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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 welldefined 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=S4\, 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, 1a = a. Addition: 3. + b: 0 —)b= —a. multiplication: for any non zeroa, 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,...) 032 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”,”
Hamil5 : 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 Dﬂauhhue 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(x04h)—;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... (Ewan12d? m. mm mm a} mum? ﬁe
41m Jﬂwﬁd :4 rd: Pant oﬁ‘ﬂumm c123: SQmAdJFW .: 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.CSoﬂe.‘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'dordermmél‘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)) = (hgHX) 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 god1744 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 fI. 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 f1 is Cl. hmse Function Theorem:
Iffgl —>R is C1 and ﬁx) atOforaﬂxE I.
s f is invertible on I (f1 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!
alfaszars
(arr=a'8
3°21 .‘ﬁ‘i Pumacities 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’mFaﬁcf {mica(e:
. duosxl =lllx)dx — (whﬂm rhnwﬂ
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— ~— —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 CompoundAngle 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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