Dietimnary M‘
`Mathemafies Terms
`
`Third Edition
`
`Douglas Downing, Ph.D.
`School of Business and Economics
`Seattle Pacific University
`
`APPLE INC.
`EXHIBIT 1008 - PAGE 0001
`
`

`
`CONTENTS
`
`Preface
`
`List of Symbols
`
`Mathematics Terms
`
`Appendix
`
`Algebra Summary
`
`Geometry Summary
`
`Trigonometry Summary
`
`Brief Table of Integrals
`
`Dedication
`This book is for Lori.
`
`Acknowledgments
`Deepest thanks to Michael Covington, Jeffrey
`Clark, and Robert Downing for their special help.
`
`© Copyright 2009 by Barron’s Educational Series. Inc.
`Prior editions (9 copyright 1995. 1987.
`
`All rights reserved.
`
`No part of this publication may be reproduced or
`disuibuted in any form or by any means without the
`written permission of the copyright owner.
`
`Ali inquiries should be addressed :0:
`Barron's Educational Series. Inc.
`250 Wireless Boulevard
`Hauppauge, New York 11788
`www.barronseduc.com
`
`ISBN-13: 978-O-754l-4139-3
`ISBN-10: 0-7641-4139-2
`
`Library of Congress Control Number: 2003931689
`PRINTEDINCHINA
`98765
`
`APPLE INC.
`EXHIBIT 1008 - PAGE 0002
`
`

`
`VARIABLE A variable is a symbol that is used to represent
`a value from a particular set. For example, in algebra it is
`common to use letters to represent values from the set of
`real numbers. (See algebra.)
`
`VARIANCE The variance of a random variable X is defined
`to be
`
`V8100 = E[(X " E00) X (X - E(X))]
`= 13[(X - ECXDZJ
`
`where E stands for “expectation.”
`The variance can also be found fiom the formula:
`
`Var(X) = E(X2) -' [E0012
`
`The variance is often written as 03. (The Greek lower-
`case letter sigma (or), is used to represent the square root
`of the variance, known as the standard deviation.)
`The variance is a measure of how widespread the
`observations of X are likely to be. If you know for sure
`what the value of X will be, then Var(X) : 0.
`For example, if X is the number of heads that appear
`when a coin is tossed five times, then the probabilities are
`given in this table:
` __
`i
`Pr(X = i)
`i X Pr(X = i)
`i3 X Pr(X = 1')
`0
`1/32
`0
`0
`1
`5/32
`5/32
`5/32
`2
`10/32
`20/32
`40/32
`3
`10/32
`30/3?.
`90/32
`4
`5/32
`20/32
`80/32
`5
`1/32
`5/32
`25l32
`
`369
`
`VECTOR
`
`The sum of column 3 [i X Pr(X = i)] gives
`E(X) = 2.5; the sum of column 4 [i2 X Pr(X = 1)] gives
`E(X2) = 7.5. From this information we can find
`
`Var(X) = E(X2) — [E(X)]2 = 7.5 — 2.52 = 1.25
`
`Some properties of the variance are as follows.
`If £1 and b are constants:
`
`Vat(aX + b) = a2Var(X)
`If X and Y are independent random variables:
`
`Var(X + Y) = Var(X) + Var-(Y)
`
`In general:
`
`Var(X + Y) = Var(X) + Var(Y) + 2Cov{X, Y)
`
`.
`
`.
`
`. x,, is given
`
`where Cov(X, Y) is the covariance.
`The variance of a list of numbers x1',Ax2,
`by either of these formulas:
`(x,—3c')2+(xg-3€)2+...+(x,,-EV
`71
`
`Var(x)
`
`._: x2 _
`
`where a bar over a quantity signifies average.
`
`VECTOR A vector is a quantity that has both magnitude
`and direction. The quantity “6O miles per hour” is a reg-
`ular number, or scalar. The quantity “6O miles per hour to
`the northwest" is a vector, because it has both size and
`direction. Vectors can be represented by drawing pictures
`of them. A vector is drawn as an arrow pointing in the
`direction of the vector, with length proportional to the
`size of the vector. (See figure 156,)
`Vectors can also be represented by an ordered list of
`numbers, such as (3,4) or (1, 0, 3). Each number in this list
`is called a component of the vector. A vector in a plane
`
`APPLE INC.
`EXHIBIT 1008 - PAGE 0003
`
`

