest periods)
and
S = 500 × (1.025)
8
S = 500 × 1.2184 = $609.20
If this annual rate were compounded
monthly,then r = 0.10/12 = 0.008333 and n =
2/(1/12) = 24,so that S becomes
S = ×
= × =
500 1 008333
500 1 22038 610 19
24
(,)
,$,
Effective Rate of Interest
When annual interest of,say,8% is compounded at
an interval such as four times per year (quarterly),
the effective yield is calculated by using the annual
rate divided by 4,thus 2% or 0.02,and the number of
compounding periods,in this case 4,Thus,
(1.02)
4
= 1.0824
155
and the effective annual rate is 0.0824,or 8.24%,In
contrast,8% is the nominal rate,Table A.7 shows the
growth of $1 for different effective annual interest
rates and numbers of years.
8,Time to Double (Your Money)
The time (in years) to double an amount deposited
depends on the annual interest rate (r) and is calcu-
lated from the following formula:
Time yrs
r r
( )
log
log( )
.
log( )
=
+
=
+
2
1
0 3010
1
example
For interest rate 6% (r = 0.06),the time in
years is
0 3010
1 06
3010
0 2531
11 89
.
log(,)
.
.
.= = yrs
Table A.8 gives the doubling time for vari-
ous annual interest rates.
9,Present Value of a Single Future Payment
If one is to receive a specified amount (A) of money at
some future time,say,n years from now,this benefit has
a present value (V) that is based on the current interest
rate (r) and calculated according to the formula
V
A
r
n
=
+( )1
156
example
You are to receive $1,000 ten years from now
and the current annual interest rate is 8% (r =
0.08) and constant,The present value of this
benefit is
V = 1000/(1.08)
10
= 1000/(2.1589) = $463.20
10,Regular Saving to Accumulate a Specified Amount
Payments at the Beginning of the Year
We wish to determine an amount P that should be
saved each year in order to accumulate S dollars in
n years,given that the annual interest rate is r,The
payment P,calculated from the formula below,is
made on a certain date and on that same date each
year,so that after n years (and n payments) the desired
amount S is available.
P
rS
r r
r
n
=
+? +
>
+
( ) ( )
( )
1 1
0
1
To make this schedule more clear,say that the pay-
ment is at the beginning of the year,then at the begin-
ning of the next year,and so on for 10 payments,the
last being made at the beginning of the 10th year,At
the end of this 10th year (and no further payments)
we have the amount S,The payment amounts P are
computed from the above formula.
example
It is desired to accumulate $20,000 for college
expenses needed 10 years hence in a savings
account that pays the constant rate of 6%
annually.
S = 20000,r = 0.06,and n = 10.
157
The quantity (1.06)
11
= 1.8983,Thus,
P =
×
=
0 06 20000
1 8983 1 06
1431 47
.
.,
.
so that $1,431.47 must be saved each year.
Payments at the End of the Year
Payments of amount P are deposited in an interest-
bearing account at the end of each year for n years so
that n such payments are made,The annual interest is
r,It is desired to have S dollars immediately after the
last payment,The annual payment P to attain this is
given by the formula
P
rS
r
r
n
=
+?
>
( )
( )
1 1
1
example
It is desired to accumulate $100,000 by making
annual deposits in amount P at the end of each
year for 40 years (say,from age 25 to 65 in
a retirement plan) on the assumption that the
interest rate is 10% per year and remains con-
stant over the entire period,P is then
P =
×
=
0 10 100000
1 10 1
225 94
40
.
(,)
$,
example
It is desired to accumulate $100,000 in 10 years
by making semiannual payments in an account
paying 4% annually,but compounded semi-
annually,i.e.,at the end of each 6-month period,
158
for 20 periods,In this case we use the interest
rate 0.04/5 = 0.02 for the compounding period,
and insert n = 20 into the above formula.
P =
×
=
0 02 100000
1 02 1
4 116
20
.
(,)
,
so that deposits of $4,116 are required every
6 months,(Result rounded to nearest dollar.)
11,Monthly Payments to Achieve a Specified Amount
It is convenient to have tables of monthly payments
for several different annual interest rates and com-
pounding periods,and these are given in Tables A.9
and A.10.
12,Periodic Withdrawals from an Interest-Bearing
Account
Balance Calculation
An account with an initial amount A is earning inter-
est at the rate r,If a fixed amount P is withdrawn
at regular intervals,then the balance B after n with-
drawals is given by
B A r P
r
r
r
n
n
= +?
+
>( )
( )
( )1
1 1
0
In a common application the withdrawals are made
monthly so that the annual interest rate r used in the
formula is the annual rate divided by 12 (with monthly
compounding),In this application the withdrawal
is made at the end of the month,(Note,Balance
decreases only if P > Ar.)
159
example
An account earning interest at 10% per year
and compounded monthly contains $25,000,
and monthly withdrawals of $300 are made
at the end of each month,How much remains
after 6 withdrawals? After 12 withdrawals?
Since the rate is 10% and withdrawals
are monthly,we use the rate r = 0.10/12 =
0.008333,with A = 25,000 and P = 300,First,
for n = 6:
B = ×? ×
25000 1 008333 300
1 008333 1
0 0083
6
6
(,)
(,)
,333
Note,(1.008333)
6
= 1.05105,Thus,
B = ×? ×
25000 1 05105 300
1 05105 1
0 008333
.
.
.
= $,24 438 (rounded)
After 12 withdrawals,
B = ×? ×
25000 1 008333 300
1 008333 1
0 00
12
12
(,)
(,)
,88333
=B $23,848 (rounded)
Figure 12.1 shows the result of depositing
$10,000 at 8% annually (0.6667% monthly)
and withdrawing a specified amount each
month,while Figure 12.2 gives the results for
$20,000 and annual interest 12%.
160
Amount on Deposit
The amount of money A,earning annual interest r,
that must be on deposit in order to withdraw amount
P at the end of each year for n years is given by
A
P
r r
r
n
=?
+
>1
1
1
0
( )
( )
example
For an annual interest rate of 6%,withdrawals
of $1,000 at the end of each of 20 years require
an amount A on deposit that is calculated as
10000
8000
6000
4000
2000
0
0 10 20 30 40 50 60 70
No,Months (withdrawals)
100 per
200 per
300 per
400 per
Balance if $10,000 deposited
and monthly withdrawal in
the specified amount is made
FiGuRe 12.1 Balance of $10,000 for specified
monthly withdrawal,Interest rate is 8% per year.
161
$
.,
$,.
1000
0 06
1
1
1 06
11 469 92
20
=
Note,If the withdrawals are monthly,then the
interest rate is r/12 (assumed monthly com-
pounding) and n is the number of months.
13,Periodic Withdrawals That Maintain the Principal
The amount of monthly withdrawals that will neither
increase nor decrease the principal,called the critical
amount,is given by
30000
25000
20000
15000
10000
5000
0
0 10 20 30 40 50 60
No,Months
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
$100/month
$200/month
$300/month
$500/month
·
×
FiGuRe 12.2 Balance of $20,000 with speci-
fied withdrawals in an account that earns 12% per
year,(Note,Withdrawals up to $200/month do not
decrease the balance.)
162
P = rA
where A is the principal and r is the interest rate.
example
Suppose an amount A = $25,000 is deposited
and r = 0.0083333 (monthly); then
P = ×
=
0 008333 25000
208 32
.
$,
so that $208.32 may be withdrawn monthly
while maintaining the original $25,000.
Figure 12.2 shows the change in principal ($20,000)
following a number of withdrawals for several differ-
ent monthly amounts in an account earning 12% per
year and compounded monthly (r = 0.01),It is note-
worthy that withdrawing less than $200 per month
(critical amount) does not decrease,but actually
increases the principal.
14,Time to Deplete an Interest-Bearing Account
with Periodic Withdrawals
If withdrawals at regular time intervals are in amounts
greater than the critical amount (see Section 13),the
balance decreases,The number of withdrawals to
depletion may be calculated as follows:
n
P r
A P r
r
P Ar=
+
>
log
log( )
( )
/
/
1
163
where
P = monthly amount
A = amount of the principal
r = interest rate
n = number of withdrawals to depletion
example
An account with principal $10,000 is earning
interest at the annual rate of 10% and monthly
withdrawals of $200 are made.
To determine the number of withdrawals
to depletion we use the monthly interest rate,
r = 0.1/12 = 0.008333,with P = 200 and A =
$10,000,The bracketed quantity is
[( /,) / ( /,)]
.
=
200 0 008333 10000 200 0 008333
1 71442
and its logarithm is 0.23406,The quantity in
parentheses is 1.008333 and its logarithm is
0.003604; hence,
n = =
0 23406
0 003604
64 94
.
.
.
Effectively this means 65 payments (months).
15,Amounts to Withdraw for a Specified Number of
Withdrawals I,Payments at the End of Each Year
Suppose an amount A has accumulated in a savings
account or pension plan and continues to earn annual
interest at the rate r,How much can one withdraw
164
each year,at the end of each year,for n years? We
denote the annual withdrawal amount by P and it is
computed from the formula below:
P
Ar
r
r
n
=
+
>
1
1
1
0
( )
( )
example
The amount in savings is $100,000 and regular
payments are desired for 20 years over which
it assumed that the annual rate of interest is
6% and payable once a year,Using r = 0.06,
n = 20,and A = 100,000 in the above gives
P =
×
+
100000 0 06
1
1
1 0 06
20
.
(,)
Note that (1.06)
20
= 3.20713 and its reciprocal
is 0.31180.
Thus,P = 6000/(1 – 0.31180) = $8,718.40.
Payments of $8,718.40 per year at the end
of each year for 20 years are possible from
this $100,000,Of course,if 10 times this,or
$1,000,000,were on hand,then 10 times this,
or $87,184 would be paid for 20 years.
example
If the same amounts above earn 8% annually
instead of 6%,the calculation is
165
P =
×
+
100000 0 08
1
1
1 0 08
20
.
(,)
Note that (1.08)
20
= 4.66096 and its reciprocal
is 0.214548,Thus,
P =? =8000 1 0 214548 10 185 22/(,) $,.
Payments of $10,185.22 are possible for
20 years from the $100,000 fund; from a
$1,000,000 fund the annual payments are
10 times this,or $101,852.20.
16,Amounts to Withdraw for a Specified Number of
Withdrawals II,Payments at the Beginning of
Each Year
An amount A has accumulated in a savings account
or pension plan and continues to earn annual interest
at the annual rate r and is payable yearly,How much
can you withdraw each year,at the beginning of each
of n years? We denote the annual withdrawal amount
by P,and it is computed from the formula below:
P
Ar
r
r
r
n
=
+?
+
>
( )
( )
( )
1
1
1
0
1
example
There is $100,000 in an account that earns 8%
annually,It is desired to determine how much can
be withdrawn (P),at the beginning of each year,
166
for 25 years,In this application,r = 0.08,n = 25
years,and A = 100,000,Thus,P is given by
P =
×
100000 0 08
1 08
1
1 08
24
.
.
(,)
Note that 1.08
24
= 6.34118 and the reciprocal
of this is 0.15770,so that P is given by
P =
8000
1 08 0 15770.,
which is $8,673.97.
example
Suppose that there is $100,000 in an account
earning 8% annually and you desire to with-
draw it at the beginning of each year for only
10 years,The amount per year P is now com-
puted as
P =
×
100000 0 08
1 08
1
1 08
9
.
.
(,)
We calculate that 1.08
9
= 1.9990 and its
reciprocal is 0.50025,so that P is given by
P =
=
8000
1 08 0 50025
13 799 05
.,
,.
Since the original amount is $100,000,this
annual withdrawal amount is 13.799% of the
original,It is convenient to have a table of the
percent that may be withdrawn for a specified
167
number of years at various interest rates,and
this is given in Table A.11,Note that the amount
just calculated can be obtained from the table
by going down to 10 years in the 8% column.
example
Find the percent of a portfolio that may be
withdrawn at the beginning of each year for
15 years if the annual average rate of interest
is 12%.
From Table A.11,in the 12% column,the
entry at 15 years is 13.109%,Thus,a portfolio of
$100,000 allows annual withdrawals of $13,109.
17,Present Value of Regular Payments
Suppose you are to receive yearly payments of a cer-
tain amount over a number of years,This occurs,for
example,when one wins a state lottery,The current
value of this stream of payments depends on the
number of years (n),the interest rate (r) that money
earns (assumed constant),and the amount (P) of the
yearly payment,The current value (V) is computed
from the formula
V
P
r r
r
n
=?
+
>1
1
1
0
( )
( )
example
The current interest rate is 7% and annual pay-
ments of $100 are to be paid for 25 years,The
current value of these payments is
168
V =?
100
0 07
1
1
1 07
25
,(,)
Note,(1.07)
25
= 5.42743; using this in the above
formula we compute
V = $,1165 36
18,Annuities
Deposits at the End of the Year
The same amount,denoted by P,is deposited in an
interest-bearing account at the end of each year,The
annual interest rate is r,At the end of n years these
deposits grow to an amount S given by
S P
r
r
r
n
=
+
>
( )
( )
1 1
0
If the deposits are made every month,the above
formula holds for the accumulated amount after n
months,In this case,the interest rate,r,is the annual
rate divided by 12 and compounded monthly.
example
The sum of $500 is deposited at the end of
every year in an account that earns 6% annu-
ally,What is the total at the end of 12 years?
P r n= = =500 0 06 12,.,and
Thus,
S = ×
+?
500
1 0 06 1
0 06
12
(,)
.
169
We must calculate (1.06)
12
,which equals
2.012196,Thus,the above becomes
S = × = ×
=
500 1 012196 0 06 500 16 8699
8434 97
,/,,
$,,
example
Monthly payments of $500 are made into a
retirement plan that has an average annual
interest rate of 12% with monthly compound-
ing,How much does this grow to in 25 years?
Because payments are made monthly,the
rate r and the value of n must be based on
monthly payments,Thus,the rate r is (0.12/12
= 0.01),and n = 25 × 12 = 300 months,Thus,
the value of S is
S = ×
+?
500
1 0 01 1
0 01
300
(,)
.
Note,(1.01)
300
= 19.7885; thus,
S = × =500 18 7885 01 939 425.,$,/
Table A.12 shows the result of depositing
$1,000 at the end of each year in an account
that earns annual interest at several different
rates (payable yearly).
Deposits at the Beginning of the Year
Amount P is deposited each time and the annual inter-
est rate is r; after n years the accumulated amount is
S given by
S
P
r
r r
n
= +? +
+
[( ) ( )]1 1
1
170
example
$1,000 is deposited at the beginning of each
year in a savings account that yields 8% annu-
ally and paid annually,At the end of 15 years
the amount is S given by
S
S
= ×?
= ×?
1000
0 08
1 08 1 08
12500 3 426
16
.
[(,),]
( ) [,1 080 12500 2 346 29325,],= × =
Thus,the amount grows to $29,325,Table
A.13 illustrates the accumulation of funds
when $1,000 is deposited at the beginning of
each year in an account that earns a specified
annual rate,Note,If interest is paid more often
than once a year,then the effective annual
interest should be used in the application of
these annuity formulas.
19,The In-Out Formula
We wish to determine the amount of money (A) to be
saved each month for a specified number of months
(M) in order that withdrawals of $1,000 monthly for
another specified time (N) may begin,It is assumed
that the interest rate (r) remains constant through-
out the saving and collecting periods and that com-
pounding occurs monthly,Thus,the interest rate,r,is
the annual interest rate divided by 12,and N and M
are in months,The monthly amount,A,which must
be saved is given by the formula
A
r
r r
r
N
N M
=
+?
+
+?
>1000
1 1
1
1
1 1
0
( )
( ) ( )
( )
171
example
The amount to be saved monthly for 15 years
(M = 15 × 12 = 180 months) is to be determined
in order that one can receive $1,000 per month
for the next 10 years (N = 10 × 12 = 120 months),
The annual interest rate is 6%; thus,r = 0.06/12
= 0.005 per month,From the above formula,
A =
1000
1 005 1
1 005
1
1 005 1
120
120 180
(,)
(,) (,)
=?
=
A
A
( )[(,) (,)]1000 0 450367 0 6877139
309..72
Thus,$309.72 must be saved each month
for 15 years in order to receive $1,000 per
month for the next 10 years.
Table A.15,for annual interest 6%,gives
the results of this calculation by reading down
to 15 years and across to 10 years,as well as
a number of different combinations of savings
years and collection years,Tables A.14 to A.17
apply to annual interest rates of 4,6,8,and
10%,The use of these tables is illustrated in
the next example.
example
For an annual interest rate of 4%,how much
should be saved monthly for 25 years in order to
collect $1,000 monthly for the next 20 years?
From Table A.14,reading down to 25 years
and across to 20 years,the table shows $320.97,
Thus,$320.97 must be saved for each $1,000
monthly collected for 20 years,If,say,$3,000
per month is to be collected,we multiply
172
$320.97 by 3 to give $962.91 as the amount to
be saved each month for 25 years.
20,Stocks and Stock Quotations
The stocks of various corporations require familiar-
ity with the terms used and the underlying calcula-
tions,Besides the high,low,and closing price,and
the change from the previous trading day,the stock
quotations,as listed in newspapers,contain addi-
tional terms that are calculated.
