univariate - ECON 5000 - I. UNIVARIATE CALCULUS I....

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Unformatted text preview: ECON 5000 - I. UNIVARIATE CALCULUS I. Unlvcrio’re Calculus Preliminaries 3 Afl’lnd'i '45-. 1. Sets A set is a well-defined collection of objects. For example we could define the set A as A a {1.2.3} or r B = {1,3,5} The numbers inside the brackets are called the elements of a set. The symbol, 6, means that a particular object belongs to the set in question while the symbol e means that an object does not belong to the set in question. ex. 16A, SEA Some examples of sets are: 9 £5: { } is called the null or empty set. 9 R is the set of real numbers. 9 U is the universal set. Notation R‘={XER:XEU} R‘is a subset of R. That is, all the elements of R,are contained in R. Another way of writing this is; man ex. {1,3} c A ECON 5000 1. UNIVARIATE CALCULUS Set Operations 60.. O 3. Uhion A U B = {x x e A or x e B} IntersectionA FIB = {x x e A and x E B} Difference A — B = {x x e A and x E B} (sometimes denoted A\B) Complement.AC=S-4\, always taken relative to some set e.g. S Rhinibers Natural numbers N {l,2,3,4,...} Integers Z = {...,—3,—2,-l,0,l,2,3,...} Rational numbers Q expansions repeating. Irrational numbers are numbers like, e, n, and 42. These numbers have infinite non—repeating decimal expansions. The set of Real numbers is given by RationaLJIrrational numbers. {a/b: a, b e Z, b i 0} The decimal of these numbers are either finite or infinite EVen numbers E = {x : x e Z : x = y/2, yeZ} Odd numbers are given by Z — E. ab (a, b are natural numbers) a = 1 Prime number m = or Inter val Notation Given two real numbers a and b, the set of all real numbers between a and b is an open interval. 0 O O 0 Open, (a,b) = {x ERH a.< x < b} Closed, [a,b] = {x 6 RH a Sx S b} Half open, (a,b] = {x 6 RW a.< x S b} Infinite interval, (a,m) = {xeRfl x > a} Notions of open and closed extend to sets more generally. 4. (SB, Sequences and Limits in R1 12.1) ECON 5000 I. UNIVARMTE CALCULUS 5. Operations Example: multiplication, and addition on real numbers. 6 Closure. aeR, bell —+ a + b, ab 6 R. Q Cmutativ‘e. a + b = b + a, ab £’ ba. 0 Associative. (a + b) + c = a + (b + c), (ab)c = a(bc) . Identity. 0 + a = a, 1-a = a. Addition: 3. + b: 0 —)b= —a. multiplication: for any non zero-a, ac = 1 -+ c l/a. II Distributive. a(b+c) = ab + ac. 6. Least Upper Bound Property This is an extremely important property of real numbers. Let, SCRandbe R. o b is an upper bound for S if a S b for all a e S. O b is a lower bound for S if a 2 b for all a e S.__ a It b is an upper bound for S and no element a < b is an upper bound for S, then b is a Least Upper Bound (LUB) . Q If b is a lower bound for S and no element a > b is a lower bound for S, then b is a Greatest Lower Bound (GLB) . LUB property: For any subset S of'R, if S has an upper bound, it has a least upper bound. GLB property: For any subset S of R, if S has a lower bound, it has a greatest lower bound. “.1 s = {0.3, 0.33, 0.333,...) 03-2 Lei- R“; K—io’é Upper bounds: 3.4, 4, 100... N LUB = 1/3 E S " ‘ Lower bounds: 0, 0.2. . . Gauge!“ (0) I) C R GLB = 0.3 e S g ECON 5000 I. UNIVARIATE CALCULUS Functions Clasp—fer Z 7. Functions on. R1 (Provisional definition) A function is a rule mapping a number from R1 to R1. f: R1 -—) R1 f(x) = x + l y = x + 1 Here we call x the independent or exogenous variable and y is the dependent or endogenous variable. 