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38 Mathematical Appendix

# 38 Mathematical Appendix - In this Appeaj'11 provide a...

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Unformatted text preview: In this Appeaj '11 provide a brief review of some of the mathematical coneepts that are 1181311 in t__he text This material is meant to serve as a reminder of ti " 'tions of various terms used 111 the text. It iS emphat- ically not a. 11111? 1:11 mathematics. The deﬁnitions given will generally be the Simpiai, _:___;t:='-Ehe.mest. rigorous. A function 5111111211; describes a relationship between numbers. For each" number—"11:, _-_ﬁn'1ct1011 assigns a unique number y aCCOrding to some rule.' This a. functi9n' can be indicated by describing the rule as take a number and square 1 ” or “take a number and multiply it by 2," and so 011. We Write tease- particular functions as y_ — 3:2 y— w 23:. Functions are sometimes referred to as. transformations. Often we want taindicate that some variable 3,: depends on some other variable :8, but we dee’t knew the speciﬁc algebraic relationship between the two Variables. 111 this case we write y = f (1:), which should be interpreted as sayiﬁg that the variable 3; depends 011 2: according to the rule f. Given a ﬁm'ction _y_= f (:L'), the number I. is often called the indepen- dent Varia_b1e,.a11d__'t_he number 3; is oft-en called the dependent variable. Figure Al A2 MAYH EMATICAL APPEN DIX The idea. is that :1: vanes independently, but the value of y depends on the value of a: Often some variable 3; depends on several other variebiliée :61, 3:2, and so on, so we write 3; = f (3:1,,352) to indicate that b0th_"variab1es tOgether determine the value of y. A.2 Graphs A graph of a function depicts the behavior 0f a functiQ'zr pmtonaﬂy Figure Al skews two graphs of. functions. In mathematics" t able IS usually depicted on the horizontal axis, and the is depicted on the vertical axis. The ng3;)h then. retirees. between the indepeﬁdent- and the dependent variables A3 Properties of Functions A continuous function is one that can be drawn wit-bout. lifting-a. pencil from the paper: there are no jumps in a continuous function-A ambath EQUATlONS AND IDENTETIES P11?- func-tion is one ithat has no “kinks" or corners. A monotonic function is one that always increases or always decreases; a. positive monotonic function always increases as :1: increases. while a. negative monotonic function always- decreases as :1: increases. AA inverse Functions Recall that a function. has the property that for each value of :1: there is a unique value of y eSsociated with it and that a monotonic function is one that. is always increasing or always decreasing This implies that for a value of; y _ We call the fame: _ that relates .7: to. y in this way an inverse function. - If you are. given in function of :1:: you can calculate the 1nverse function just by Solving . asa function {if 1;. If y— — 2:1: then the inverse function is :12 z y/Z. If. y 9:2 then there' 1s no inverse function; given any y. both :17— 11 +W and :1: 1. '—.