Yield,The dividend or other cash distribution that
is paid on the security and usually expressed as a
percentage of the closing price,The dollar amount
of the distribution divided by the closing price,
when multiplied by 100,gives the yield,Thus,a
dividend of $3.50 for a stock selling for $40.75
has a yield of
100 3 50 40 75 8 6× =(,,),%/ (rounded)
Price-earnings ratio (P/e),The closing price
divided by the earnings per share (for the most
recent four quarters); for example,if annual earn-
ings = $2.25 for the above stock,priced at $40.75,
then P/E = 40.75/2.25 = 18.1.
Volume,The volume traded,usually on a daily basis,is
quoted in units of 100,For example,a volume figure
of 190 means 190 × 100 = 19,000 shares traded.
A listing might look as follows:
Stock Div Yield Vol Hi Lo Close Change
XYZ 3.50 8.6 190 42
1
/4 40
1
/8 40
3
/4 +
1
/2
which means that this stock attained daily highs
and lows of 42
1
/4 and 40
1
/8,respectively,and
173
closed at
1
/2 above the previous day’s closing price
of 40
1
/4.
21,Bonds
Bonds are issued by many corporations (and govern-
ments),usually with a par value or face value of
$1,000,and mature at a specified time that is part of
the quotation information found in newspapers,The
corporation (or government) thus promises to pay the
face value of $1,000 at maturity and also pays interest
to the bond holder,The quotation also includes this
annual interest expressed in percent,Although the
face value of the bond may be $1,000,the price that
purchases it is based on units of $100; for example,
the quoted purchase price,such as $95,means that
the bond costs 10 times this,or $950,whereas a pur-
chase listing of $110 would mean that it costs $1,100,
Thus,XYZ corporation bonds that pay interest at
8.5% and mature in 1998 would be listed as
XYZ 8? 98
If the purchase price is $110,then the cost (without
commission) is 10 × $110 = $1,100 but pays interest
of 8.5% of the face value of $1,000,or $85,This is
the amount paid annually regardless of the purchase
price,Thus,the effective yield is computed from this
earned interest and the purchase price:
100 × (85/1100) = 7.7%
The listing,as published in newspapers,might look
as follows:
Bond Current Yield Close Net Change
XYZ 8? 98 7.7 110 +?
174
The last column,“Net Change,” means that the clos-
ing price on the previous trading day was 109?,The
quotation might also include the sales volume (usu-
ally in units of $1,000) as well as the high and low
prices of the bond during the trading day.
Bond Value
The value of a bond is determined from the number
of years to maturity and the amount of the annual
coupon payments paid each year until the bond
matures,The face value (par value) of most bonds is
$1,000.00,The current value uses the current interest
rate,e.g.,7%,to compute the current value of $1,000
at 7% for the number of years to maturity,such as 30,
This is given by 1000/(1 + 0.07)
30
= $131.37,This is
the first part of the computation,The next part uses
the amount of the coupon payment,e.g.,$70 per year
for 30 years,This is calculated from the product of
$70 and the factor [1 – 1/(1.07)
30
]/0.07,This factor is
12.4090 and when multiplied by $70 gives $868.63,
This is the second part of the calculation,When these
parts are added,$131.37 + $863.63,the sum is $1,000,
Accordingly,this bond is presently worth $1,000,i.e.,
a bond with face value of $1,000 that pays $70 per
year for 30 years should have a current selling price
of $1,000 (assuming safety) based on the current
interest rate of 7%.
The two parts of the calculation are based on the for-
mulas below,in which r is the annual interest rate and
N is the number of years:
Z face value r
N
= +( ) ( )/ 1
175
The second part uses these values and the annual
payment C:
T
C
r r
N
=?
+
1
1
1( )
example
The previously illustrated 30-year bond pays
$70 per year,but the current interest rate is now
only 6%,For this calculation we need (1.06)
30
,
which is 5.7435,Thus,Z = $1,000/5.7435 and
T = (70/0.06) × (1 – 1/5.7435) = $963.54,Add-
ing the two parts,$174.11 + $963.54,gives
$1,137.65,Note,The bond value has increased
as a result of this interest drop.
22,Tax-Free Yield
Certain securities such as municipal bonds may be
purchased tax-free,The relationship between the tax-
free yield (F) and the tax-equivalent yield (T) depends
on one’s tax rate (R) according to the formula
F = T(1 – R)
example
If one is in the 28% tax bracket,i.e.,R = 0.28,
then the tax-free equivalent of a corporate
bond paying 6.5% is
F = 0.065 × (1 – 0.28) = 0.0468,or 4.68%
(The tax rate is taken to be the total of the
federal and effective state rates.)
176
23,Stock Options (Puts and Calls)
Various stock exchanges permit the purchase of stock
options such as,puts” and,calls.” Each of these has
an exercise price and an expiration date,The call
option is the right to buy shares at the exercise price
at any time on or before the expiration date,The put
option is the right to sell shares at the exercise price,
Thus,if the stock of XYZ corporation is currently
trading at 52? ($52.50) and the exercise price is
$50 with an expiration date 3 weeks hence,the call
provides a guarantee of $2.50 if sold now (less com-
missions),Thus,the call has a value of at least that
amount and would sell for even more since the stock
price might increase even further,The price of the
call might thus be $3.25,In contrast,the put,if exer-
cised now,would lose $2.50,a negative value,But
because the exercise date is still weeks away,the put
still has worth since the stock price could fall below
$50 (the exercise price),giving the option some value,
such as
3
/8 (37
1
/2 cents),As the time of expiration gets
nearer,this value would dwindle to zero,The listing
of these options (in early March 1997) would appear
as follows:
XYZ C 521?2
Date Strike Call Put
March 97 50 3
1
/4
3
/8
(C is a code for the exchange.)
If the expiration is a month later,April 1997,the
call and put prices would be greater,say 4 and 1
1
/8,
respectively,because of the time to expiration (the
third Friday of the month).
177
24,Market Averages
The simple average of a set of n numbers,also called
the arithmetic mean,is computed by summing the
numbers and dividing by n,The closing prices of
groups of stocks,such as the stocks of 30 large compa-
nies that comprise the well-known Dow Jones Indus-
trials,provide an average,Because corporations often
split shares,thereby changing their price per share,
and because some of the corporations on the list of 30
may change over time,the simple formula for getting
these averages is modified,For example,in the sum-
mer of 1997 the total of the 30 prices was divided by
0.26908277 to get the average (or average change),For
example,if each gained 1 point,the sum 30 divided
by 0.26908277 is $111.49,a gain in the average,Thus,
even over several years,with stock splits (and even
some different corporations),a change in the average
is a useful indicator of performance.
Other popular averages such as Standard & Poor’s
and the New York Stock Exchange are comprised of
different groups of stocks in segments such as trans-
portation,utilities,etc.,as well as broad,composite
averages,Each group has its own divisor.
25,Mutual and Quotations
Mutual funds are usually listed in newspapers with
values of the net asset value (NAV) of a share,the
buy price of a share,and the change in net asset value
from the previous day’s closing price,The net asset
value is computed as the total of securities and cash
in the fund divided by the number of shares,When
the buy price is greater than the NAV,the difference
is known as the load or cost (commission) of buying
178
the fund,The percent as commission is computed as
100 × load/NAV.
example
The XYZ fund is listed as follows:
Fund NAV Buy Change
XYZ 18.40 19.52 –0.03
The load is 100 × (19.52 – 18.40)/(18.40) = 6.087%.
The listing also indicates that on the pre-
vious trading day the NAV was $18.43,If the
fund is sold without a load,the symbol,NL”
(no load) appears in the buy column,Total
return may be computed from the difference
between your cost (buy price) and the NAV
when you sell and will also include dividend
and distributions that the fund may pay.
example
The fund above,which was purchased at $19.52
per share,attains a net asset value of $22 eight
months later,It also declares a dividend (D) of
25 cents and a capital gain distribution (CG) of
40 cents during that time,These are added to
the difference between the net asset value and
the buy price,and this quantity is divided by
the buy price to give the proportional return
(PR); percent return is 100 × PR:
PR
D CG NAV Buy
Buy
PR
=
+ +?
=
+ +?
( )
.,(,)0 25 0 40 22 19 52
119 52
3 13
19 52
0 1603
.
.
.
.= =
179
Thus,the percent return is 16.03%,Because
this was attained in only 8 months,it is equiva-
lent to a 12-month return obtained by multiply-
ing by
12
8
,or 1.5,Thus,the annual percent
return is 1.5 × 16.03%,or 24.04%.
26,Dollar Cost Averaging
The share price of a stock or mutual fund varies so
that regular investment of a fixed amount of money
will buy more shares when the price is low and fewer
shares when the price is high,The table below illus-
trates the results of investing $100 each month for
9 months in a stock whose price is initially $15.00
and which fluctuates over the 9-month period but
returns to $15.00 per share,The same $100 divided
by the share price gives the number of shares pur-
chased each month,The total number of shares accu-
mulated is 62.742 and has a price of $15 at the end of
9 months so that the total is worth $941.13,This is
a gain of $41.13,even though the share price is the
same at the beginning and end of the time period.
Month Price/Share No,of Shares
1 15.00 6.6667
2 14.50 6.8966
3 14.00 7.1429
4 14.00 7.1429
5 13.50 7.4074
6 14.00 7.1429
7 14.50 6.8966
8 14.75 6.7797
9 15.00 6.6667
Total shares 62.7424
Value = $15.00 × 62.7424 = $941.14
180
27,Moving Average
Stocks,bonds,mutual funds,and other instruments
whose prices change are sometimes plotted along with
their moving average over some specified time interval,
For example,suppose the closing prices of a mututal
fund for a sequence of days were as shown below:
14.00,14.25,14.35,15.02,14.76,14.81,14.92,14.99,
15.32,15.45,15.32,15.05,…,17.45
Illustrated here is the 10-day moving average,The
average of the first 10 prices is the sum (14.00 + 14.25
+ … + 15.45) divided by 10,which is 14.79,The next
average is obtained from day 2 to day 11,that is,
drop 14.00,which is day 1’s price,and average by
summing to day 11 (14.25 + 14.35 + … + 15.32) and
dividing by 10,which gives 14.92,These numbers,
computed on days 10,11,etc.,are the 10-day moving
average values,They are plotted,along with the daily
prices,in the graph in Figure 12.3.
19
18
17
16
15
14
13
0 10 20 30 40 50 60
Day
Price
10-Day Moving Average
Average
FiGuRe 12.3 The moving average.
181
While the daily prices fluctuate considerably,the
moving average has much lower fluctuation,as seen
by the smoother curve,The usefulness of a moving
average is that it indicates the main trend in prices,
Whereas this example uses the 10-day moving
average,other time intervals may be used,such as
30-day,200-day,etc,Some mutual funds use a
39-week moving average.
182
Table of Derivatives
In the following table,a and n are constants,e is the
base of the natural logarithms,and u and v denote
functions of x.
1,
d
dx
a( ) 0=
2,
d
dx
x( ) 1=
3,
d
dx
au a
du
dx
( ) =
4,
d
dx
u v
du
dx
dv
dx
( )+ = +
5,
d
dx
uv u
dv
dx
v
du
dx
( ) = +
6,
d
dx
u v
v
du
dx
u
dv
dx
v
( )/ =
2
7,
d
dx
u nu
du
dx
n n
( ) =
1
8,
d
dx
e e
du
dx
u u
=
9,
d
dx
a a a
du
dx
u
e
u
= (log )
10,
d
dx
u u
du
dx
e
log (1/ )=
11,
d
dx
u e u
du
dx
a a
log (log )(1/ )=
183
12,
d
dx
u vu
du
dx
u u
dv
dx
v v v
e
= +
1
(log )
13,
d
dx
u u
du
dx
sin cos=
14,
d
dx
u u
du
dx
cos sin=?
15,
d
dx
u u
du
dx
tan sec
2
=
16,
d
dx
u u
du
dx
ctn csc
2
=?
17,
d
dx
u u u
du
dx
sec sec tan=
18,
d
dx
u u
du
dx
csc csu ctn=?
19,
d
dx
u
u
dx
dx
usin,( sin )
1
2
=
≤ ≤
1
2
1 1
2
1
1
π π
20,
d
dx
u
u
du
dx
ucos,(0 cos )
=
≤ ≤
1
2
1
1
1
π
21,
d
dx
u
u
du
dx
tan
=
+
1
2
1
1
22,
d
dx
u
u
du
dx
ctn
=
+
1
2
1
1
23,
d
dx
u
u u
du
dx
u πsec,( sec ;
0 sec
=
≤ <?
≤
1
2
1 1
2
1
1
π
1 1
2
u π< )
184
24,
d
dx
u
u u
du
dx
π u πcsc,( csc ;
0
=
< ≤?
1
2
1 1
2
1
1
< ≤
csc )
1 1
2
u π
25,
d
dx
u u
du
dx
sinh cosh=
26,
d
dx
u u
du
dx
cosh sinh=
27,
d
dx
u u
du
dx
tanh sech
2
=
28,
d
dx
u u
du
dx
ctnh csch
2
=?
29,
d
dx
u u u
du
dx
sech sech tanh=?
30,
d
dx
u u u
du
dx
csch csch ctnh=?
31,
d
dx
u
u
du
dx
sinh
=
+
1
1
2
1
32,
d
dx
u
u
du
dx
cosh
=
1
2
1
1
33,
d
dx
u
u
du
dx
tanh
=
1
2
1
1
34,
d
dx
u
u
du
dx
ctnh
=
1
2
1
1
35,
d
dx
u
u u
du
dx
sech
1
2
=
1
1
185
36,
d
dx
u
u u
du
dx
csch
=
+
1
2
1
1
Additional Relations with Derivatives
d
dt
f x dx f t
d
dt
f x dx f t
x f y
a
t
t
a
∫
∫
=
=?
=
( ) ( )
( ) ( )
If ( ),then
dy
dx dx
dy
=
1
If y = f(u) and u = g(x),then
dy
dx
dy
du
du
dx
=?
(chain rule)
If x = f(t) and y = g(t),then
dy
dx
g t
f t
=
′
′
( )
( )
,
and
d y
dx
f t g t g t f (t
f t
2
2
=
′ ′′? ′ ′′
′
( ) ( ) ( ) )
[ ( )]
3
(Note,exponent in denominator is 3.)
Table of Integrals,
Indefinite and Definite Integrals
188
Table of Indefinite Integrals
Basic Forms (all logarithms are to base e)
1,dx x C= +
∫
2,x dx
x
n
C n
n
n
=
+
+ ≠
+
∫
1
1
1,( )
3,
dx
x
∫
= +logx C
4,e dx e C
x x
= +
∫
5,a dx
a
C
x
x
= +
∫
loga
6,sin cosxdx x C=? +
∫
7,cos sinx dx x C= +
∫
8,tan log cosxdx x C=? +
∫
9,sec tan
2
x dx = +
∫
x C
10,csc ctn
2
x dx x C=? +
∫
11,sec tanx x dx= +
∫
sec x c
12,sin
1
2
1
2
sin cos
2
x dx x
∫
=? +x x C
Table of Indefinite Integrals
189
13,cos
2
x dx x x C= + +
∫
1
2
1
2
sin cosx
14,log logxdx x=? +
∫
x x C
15,a a C a
x x
log a dx= + >
∫
,( )0
16,
dx
a x a
C
2 2
1
+
= +
∫
arc tan
x
a
17,
dx
x a a
C x a
x
a
2 2
2 2
1
2?
=
+
+ >
=
+
∫
log
1
2a
log
-
x a
x a
a
,( )
x
C x a+ <,( )
2 2
18,
dx
x a
a C
2 2
2
+
= + +
( )
+
∫
log x x
2
19,
dx
x a
a C
2 2
2
= +?
( )
+
∫
log x x
2
20,
dx
a x
x
a
2 2
= +
∫
arcsin C
21,a x dx x a
a
C
2 2 2 2
=? +
+
∫
1/2 arcsinx a
x
2
22,a x dx
x a x a x C
2 2
2 2 2 2
+
= + + + +
{ }
+
∫
1/2 log )x a
2
(
23,x a dx
a a x x a C
2 2
2 2 2 2
= +?
{ }
+
∫
1/2 logx x )
2
(
Table of Indefinite Integrals
190
Form ax + b
In the following list,a constant of integration C
should be added to the result of each integration.
24,( )
( )
( )
ax b dx
ax b
a m
m
m
m
+ =
+
+
≠
+
∫
1
1
1( )
25,x ax b dx
ax b
a m
b ax b
a m
m
m m
( )
( )
( )
( )
(
+ =
+
+
+
+
+ +2
2
1
2
2 1
1 2
)
,
(,)
∫
≠m
26,
dx
ax b a
ax b
+
= +
∫
1
log( )
27,
dx
ax b a ax b( ) ( )+
=?
+
∫ 2
1
28,
dx
ax b a ax b( ) ( )+
=?