1 is a parameter. A parameter can define a family of functions: y=x+a A monomial is a function that can be written in the form; f(x) = axk , k is a non—negative integer Where a, k are parameters, k is the degree of the monomial (polynomial), a is the coefficient. A general polynomial can be written as a sum of monomials such as; f(x) x3 + 2x2 The degree of a polynomial is the degree of the monomial of highest degree. In this case it is (x3 is 3. Some other functions: 0 Rational functions, such as y = (xfiJJ/(x—l) ; x¢l 0 EXponential function, such as y = 3b 0 Trigonometric function, such as y = sin x. ECON 5000 I. UNIVARIATE CALCULUS 8. Domain and Range Some functions are defined only on proper subsets of R1. Ex. y = 1/3: is not defined for x = O. The domain of this function is the subset R1 - {0}. Some functions only map into proper subsets of R1. Ex. y - x2. The range of this function is {y:yeR, 3'20} 9. Graphs A number line (with origin) can represent RI. Since a function maps from R1 to R1, a representation of functions will require two number lines. Graphs represent a function in the Cartesian plane, {(X.y) :y f(X)}. 10. Increasing and Decreasingfunctions O A function, f, is said to be increasing if x12»):2 —) f(xl) > flag). 6 A function, f, is said to be decreasing if x1>x2 —> f(x1) < f(x2) . A function if changes from decreasing to increasing at a local minimum (x0,f(xo) ) . Note, if f(x) > f(xo) for all x at xu then we have a global minimum. A function f changes from increasing to decreasing at a local maximum (x*,f(x*)) . If f(x*) > f(x) for all x at x*, then we have a global maximum. 1 1. Concavity and Convexity Defn: A function is called convex (concave) if over the interval [a.b]. (l—t)f(a) + tf(b) 2(S) f((l—t)a+tb)) t E[O,l] ECONSfln LUNHMRMJECMLCUIUS 12. Linear Functions Polynomials of degree 0 are constant functions, i.e. f(x) = b Polynomials of degree 1 are linear functions, In. woody. ‘ . (ilSlofe: (Renae In (mm Af(x) yryo m,—b—mxo+b «no than“: 5? land Slope = —=——=————=m -. - \ h '3'. Ax xI—xo xI—xo 010‘) (g x, Mc‘MACQS‘ cm: 9 en Slope. 9" Immgc M -F ditfi‘?!‘ La h For a linear function the slope is independent of x, i.e. it is the same everywhere. i.e.y=f(x)=mx+b The second parameter (b) is the y intercept (f(0)=b) . We can compute the equation of a line from: EL. 'm‘he slope and a point 2. Two points 1 In applications slope is a rate of change, velocity, marginal cost etc. - 7 The slope of a line perpendicular to the line y = mx + b is —l/m. . A concept that is often used in economics is elasticity; Ay £35- . . y Blast or: — 1c: Ax y A% 'the advantage of this measure is that it is independent of units of measure while SIOpe is not. We: gm cw fut {he 54,,“ “(a {.53 we“ 7”,” Hamil-5 : slave}; Mt Mix free ECON 5000 I. UNIVARMTE CALCULUS Derivatives and Differentials 13. The Slope of Nonlinear functions and Definition of a Derivative The slope of a nonlinear function is not constant. The natural measure of the slope of f(x) at xois the slope of the tangent to f(xo), which closely approximates f(x) at xa. We call this slope the derivative of f(x) at XV This is represented as: f’fxo) or fibre) We formally define the derivative as the limit of a sequence of secants m Him flxo + h)-f(xa) 11—»? h if this limit exists. We say that f is differentiable at xoif this limit exists. 14. Rulesfor Derivatives (The proofs can be done via limits). 0 f(xu) = xk, where k is a constant, f'(xu) = kxH O (f + g)’ (x0) = f’(xo) + g'(xu) o (kf)'(xa) = k(f'(x.,)) O (f - g)‘(xu) = f’(xo)g(xo) + f(xo)g'(xa) o (f/g)'(xo) = [ f‘(x.,)g(x.) — f(xa)g'(xn)]/g(xu)2 O (f(X)“)' = n(f(X))“"-f'(X) ECON 5000 I. UNIVARIATE CALCULUS l5. Differentiability and Continuity Defn. A function is differentiable if it is differentiable at every point xaeD. A function is differentiable at X“ if; .fl%+hhfiu Im— A. —)0 h” iexists and is the same for all {h} sequences converging to 0. Intuition: Differentiable <=> smooth ex: y =Ix[+1. This is a non—differentiable function. The above limit exists x0, but is not the same for all sequences {ha}. when x50. Consider the sequences h_ { +.1, +.1‘, +.1’....