—‘/' have the property that their square is equai to “11 Thus there is not a intrigue value of :1: associated with each mine of y. as is required by the deﬁnition of a function. A5 Equations; and Identities An eqHation- asks"when a function is equal to some particular number. Examples of equations are 2.13:8 .229 fiw)=0 The sedation to an equation is a value of m that satisﬁes the equation. The ﬁrst equatlon has a solution of c z 4. The second equation has two solutions, .11]: 3 anti :1: == —3. The third equation is just a. general equation We don t knoW its solution until we know the actual rule that .f stands for. but we can denote its solution by 112* .This simply means that at“ is a nnniher such that {(23“). = D. Vie say that 3:* satisﬁes the equation f(. (1:—) *- 0. An identity 1s a relationship between variables that holds for all values of the variables; Here are some examples of identities: (:1: + 3})2 3:172 + 21:11 + 1,12 2(1: + 112 2:1: + 2. The special symbol E means that. the left-hand side and the right-hand side are equal for all Values of the variables. An equation only holds for some values of the variables. whereas an identity is true for all values of the variables. Often an identity is true by the deﬁnition of the terms involved. A4 MATHEMATiCAL APPENDIX 131.6 Linear Functions A linear function is a function of the form 1,: = as: + b, where a and b are constants. Examples of linear functions are y=2ic+3‘ yin—499 Strictly speaking, a- function of the for-.111 y' = as: +‘ afﬁne function, anti Only seasons of. the form :1; _- linear functions. However, we will not inSist on this (113\$: Linear functions can also be expressed implicitly 111 forms git:ei one-by = c In such a case we often like to solve for y as a function} of :1: to cenﬁert this to the standard” form: c a. y=—ﬁ—x. £131 A. 7 Changes and Rates of Change The notation Ax IS read as “the. Change in 51:. ” it does-111T" .If m Changes from :17" . to :12”, then the. change 111 a re 'Ae my,“ _gc: We canalsowrite . _ _ _ to indicate that 112*" ifs 517* plus a; change in :15, " ' Typically As will refer to a 3111,1111 change in .127 this by saying that. Am represents a marginal change A rate of change IS the ratio of two changes. If- i -- given by 1;: ﬁx) then the rate of change of y with respect to a; is {toasted _ b1 _A_y : 11:1 '+ Ax) — 111:) .5351: A21: ' The rate of change measures how y changes as :5 changes._: A linear function has the property that the rate of change of y with respect to x is constant. To prove this, note that if y -_.- (1 +631: then ' Ay _ a + b(e + my a, :_ be * NHL; [5.2: A:.-: As: ' Sit—)l’FS AM.)!1\' TFRU'P'iS A? For I‘lOnliIiBai‘_ﬁlIiCi-lﬂiih. 1111: rate of 1‘l11111ge 1115 1111: 1111311111111 will 1l1.-p=:.‘111.l 1.111 the value of 11:. Consider. i111 15.111111111111. the .l'11111’1i1111 1,1 —- 1‘3. F111 This function __A_y _ (3: + AIJB— :r2 {1 21131.11 {\$.11}: .13 .; . 1 '3 1. 13—1 ““7121“ W "m A?“ H “J “ Here the rat-e of. 0113.1ng 1111-111 .1? 1:11 .‘1’ —1'— A1“ 1103113111115 1111 the mine of .1' 1111:.l 011 the Size. Of the 0111111111.). \$1.17. B111. if we. consider V01"): Hillé-lll Changes 1'11 .1'. A11 Will be nearly zero. 15.11 the. rate 11f change of 1; wi1l1 respect 1.11 :1' will be approximately 2:5. A.8 Slopes. anid' intercepts The rate of change of a function can be interpreted graphically; as the slope of the ftmetion In } 1g111e A. 2A we hate depic: led a. linear {unminu y :7 ~23 + 4,-1‘1168 vertical intercept of this 111111113011 is the value of 1,1 when .1: I G, which if; y = 4. The horizontal intercept is the value {if :1? when 3; = 0, which is :11 : 2. The slope of the {1111011011 is the 11.1.1.1? 11l’11l1111'1g11 of y as :1“: changes. In this C1151). the. 5101.19 of the function is —2. tercepts "Pa. 191 A depicts. 111a funetioi1 1 ~ 1.1111191 B depn‘ts 1;}19 £111113111311H.111Ia.~..;:1i'.2 In generah if a. line-.111: 111111111111 11215: the 11.11111 1; :2 11.11 +11. 