+
∫ 3 2
1
2
29,
x dx
ax b
x
a
b
a
ax b
+
=? +
∫ 2
log( )
30,
x dx
ax b
b
a ax b a
ax b
( ) ( )
log( )
+
=
+
+ +
∫ 2 2 2
1
31,
x dx
ax b
b
a ax b a ax b( ) ( ) ( )+
=
+
+
∫ 3 2 2 2
2
1
32,x ax b dx
a
ax b
m
b ax b
m
m
m m
2
3
1
( )+
=
+
+
+
+
∫
+ +
( )
3
2 ( )
2
3 2
+
+
+
≠
+
b ax b
m
m,,
m2 1
( )
1
( )1 2 3
Form ax + b
191
33,
x dx
ax b a
ax b b ax b b ax b
2
3
2 2
1 1
2
2
+
= +? + + +
( ) ( ) log( )
∫
34,
x dx
ax b a
ax b
b
ax b
b ax b
2
2 3
2
1
2
( )
( ) log( )
+
= +?
+
+
∫
35,
x dx
ax b a
ax b
b
ax b
b
ax b
2
3 3
2
2
1 2
2( )
log( )
( )+
= + +
+
+
∫
36,
dx
x ax b b
x
ax b( )
log
+
=
+
∫
1
37,
dx
x ax b bx
a
b
ax b
x
3 2
1
( )
log
+
=? +
+?
∫
38,
dx
x ax b b ax b b
ax b
x( ) ( )
log
+
=
+
+?
∫ 2 2
1 1
39,
dx
x ax b
ax b
b x ax b
a
b
ax b
x
2 2 2 3
2 2
( ) ( )
log
+
=?
+
+
+
+?
∫
40,x ax b dx
a m n
x ax b mb x ax
m n
m n m
( )
( )
( ) (
+
=
+ +
+?
∫
+?
1
1
1 1
+
=
+ +
+ + +
∫
+?
b dx
m n
x ax b nb x ax b
n
m n m n
)
( ) ( )
1
1
1 1
0 1 0
dx
m m n
∫
> + + ≠(,)
Forms ax + b and cx + d
41,
dx
ax b cx d bc ad
cx d
ax b( )( )
log
+ +
=
+
+
∫
1
Forms ax + b and cx + d
192
42,
x dx
ax b cx d
bc ad
b
a
ax b
d
c
cx
( )( )
log( ) log(
+ +
=
+?
∫
1
+
d)
43,
dx
ax b cx d
bc ad ax b
c
bc ad
cx d
a
( ) ( )
log
+ +
=
+
+
+
∫ 2
1 1
x b+
44,
x dx
ax b cx d
bc ad
b
a ax b
d
bc ad
( ) ( )
( )
log
+ +
=
+
∫ 2
1 cx d
ax b
+
+
Forms with ax b cx d and ax b+ + +,,
45,
x dx
ax b cx d
b
a bc ad ax b
bc a
2
2
2
2
1
( ) ( ) ( )( )
(
+ +
=
+
+
∫
d
d
c
cx d
b bc ad
a
ax b
)
log | |
( )
log ( )
2
2
2
2
+ +
+
46,
ax b
cx d
dx
ax
c
bc ad
c
cx d
+
+
= +
+
∫ 2
log( )
47,( ) ( )
( )
( ) ( )
ax b cx d dx
a m n
ax b cx d
m n
m n
+ +
=
+ +
+ +
∫
+
1
1
1
+ +
∫
n bc ad ax b cx d dx
m n
( ) ( ) ( )
1
Forms with
a
x
b
c
x
d
a
n
d
a
x
b
+
+
+
,
,
193
Forms with ax b+
48,ax b dx
a
ax b+ = +
∫
2
3
3
( )
49,x ax b dx
ax b
a
ax b+ =
+
∫
2 3 2
15
2
3
( )
( )
50,x ax b dx
a x abx b
a
ax b
2
2 2 2
3
3
2 15 12 8
105
+ =
+
+
∫
(
( )
)
51,x ax b dx
a m
x ax b mb x ax b dx
m
m m
+
=
+
+? +
∫
∫
2
2 3
3 1
( )
( )
52,
( )
( )
( )ax b dx
x
a ax b dx b
ax b dx
x
m
m
m
+
= + +
+
∫∫∫
2
2
2
2
2
53,
dx
x ax b
b
dx
x ax b
a
b
dx
ax b
m m m
( ) ( ) ( )+
=
+
+
∫∫ ∫
2
2
2 2
1
54,
ax bdx
cx d
ax b
c
c
bc ad
c
c ax b bc ad
c
+
+
=
+
+
+
∫
2
1
log
(
( )
(,)
ax b bc ad
c bc ad
+ +?
> >0
55,
ax bdx
cx d
ax b
c
c
ad bc
c
c ax b
ad
+
+
=
+
+
∫
2
2
arc tan
( )
> <
bc
c bc ad(,)0
Forms with
a
x
b
+
194
56,
( )
( )
cx d dx
ax b
a
ad bc acx ax b
+
+
=? + +
∫
2
3
3 2
2
57,
dx
cx d ax b c ad bc
c ax b
ad bc
c
( )
(
+ +
=
+
∫
2
arc tan
( > <0,)bc ad
58,
dx
cx d ax b
c bc ad
c ax b bc ad
c ax b
( )
log
( )
(
+ +
+
+
∫
=
1
) +?
> >
bc ad
c bc ad( 0,)
59,ax b cx d dx acx ad bc x bddx+ + = + + +
∫ ∫
2
( )
(see 154)
60,
ax b dx
x
ax b b
ax b b
ax b b
b
+
= + +
+?
+ +
>
∫
2
0
log
( )
61,
ax bdx
x
ax b b
ax b
b
b
+
= +
+
<
∫
2 2
0
arctan
( )
62,
ax b
x
dx
ax b
x
a dx
x ax b
+
=?
+
+
+
∫ ∫2
2
Forms with
a
x
b
+
195
63,
ax bdx
x
m b
ax b
x
m a ax b d
m
m
+
=?
+
+
+
∫
1
1
2 5
2
3
1
( )
( ) ( ) x
x
m
m?∫
≠
1
1( )
64,
dx
ax b
ax b
a
+
=
+
∫
2
65,
x dx
ax b
ax b ax b
a
+
=
+
∫
2 2
3
2
( )
66,
x dx
ax b
a x abx b ax b
a
2 2 2 2
3
2 3 4 8
15
+
=
+ +
∫
( )
67,
x dx
ax b
a m
x ax b mb
x dx
ax b
m
m
m
+
=
+
+?
+
∫ ∫
2
2 1
1
( )
≠?( )m
1
2
68,
dx
x ax b b
ax b b
ax b b
b
+
=
+?
+ +
>
∫
1
0log ( )
69,
dx
x ax b b
ax b
b
b
+
=
+
<
∫
2
0arctan ( )
Forms with ax b+ and ax
2
+ c
70,
dx
x ax b
ax b
bx
a
b
dx
x ax b
2
2
+
=?
+
+
∫ ∫
Forms with
a
x
b
+
and ax
2
+ c
196
71,
dx
x ax b
ax b
m bx
m a
m b
dx
x
m
m
+
=?
+
∫
( )
( )
( )1
2 3
2 2
1
m
ax b
m
+
≠
∫
1
1( )
72,( )
( )
( )
ax b dx
ax b
a m
m
m
+ =
+
±
±
±
∫
2
2
2
2
2
73,x ax b dx
a
ax b
m
b ax b
m
m m
( )
( ) ( )
+ =
+
±
+
±
±
± ±
2
2
4 2
2
4
2
2
2
m
∫
Form ax
2
+ c
74,
dx
ax c
ac
x
a
c
a c
2
1
0 0
+
=
> >
∫
arc tan (,)
75,
dx
ax c
ac
x a c
x a c
a c
2
1
2
0 0
+
=
+?
> <
∫
log (,)
76,
dx
ax c
ac
c x a
c x a
a c
2
1
2
0 0
+
=
+?
< >
∫
log (,)
77,
x dx
ax c a
ax c
2
2
1
2+
= +
∫
log( )
78,
x dx
ax c
x
a
c
a
dx
ax c
2
2 2
+
=?
+
∫ ∫
79,
x dx
ax c
x
a m
c
a
x dx
ax c
m
m m m
2
1 2
2
1
1
+
=
+
≠
∫ ∫
( )
( )
80,
dx
x ax c c
ax
ax c( )
log
2
2
2
1
2+
=
+
∫
Form ax
2
+ c
197
81,
dx
x ax c cx
a
c
dx
ax c
2 2 2
1
( )+
=
+
∫ ∫
82,
dx
x ax c c m x
a
c
dx
x ax c
m m m
( ) ( ) ( )
(
2 1 2 2
1
1+
=?
+
∫∫
m ≠1)
83,
dx
ax c m c
x
ax c
m
m
m m
( ) ( ) ( )
( )
2 2 1
1
2 1
2 3
2 1
+
=
+
+
∫
c
dx
ax c
m
m
( )
( )
2 1
1
+
≠
∫
Forms ax cand ax bx c
2 2
+ + +
84,
x dx
ax c a m ax c
m
m m
( ) ( )( )
( )
2 2 1
1
2 1
1
+
=?
+
≠
∫
85,
x dx
ax c
x
a m ax c
a m
d
m m
2
2 2 1
2 1
1
2 1
( ) ( )( )
( )
+
=?
+
+
∫
x
ax c
m
m
( )
( )
2 1
1
+
≠
∫
86,
dx
x ax c c m ax c
c
dx
x ax c
m m
( ) ( )( )
(
2 2 1
2
1
2 1
1
+
=
+
+
+
∫
)
( )
m
m
≠
∫ 1
1
87,
dx
x ax c c
dx
x ax c
a
c
dx
ax c
m m m2 2 2 2 1 2
1
( ) ( ) ( )+
=
+
+
∫? ∫∫
(see 82 and 83)
Forms
a
x
c
a
n
d
a
x
b
x
c
2
2
+
+
+
198
Form ax
2
+ bx + c
88,
dx
ax bx c
b ac
ax b b ac
ax b b
2
2
2
2
1
4
2 4
2
+ +
=
+
+ +?
∫
log
4
4
2
ac
b ac( )>
89,
dx
ax bx c
ac b
ax b
ac b
b ac
2
2
1
2
2
2
4
2
4
4
+ +
=
+
<
∫
tan ( )
90,
dx
ax bx c ax b
b ac
2
2
2
2
4
+ +
=?
+
=
∫
,( )
91,
dx
ax bx c
ax b
n ac b ax bx c
n
n
( )
( )( )
2 1
2 2
2
4
+ +
=
+
+ +
+
+∫
2 2 1
4
2 2
( )
( ) ( )
n a
n ac b
dx
ax bx c
n
+ +
∫
92,
xdx
ax bx c
a
ax bx c
b
a
dx
ax bx c
2
2
2
1
2 2
+ +
= + +?
+ +
∫
log ( )
∫
93,
x dx
ax bx c
x
a
b
a
ax bx c
b ac
a
2
2 2
2
2
2
2
2
+ +
=? + +
+
∫
log( )
2 2
dx
ax bx c+ +
∫
Form ax
2
+ bx + c
199
94,
x dx
ax bx c
x
n a
c
a
x dx
ax bx c
n n n
2
1 2
2
1+ +
=
+ +
∫ ∫
( )
b
a
x dx
ax bx c
n?
+ +
∫
1
2
Forms with ax bx c and ax x
2 2
2+ +?
95,
xdx
ax bx c
c bx
n ac b ax bx
n
( )
( )
( )(
2 1 2 2
2
4+ +
=
+
+ +
+∫
c
b n
n ac b
dx
ax bx c
n
n
)
( )
( ) ( )
+ +
∫
2 1
4
2 2
96,
dx
x ax bx c c
x
ax bx c
b
c
dx
( )
log
2
2
2
1
2
2
+ +
=
+ +
∫
( )ax bx c
2
+ +
∫
97,
dx
x ax bx c
b
c
ax bx c
x cx
2 2 2
2
2
2
1
( )
log
+ +
=
+ +
∫
+?
+ +
∫
b
c
a
c
dx
ax bx c
2
2 2
2 ( )
Forms with 2
2
ax x?
98,2
2
2
2
2 2
2
ax x dx
x a
ax x
a x a
a
=
+
∫
arcsin
Forms with
a
x
b
x
c
a
n
d
a
x
x
2
2
2
+
+
200
99,x ax x dx
a ax x
ax x
a x a
a
2
3 2
6
2
2
2
2 2
2
3
=?
+?
+
∫
arcsin
100,x ax x dx
x ax x
m
a m
m
x
m
m
m
2
2
2
2 1
2
2
1 2 3
=?
+
+
+
+
∫
( )
( )
1 2
2ax x dx?
∫
101,
2
2
2
2
ax x dx
x
ax x a
x a
a
=? +
∫
arcsin
102,
2 2
2 3
3
2 3
2
2 2 3
ax x dx
x
ax x
a m x
m
a m
m m
=?
+
∫
( )
( )
( )
aax x dx
x
m
∫
2
1
103,
dx
ax x
x a
a
2
2
=
∫
arcsin
104,
x dx
ax x
ax x a
x a
a
2
2
2
2
= +
∫
arcsin
105,
x dx
ax x
x ax x
m
a m
m
x dx
ax
m m
m
2
2
2 1
2
2
1 2
1
=?
+
∫
( )
x
2
∫
Forms with
2
2
a
x
x
201
Forms with 2
2
ax x? and
Forms a x
2 2
106,
dx
x ax x
ax x
ax
2
2
2
2
=?
∫
107,
dx
x ax x
ax x
a m x
m
a m
dx
x
m
m
m
2
2
2 1
1
2 1
2
2
=?
+
∫
( )
( )
∫
1 2
2ax x
Forms with a x
2 2
108,a x dx x a x a
x
a
2 2 1
2
2 2 2
=? +
∫
arcsin
109,x a x dx a x
2 2 1
3
2 2 3
=
∫
( )
110,x a x dx
x
a x
a
x a x a
x
a
2 2 2 2 2 3
2
2 2 2
4
8
=
+? +
∫
( )
arcsin
111,x a x dx x a a x
3 2 2 1
5
2 2
15
2 2 2 3
=
∫
( ) ( )
112,
a x dx
x
a x a
a a x
x
2 2
2 2
2 2
=
+?
∫
log
Forms with
2
2
a
x
x
and Forms
a
x
2
2
202
113,
a x dx
x
a x
x
x
a
2 2
2
2 2
=?
∫
arcsin
114,
a x dx
x
a x
x a
a a x
x
2 2
3
2 2
2
2 2
2
1
2
=?
+
+?
∫
log
115,
dx
a x
x
a
2 2
=
∫
arcsin
116,
x dx
a x
a x
2 2
2 2
=
∫
117,
x dx
a x
x
a x
a x
a
2
2 2
2 2
2
2 2
= +
∫
arcsin
118,
x dx
a x
a x a a x
3
2 2
1
3
2 2 3 2 2 2
=
∫
( )
119,
dx
x a x
a
a a x
x
2 2
2 2
1
=?
+?
∫
log
120,
dx
x a x
a x
a x
2 2 2
2 2
2
=?
∫
121,
dx
x a x
a x
a x a
a a x
x
3 2 2
2 2
2 2 3
2 2
2
1
2
=?
+?
∫
log
Forms with x a
2 2
122,x a dx
x
x a
a
x x a
2 2 2 2
2
2 2
2 2
= +?
∫
log
Forms with
2
2
a
x
x
and Forms
a
x
2
2
203
123,x x a dx x a
2 2 1
3
2 2 3
=?
∫
( )
124,x x a dx
x
x a
a x
x a
a
x x
2 2 2 2 2 3
2
2 2
4
3
4 8
8
=? +?
+?
∫
( )
log a
2
125,x x a dx x a
a
x a
3 2 2 2 2 5
2
2 2 3
1
5 3
=? +?
∫
( ) ( )
126,
x a dx
x
x a a
a
x
2 2
2 2
=
∫
arccos
127,
x a
x
dx
x
x a x x a
2 2
2
2 2 2 2
1?
=
+ +?
∫
log
128,
x a dx
x
x a
x a
a
x
2 2
3
2 2
2
2
1
2
=?
+
∫
arccos
129,
dx
x a
x x a
2 2
2 2
= +?
∫
log
130,
x dx
x a
x a
2 2
2 2
=?
∫
131,
x dx
x a
x
x a
a
x x a
2
2 2
2 2
2
2 2
2 2
=? + +?
∫
log
132,
x dx
x a
x a a x a
3
2 2
2 2 3 2 2 2
1
3
=? +?
∫
( )
Forms x a and a x
2 2 2 2
+
133,
dx
x x a
a
a
x
2 2
1
=
∫
arccos
Forms with
x
a
2
2
204
134,
dx
x x a
x a
a x
2 2 2
2 2
2
=
∫
135,
dx
x x a
x a
a x a
a
x
2 2 2
2 2
2 2 2
2
1
2
=
+
∫
arccos
Forms with a x
2 2
+
136,a x dx
x
a x
a
x a x
2 2 2 2
2
2 2
2 2
+ = + + + +
( )∫
log
137,x a x dx a x
2 2 2 2 3
1
3
+ = +
∫
( )
138,x a x dx
x
a x
a x
a x
a
x a
2 2 2 2 2 3
2
2 2
4
2
4 8
8
+ = +? +
+ +
∫
( )
log x
2
( )
139,x a x dx x a a x
3 2 2 1
5
2 2
15
2 2 2 3
+ =?