} and h“ {—.1. —.12, —.1’,...} {late that both converge to zero). Evaluating the relevant limit at xo=0 with the former gives +1,- with the later —1. Note that the former is “taking the limit from the right' and the later is “taking the limit from the left”. Defn: f: D—)R1 is continuous at x0 6 D if for any sequence {xn} which converges to x in D, f(x) converges to f(x) . Intuition: A continuous function is one with no breaks. If f'(x) of x is a continuous function, f is continuously differentiable (C1). Nofei D-flauhhue M (We Mfigms bwf' MZW {Lawn um: Jn‘P—rmtble- u «n "de g fmhhaétk ECON 5000 I. UNIVARIATE CALCULUS 1 6. Higher Order Derivatives Let the function f be C2 the derivative of f'() is called the second derivative of f and can be written, f'(x0). igfi), d f (x0) dx dx d? If f” is continuous, f is said to be twice continuously differentiable (CW. Generally, d‘flx) dxk f *(x)= (x) If the kEh derivative is continuous then the function is said to be c". 1 7. Approximation by Differentials Recall the slope of the tangent to f at (xu,f(xo)) can be approximated by the slope of a secant line through xw f(x04-h)—;f(x0) h =f’(xo) when h is small. Note that the converse is true. and that is really the slope of the secant (or at least f(xfih)—f(x)) that we are interested in. f(xo+h) — f(xo) = f’(xa)h So we can approximate the change in f() due to an additional (marginal) increase in x from xtr The important point is that this is only an approximation unless h is infinitesimally small (or f is linear). Of course, we never really make infinitesimally small changes in x. (Imagine that f models the relationship between a policy and an outcome),- in reality h is typically discrete. ECO” 5000 I. UNIVARIA TE CALCULUS We use differentials because they: 9 are sometimes easier to evaluate than f(xuln - f(xg 9 sometimes provide a useful summary of the qualitative nature of f (see the section on graphing). More notation: Let h = Ax. Ay = f(xo+Ax) — f(x°) = f'(xo)Ax f(x5 + Ax) = f(x5) + f'Ug)Ax &y is an exact change, the smaller is Ax the better the approximation. Define: See (0 a— 9 dy = f'hg)Ax (the change in y along the tangent line) odx=Ax. ‘ dy, dx are differentials. Ay = dy = £'(x,)dx The bold section above is called the equation of the differential. O Sacond Differentials 7L2 Mammy A? Camk mqa n‘ a» a Mai 16 was. an... (Ewan-12d? m. mm mm a} mum? fie 41m Jflwfid :4 rd: Pant ofi‘flumm c123: SQmAdJ-FW .-: déflcificfl‘ffirufilfl) .475 oh I :: fi “@76) m game}? afi Dcflfuhm 1‘. 130m Juflerewhls d O : ECON 5000 I. UNIVARIATE CALCULUS 18. Taylor's Series: the One Variable Case Let f(x) be a real valued function on the closed interval [a,b] such that for all x 6 [a,b], f', f', ...,fflfmlall exist and the n+1st derivative f "” is continuous on [a,b]. Then f(x) can be written: 2. f(x) = f(a) + f'(a)(x—a)/1! + f”(a)(x—a)¢/2! + ... + f“(a)(x—aJ“/n! + R’Jx) A {3 where: x EIa,b]. RM is called the remainder term with: lim Rn+1(x)=0 n—)°o R%d{x} can be written in many forms: The Lagrange form is fn+l&) __ n+r "+100: (n+1)! (x a) For some c E [a,b]. The nice thing about this form of the remainder is that you can determine the-“worst case” size of Rmfixj by choosing c to maximize Rm1(x). Sometimes Taylor's formula is used to approximate f(x+Ax). Assuming that f(x) satisfies the conditions of Taylor's Theorem over the interval [x, x+Ax] we can apply the Theorem to obtain; _f(x+Ax) = f(x) + f'(x)Ax/1! + f"(x)(Ax)'/2! + Often what happens in practice is that the series is truncated after a few terms (2?) And the remainder is ignored. This approach will give better results the more terms you add. The truly remarkable property of Taylor's Theorem is how much you can learn about f(x+Ax) from the properties of f(x) and its' derivatives at x. When x:=: D a Taylor series is sometimes called a MacLauren series. 