1i11- 1-111-111111 intercept. will be y“ : 1’1 and the l101‘1x1111111l 1111011111111. will 1111 .1'" —— 313511. ll" 11. linear function is EEK}.'1J'1:’1"-h(‘.1'l 111 H111 1111111 1.11.11 + 1.13.132 2:. 1‘. ' Fign re 11.2 A6 MATHEMATICAL APPENDIX then the horizontal int'elcept will be the urine of .1. 1 when 51:;— — D,- which is ;:'1 1 _ 5/111 and the tertical intercept v.11] occur when 3:1 — O which means 1 7'; __ :c/ag The slope of this function is urn/11,2. A nonlinear function has 1he property that its slope changes as 9: changes. A tangent to a function at some point :r is a linear function that has the same slope. In Figine A. 2B Wt have depicted the function 3:2 and the ' tangent line at I— __ 1. If: y incmases whenever :11 increases. then Ag; will always have the same sigma as £1.11" so that the slope of the function will be positive If on the other hand y decreases when :1: increases. or 3; increases when :1: decreases, ._ Ag; and As; will have opposite signs, so that the slope of the function will be negative. I A.9 Absolute Values and Logarithms T he absolute value of a number 1s a function f(.r) defined by the following rule: ' (3+ 2? imeU if :r < 0. Thus the absolute value of a number can be found by dropping the sign of the number. The absolute value function is usuall}, written as Is]. The (natural) logarithm or log of :1: describes a particular functioniof at, which we write as 1; 2 ins or y = 111(3). The logarithmfanctionsis the ' _ unique function that has the properties may) = 11.1..) +1n1y) for all positive numbers :1: and y and .. 111(6) = 1. (in this last equation e is the base of natural logarithms Winch 18 equal to _ 2. 7183.. .J In words the log of the preduct of two numbers is the sum of _ the individual logs. This property implies another important property of logal ithms: inﬁri’) : yln(:r), which says that the log of :17 raised to the power y is'eqnal' to :1; times the log of :11. ' I 151.10 Derivatives The derivative of a function y = f (.11) is deﬁned to be tif(\$)_ . flT+A1'l—f{-I?l 11131 W (3:1: .51"- -—+0 131‘ SECOND DERIVATIVES A." In Werds, the derivative is the limit of the rate of change of y with respect to 1: as the change .111 :1: goes to zero The derivative gives precise meaning . to the phrase ‘t rate of change of y with respect to 1: for small changes in 1:.” "The ad is veef ﬁx) with respect to 3 is also denoted by f’ (1:). We have shady-”seen that the rate of change of a linear function y— _— as: ”+ b is 110th Thus for this linear function _tien the rate ei change pi y with respect2 to 1: will Wesaw that in the” case of _.j(3-) = 1:2 we had Applying the”. deﬁnition of the derivative 03(3) 11111-23 + A3— ‘1 21: 111:”- Ass-+11 . .. Thus the is? With respect to is is is. ' ”113111.21 ”mere” advanced methods that if y: 1111;, then tire) 2 1 - ' d1: . 3 : _ 11113 of a'function is the derivative of the derivative of _f(:1:), the second derivative of f (1:) with respect to 1: " T111151 ' (12(21):) _ (1(2) ..., dis? ““ “E; ‘ (12(3”) d(21) _2 (£32 _ do: ' The second dai'ivative measures the curvature of a function. A function with a negative second derivativa at some point 1s concave near that point; ' its slope is decreasing A function with ”a positive second derivative at a __ point is convex near that point; its slope lS increasing. A function with a zere. second derivative at a point is ﬂat near that point. is; MATHEMATICAL APPENDIX ' A. \$2 The Product Rule and the Chain Rule Suppose that 9(13) and 11(3) are both functions of a: -We ._gean define the function f (2:) that represents their product by f (1;) g(m)h(\$) Then the derivative of f (:13) is given by ' (WI) 691(3) dyiiﬁ) 07.1: + Mm) .. = 90"?) d ﬁtnettoms Given two functions y— ”9(3) and z — My) the co Km): h(9(\$)) I' _ ' For exampie, if 9(12) : 51:2 and My) = 23; + 3, then thef-’eezn%eeite {sateen '._. is - - ' f (3:) = 2:52 + 3. I The chain rule says. that the derivative of a compomtefnnction,f(:t), with respect to .12 is given by em =- M age). 