( )
+
∫
( )
140,
a x dx
x
a x a
a a x
x
2 2
2 2
2 2
+
= +?
+ +
∫
log
141,
a x dx
x
a x
x
x a x
2 2
2
2 2
2 2
+
=?
+
+ + +
( )∫
log
142,
a x dx
x
a x
x a
a a x
x
2 2
3
2 2
2
2 2
2
1
2
+
=?
+
+ +
∫
log
143,
dx
a x
x a x
2 2
2 2
+
= + +
( )∫
log
Forms with
a
x
2
2
+
205
144,
x dx
a x
a x
2 2
2 3
+
= +
∫
145,
x dx
a x
x
a x
a
x a x
2
2 2
2 2
2
2 2
2 2
+
= +? + +
( )∫
log
Forms a x and ax bx c
2 2 2
+ + +
146,
x dx
a x
a x a a x
3
2 2
1
3
2 2 3 2 2 2
+
= +? +
∫
( )
147,
dx
x a x
a
a a x
x
2 2
2 2
1
+
=?
+ +
∫
log
148,
dx
x a x
a x
a x
2 2 2
2 2
2
+
=?
+
∫
149,
dx
x a x
a x
a x a
a a x
x
3 2 2
2 2
2 2 3
2 2
2
1
2
+
=?
+
+
+ +
∫
log
Forms with ax bx c
2 2
+ +
150,
dx
ax bx c
a
ax b ax bx c a
2
2
1
2 2 0
+ +
= + + + +
( )
>
∫
log,a
151,
dx
ax bx c a
ax b
b ac
a
2
1
2
1 2
4
0
+ +
=
<
∫
sin,
152,
x dx
ax bx c
ax bx c
a
b
a
dx
ax bx c
2
2
2
2
+ +
=
+ +
+ +
∫ ∫
Forms
a
x
a
n
d
a
x
b
x
c
2
2
2
+
+
+
206
153,
x dx
ax bx c
x
an
ax bx c
b n
an
x
n n
n
2
1
2
1
2 1
2
+ +
= + +
∫
( ) ddx
ax bx c
c n
an
x dx
ax bx c
n
2
2
2
1
+ +
+ +
∫
∫
( )
154,ax bx c dx
ax b
a
ax bx c
ac b
a
dx
ax b
2 2
2
2
2
4
4
8
+ + =
+
+ +
+
+
∫
x c+
∫
155,x ax bx c dx
ax bx c
a
b
a
ax bx c dx
2
2
2
3
2
3
2
+ + =
+ +
+ +
∫
∫
( )
156,x ax bx c dx x
b
a
ax bx c
a
2 2
2
5
6 4
5
3
2
+ + =?
+ +
+
∫
( )
( b ac
a
ax bx c dx
2
2
2
4
16
+ +
∫
)
Form
ax bx c
2
+ +
157,
dx
x ax bx c
c
ax bx c c
x
b
c
c
2
2
1
2
+ +
=?
+ + +
+
>
∫
log,0
Form
a
x
b
x
c
2
+
+
207
158,
dx
x ax bx c c
bx c
x b ac
c
2
1
2
1 2
4
0
+ +
=
+
<
∫
sin,
159,
dx
x ax bx
bx
ax bx c
2
2
2
0
+
=? + =
∫
,
160,
dx
x ax bx c
ax bx c
c n x
b n
c
n
n
2
2
1
1
3 2
2
+ +
=?
+ +
+
∫?
( )
( )
( )
( )
( )
n
dx
x ax bx c
a n
c n
dx
x ax
n
n
+ +
+
∫
1
2
1
1 2
2 2
+ +
∫
bx c
161,
dx
ax bx c
ax b
b ac ax bx c
b
( )
( )
( )
,
2 2 2
2
3
2
2 2
4+ +
=?
+
+ +
≠
∫
4ac
162,
dx
ax bx c a x b a
b ac
( ) ( / )
,
2 3 2
2
3
2
1
2 2
4
+ +
=?
+
=
∫
Miscellaneous Algebraic Forms
163,
a x
b x
dx a x b x
a b a x b x
a x
+
+
= + +
+? + + +
( )
+ >
∫
( )( )
( )log
( 0 0andb x+ > )
164,
a x
b x
dx a x b x a b
b x
a b
+
=? + +
+
∫
( )( ) ( )arcsin
Miscellaneous Algebraic Forms
208
165,
a x
b x
dx a x b x a b
b x
a b
+
=? + + +
+
+
∫
( )( ) ( )arcsin
166,
1
1
1
2
+
= +
∫
x
x
dx x xarcsin
167,
dx
x a b x
x a
b a
( )( )
arcsin
=
∫
2
168,
dx
ax b
k
b
x k
k
k x
x kx k
3
2 2
3
3
2
3
+
=
+
+
+
∫
arctan log
≠ =
b k
b
a
0
3
,
Form ax bx c
2
+ + and
Miscellaneous Algebraic Forms
169,
x dx
ax b
ak
x k
k
k x
x kx k
3
2 2
1
3
3
2
3
+
=
+
+
∫
arctan log
≠ =
b k
a
b
0
3
,
170,
dx
x ax b bm
x
ax b
b
m
m
m
( )
log ( )
+
=
+
≠
∫
1
0
171,
dx
ax x
x a
a ax x2 2
2 3 2 2
=
∫
)
Form
a
x
b
x
c
2
+
+
and Miscellaneous Algebraic Forms
209
172,
x dx
ax x
x
a ax x
( )
)
2
2
2 3
2
=
∫
173,
dx
ax x
x a ax x
2
2
2
2
+
= + + +
∫
log
174,
cx d
ax b
dx
ax b cx d
a
ad bc
a
dx
ax b cx d
+
+
=
+? +
+
+? +
∫
( )
2
∫
Trigonometric Forms
175,(sin ) cosax dx
a
ax=?
∫
1
176,(sin ) cos sin
sin
2
1
2
1
2
1
2
1
4
2
ax dx
a
ax ax x
x
a
a
∫
=? +
=? x
177,(sin ) (cos )(sin )
3 2
1
3
2ax dx
a
ax ax=? +
∫
178,(sin )
sin sin
4
3
8
2
4
4
32
∫
=? +ax dx
x ax
a
ax
a
179,(sin )
sin cos
(sin )
n
n
n
ax dx
ax ax
na
n
n
ax d
∫
= +
1
2
1
x
∫
180,
dx
ax
ax dx
a
ax
sin
(csc ) cot
2
2
1
= =?
∫∫
Trigonometric Forms
210
181,
dx
ax
ax dx
m a
ax
a
m
m
m
sin
(csc )
( )
cos
sin
∫ ∫
=
=?
1
1
1
x
m
m
dx
ax
m
+
∫
2
1
2
sin
182,sin( ) cos( )a bx dx
b
a bx+ =? +
∫
1
183,
dx
ax a
ax
1
1
4 2±
=
∫
sin
tan
π
184,
sin
sin
tan
ax
ax
dx x
a
ax
1
1
4 2±
=± +
∫
π
185,
dx
ax ax a
ax
a(sin )( sin )
tan log
1
1
4 2
1
±
=
+
∫
π
tan
ax
2
186,
dx
ax a
ax
a
( sin )
tan
tan
1
1
2 4 2
1
6 4
2
3
+
=
∫
π
π
ax
2
187,
dx
ax a
ax
a( sin )
cot cot
1
1
2 4 2
1
6 4
2
3
=?
+?
∫
π π aax
2
188,
sin
( sin )
tan t
ax
ax
dx
a
ax
a
1
1
2 4 2
1
6
2
+
=
+
∫
π
an
3
4 2
π
ax
Trigonometric Forms
211
189,
sin
( sin )
cot c
ax
ax
dx
a
ax
a
1
1
2 4 2
1
6
2
=
+
∫
π
oot
3
4 2
π
ax
190,
sin
sin sin
x dx
a b x
x
b
a
b
dx
a b x+
=?
+
∫ ∫
191,
dx
x a b x a
x b
a
dx
a b x(sin )( sin )
logtan
sin+
=?
+
∫ ∫
1
2
192,
dx
a b x
b x
a b a b x
a
a b
d
( sin )
cos
( )( sin )
+
=
+
+
∫ 2
2 2 2 2
x
a b x+
∫
sin
193,
sin
( sin )
cos
( )( sin )
x dx
a b x
a x
b a a b x
b
b
+
=
+
+
∫ 2
2 2 2
+
∫
a
dx
a b x
2
sin
194,1 2
2 2
+ =±?
∫
sin sin cosx dx
x x
use if (8
otherwise ;
+? < ≤ +
k x k
k
1
2
8 3
2
) ( ),
π π
an integer?
Trigonometric Forms
212
195,
1 2
2 2
=± +
∫
sin sin cosx dx
x x
use if (8
otherwise ;
+? < ≤ +
k x k3
2
8 1
2
) ( ),
π π
k an integer?
196,(cos ) sinax dx
a
ax=
∫
1
197,(cos ) sin cos
sin
2
1
2
1
2
1
2
1
4
2
ax dx
a
ax ax x
x
a
ax
∫
= +
= +
198,(cos ) (sin )(cos )
3 2
1
3
2ax dx
a
ax ax= +
∫
199,(cos )
sin sin
4
3
8
2
4
4
32
ax dx
x ax
a
ax
a
= + +
∫
200,(cos )
cos sin (cos )
n
n n
ax dx
na
ax ax
n
n
ax
∫
= +
1 1
1 2
ddx
∫
201,(cos )
sin ( )!( !)
( )!(
2
2
2 2
2
2 2 1
m
m r
ax dx
ax
a
m r
r
∫
=
+
m
ax
m
m
x
r
m
r
m
!)
cos
( )!
( !)
2
2 1
2 2
0
1
2
2
+
=
+
∑
202,(cos )
sin ( !) ( )!
( )
2 1
2 2 2
2 2
2 1
m
m r
ax dx
ax
a
m r
m
+
∫
=
+ !( !)
cos
r
ax
r
m
r
2
0
2
=
∑
Trigonometric Forms
213
203,
dx
ax
ax dx
a
ax
cos
(sec ) tan
2
2
1
= =
∫∫
204,
dx
ax
ax dx
n a
ax
ax
n
n
n
cos
(sec )
( )
sin
cos
∫ ∫
=
=
1
1
1
+
∫
n
n
dx
ax
n
2
1
2
cos
205,cos( ) sin( )a bx dx
b
a bx+ = +
∫
1
206,
dx
ax a
ax
1
1
2+
=
∫
cos
tan
207,
dx
ax a
ax
1
1
2?
=?
∫
cos
cot
208,
dx
a b x
a b
a b
x
a b
a
+
=
+
∫
cos
tan
tan
2
2
2 2
1
2 2
2
or
1
b
2
log
tan
tan
b a
x
a b
b a
x
a b
2 2
2 2
2
2
+ +
209,
cos
cos
tan
ax
ax
dx x
a
ax
1
1
2+
=?
∫
210,
cos
cos
cot
ax
ax
dx x
a
ax
1
1
2?
=
∫
Trigonometric Forms
214
211,
dx
ax ax a
ax
a(cos )( cos )
logtan
1
1
4 2
1
+
= +
∫
π
tan
ax
2
212,
dx
ax ax a
ax
a(cos )( cos )
logtan
1
1
4 2
1
= +
∫
π
cot
ax
2
213,
dx
ax a
ax
a
ax
( cos )
tan tan
1
1
2 2
1
6 2
2
3
+
= +
∫
214,
dx
ax a
ax
a
ax
( cos )
cot cot
1
1
2 2
1
6 2
2
3
=
∫
215,
cos
( cos )
tan tan
ax
ax
dx
a
ax
a
ax
1
1
2 2
1
6 2
2
3
+
=?
∫
216,
cos
( cos )
cot cot
ax
ax
dx
a
ax
a
ax
1
1
2 2
1
6 2
2
3
=?
∫
217,
cos
cos cos
x dx
a b x
x
b
a
b
dx
a b x+
=?
+
∫∫
218,
dx
x a b x
a
x b
a
dx
(cos )( cos )
logtan
+
= +
∫
1
2 4
π
a b x+
∫
cos
219,
dx
a b x
b x
b a a b x
a
b a
d
( cos )
sin
( )( cos )
+
=
+
∫ 2
2 2 2 2
x
a b x+
∫
cos
Trigonometric Forms
215
220,
cos
( cos )
sin
( )( cos )
x
a b x
dx
a x
a b a b x
a
a
+
=
+
∫ 2
2 2 2
+
∫
b
dx
a b x
2
cos
221,1
2
1
2 2
2
=?
=?
∫
cos
sin
cos
cosax dx
ax
a ax
a
ax
222,1
2
1
2 2
2
+ =
+
=
∫
cos
sin
cos
sinax dx
ax
a ax
a
ax
223,
dx
x
x
1
2
4
=±
∫
cos
logtan,
[use + if 4kπ < x < (4k + 2)π,otherwise –;
k an integer]
224,
dx
x
x
1
2
4
+
=±
+?
∫
cos
logtan,
π
[use + if (4k – 1)π < x < (4k +1)π,otherwise –;
k an integer]
225,(sin )(sin )
sin( )
( )
sin( )
mx nx dx
m n x
m n
m n x
∫
=
+
2 2
2 2
( )
,
( )
m n
m n
+
≠
226,(cos )(cos )
sin( )
( )
sin( )
mx nx dx
m n x
m n
m n x
∫
=
+
+
2 2
2 2
( )
,
( )
m n
m n
+
≠
227,(sin )(cos ) sinax ax dx
a
ax
∫
=
1
2
2
Trigonometric Forms
216
228,(sin )(cos )
cos( )
( )
cos( )
mx nx dx
m n x
m n
m n
∫
=?
+
2
x
m n
m n
2
2 2
( )
,
( )
+
≠
229,
(sin )(cos ) sin
2 2
1
32
4
8
ax ax dx
a
ax
x
∫
=? +
230,(sin )(cos )
cos
( )
ax ax dx
ax
m a
m
m
∫
=?
+
+1
1
231,(sin )(cos )
sin
( )
m
m
ax ax dx
ax
m a
=
+
+
∫
1
1
232,
sin
cos cos
secax
ax
dx
a ax
ax
a
2
1
∫
= =
233,
sin
cos
sin logtan
2
1 1
4 2
ax
ax
dx
a
ax
a
ax
∫
=? + +
π
234,
cos
sin sin
cscax
ax
dx
a ax
ax
a
2
1
∫
=? =?
235,
dx
ax ax a
ax
(sin )(cos )
logtan
∫
=
1
236,
dx
ax ax a
ax
ax
(sin )(cos )
sec logtan
2
1
2
= +
∫
237,
dx
ax ax
a n ax
dx
ax
n
n
(sin )(cos )
( )cos (sin
∫
=
+
1
1
1
)(cos )
n
ax
∫ 2
238,
dx
ax ax a
ax
a
ax
(sin )(cos )
logtan
2
1 1
4 2
∫
=? + +
csc
π
Trigonometric Forms
217
239,
dx
ax ax a
ax
(sin )(cos )
cot
2 2
2
2
∫
=?
240,
sin
cos
log( cos )
ax
ax
dx
a
ax
1
1
1
±
= ±
∫
241,
cos
sin
log( sin )
ax
ax
dx
a
ax
1
1
1
±
=± ±
∫
242,
dx
ax ax
a ax a
(sin )( cos )
( cos )
logtan
1
1
2 1
1
2
±
=±
±
+
∫
aax
2
243,
dx
ax ax
a ax a
(cos )( sin )
( sin )
logtan
1
1
2 1
1
2
±
=
±
+
∫
π
4 2
+
ax
244,
sin
(cos )( cos )
log(sec )
ax
ax ax
dx
a
ax
1
1
1
±
= ±
∫
245,
cos
(sin )( sin )
log(csc )
ax
ax ax
dx
a
ax
1
1
1
±
=? ±
∫
246,
sin
(cos )( sin )
( sin )
lo
ax
ax ax
dx
a ax a
1
1
2 1
1
2
±
=
±
±
∫
gtan
π
4 2
+
ax
247,
cos
(sin )( cos )
( cos )
l
ax
ax ax
dx
a ax a
1
1
2 1
1
2
±
=?
±
±
∫
oogtan
ax
2
Trigonometric Forms
218
248,
dx
ax ax
a
ax
sin cos
logtan
±
= ±
∫
1
2
2 8
π
249,
dx
ax ax a
ax
(sin cos )
tan
±
=
∫ 2
1
2 4
π
250,
dx
ax ax a
ax
1
1
1
2+ ±
=± ±
∫
cos sin
log tan
251,
dx
a cx b cx abc
b cx a
b cx
2 2 2 2
1
2cos sin
log
tan
tan?