11 E CON 500‘) I. UNIVARIA TE CALCULUS 1 9. Using Derivatives for Graphing 0 If f'(xn) > 0 there is an open interval containing x5 on which f is increasing. § If f'(x5) < 0 there is an open interval containing x5 on which f is decreasing. Let f be a continuous differentiable function on domain D C Rh i t' > 0 on (a,b) c D —9 f is increasing on (a,b). i f' < 0 on (a,b) C D —+ f is decreasing on (a,b). Critical points occur' when f'(xJ = 0. f' is :> or‘ < zero on intervals between these points. 20. Second Derivatives and Convexity We can use the second derivative of a function to learn more about a function's curvature. Suppose we know that the function y = f(x) is increasing over the range x 6(0, b). In the previous section we learned that, f‘(x) > 0 over this range. The problem is that we don't learn much about the 'shape of f from this information. To learn more about what the function is doing over (0,b) we use the second derivative, f'(x). Consider, [figure] o a b 9" 0 Over the range, (0,a) the function increases at an increasing rate, over this range f'(x) > 0. 0 Over the range, (a,b) the function increases at a decreasing rate, over this range f'(x) < 0. At, the point a, f'(x) = 0, this is called an inflection point.CSofl-e.‘flmq9 The function is conVEX'on [0,a] and concave on [a,b]. 12 ECON 54113 I. UNIVARIATE CALCULUS 2}. Maxima and Minima He can use the second derivative of a function to check for (local) maxima and/or minima. Ex. Suppose we have f'(x) = O at some value x*. To see whether this is a local max. or min. check f'(x*). If f'(x*) < 0 we have a local maximum, if f'(x*) > 0 we have a local minimum. More on Functions Excin Kales (D H? 011 We'd-ordermmél‘o dx‘MModdMW>l v . . daflwnimfiLy {:0 iii? fibfiu (SB 3 4) ix} “Fofhfoe 2'2. Asymptotes and tails 23. Composite functions Let fix} = h(g(X)) = (h-gHX) Ex. Profit function nty) Production function y = f(L) P(L) = (mf)L Chain rule: d)’ (1de dx dzdx Alternative Notation: Let y = h(x) and z - g (x). %(h(g(x»)=h'(g(x»— gm 24. Monotonicity Detn: (for Clfiunctions) f is monotonically increasing (decreasing) on E if f’(x) > O (< 0) for all x e E. 13 DUWs god-1744 also 7 Q ECON 5000 25. Inverse Functions Consider: f:El—)R1E1CR1 l 1 gzfiz—lREch g is the inverse of f if I. UNIVARIATE CALCULUS t gifixH = x for all x e E1 J‘ufffl.‘ 9 ttgiz)) = z for all z e E2 g; "| I. :1 Defn: If x1 at x2 and fur) at f(x,) xv}:2 e E, f is a one—to—one or injective mapping on E. s For a [unction to be invertible, it must be one to one. s N a finnction f is one to one, its inverse exists._ i N f is invertible on its domain, then its inverse is unique. The imrerse of f is denoted f-I. 7 o f is one to one and hence has an inverse on E if and only if f is monotonically increasing or monotonically decreasing on E. n The hmrse of a function is its' reflection through the 450 line {x=y}. I Ma function fis Cl then f-1 is Cl. hmse Function Theorem: Iffgl —>R| is C1 and fix) atOforaflxE I. s f is invertible on I (f-1 exists). 9 [-l is Cl on fll). 9 lioraze flI): , 7 1 g (I) f'(g(z)) [See proof on p. 80. In Simon and Blume) 14 ECON 5000 I. UNIVARIATE CALCULUS 26. Exponential Functions x f(x) = a = a a a..... ‘x’ times. a is the base and x is the exponent. y = af increases faster than any polynomial — exponentially fast note: 6 a°= l a”‘=fla o a"y=(%/E)‘ 1 x<0, a" =T a The most common base for exponential functions is e. I 1'! eElim[1 + —] n—Mn n etzlim [I + E] n—>=a I! Consider the return with interest rate r on an investment over the time period, 15 ECGN 5000 I. UNIVARIATE CALCULUS Logaritth are the inverse functions of exponential functions. 