'da: ' dy "da: I In our example, dh(y)/d-:.=‘y 2, and dg(a:)/ds:— — 23: so theeliem rote says I- -. that dfCa: m)/d\$ = 2 x 22: = 4:1: Direct caleulation ver Lithét tthie is" the f' derivative of the function f (to) 2x2 7i:- -.3 ' ' " A.1 3 Partial-Derivatives _- _: ”if Suppose that y depends on both as; {MIMI so that i _.____ the partial derivative 0ff(i151,-' 22'} with-'irespeet toe; 31061713332) 411211 ﬂail +A\$I:\$2)~ fi if _Bxl _ MIN) WW Ax; .. . _.._. i 'I The partial derivative of 3' (3:1, 2:2) with respect to 3:1 is'1net- the derivative " of the function with respect to 3:1, holding 9:; ﬁxed. Smiarly, the partial ' derivative with respect to 3:2 is . LWW s hm 2W2 61:2 Ame-HO A322 " ‘ Partial derivatives have exactly the same properties as etdim' derivatives; " only the name has been changed to. protect the innocent _,§that is, people who haven t seen the a symbol). OPTIMiZATION in particular, partial clerivaiive'zs obey the. chain rule, but with an extra twiSt. Suppose that an and I: both depend on some veriaMe t and that we-deﬁne the function eff.) b}: 5(1) : f{_1'1(t‘,1.333(t)). Then the derivative of git} with respect to f. is given by dgﬁt) __ 8f(_:r:1._;i!2} afxrﬁt} ‘ 5f{rr;,.rz_} digit) alt 51‘ 1 (it 811:2 (if When It changes}, it affects both QUE) and 9:26.). Therefore, we neeé to calculate the derivative of f {'_ 3:11 3:2) with respect to each of those changes. A."1':_4 Optimiiation If y = ﬁe) then ﬂax) achieves a maximum at cc“ if f{.r"‘) 2 f(:E) for all 2:. It can beg-shown that if f (.r) is a smooth function that achieves its maximum values-at :3“, then are“) _ (LI; ’0 2 at L “11’ l < 0. d3?“ "' These expressions are referred to as the ﬁrst-order condition and the second-order Condition for a maximum. The first—order condition says that the function is ﬂat at 1*; while the second-order condition says that the-function is concave near r* Clearly both of these properties have to hold if 3:“ is indeed a maximum. We say that ﬂat) achieves its minimum value at :r’“ if f (3") g f (I) for ails: If ffas) isle. smooth function that achieves its minimum at 3"“, then dféx ) :0 TC 2 I. :9: El—ﬂ‘f ) 20. an. ap- The ﬁl‘shorder condition again says that the function is flat. at .15 , the second-order condition now says «that. the function is convex near 31*. If I-y 2 f(;E1,j;132) is :1. smooth function that achieves its inaxitniun or minimum at some point {ﬁn-3]. then we must satisfy creme while 1 _ I U (33.? J. L) : U' 53.12 ‘ These are referred to as the ﬁrst-order conditions. There are also second- order conditions for this probiern: but they are more difﬁcult to describe. A10 MATHEMATiCAL APPENDIX AJS Constrained Optimization Often we want to consider the maximum or minimum ofﬁﬂiiieieinctim ever some restricted values of (2:1, 9:2). "The notation max f (5'11, 932) 31 1972 ' " ﬁnch" they etch-7&1). I: Cir . means ﬁnd {31 and 532 511911 that 11313322} > f(\$1,£2) far all: that satisfy the equatioﬁ g-(atl, \$2) - c - The function f (:51, 2:2) is caﬂedthe'abjéctive- funct tion 9(x1,\$2]—- - c is called tﬁe'cbrrﬁtfdiﬁt. Metﬁods \ of constrained maximization problem are” described Chapter 5. ...
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