=
+
∫
a
252,
cos
sin
log sin sin
ax
b ax
dx
ab
b ax b ax
1
1
1
2 2
2 2
+
= + +
(
∫
)
253,
cos
sin
sin ( sin )
ax
b ax
dx
ab
b ax
1
1
2 2
1
=
∫
254,(cos ) sin
sin
sinax b ax dx
ax
a
b ax
ab
1
2
1
1
2
2 2 2 2
+ = +
+
∫
log sin sinb ax b ax+ +
( )
1
2 2
255,(cos ) sin
sin
sin
ax b ax dx
ax
a
b ax
ab
1
2
1
1
2
2 2
2 2
=? +
∫
sin ( sin )
1
b ax
256,(tan ) logcos logsecax dx
a
ax
a
ax=? =
∫
1 1
257,(cot ) logsin logcscax dx
a
ax
a
ax= =?
∫
1 1
Trigonometric Forms
219
258,(sec ) log(sec tan )
logtan
ax dx
a
ax ax
a
ax
∫
= +
= +
1
1
4
π
2
259,(csc ) log(csc cot ) logtanax dx
a
ax ax
a
ax
=? =
∫
1 1
2
260,(tan ) tan
2
1
ax dx
a
ax x=?
∫
261,(tan ) tan logcos
3 2
1
2
1
ax dx
a
ax
a
ax= +
∫
262,(tan )
tan
tan
4
3
3
1
ax dx
ax
a a
x x
∫
=? +
263,(tan )
tan
( )
(tan )
n
n
n
ax dx
ax
a n
ax dx
∫ ∫
=
1
2
1
Forms with Trigonometric Functions
and Inverse Trigonometric Functions
264,(cot ) cot
2
1
ax dx
a
ax x=
∫
265,(cot ) cot logsin
3 2
1
2
1
ax dx
a
ax
a
ax=
∫
266,(cot ) cot cot
4 3
1
3
1
ax dx
a
ax
a
ax x=? + +
∫
267,(cot )
cot
( )
(cot )
n
n
n
ax dx
ax
a n
ax dx=?
∫ ∫
1
2
1
Forms with Trigonometric Functions and Inverse Trigonometric Functions
220
Forms with Inverse
Trigonometric Functions
268,(sin ) sin
= +
∫
1 1
2 2
1
ax dx x ax
a x
a
269,(cos ) cos
=?
∫
1 1
2 2
1
ax dx x ax
a x
a
270,(tan ) tan log( )
∫
=? +
1 1 2 2
1
2
1ax dx x ax
a
a x
271,(cot ) cot log( )
∫
= + +
1 1 2 2
1
2
1ax dx x ax
a
a x
272,(sec ) sec log
=? +?
( )∫
1 1 2 2
1
1ax dx x ax
a
ax a x
273,(csc ) csc log
= + +?
( )∫
1 1 2 2
1
1ax dx x ax
a
ax a x
274,x ax dx
a
a x ax ax
[sin ( )]
( )sin ( )
∫
=? +?
1
2
2 2 1
1
4
2 1 1 a x
2 2
275,x ax dx
a
a x ax ax
[cos ( )]
( )cos ( )
∫
=
1
2
2 2 1
1
4
2 1 1 a x
2 2
Mixed Algebraic and
Trigonometric Forms
276,x ax dx
a
ax
x
a
ax(sin ) sin cos=?
∫
1
2
Forms with Inverse Trigonometric
Functions
221
277,x ax dx
x
a
ax
a x
a
ax
2
2
2 2
3
2 2
(sin ) sin cos
∫
=?
278,x ax dx
a x
a
ax
a x x
a
ax
3
2 2
4
2 3
3
3 6 6
(sin ) sin cos
∫
=
279,x ax dx
a
ax
x
a
ax(cos ) cos sin= +
∫
1
2
280,x ax dx
x ax
a
a x
a
ax
2
2
2 2
3
2 2
(cos )
cos
sin
∫
= +
281,x ax dx
a x
a
ax
a x x
a
ax
3
2 2
4
2 3
3
3 6 6
(cos ) cos sin=
+
∫
282,x ax dx
x x ax
a
ax
a
(sin )
sin cos
2
2
2
4
2
4
2
8
=
∫
283,x ax dx
x x
a a
ax
x
2 2
3 2
3
6 4
1
8
2
(sin )
sin
co
∫
=
s2
4
2
ax
a
284,x ax dx
x ax
a
ax
a
x ax
(sin )
cos sin cos
3
2
3
12
3
36
3
4
∫
=
a
ax
a
+
3
4
2
sin
285,x ax dx
x x ax
a
ax
a
(cos )
sin cos
2
2
2
4
2
4
2
8
= + +
∫
286,x ax dx
x x
a a
ax
x
2 2
3 2
3
6 4
1
8
2
(cos )
sin
co
∫
= +?
+
s2
4
2
ax
a
Mixed Algebraic and Trigonometric
Forms
222
287,
x ax dx
x ax
a
ax
a
x ax
(cos )
sin cos sin
3
2
3
12
3
36
3
4
∫
= + +
a
ax
a
+
3
4
2
cos
288,
sin sin
( )
cosax
x
dx
ax
m x
a
m
ax
x
dx
m m m∫ ∫
=?
+
1 1
1 1
289,
cos cos
( )
sinax
x
dx
ax
m x
a
m
ax
x
dx
m m m∫ ∫
=?
1 1
1 1
290,
x
ax
dx
x ax
a ax a
a
1 1
1
1
2
±
=
±
+ ±
∫
sin
cos
( sin )
log( sin x)
291,
x
ax
dx
x
a
ax
a
ax
1 2
2
2
2
+
= +
∫
cos
tan logcos
292,
x
ax
dx
x
a
ax
a
ax
1 2
2
2
2
=? +
∫
cos
cot logsin
293,
x x
x
dx x
x+
+
=
∫
sin
cos
tan
1 2
294,
x x
x
dx x
x?
=?
∫
sin
cos
cot
1 2
295,
x
ax
dx x ax dx
x ax
a a
a
sin
(csc )
cot
logsin
2
2
2
1
∫ ∫
=
=? + x
Mixed Algebraic and Trigonometric
Forms
223
296,
∫ ∫
=
=?
x
ax
dx x ax dx
x ax
a n
n
n
n
sin
csc )
cos
( sin
(
)1
1
aax
a n n ax
n
n
x
n
n
+
∫
1
1
2
1
2 2
( )( 2)sin
( )
( ) sin
2
ax
dx
297,
∫ ∫
=
= +
x
ax
dx x ax dx
a
x ax
a
cos
sec )
tan logcos
2
2
2
1 1
(
aax
298,
∫ ∫
= =
x
ax
dx x ax dx
x ax
a n a
n
n
n
cos
sec )
sin
( ) cos
1
(
1 x
a n n ax
n
n
x
ax
dx
n
n
+
∫
1
1
2
1
2 2
2
( ) ( 2) cos
cos
Logarithmic Forms
299,
∫
=?(log ) logx dx x x x
300,
∫
=?x x dx
x
x
x
(log ) log
2 2
2 4
Logarithmic Forms
224
Mixed Algebraic and Trigonometric
Forms and Logarithmic Forms
301,
∫
=?x x dx
x
x
x
2
3 3
3 9
(log ) log
302,
∫
=
+
+
+ +
x ax dx
x
n
ax
x
n
n
n n
(log ) log
( )
2
1 1
1 1
303,
∫
=? +(log ) log ) log 2
2
x dx x x x x x(
2
2
304,
∫
=
+
+
(log )
(log )
x
x
dx
n
x
n
n
1
1
1
305,
∫
= + +
+
dx
x
x x
x x
log
log (log ) log
log
!
(log )
3
( )
2
2 2 3?
+
…
3!
306,
∫
=
dx
x x
x
log
log (log )
307,
∫
=?
dx
x x n x
n n
(log ) ( 1) (log )
1
1
308,[
∫
+ =
+
+?log( )] log ( )ax b dx
ax b
a
ax b x
309,
∫
+
=?
+
+
log(
log log ( )
ax b
x
dx
a
b
x
ax b
bx
ax b
)
2
310,
∫
+
= + +
log
( ) log( ) ( log (
x a
x a
dx
x a x a x a x)?a)
Mixed Algebraic and Trigonometric Forms and Logarithmic Forms
225
311,
∫
∫
=
+
+
+
+ +
x X dx
x
n
X
c
n
x
X
dx
b
n
n
n n
(log )
log
1 2
1
2
1 1
∫
+
= + +
x
X
dx
X a bx cx
n 1
2
where
312,
∫
+
= +? +
[log( )]
log ( ) tan
1
x a dx
x x a x a
x
a
2 2
2 2
2 2
313,
∫
= +
+
[log( )]
log ( ) log
x a dx
x x a x a
x a
x a
2 2
2 2
2
314,
∫
±
= ± ±?
x x a dx
x a x a x
[log( )]
( ) log ( )
2 2
1
2
2 2 2 2 1
2
2
315,log
log
x x a dx
x x x a x a
+ ±
( )
= + ±
( )
±
∫
2 2
2 2 2 2
316,x x x a dx
x a
x x
log
log
+ ±
( )
= ±
+
∫
2 2
2 2
2
2 4
±
( )
±
a
x x a
2
2 2
4
317,x x x a dx
x
m
x x a
m
m
log
log
+ ±
( )
=
+
+ ±
( )
∫
+
2 2
1
2 2
1
+
±
∫
+
1
1
1
2 2
m
x
x a
dx
m
Mixed Algebraic and Trigonometric Forms and Logarithmic Forms
226
318,
log
log
log
2
x x a
x
dx
x x a
x a
a x
+ +
( )
=?
+ +
( )
+ +
∫
2 2
2 2
2
1 a
x
2
319,
log
log
sec
2
x x a
x
dx
x x a
x a
x
+?
( )
=?
+?
( )
+
∫
2 2
2 2
1
1
| | a
Exponential Forms
320,
∫
=e dx e
x x
321,
∫
=?e dx e
x x
322,
∫
=e dx
e
a
ax
ax
323,
∫
=?x e dx
e
a
ax
ax
ax
2
1( )
324,
∫ ∫
=?
+
e
x
dx
m
e
x
a
m
e
x
dx
ax
m
ax
m
ax
m
1
1 1
1 1
325,
∫ ∫
=?e x dx
e x
a a
e
x
dx
ax
ax ax
log
log 1
326,
∫
+
=? + =
+
dx
e
x e
e
e
x
x
x
x
1 1
log (1 ) log
Exponential Forms
227
327,
∫
+
=? +
dx
a be
x
a ap
a be
px
px
1
log ( )
328,
∫
+
=
> >
dx
ae be
m ab
e
a
b
a b
mx mx
mx
1
0 0
1
tan,
(,)
329,
∫
=
+
( )
log
a a dx
a a
a
x x
x x
330,
∫
+
= +
e
b ce
dx
ac
b ce
ax
ax
ax
1
log ( )
331,
∫
+
=
+
x e
ax
dx
e
a ax
ax ax
(1 ) (1 )
2 2
332,x e dx e
x x
∫
=?
2 2
1
2
333,
∫
=
+
e bx dx
e a bx b bx
a b
ax
ax
[sin( )]
[ sin( ) cos( )]
2 2
334,
∫
=
e bx cx dx
e b b c x
ax
ax
[sin( )] [sin( )]
[( c) sin ( ) +?
+?
+ +
a b c x
a b c
e b c b c
ax
cos ( ) ]
[ ( ]
[( ) sin ( )
2
2 2
)
x a b c x
a b c
+ +
+ +
cos ( ) ]
[ ( ]2
2 2
)
Exponential Forms
228
335,
∫
=
+
+e bx dx
e
a b
a bx b bx
ax
ax
[cos ( )] [ cos ( ) sin( )]
2 2
336,
∫
=
e bx cx dx
e b b c x
ax
ax
[cos ( )] [cos(
[( c) sin ( )
)]
+?
+?
+
+ +
a b c x
a b c
e b c b c
ax
cos ( ) ]
[ ( ]
[( ) sin ( )
2
2 2
)
x a b c x
a b c
+ +
+ +
cos ( ) ]
[ ( ]2
2 2
)
337,
∫
=
+
e bx dx
a n b
a bx nb bx e
ax n
ax
[sin ]
( sin cos s
1
2 2 2
) in
( ) sin
n
ax n
bx
n n b e bx dx
+?
∫
1
2 2
1 [ ]
Hyperbolic Forms
338,
∫
=
+
+
e bx dx
a n b
a bx nb bx e
ax n
ax
[cos ]
( cos sin c
1
2 2 2
) oos
( ) [cos ]
n
ax n
bx
n n b e bx dx
+?
∫
1
2 2
1
339,
∫
=
+
x e bx dx
x e
a b
a bx b bx
e
ax
ax
a
(sin ]
( sin cos )
)
2 2
x
a b
a b bx ab bx
( )
[( )sin cos ]
22 2
2 2
2
+
Hyperbolic Forms
229
340,
∫
=
+
+
x e bx dx
x e
a b
a bx b bx
e
ax
ax
a
(cos ]
( cos sin )
)
2 2
x
a b
a b bx ab bx
( )
[( ) cos sin ]
22 2
2 2
2
+
+
341,
∫
=(sinh ) coshx dx x
342,
∫
=(cosh ) sinhx dx x
343,
∫
=(tanh ) log coshx dx x
344,
∫
=(coth ) log sinhx dx x
345,
∫
=
(sech ) tan (sinh )x dx x
1
346,
∫
=
csch log tanhx dx
x
2
347,
∫
=?x x dx x x x(sinh ) cosh sinh
348,
∫ ∫
=?
x x dx x x n x x dx
n n n
(sinh ) cosh (cosh )
1
349,
∫
=?x x dx x x x(cosh ) sinh cosh
350,
∫ ∫
=?
x x dx x x n x x dx
n n n
(cosh ) sinh (sinh )
1
351,
∫
=?( (sech ) tanh ) sechx x dx x
352,
∫
=?( )csch ) (coth cschx x dx x
Hyperbolic Forms
230
353,
∫
=?(sinh )
sinh
2
x dx
x x2
4 2
354,
∫
=?(tanh ) tanh
2
x dx x x
355,
∫ ∫
=?
+
≠
( (tanh )
tanh
tanh ),
( 1)
2n
n
n
x dx
n
x dx
n
1
1
356,
∫
=(sech ) tanh
2
x dx x
357,
∫
= +(cosh )
sinh 2
2
x dx
x x
4 2
358,
∫
=?(coth ) coth
2
x dx x x
359,
∫ ∫
=?
+
≠
(coth )
coth
coth,
( 1)
2n
n
n
x dx
x
n
x dx
n
1
1
Hyperbolic Forms
231
Table of Definite Integrals
360,
1
1
1
∞
∫
=
>
dx
x m
m
m
,[ 1]
361,
0
1
∞
∫
+
= <
dx
x x
p p
p
(
csc,[ 1]
)
π π
362,
0
1
∞
∫
=? <
dx
x x
p p
p
(
cot,[ 1]
)
π π
363,
0
1
1
1
∞
∫
+
= =? =?
<
x dx
x p
p p p p
p
π
πsin
B (,) ( ) (1 ),
[0
Γ Γ
p <1]
364,
0
1
1
∞
∫
+
= < <
x dx
x
n
n
m n
m
n
π
π
sin
m
,[0 ]
365,
0
1
∞
∫
+
=
dx
x x( )
π
366,
a dx
a x
a a
2 2
0
2 2+
= > =?
∞
∫
π π
,if 0; 0,if 0;
if
,
a < 0
367,
0
1
0
∞
∫
= >e dx
a
a
ax
,( )
368,
0
0
∞
∫
= >
e e
x
dx
b
a
a b
ax bx
log,(,)
Table of Definite Integrals
232
369,x e dx
n
a
n a
n
a
a
n ax
n
n
+
+
=
+
>? >
>
Γ( )
,(,)
or
!
,(
1
1 0
1
1
0
0
,positive integer)n
∞
∫
370,x ax dx
k
pa
n p a k
n p
k
exp ( )
( )
,
,
=
>? > > =
∞
∫
Γ
0
1 0 0,,
n
p
+?
1
371,e dx
a a
a
a x?
∞
∫
= =
>
2 2
0
1
2
1
2
1
2
0π,( )Γ
372,xe dx
x?
∞
∫
=
2
0
1
2
373,x e dx
x2
0
2
4
∞
∫
=
π
374,x e dx
n
a a
n ax
n n
2
0
1
2 1 3 5 1
2
∞
+∫
=
…(2 ) π
375,x e dx
n
a
a
n ax
n
2 1
0
1
2
0
+
∞
+∫
= >
!
2
,( )
376,x e dx
m
a
e
a
r
m ax
m
a
r
r
m
0
1
1
0
∫
∑
+
=
=?
!
1
!
377,
0
22
2
2
2
0
∞
∫
= ≥e dx
e
a
x
a
x
a
( )
π
,( )
Table of Definite Integrals
233
378,e x dx
n n
nx?