3! = loga(z) air = z antz)=z Cmmnon bases for logs are 10, and e. The base e log is often written In. e'lze. inezl. eflzl, lnl=0. Wrties of exponentials: araSEafls awful/a! alfaszar-s (arr=a'8 3°21 .‘fi‘i Puma-cities OI logs: 7 Md=hslfl+loglsl malavtlnfie. balls! 2 - log 8 Ioglrfsb = log(r) - log{s) log 1'8 = s logir) he 1 = o Derivatives; - o (is) new; / PM \e’m-Faficf- {mica-(e: . duosxl =lllx)dx -—- (whflm rhnwfl Wider: 10g Y = a log x This function exhibits constant elasticity which is useful in certain applications such as econometrics. ‘0‘“. {.m‘ 7- 16 EMA Dfin'h ' demo Fee) 4"" ‘an , I “:9 ‘pGQ- EF '5 ‘5'” ' «5‘ 0 .3r 5 fl unwrr M, mo“ u.+“"‘°’ anr?‘ “I. .61 {who = {-6. -')€ + (‘7‘) (“)3- '91:] ‘ 7 : [xi'fl ' 43¢ aw @Mm : iW/a, .1 74d anre 0‘ D-cfiE/M¢finl [fiam- ‘ .fix)J%M-JW Mm; gm dc ( aw) we 52% mm may #«2‘ %M _ 1fl[%)) 40m: ' 71M, fizmc Mac 51$; £47; : 7/531) 3”; WC); LL0+9¢Q+3 gag. MM, ,4» .24 36x): gémxi) 4-K M L»; W MW. . 7 Arc/Zach? aflmaz) thra’fl») : 3,: M amimmam 0’99 (+11 2 ’<. frfl’éi 34%): éLU—Drf) 1-K 7 £25ch five—4%” ' 39¢),- MIW %Qfl)*?{x) 72AM,an Méo 99a; "E. " ', ...ar%z)-4-_e-"*7542¢ 2%,, WW 6‘1 2co<2c,<xL< (29,25 be. Mr: Arm: gag—x“) p.21) ,vh jib W WW4». fab; M #4 M #14722 UAW/.96 9:2” A0 70%? ELL-q, é x: 590,; {CM 412 m 7%; M [(74) at WpW’fé " £77m mt. 97L WW % flocww paw [25.4%.]. 11’“me ’ ac Mg; m 15(74): mg 7%” at W: __M,_A7.L,_ *ML491; + Nb A20” A) xL = t4z+r79+23£= +23% :9” 3 2 m T‘ZEQi—ZSO +290 .— €34 . a funcj (TEX-93% 4‘15? a OLE? b S. 284x _ n,,c'n+b(2nrn>_f 2. .1 fm , u if " ° _ _ .__n->W .8 n Pm‘ Maw 7n» KO» W m W)»; m filmfiw ('5- " my mm M W [(2);],7244) 7, h __ _JJ flood» = 7365) W _ _ ,, 5’4 __,, a , . W )W} £4 a, L-I n g’Lf 2439;; =— 19mg, 1rd, +x1-x,+ + 5—29“, [3| puff My» ’le ‘ >0 M, 7¢+Ax Wm M x+~4>¢ £11k;— f’LL W [(112] 9134'» 3‘5 3°.) flxwfl’flx}; L #90415 —. #(éxd’c a. ’- xwx r" I FéeHJc X. 1’ Cm: 5'1’= 47.) CL, {rm m 4= we a»: 3M" * 'F (var-4x) w F650 — 75w) Ave (2) W 715w» Hair-3 14%) * 5») fl“! FM 2 A, reek flea An ~90 A Em MTW bu .fl (Ara/bar 7&4— }:(x) — W m1_m;ca,é_ C; q _ 6(1) ; .fifixgaadt W- _ _ j: jaw: _ 7 . C25 'flélw Wéficgfifi) -2Kn'o<:£%/i _ _ _ «Lé @925 Wdéfé/‘S. (bufflx‘)? if ) 13509) f: 14'" f u ' @0911“ 7 Cid/- . 'chv‘aa “War—FCWD Notim ’ Mg) 2 HM» = F65) , , . , _ Moo: Ham = F0.) , ’ £0 .fmex 2 F(b)—F(a): wam , HfFVHCé): fgfdm L WWW?th , f; ._. H _ .1 .— Hw wfl_._.. _ ,_ v a. _m _._..u._ «v __ --—\—»V7 i v“ .- .‘ ...__., . 17 _k m __ _ — ~— —-r —A — - ,— an. — i v.0..." Wfi—wn—HW mrm m v.4“: .—_...u — .2 I __._.,,,H ,m.n$_héw4éyyflmecmfim.wiéfimlh“MW” _ mm“_hm_m..,_.fl,. «- .— j—. q ...7 “w __ .w -mfiwmflr _ mwmww—hmk .- fi—r WWW—.1 i— -_—.u.._. ——. Section 13.5 Trigonometric Identities Compound-Angle Identities To establish formulas for the sine, cosine, and tangent func— tionsof the sum (9 + tp) and difference (9 -— ¢) of angles 9 and ¢ in terms of functions of the constituent angles, we refer to the compound figure below. sin(9+¢) = sin9cos¢ + cosesinmp. Replace ¢in(1)by(—¢) toobtain sin(9-¢) = sinecosq‘r — c0895in¢. Replace sin (2) by G- a) to obtain . 1r , 1: m(§—9—¢)=sm(-2-- OJCOBCp , 7 fl . -cos(:z— 9)s1n¢ ' cos(9+¢)= cosBcos¢—sinesin¢. Replace ¢ (3) by (-4011) obtain cos(9—V¢) = case cos ¢+sin9 sine. ...
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This note was uploaded on 11/20/2009 for the course ECON 5000 taught by Professor Smith during the Summer '09 term at York University.

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univariate - ECON 5000 - I. UNIVARIATE CALCULUS I....

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