`
`a+b
`
`Figure 157 Adding vectors
`
`To find two different ways of multiplying vectors,‘ see
`dot product and cross product.
`
`VECTOR FIELD A two—dimensional vector field f trans-
`forms a vector (x, y) into another vector f(;c, y) = [];(x, y),
`j;(x, y)]. Here ]‘;(x, y) andfy(x, y) are the two components
`of the vector field; each is a scalar function of two var-i~
`ables. An -example of a vector field is:
`x
`—wry
`,———-—x2+ ya
`,——————x2+ yz,
`f(x,y) =
`If we evaluate this vector field at (3,4) we find:
`~—4 3
`f(3,4) — [ 5 ,5]
`In this particular case, the output of the vector field is
`perpendicular to the input vector.
`The same concept can be generalized to higher-
`dimensional vector fields. For examples of calculus oper-
`ations on vector fields, see divergence; curl;
`line
`integral; surface integral; Stokes’ theorem; Maxwell’s
`equations.
`‘
`
`VECTOR PRODUCT This is a synonym for cross
`product.
`
`VELOCITY The velocity vector represents the rate of
`change of position of an object. To specify a velocity, it is
`necessary to specify both a speed and a direction (for
`
`Figure 156 Vector
`
`A vector in space (three dimensions) can be represented
`as an ordered triple.
`'
`Vectors are symbolized in print by boldface type, as
`a.” A vector can also be symbolized by plac-
`ing an arrow over it: 3.
`The length, or magnitude or norm, of a vector a is
`written as “all
`Addition of vectors is defined as follows: Move the
`tail of the second vector so that it touches the head of
`the first vector, and then the sum vector (called the
`resultant) stretches from the tail of the first vector to the
`head of the second vector. (See figure 157.) For vectors
`expressed by components, addition is easy: just add the
`components:
`
`(3, 21+ (4, 1) = (7, 3)
`(a.b)+{c.a')~“~“(a+c,l7+d)
`
`To multiply a scalar by a vector, multiply each com—
`ponent by that scalar:
`
`lO(3, 2) = (30, 20)
`ram, 19) = (rm, rtb)
`
`APPLE INC.
`EXHIBIT 1008 - PAGE 0004
`
`

`
`VENN DIAGRAM
`
`If the motion is in one dimension, then the velocity is
`the derivative of the function that gives the position of the
`object as a function of time. The derivative of the veloc-
`ity is called the acceleration.
`If the vector [x(z), y(t), z{z)] gives the position of the
`object in three dimensional space, where each component
`of the vector is given as a function of time, then the veloc-
`ity vector is the vector of derivatives of each component:
`:1
`velocity = Jdr ’ dr ’dr
`VENN DIAGRAM A Venn diagram (see figure 158) is a pic—
`ture that illustrates the relationships between sets. The uni-
`versal set .you are considering is represented by a
`rectangle, and sets are represented by circles or ellipses.
`The possible relationships between two sets A and B are as
`follows:
`
`Set B is a subset of set A, or set A is a subset of set B.
`Set A and set B are disjoint (they have no elements in
`common).
`
`Set/L and set B have some elements in common.
`
`00
`
`AandBare
`disjoint
`
`AandBhave
`some elements
`in common
`
`Bis a
`subset ofA
`
`VERTICAL ANGLES
`
`Figure 159 is a Venn diagram for the universal set of
`complex numbers.
`
`Complex Numbers
`
`integers
`
`rational
`numbers
`
`transcendental
`numbers
`
`1T‘
`
`Figure 159 Venn diagram
`
`VERTEX The vertex of an angle is the point where the two
`sides of the angle intersect.
`
`VERTICAL ANGLES Two pairs of vertical angles are
`formed when two lines intersect. In figure 160, angle 1
`and angle 2 are a pair of vertical angles. Angle 3 and
`angle 4 are anotherzpair of vertical angles. The two angles
`in a pair of vertical angles are always equal in measure.
`
`APPLE INC.
`EXHIBIT 1008 - PAGE 0005