∞
∫
=
0
1
2
π
379,
0
∞
∫
=
e
x
dx
n
nx
π
380,
0
2 2
0
∞
∫
=
+
>e mx dx
a
a m
a
ax
(cos ),( )
381,
0
2 2
0
∞
∫
=
+
>e mx dx
m
a m
a
ax
(sin ),( )
382,xe bx dx
ab
a b
a
ax
0
2 2
2
0
∞
∫
=
+
>[sin ( )]
( )
,( )
2
383,xe bx dx
a b
a b
a
ax
0
2 2
2 2
0
∞
∫
=
+
>[cos ( )]
( )
,( )
2
384,x e bx dx
n a ib a ib
n ax
n n
0
1
∞
+ +
∫
=
+
[sin ( )]
![( ) ( )
1
1
2
]
2 ( )
,
(,)
i a b
i a
n2 2
1 0
+
=? >
+
385,x e bx dx
n a ib a ib
n ax
n n
0
1
∞
+ +
∫
=
+ +
[cos ( )]
![( ) ( )
1
1
2
]
2 ( )
,(,)
a b
i a
n2 2
1 0
+
=? >
+
386,
0
0
∞
∫
= >
e x
x
dx a a
ax
sin
cot,( )
1
387,e bx dx
a
b
a
ab
a x?
∞
∫
=?
≠
2 2
0
2
2
2 4
0cos exp,(
π
)
Table of Definite Integrals
234
388,
0
1
∞
∫
=e t t dt b
t bcos
sin ( sin )] [ ( )] sin
φ
φ Γ ( ),b
b,
π π
φ
φ>? < <
0
2 2
389,
0
1
∞
∫
=e t t dt b
t bcos
[cos ( sin )] [ ( )] co
φ
φ Γ s ( ),b
b
φ
π
φ
π
>? < <
0
2 2
,
390,
0
1
2
∞
∫
=
< <
t t dt b
b
b
b
cos [ ( )] cos,
(0 1
Γ
π
)
391,
0
1
2
∞
∫
=
< <
t t dt b
b
b
b
(sin ) [ ( )] sin,
(0
Γ
π
1)
392,
0
1
1
∫
( )
=log ( !x dx n
n
n
)
393,
0
1
1
2
2
∫
=log
1
x
dx
π
394,
0
1
1
2
∫
=
log
1
x
dx π
395,
0
1
∫
=log
1
!
x
dx n
n
396,
0
1
3
4∫
=?x x dxlog(1 )
Table of Definite Integrals
235
397,
0
1
1
4∫
+ =x x dxlog(1 )
398,
0
1
1
1
1 0
∫
=
+
>? =
+
x x dx
n
m
m n
m n
n
n
(log )
( 1) !
( )
,
,1,,2,…
If n ≠ 0,1,2,…,replace n! by Γ( ).n+1
399,
0
2 2
1
∞
∫
= < <
sin
( ) sin ( / )
,0
x
x
dx
p p
p
p
π
πΓ
400,
0
2 2
1
∞
∫
= < <
cos
( ) cos ( / )
,0
x
x
dx
p p
p
p
π
πΓ
401,
0
2
2
∞
∫
=
1 cos px
x
dx
pπ
402,
0
0 0
2
0
∞
∫
= > > > >
sin cos;,;
4
,
px qx
x
dx
q p p q,
π π
p q= >
0
403,
0
2 2
∞
∫
+
=
cos ( )
2|a|
| |
mx
x a
dx e
ma
π
404,
0
2
0
2
1
2 2
∞ ∞
∫ ∫
= =cos( ) sin( )x dx x dx
π
405,
0
1
1
1
∞
∫
= >sin (1/ ) sin
2
,ax dx
na
n
n
n
n
n/
Γ
π
406,
0
1
1
1
∞
∫
= >cos (1/ ) cos
2
,ax dx
na
n
n
n
n
n/
Γ
π
Table of Definite Integrals
236
407,
0 0
2
∞ ∞
∫ ∫
= =
sin cosx
x
dx
x
x
dx
π
408,(a)
0
4
∞
∫
=
sin
3
x
x
dx
π
(b)
0
2
3
4
∞
∫
=
sin
log 3
3
x
x
dx
409,
0
3
3
8
∞
∫
=
sin
3
x
x
dx
π
410,
0
4
3
∞
∫
=
sin
4
x
x
dx
π
411,
0
2
1
2
1
1
1
π/
∫
+
=
<
dx
a x
a
a
a
cos
cos
,( )
412,
0 2 2
0
π
π
∫
+
=
> ≥
dx
a b x
a b
a b
cos
,( )
413,
0
2
2
2
1
2
1
1
π
π
∫
+
=
<
dx
a x
a
a
cos
,( )
414,
0
∞
∫
=
cos cos
log
ax bx
x
dx
b
a
415,
0
2
2 2
2
π
π
/
∫
+
=
dx
a x b x absin cos
2 2
Table of Definite Integrals
237
416,
0
2
0
2
1
π
π
/
/
∫
∫
=
(sin )
(cos )
or
1 3 5 7…(
n
x dx
x dx
n
n
)
2 4 6 8…( ) 2
,
( an even integer,)
or
≠
n
n n
π
0
2 4 6 8…( )
1 3 5 7…( )
,
( an odd integer
n
n
n
1
,)
or
,( )
n
n
n
n
≠
+?
+
>?
1
2
1
2
2
1
1
π
Γ
Γ
417,
0
2
0
2
∞
∫
= > =?
sin
,if 0; 0,if ;,i
mx dx
x
m m
π π
f m < 0
418,
0
∞
∫
= ∞
cos x dx
x
419,
0
2
∞
∫
=
tan x dx
x
π
Table of Definite Integrals
238
420,
0 0
0
π π
∫ ∫
=? =
≠
sin sin cos,
(
ax bx dx ax bx dx
a
cos
b a b;,integers)
421,[sin ( )] [cos ( )]
[sin ( )]
0
0
π
π
/a
ax ax dx
ax
∫
∫
= [cos ( )]ax dx = 0
422,[sin ( )] [cos ( )]
,if i
0
2 2
2
π
∫
=
ax bx dx
a
a b
a b s odd,or 0 if is evena b?
423,
0
0
1 1
∞
∫
=
<? >
sin cos
,
if or ;
4
,if
x mx dx
x
m m
π
m m=± <1 1
2;
2
,if
π
424,
sin sin
,( )
ax bx
x
dx
a
a b
2
0
2
∞
∫
= ≤
π
425,sin cos
2 2
mx dx mx dx
0 0
2
π π
π
∫ ∫
= =
426,
sin ( )
2
px
x
dx
p
2
0
2
∞
∫
=
π
427,
log x
x
dx
1 12
0
1
2
+
=?
∫
π
428,
log x
x
dx
1 6
0
1
2
=?
∫
π
Table of Definite Integrals
239
429,
log( )1
12
0
1
2
+
=
∫
x
x
dx
π
430,
log( )1
6
0
1
2
=?
∫
x
x
dx
π
431,
0
1
2
2 2
12
∫
+ =(log )[log (1 )] log 2x x dx
π
432,
0
1
2
2
6
∫
=?(log )[log (1 )]x x dx
π
433,
0
1
2
2
1 8
∫
=?
log x
x
dx
π
434,
0
1
2
4
∫
+
=log
1
1
x
x
dx
x
π
435,
0
1
2
1
2
∫
=?
log
log2
x dx
x
π
436,
0
1
1
1
1
∫
=
+
+
+
x
x
dx
n
m
m
n
n
log
1 ( )
( )
,
Γ
if,m n+ > + >1 0 1 0
437,
0
1
1
1
1 0
∫
=
+
+
+ >
( )
log
log,
(
x x dx
x
p
q
p q
p q
,+ >1 0)
438,
0
1
∫
=
dx
x
log
1
π
Table of Definite Integrals
240
439,
0
2
1
1 4
π
π
∫
+
=log
e
e
dx
x
x
440,
0
2
0
2
2
π π
π
/ /
∫ ∫
= =?(log sin ) log cos logx dx x dx 2
441,
0
2
0
2
2
π π
π
/ /
∫ ∫
= =(log sec ) log csc log 2x dx x dx
442,x x dx
0
2
2
π
π
∫
=?(log sin ) log 2
443,
0
2π/
∫
=?(sin ) (log sin ) log2 1x x dx
444,
0
2
0
π/
∫
=(log tan )x dx
445,
0
2 2
2
π
π
∫
±
=
+?
log ( cos )
log,(
a b x dx
a a b
a b≥ )
446,
0
2 2
2 0
π
π
∫
+
=
≥ >
log ( 2 cos )
log,
a ab x b dx
a a b
2 0π log,b b a≥ >
447,
0
2 2
∞
∫
=
sin
sinh
tanh
ax
bx
dx
b
a
b
π π
448,
0
2 2
∞
∫
=
cos
cosh
sech
ax
bx
dx
b b
π απ
Table of Definite Integrals
241
449,
0
2
∞
∫
=
cosh
dx
ax a
π
450,
0
2
2
4
∞
∫
=
x dx
ax asinh
π
451,
0
2 2
∞
∫
=
≤ <e bx dx
a
a b
b a
ax
(cosh ),(0 | )|
452,
0
2 2
∞
∫
=
≤ <e bx dx
b
a b
b a
ax
(sinh ),(0 | )|
453,
0
1 2
1
2
∞
∫
+
=?
sinh
csc
ax
e
dx
b
a
b a
bx
π π
454,
0
1
1
2 2
∞
∫
=?
sinh
cot
ax
e
dx
a b
a
b
bx
π π
455,
0
2
2
2
2
1
2
1
1
2
1 3
2 4
π
π
/
∫
= +
+
dx
k x
k
sin
2
+
+
…
2
4
2
6
2
1 3 5
2 4 5
k k
k
,
if <1
456,
0
2
2
2
2
1
2
1
1
2
1 3
2 4
π
π
/
∫
=?
k x dx
k
sin
2
…
2
4
2
6
3
1 3 5
2 4 6 5
k k
,
if k
2
1<
457,
0
0 5772157
∞
∫
=? =?e x dx
x
log,…γ
Table of Definite Integrals
242
458,
0
2
4
∞
∫
=? +e x dx
x
log ( 2 log 2)
π
γ
Table of Definite Integrals
243
Appendix
TabLe a.1 areas under
the standard Normal Curve
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
Reprinted from Tallarida,R,J,and Murray,R,B.,Manual of Pharmacologic Calculations
with Computer Programs,2nd ed.,1987,With permission of Springer-Verlag,New York.
244
T
ab
L
e
a
.2
P
oisson
d
istrib
ution
Each number in this table represents the probability of obtaining at least
X
successes,or the
area under the histogram to the right of and including the rectangle whose center is at X.
m
X
= 0
X
= 1
X
= 2
X
= 3
X
= 4
X
= 5
X
= 6
X
= 7
X
= 8
X
= 9
X
= 10
X
= 11
X
= 12
X
= 13
X
= 14
.10
1.000
.095
.005
.20
1.000
.181
.018
.001
.30
1.000
.259
.037
.004
.40
1.000
.330
.062
.008
.001
.50
1.000
.393
.090
.014
.002
.60
1.000
.451
.122
.023
.003
.70
1.000
.503
.156
.034
.006
.001
.80
1.000
.551
.191
.047
.009
.001
.90
1.000
.593
.228
.063
.063
.013
.002
1.00
1.000
.632
.264
.080
.019
.004
.001
1.1
1.000
.667
.301
.100
.026
.005
.001
1.2
1.000
.699
.337
.120
.034
.008
.002
1.3
1.000
.727
.373
.143
.043
.011
.002
1.4
1.000
.753
.408
.167
.054
.014
.003
.001
1.5
1.000
.777
.442
.191
.066
.019
.004
.001
1.6
1.000
.798
.475
.217
.079
.024
.006
.001
1.7
1.000
.817
.517
.243
.093
.030
.008
.002
1.8
1.000
.835
.537
.269
.109
.0.36
.010
.003
.001
1.9
1.000
.850
.566
.296
.125
.044
.013
.003
.001
245
m
X
= 0
X
= 1
X
= 2
X
= 3
X
= 4
X
= 5
X
= 6
X
= 7
X
= 8
X
= 9
X
= 10
X
= 11
X
= 12
X
= 13
X
= 14
2.0
1.000
.865
.594
.323
.143
.053
.017
.005
.001
2.2
1.000
.889
.645
.377
.181
.072
.025
.007
.002
2.4
1.000
.909
.692
.430
.221
.096
.036
.012
.003
.001
2.6
1.000
.926
.733
.482
.264
.123
.049
.017
.005
.001
2.8
1.000
.939
.769
.531
.308
.152
.065
.024
.008
.002
.001
3.0
1.000
.950
.801
.577
.353
.185
.084
.034
.012
.004
.001
3.2
1.000
.959
.829
.620
.397
.219
.105
.045
.017
.006
.002
3.4
1.000
.967
.853
.660
.442
.256
.129
.058
.023
.008
.003
.001
3.6
1.000
.973
.874
.697
.485
.294
.156
.073
.031
.012
.004
.001
3.8
1.000
.978
.893
.731
.527
.332
.184
.091
.040
.016
.006
.002
4.0
1.000
.982
.908
.762
.567
.371
.215
.111
.051
.021
.008
.003
.001
4.2
1.000
.985
.922
.790
.605
.410
.247
.133
.064
.028
.011
.004
.001
4.4
1.000
.988
.934
.815
.641
.449
.280
.156
.079
.036
.015
.006
.002
.001
4.6
1.000
.990
.944
.837
.674
.487
.314
.182
.095
.045
.020
.008
.003
.001
4.8
1.000
.992
.952
.857
.706
.524
.349
.209
.113
.056
.025
.010
.004
.001
5.0
1.000
.993
.960
.875
.735
.560
.384
.238
.133
.068
.032
.014
.005
.002
.001
Reprinted from
Alder
,H,L,and Roessler
,E,B.,
Intr
oduction to Pr
obability and Statistics
,6th ed.,1977,
W
ith permission of
W
,H,Freeman,Ne
w
Y
ork.
246
TabLe a.3 t-distribution
Deg,Freedom,f
90%
(P = 0.1)
95%
(P = 0.05)
99%
(P = 0.01)
1 6.314 12.706 63.657
2 2.920 4.303 9.925
3 2.353 3.182 5.841
4 2.132 2.776 4.604
5 2.015 2.571 4.032
6 1.943 2.447 3.707
7 1.895 2.365 3.499
8 1.860 2.306 3.355
9 1.833 2.262 3.250
10 1.812 2.228 3.169
11 1.796 2.201 3.106
12 1.782 2.179 3.055
13 1.771 2.160 3.012
14 1.761 2.145 2.977
15 1.753 2.131 2.947
16 1.746 2.120 2.921
17 1.740 2.110 2.898
18 1.734 2.101 2.878
19 1.729 2.093 2.861
20 1.725 2.86 2.845
21 1.721 2.080 2.831
22 1.717 2.074 2.819
23 1.714 2.069 2.807
24 1.711 2.064 2.797
25 1.708 2.060 2.787
26 1.706 2.056 2.779
27 1.703 2.052 2.771
28 1.701 2.048 2.763
29 1.699 2.045 2.756
inf,1.645 1.960 2.576
Reprinted from Tallarida,R,J,and Murray,R,B.,Manual of
Pharmacologic Calculations with Computer Programs,2nd ed.,
1987,With permission of Springer-Verlag,New York.
247
TabLe a.4 χ
2
-distribution
v 0.05 0.025 0.01 0.005
1 3.841 5.024 6.635 7.879
2 5.991 7.378 9.210 10.597
3 7.815 9.348 11.345 12.838
4 9.488 11.143 13.277 14.860
5 11.070 12.832 15.086 16.750
6 12.592 14.449 16.812 18.548
7 14.067 16.013 18.475 20.278
8 15.507 17.535 20.090 21.955
9 16.919 19.023 21.666 23.589
10 18.307 20.483 23.209 25.188
11 19.675 21.920 24.725 26.757
12 21.026 23.337 26.217 28.300
13 22.362 24.736 27.688 29.819
14 23.685 26.119 29.141 31.319
15 24.996 27.488 30.578 32.801
16 26.296 28.845 32.000 34.267
17 27.587 30.191 33.409 35.718
18 28.869 31.526 34.805 37.156
19 30.144 32.852 36.191 38.582
20 31.410 34.170 37.566 39.997
21 32.671 35.479 38.932 41.401
22 33.924 36.781 40.289 42.796
23 35.172 38.076 41.638 44.181
24 36.415 39.364 42.980 45.558
25 37.652 40.646 44.314 46.928
26 38.885 41.923 45.642 48.290
27 40.113 43.194 46.963 49.645
28 41.337 44.461 48.278 50.993
29 42.557 45.722 49.588 52.336
30 43.773 46.979 50.892 53.672
Reprinted from Freund,J,E,and Williams,F,J.,Elementary Busi-
ness Statistics,The Modern Approach,2nd ed.,1972,With permis-
sion of Prentice Hall,Englewood Cliffs,NJ.
248
TabLe a.5 Variance Ratio
F(95%)
n
1
n
3
1 2 3 4 5 6 8 12 24 ∞
1 161.4 199.5 215.7 224.6 230.2 234.0 238.9 243.9 249.0 254.3
2 18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.41 19.45 19.30
3 10.13 9.55 9.28 9.12 9.01 8.94 8.84 8.74 8.64 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.91 5.77 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.68 4.53 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.00 3.84 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.57 3.41 3.23
8 5.32 4.46 4.07 3.84 3.69 33.58 3.44 3.28 3.12 2.93
9 5.12 4.26 3.86 3.63 3.48 36.37 3.23 3.07 2.90 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.91 2.74 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.79 2.61 2.40
12 4.75 3.88 3.49 3.26 3.11 3.00 2.85 2.69 2.50 2.30
13 4.67 3.80 3.41 3.18 3.02 2.92 2.77 2.60 2.42 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.53 2.35 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.64 2.48 2.29 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.42 2.24 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.55 2.38 2.19 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.34 2.15 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.48 2.31 2.11 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.38 2.08 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.42 2.25 2.05 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.40 2.23 2.03 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.38 2.20 2.00 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.36 2.18 1.98 1.73
25 4.24 3.38 2.99 2.76 2.60 2.49 2.34 2.16 1.96 1.71
26 4.22 3.37 2.98 2.74 2.59 2.47 2.32 2.15 1.95 1.69
27 4.21 3.35 2.96 2.73 2.57 2.46 2.30 2.13 1.93 1.67
28 4.20 3.34 2.95 2.71 2.56 2.44 2.29 2.12 1.91 1.65
29 4.18 3.33 2.93 2.70 2.54 2.43 2.28 2.10 1.90 1.64
30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.09 1.89 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.00 1.79 1.51
60 4.00 3.15 2.76 2.52 2.37 2.25 2.10 1.92 1.70 1.39
120 3.92 3.07 2.68 2.45 2.29 2.17 2.02 1.83 1.61 1.25
∞ 3.84 2.99 2.60 2.37 2.21 2.10 1.94 1.75 1.52 1.00
249
TabLe a.5 (continued) Variance Ratio
F(99%)
n
1
n
2
1 2 3 4 5 6 8 12 24 ∞
1 4.052 4.999 5.403 5.625 5.764 5.859 5.982 6.106 6.234 6.366
2 98.50 99.00 99.17 99.25 99.30 99.33 99.37 99.42 99.46 99.50
3 34.12 30.82 29.46 28.71 28.24 27.91 27.49 27.05 26.60 26.12
4 21.20 18.00 16.69 15.98 15.52 15.21 14.80 14.37 13.93 13.46
5 16.26 13.27 12.06 11.39 10.97 10.67 10.29 9.89 9.47 9.02
6 13.74 10.92 9.71 9.15 8.75 8.47 8.10 7.72 7.31 6.88
7 12.25 9.55 1.45 7.85 7.46 7.19 6.84 6.47 6.07 5.65
8 11.26 8.65 7.59 7.01 6.63 6.37 6.03 5.67 5.28 4.86
9 10.56 8.02 6.99 6.42 6.06 5.80 5.47 5.11 4.73 4.31
10 10.04 7.56 6.55 5.99 5.64 5.39 5.06 4.71 4.33 3.91
11 9.65 7.20 6.22 5.67 5.32 5.07 4.74 4.40 4.02 3.60
12 9.33 6.93 5.95 5.41 5.06 4.82 4.50 4.16 3.78 3.36
13 9.07 6.70 5.74 5.20 4.86 4.62 4.30 3.96 3.59 3.16
14 8.86 6.51 5.56 5.03 4.69 4.46 4.14 3.80 3.43 3.00
15 8.68 6.36 5.42 4.39 4.56 4.32 4.00 3.67 3.29 2.87
16 8.53 6.23 5.29 4.77 4.44 4.20 3.89 3.55 3.18 2.75
17 8.40 6.11 5.18 4.67 4.34 4.10 3.79 3.45 3.08 2.65
18 8.28 6.01 5.09 4.58 4.25 4.01 3.71 3.37 3.00 2.57
19 8.18 5.93 5.01 4.50 4.17 3.94 3.63 3.30 2.92 2.49
20 8.10 5.85 4.94 4.43 4.10 3.87 3.56 3.23 2.86 2.42
21 8.02 5.78 4.87 4.37 4.04 3.81 3.51 3.17 2.80 2.36
22 7.94 5.72 4.82 4.31 3.99 3.76 3.45 3.12 2.75 2.31
23 7.88 5.66 4.76 4.26 3.94 3.71 3.41 3.07 2.70 2.26
24 7.82 5.61 4.72 4.22 3.90 3.67 3.36 3.03 2.66 2.21
25 7.77 5.57 4.68 4.18 3.86 3.63 3.32 2.99 2.62 2.17
26 7.72 5.53 4.64 4.14 3.82 3.59 3.29 2.96 2.58 2.13
27 7.68 5.49 4.60 4.11 3.78 3.56 3.26 2.93 2.55 2.10
28 7.64 5.45 4.57 4.07 3.75 3.53 3.23 2.90 2.52 2.06
29 7.60 5.42 4.54 4.04 1.73 3.50 3.20 2.87 2.49 2.03
30 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.84 2.47 2.01
40 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.66 2.29 1.80
60 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.50 2.12 1.60
120 6.85 4.79 3.95 3.48 3.17 2.96 2.66 2.34 1.95 1.38
∞ 6.64 4.60 3.78 3.32 3.02 2.80 2.51 2.18 1.79 1.00
Reprinted from Fisher,R,A,and Yates,F.,Statistical Tables for Biological Agricultural
and Medical Research,The Longman Group Ltd.,London,With permission.
250
TabLe a.6 monthly Payments per $1,000
of Loan Value
Annual
Rate (%)
Payment ($)
Monthly
Annual
Rate (%)
Payment ($)
Monthly
3-Year Loan
4.00 29.52 9.500 32.03
4.25 29.64 9.750 32.15
4.50 29.75 10.00 32.27
4.75 29.86 10.25 32.38
5.00 29.97 10.50 32.50
5.25 30.08 10.75 32.62
5.50 30.20 11.00 32.74
5.75 30.31 11.25 32.86
6.00 30.42 11.50 32.98
6.25 30.54 11.75 33.10
6.50 30.65 12.00 33.21
6.75 30.76 12.25 33.33
7.00 30.88 12.50 33.45
7.25 30.99 12.75 33.57
7.50 31.11 13.00 33.69
7.75 31.22 13.25 33.81
8.00 31.34 13.50 33.94
8.25 31.45 13.75 34.06
8.50 31.57 14.00 34.18
8.75 31.68 14.25 34.30
9.00 31.80 14.50 34.42
9.25 31.92 14.75 34.54
15.00 34.67
5-Year Loan
4.00 18.42 9.500 21.00
4.25 18.53 9.750 21.12
4.50 18.64 10.00 21.25
4.75 18.76 10.25 21.37
5.00 18.87 10.50 21.49
5.25 18.99 10.75 21.62
5.50 19.10 11.00 21.74
5.75 19.22 11.25 21.87
6.00 19.33 11.50 21.99
6.25 19.45 11.75 22.12
6.50 19.57 12.00 22.24
6.75 19.68 12.25 22.37
7.00 19.80 12.50 22.50
7.25 19.92 12.75 22.63
7.50 20.04 13.00 22.75
7.75 20.16 13.25 22.88
8.00 20.28 13.50 23.01
8.25 20.40 13.75 23.14
8.50 20.52 14.00 23.27
8.75 20.64 14.25 23.40
9.00 20.76 14.50 23.53
9.25 20.88 14.75 23.66
15.00 23.79
251
TabLe a.6 (continued) monthly
Payments per $1,000 of Loan Value
Annual
Rate (%)
Payment ($)
Monthly
Annual
Rate (%)
Payment ($)
Monthly
10-Year Loan
4.00 10.12 9.500 12.94
4.25 10.24 9.750 13.08
4.50 10.36 10.00 13.22
4.75 10.48 10.25 13.35
5.00 10.61 10.50 13.49
5.25 10.73 10.75 13.63
5.50 10.85 11.00 13.78
5.75 10.98 11.25 13.92
6.00 11.10 11.50 14.06
6.25 11.23 11.75 14.20
6.50 11.35 12.00 14.35
6.75 11.48 12.25 14.49
7.00 11.61 12.50 14.64
7.25 11.74 12.75 14.78
7.50 11.87 13.00 14.93
7.75 12.00 13.25 15.08
8.00 12.13 13.50 15.23
8.25 12.27 13.75 15.38
8.50 12.40 14.00 15.53
8.75 12.53 14.25 15.68
9.00 12.67 14.50 15.83
9.25 12.80 14.75 15.98
15.00 16.13
15-Year Loan
4.00 7.39 9.500 10.44
4.25 7.52 9.750 10.59
4.50 7.65 10.00 10.75
4.75 7.78 10.25 10.90
5.00 7.91 10.50 11.05
5.25 8.04 10.75 11.21
5.50 8.17 11.00 11.37
5.75 8.30 11.25 11.52
6.00 8.44 11.50 11.68
6.25 8.57 11.75 11.84
6.50 8.71 12.00 12.00
6.75 8.85 12.25 12.16
7.00 8.99 12.75 12.49
7.50 9.27 13.00 12.65
7.75 9.41 13.25 12.82
8.00 9.56 13.50 12.98
8.25 9.70 13.75 13.15
8.50 9.85 14.00 14.32
8.75 9.99 14.25 13.49
9.00 10.14 14.50 13.66
9.25 10.29 14.75 13.83
15.00 14.00
continued
252
TabLe a.6 (continued) monthly
Payments per $1,000 of Loan Value
Annual
Rate (%)
Payment ($)
Monthly
Annual
Rate (%)
Payment ($)
Monthly
20-Year Loan
4.00 6.06 9.50 9.32
4.25 6.19 9.75 9.49
4.50 6.33 10.00 9.65
4.75 6.46 10.25 9.82
5.00 6.60 10.50 9.98
5.25 6.74 10.75 10.15
5.50 6.88 11.00 10.32
5.75 7.02 11.25 10.49
6.00 7.16 11.50 10.66
6.25 7.31 11.75 10.84
6.50 7.46 12.00 11.01
6.75 7.60 12.25 11.19
7.00 7.75 12.50 11.36
7.25 7.90 12.75 11.54
7.50 8.06 13.00 11.72
7.75 8.21 13.50 12.07
8.25 8.52 13.75 12.25
8.50 8.68 14.00 12.44
8.75 8.84 14.25 12.62
9.00 9.00 14.50 12.80
9.25 9.16 14.75 12.98
15.00 13.17
25-Year Loan
4.00 5.28 9.500 8.74
4.25 5.42 9.750 8.91
4.50 5.56 10.00 9.09
4.75 5.70 10.25 9.26
5.00 5.85 10.50 9.44
5.25 5.99 10.75 9.62
5.50 6.14 11.00 9.80
5.75 6.29 11.00 9.80
5.75 6.29 11.25 9.98
6.00 6.44 11.50 10.16
6.25 6.60 11.75 10.35
6.50 6.75 12.00 10.53
6.75 6.91 12.25 10.72
7.00 7.07 12.50 10.90
7.25 7.23 12.75 11.09
7.50 7.39 13.00 11.28
7.75 7.55 13.25 11.47
8.00 7.72 13.50 11.66
8.25 7.88 13.75 11.85
8.50 8.05 14.00 12.04
8.75 8.22 14.25 12.23
9.00 8.39 14.50 12.42
9.25 8.56 14.75 12.61
15.00 12.81
253
TabLe a.6 (continued) monthly
Payments per $1,000 of Loan Value
Annual
Rate (%)
Payment ($)
Monthly
Annual
Rate (%)
Payment ($)
Monthly
30-Year Loan
4.00 4.77 9.500 8.41
4.25 4.92 9.750 8.59
4.50 5.07 10.00 8.78
4.75 5.22 10.25 8.96
5.00 5.37 10.50 9.15
5.25 5.52 10.75 9.34
5.50 5.68 11.00 9.52
5.75 5.84 11.25 9.71
6.00 6.00 11.50 9.90
6.25 6.16 11.75 10.09
6.75 6.49 12.25 10.48
7.00 6.65 12.50 10.67
7.25 6.82 12.75 10.87
7.50 6.99 13.00 11.06
7.75 7.16 13.25 11.26
8.00 7.34 13.50 11.45
8.25 7.51 13.75 11.65
8.50 7.69 14.00 11.85
8.75 7.87 14.25 12.05
9.00 8.05 14.50 12.25
9.25 8.23 14.75 12.44
15.00 12.64
The number of thousands borrowed is multiplied by the listed
monthly payment for the indicated annual interest rate,The product
is the total monthly payment,Due to rounding,this may be off by a
few cents from the actual.
254
TabLe a.7 The Growth of $1 at Various annual
interest Rates and specified Number of Years
Years 3% 4% 5% 6% 7%
1 1.0300 1.0400 1.0500 1.0600 1.0700
2 1.0609 1.0816 1.1025 1.1236 1.1449
3 1.0927 1.1249 1.1576 1.1910 1.2250
4 1.1255 1.1699 1.2155 1.2625 1.3108
5 1.1593 1.2167 1.2763 1.3382 1.4026
6 1.1941 1.2653 1.3401 1.4185 1.5007
7 1.2299 1.3159 1.4071 1.5036 1.6058
8 1.2668 1.3686 1.4775 1.5938 1.7182
9 1.3048 1.4233 1.5513 1.6895 1.8385
10 1.3439 1.4802 1.6289 1.7908 1.9672
11 1.3842 1.5395 1.7103 1.8983 2.1049
12 1.4258 1.6010 1.7959 2.0122 2.2522
13 1.4685 1.6651 1.8856 2.1329 2.4098
14 1.5126 1.7317 1.9799 2.2609 2.5785
15 1.5580 1.8009 2.0789 2.3966 2.7590
20 1.8061 2.1911 2.6533 3.2071 3.8697
25 2.0938 2.6658 3.3864 4.2919 5.4274
30 2.4273 3.2434 4.3219 5.7435 7.6123
35 2.8139 3.9461 5.5160 7.861 10.677
40 3.2620 4.8010 7.0400 10.286 14.974
45 3.7816 5.8412 8.9850 13.765 21.002
50 4.3839 7.1067 11.467 18.420 29.457
Years 8% 9% 10% 11% 12%
1 1.0800 1.0900 1.1000 1.1100 1.1200
2 1.1664 1.1881 1.2100 1.2321 1.2544
3 1.2597 1.2950 1.3310 1.3676 1.4049
4 1.3605 1.4116 1.4641 1.5181 1.5735
5 1.4693 1.5386 1.6105 1.6851 1.7623
6 1.5869 1.6771 1.7716 1.8704 1.9738
7 1.7138 1.8280 1.9487 2.0762 2.2107
8 1.8509 1.9926 2.1436 2.3045 2.4760
9 1.9990 2.1719 2.3579 2.5580 2.7731
10 2.1589 2.3674 2.5937 2.8394 3.1058
11 2.3316 2.5804 2.8531 3.1518 3.4785
12 2.5182 2.8127 3.1384 3.4985 3.8960
13 2.7196 3.0658 3.4523 3.8833 4.3635
14 2.9372 3.3417 3.7975 4.3104 4.8871
15 3.1722 3.6425 4.1772 4.7846 5.4736
20 4.6610 5.6044 6.7275 8.0623 9.6463
25 6.8485 8.6231 10.835 13.585 17.000
30 10.063 13.268 17.449 22.892 29.960
35 14.785 20.414 28.102 38.575 52.800
40 21.725 31.409 45.259 65.001 93.051
45 31.920 48.327 72.890 109.53 163.99
50 46.902 74.358 117.39 184.56 289.00
255
TabLe a.8 doubling Time for Various annual
interest Rates
Rate (%) Years
1 69.7
2 35.0
3 23.4
4 17.7
5 14.2
6 11.9
7 10.2
8 9.01
9 8.04
10 7.27
11 6.64
12 6.12
13 5.67
14 5.29
15 4.96
256
TabLe a.9 monthly savings to Produce $1,000
in the specified Number of Years at the Given
annual interest Rate (Compounded monthly)
Years 3% 4% 5% 6% 7%
1 82.19 81.82 81.44 81.07 80.69
2 40.48 40.09 39.70 39.32 38.94
3 26.58 26.19 25.80 25.42 25.04
4 19.63 19.25 18.86 18.49 18.11
5 15.47 15.08 14.70 14.33 13.97
6 12.69 12.31 11.94 11.57 11.22
7 10.71 10.34 9.97 9.61 9.26
8 9.23 8.85 8.49 8.14 7.80
9 8.08 7.71 7.35 7.01 6.67
10 7.16 6.79 6.44 6.10 5.78
15 4.41 4.06 3.74 3.44 3.16
20 3.05 2.73 2.43 2.16 1.92
25 2.24 1.94 1.68 1.44 1.23
30 1.72 1.44 1.20 0.99 0.82
35 1.35 1.09 0.88 0.71 0.56
40 1.08 0.85 0.66 0.50 0.38
Years 8% 9% 10% 11% 12%
1 80.32 79.95 79.58 79.21 78.85
2 38.56 38.18 37.81 37.44 37.07
3 24.67 24.30 23.93 23.57 23.21
4 17.75 17.39 17.03 16.68 16.33
5 13.61 13.26 12.91 12.58 12.24
6 10.87 10.53 10.19 9.87 9.55
7 8.92 8.59 8.27 7.96 7.65
8 7.47 7.15 6.84 6.54 6.25
9 6.35 6.04 5.74 5.46 5.18
10 5.47 5.17 4.88 4.61 4.35
15 2.89 2.64 2.41 2.20 2.00
20 1.70 1.50 1.32 1.16 1.01
25 1.05 0.89 0.75 0.63 0.53
30 0.67 0.55 0.44 0.36 0.29
35 0.44 0.34 0.26 0.20 0.16
40 0.29 0.21 0.16 0.12 0.08
257
TabLe a.10 monthly savings to Produce $1,000
in specified Number of Years at the Given annual
interest Rate (Compounded annually)
Years 3% 4% 5% 6% 7%
1 83.33 83.33 83.33 83.33 83.33
2 41.05 40.85 40.65 40.45 40.26
3 26.96 26.70 26.43 26.18 25.92
4 19.92 19.62 19.33 19.05 18.77
5 15.70 15.39 15.08 14.78 14.49
6 12.88 12.56 12.25 11.95 11.65
7 10.88 10.55 10.223 9.93 9.63
8 9.37 9.04 8.73 8.42 8.12
9 8.20 7.87 7.56 7.25 6.96
10 7.27 6.94 6.62 6.32 6.03
15 4.48 4.16 3.86 3.58 3.32
20 3.10 2.80 2.52 2.26 2.03
25 2.29 2.00 1.75 1.52 1.32
30 1.75 1.49 1.25 1.05 0.88
35 1.38 1.13 0.92 0.75 0.60
40 1.10 0.88 0.69 0.54 0.42
Years 8% 9% 10% 11% 12%
1 83.33 83.33 83.33 83.33 83.33
2 40.06 39.87 39.68 39.49 39.31
3 25.67 25.42 25.18 24.93 24.70
4 18.49 18.22 17.96 17.69 17.44
5 14.20 13.92 13.65 13.38 13.12
6 11.36 11.08 10.80 10.53 10.27
7 9.34 9.06 8.78 8.52 8.26
8 7.83 7.56 7.29 7.03 6.78
9 6.67 6.40 6.14 5.88 5.64
10 5.75 5.48 5.23 4.98 4.75
15 3.07 2.84 2.62 2.42 2.23
20 1.82 1.63 1.45 1.30 1.16
25 1.14 0.98 0.88 0.73 0.63
30 0.74 0.61 0.51 0.42 0.35
35 0.48 0.39 0.31 0.24 0.19
40 0.32 0.25 0.19 0.14 0.11
258
TabLe a.11 Percentage of Funds That may be
Withdrawn each Year at the beginning of the Year
at different annual interest Rates
Years 4% 5% 6% 7% 8%
1 100.000 100.000 100.000 100.000 100.000
2 50.980 51.220 51.456 51.691 51.923
3 34.649 34.972 35.293 35.612 35.929
4 26.489 26.858 27.226 27.591 27.956
5 18.343 18.764 19.185 19.607 20.029
6 16.020 16.459 16.900 17.341 17.784
7 14.282 14.735 15.192 15.651 16.112
8 12.932 13.399 13.870 14.345 14.822
9 11.855 12.334 12.818 13.306 13.799
10 8.6482 9.1755 9.7135 10.261 10.818
15 7.0752 7.6422 8.2250 8.8218 9.4308
20 6.1550 6.7574 7.3799 8.0197 8.6740
25 5.5606 6.1954 6.8537 7.5314 8.2248
30 5.1517 5.8164 6.5070 7.2181 7.9447
35 4.8850 5.5503 6.2700 7.0102 7.7648
40 4.6406 5.3583 6.1038 6.8691 7.6470
45 4.6406 5.3583 6.1038 6.8691 7.6470
50 4.4760 5.2168 5.9853 6.7719 7.5688
Years 9% 10% 11% 12% 13%
1 100.000 100.000 100.000 100.000 100.000
2 52.153 52.381 52.607 52.830 53.052
3 36.244 36.556 36.866 37.174 37.480
4 28.318 28.679 229.038 29.396 29.752
5 23.586 23.982 24.376 24.769 25.161
6 20.451 20.873 21.295 21.717 22.137
7 18.228 18.673 19.118 19.564 20.010
8 16.576 17.040 17.506 17.973 18.441
9 15.303 15.786 16.270 16.757 17.245
10 14.295 14.795 15.297 15.802 16.309
15 11.382 11.952 12.528 13.109 13.694
20 10.050 10.678 11.313 11.953 12.598
25 9.3400 10.015 10.697 11.384 12.073
30 8.9299 9.6436 10.363 11.084 11.806
35 8.6822 9.4263 10.174 10.921 11.666
40 8.5284 9.2963 10.065 10.831 11.592
45 8.4313 9.2174 10.001 10.780 11.552
50 8.3694 9.1690 9.9639 10.751 11.530
259
TabLe a.12 Growth of annual
deposits of $1,000 at the end of the
Year at specified annual interest Rates
Years 6% 8% 10%
1 1000 1000 1000
2 2060 2080 2100
3 3183.60 3246.4 3310
4 4374.62 4506.11 4641
5 5637.09 5866.60 6105.11
6 6975.32 7335.93 7715.61
7 8393.84 8922.80 9487.17
8 9897.47 10636.63 11435.89
9 11491.32 12487.56 13579.48
10 13180.79 14486.56 15937.42
11 14971.64 16645.49 18531.17
12 16869.94 18977.13 21384.28
13 18882.14 21495.30 24522.71
14 21015.07 24214.92 27974.98
15 23275.97 27152.11 31772.48
20 36785.59 45761.96 57275.00
25 54864.51 73105.94 98347.06
30 79058.19 113283.21 164494.02
35 111434.78 172316.8 271024.38
40 154761.97 259056.52 442592.56
260
TabLe a.13 Growth of annual
deposits of $1,000 at the beginning of the
Year at specified annual interest Rates
Years 6% 8% 10%
1 1060.00 1080.00 1100.00
2 2183.60 2246.40 2310.00
3 3374.62 3506.11 3641.00
4 4637.09 4866.60 5105.10
5 5975.32 6335.93 6715.61
6 7393.84 7922.80 8487.17
7 8897.47 9636.63 10435.89
8 10491.32 11487.56 12579.48
9 12180.79 13486.56 14937.42
10 13971.64 15645.49 17531.17
11 15869.94 17977.13 20384.28
12 17882.14 20495.30 23522.71
13 20015.07 23214.92 26974.98
14 22275.97 26152.11 30772.48
15 24672.53 29324.28 34949.73
20 38992.73 49422.92 63002.50
25 58156.38 78954.41 108181.77
30 83801.68 122345.87 180943.42
35 118120.87 186102.14 298126.81
40 164047.69 279781.03 486851.81
261
TabLe a.14 monthly amount That must be saved
for the Years indicated (down) in Order to Collect
$1,000 per month Thereafter (across) at 4% annual
interest Compounded monthly
Years
Saving
Years Collecting
5 10 15 20 25
5 819.00 1489.80 2039.10 2489.10 2857.50
10 368.75 670.77 918.11 1120.69 1286.61
15 220.65 401.36 549.36 670.57 769.85
20 148.04 269.29 368.60 449.93 516.54
25 105.61 192.11 262.95 320.97 368.49
30 78.24 142.31 194.79 237.77 272.97
35 59.43 108.10 147.96 180.60 207.34
40 45.94 83.56 114.38 139.62 160.29
TabLe a.15 monthly amount That must be saved
for the Years indicated (down) in Order to Collect
$1,000 per month Thereafter (across) at 6% annual
interest Compounded monthly
Years
Saving
Years Collecting
5 10 15 20 25
5 714.37 1291.00 1698.50 2000.60 2224.55
10 315.63 549.63 723.11 851.73 947.08
15 177.86 309.72 407.48 479.96 533.69
20 111.95 194.95 256.48 302.10 335.92
25 74.64 129.98 171.00 201.42 223.97
30 51.49 89.67 117.97 138.95 154.51
35 36.31 63.22 83.18 97.97 108.94
40 25.97 45.23 59.50 70.09 77.94
262
TabLe a.17 monthly amount That must be saved
for the Years indicated (down) in Order to Collect
$1,000 per month Thereafter (across) at 10% annual
interest Compounded monthly
Years
Saving
Years Collecting
5 10 15 20 25
5 607.79 977.20 1201.72 1338.18 1421.12
10 229.76 369.41 454.28 505.87 537.22
15 113.56 182.57 224.52 250.02 265.51
20 61.98 99.65 122.55 136.46 144.92
25 35.47 57.03 70.13 78.10 82.94
30 20.82 33.48 41.17 45.84 48.68
35 12.40 19.93 24.51 27.29 28.99
40 7.44 11.97 14.71 6.39 17.40
TabLe a.16 monthly amount That must be saved
for the Years indicated (down) in Order to Collect
$1,000 per month Thereafter (across) at 8% annual
interest Compounded monthly
Years
Saving
Years Collecting
5 10 15 20 25
5 671.21 1121.73 1424.13 1627.10 1763.34
10 269.58 450.52 571.98 653.49 708.21
15 142.52 238.19 302.40 345.49 374.42
20 83.73 139.93 177.65 202.97 219.97
25 51.86 86.67 110.03 125.71 136.24
30 33.09 55.30 70.21 80.22 86.94
35 21.50 35.93 45.62 52.12 56.48
40 14.13 23.61 29.97 34.25 37.11
263
Index
A
Abscissa,33
Acceleration,90
Adjoint matrix,21–22
Algebra,1–9
Algebraic equations,8–9
Analytic geometry,32–56
Angle of intersection,69
Annuities,168–170
Arc length,78–79
Area
in rectangular coordinates,
78
in polar coordinates,79
of surface,80
Associative laws,1
Asymptotes of hyperbola,
43–44
Auxiliary equation,113–114
B
Balance calculation,158
Base of logarithms,2–3
Bayes’ rule,118–120
Beatty theorem,64
Bessel functions,103–105
Binomial distribution,120
Binomial theorem,3–4
Bernoulli numbers,57–62
Bonds,173–175
Boyle’s Law,75
C
Cartesian coordinates,
see Rectangular
coordinates
Cauchy’s form of remainder,72
Centroid,83–85
table of,85
Charles’ Law,75
Chi square,129–131
Circle,11,37,43
Coefficient of determination,
132
Coefficient of variation,118
Cofactors,16–17
Collatz conjecture,67
Combinations,7
Commutative laws,1
Complex numbers,5–7
Components of vector,
see Vector
Compound interest,153–154
Concavity,70–71
Cone,13
Confidence interval,124
Conformable matrices,19
Contingency table,129–130
Convergence,interval of,58
Correlation coefficient,133
Cosecant of angle,27
Cosh,see Series of functions
Cosine of angle,27
264
Cosines,law of,25
Cotangent of angle,27
Cramer’s rule,23
Critical value,70
Csch,see Series of functions
Ctnh,see Series of functions
Cubic equation,8–9
Curl,89–90
Curves and equations,50–56
Cylinder,12–13
Cylindrical coordinates,80–81
D
Definite integrals,table of,
231–242
Degree of differential equation,
110
Degrees of freedom,125,
127–128
Degrees and radians,27
Degree two equation,general,
47
Deposit amount,160
Determinants,15–18,22–23
Derivatives,68
Derivatives,table of,182–185
Differential calculus,68–76
Differential equations,110–115
Directrix,37
Distance between two points,
33
Distance from point to line,37
Distributive law,1
Divergence,88–89
Division by zero,1
Dollar cost average,179
Double integration,82–83
Drug kinetics,111–112
E
Eccentricity,41
Ellipse,39,41–42
Empirical distributions,123
Error function,63–64
Estimation,123
Euler numbers,57–62
Even permutation,15–16
Exact differential equation,
110–111
Expected value,120
Exponential function,51,56
Exponents,2
F
Factorials,3,93
Factors and expansions,4
F-distribution,132,138–139
Fermat
little theorem,64
last theorem,64
near misses (cubic form),
64–65
Focus,37–43
Fourier series,100–103
Functions of two variables,
75–76
Fundamental theorem of
integral calculus,
77–78
G
Gamma function,93,139
Gas constant,75
Gas laws,75
Geometric figures,9–14
Geometric mean,116
265
Geostationary satellite orbit,
90–91
Goldbach conjecture,66
Gradient,88
H
Half-life,56
Half wave rectifier,103
Hermite polynomials,108
Homogeneous differential
equation,110
Homogeneous functions of x,y,
110
Horizontal line equation,35
Hyperola,43–45
Hyperbolic functions,92–93
Hypothesis testing,124–128
I
Identity laws,1
Imaginary part of complex
number,5–7
Inclination,angle of,34
Indeterminant forms,72
In-Out formula,170–172
Integral calculus,77–85
Integral,definite,77–78
Integral,indefinite,77
Integral tables,187–242
Interest,146–147
Interest rate,effective,154
Intersection,angle of,69
Inverse laws,1
Inverse matrix,21–22
Inverse trigonometric
functions,31
Inversions of permutations,7
L
Laguerre polynomials,107–108
Laplace transforms,94–97
Lease squares regression,
131–134
Legendre polynomials,105–107
L’Hopital’s rule,72–73
Linear differential equation,
111–112
Linear system of equations,
23–24
Lines,equations,34–37
Loan balance,149
Loan payment,147,148
Logarithms,2–3
Logistic equation,56
Lump sum payment,152
M
Major axis of ellipse,41–43
Market average (stock),177
Matrix,18
Matrix operations,18–19
Maxima of functions,70
Mean,116–117,120–121,123,
128,138–141
Mean value of function,79
Median,116,123
Midpoint of line segment,33
Minimum of function,70
Minor axis of ellipse,41
Minor of matrix,16
Mode,116,123
Moving average,180–181
Multiple regression,136–138
Mutual funds,177
266
N
Newton’s method for roots of
equations,73
Nonlinear regression,134–136
Nonsingular matrix,21
Normal distribution,121–122
Normal form of straight line,
35–36
Normal line,69
Null hypothesis,124,126–127
Numbers,real,1,5
Numerical methods,73–74
O
Odd permutation,7,15–16
Order of differential equation,
110–113
Ordinate,33
Origin,32,47
Orthogonality,108–109
P
Pairing in t-test,127
Parabola,37–39
Parallel lines,34
Parallelogram,10
Partial derivatives,76
Partition (Beatty theorem),65
Payment
Accelerated,150
Loan,147–150
lump sum,152
monthly,158
schedules,148–149
Permutations,7,15–16
Perpendicular lines,34
Poisson distribution,122–123
Polar coordinates,47–50,79
Polar form of complex number,
6–7
Polygon,12
Population,standard deviation
of,117
Population,variance of,117
Power,142–145
Powers of complex numbers,6
Present value,155,167
Probability,118–120
Probability curve,121
Probability distributions,
139–141
Prime number,66,67
Prism,13
Progressions,4–5
Pythagorean theorem,9
Q
Quadrants,26
Quadratic equation,8
R
Radians,27
Radius of curvature,69
Rectangle,10
Rectangular coordinates
(Cartesian
coordinates),
32–33,78–79
Rectifier,half wave,103
Reduced cubic equation,8–9
Regression,131–132,134–137
Regular saving,156
beginning of year,156
end of year,157
267
Rodrigues formula,107
Runge-Kutta method,114,115
S
Sample,117–118
Sample size
sample mean,143
single proportion,142
two means,144
two proportions,143
Sample standard deviation,118,
126–128
Satellite orbit,90–91
Sawtooth wave,102
Scalar multiplication
of vectors,87
of matrices,18
Scalar product of vectors,87
Secant,27
Sech,see Series of functions
Second derivative,68,70–71
Second derivative test,70
Sector of circle,11
Segment of circle,11
Separable differential equation,
110
Series of functions,58–64
Sine,25,27
Sines,law of,25
Sinh,see Series of functions
Skewness,123
Slope,33–34,68–69
Sphere,14
Spherical coordinates,80–82
Spherical harmonics,105
Standard deviation,117–118,
121,126–128
Standard error,124,133–134
Standard error of estimate,133
Standard normal curve,121–122
Statistics,116–145
Stirling’s approximation,3
Stock
options,176
yield,172
Stocks,172
Sum of matrices,18–19
Sum of progression(s),4–5
Sum of vectors,87
Surface area by double
integration,83
Surface area of revolution,80
Symmetric matrix,20
T
Tangent of angle,25,27
Tangent line,68
Tangents,law of,25
Tanh,see Series of functions
Taylor’s formula,71–72
t-distribution,125–128
Translation of axes,45–46
Transpose of matrix,20
Trapezoid,11
Trapezoidal rule,74
Trigonometric functions of
angles,26–27
Trigonometric identities,28–30
Twin primes,67
V
Variance,117,137,139–141
analysis of,131,138–139
Vector,86
Vector product,87
Velocity,90
268
Vertical line equation,34
Volume by double integration,83
Volume of revolution,79–80
W
Withdrawals
Amounts,151,153
Periodic,161–163,165–167
Work,80
Y
Yield
Stock,172,176
tax-free,175
Z
Z-transform,97–100
properties of,98–99
table of,99–100
