Matematika Ekonomi2

7/24/2019 Matematika Ekonomi2

1/72

MathematicalMathematicalEconomicsEconomics

Week 2Week 2


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22

Relation and FunctionRelation and Function


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33

RelationRelation

A relation: (between two variables) is aset of ordered pairs of real numbers

Unordered sets or pairs NOT a relation Example: nalist of !ndonesia !dol "(#i$e% &)%

('udi$a%2)% (irman%3) * "('udi$a%&)% (irman%2)%(#i$e%3) * "(irman%&)% (#i$e%2)% ('udi$a%3)

Ordered sets or pairs a relation Example: 'uara !ndonesia !dol diurut$an

berdasar$an poolin+ ,#, -an+ masu$ "(#i$e%&)% ('udi$a%2)% (irman%3)


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..

Domain and RangeDomain and Range

/omain: T0e setof all rst

elements of a+iven relation

1an+e of t0erelation: T0e set

of all seondelements of a+iven relation

'uara!ndonesia

!dol(/omain)

oolin+ ,#,(1an+e)


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44

Relation: exampleRelation: example

E$$i

5a+d6a

#utiara

e-7a

!ntan

8ubun+an A-a0 9 Ana$

E$$i

5a+d6a

Ari

awan

1a0mat

a0mi

8ubun+an ertemanan


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;;

RelationRelation

Anot0er example% t0e set"(x%-):-x -*x

x

-


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>>% we an write

- * f (x) w0i0 is read a ?- e=uals f of x?or ?- is t0e ima+e of x under t0e funtion f ?>


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@@

Function: exampleFunction: example

#atemati$a

E$> #a$ro

,tatisti$a

E$onometri$a

Ari

err-

Amel

#aman

un+si #ata ulia0 9 /osen

&B2B

3B.B4B;B


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CC

FunctionFunction

!n t0e funtion - * f(x)% x is alledt0e ar+ument of t0e funtion and

- is t0e value of t0e funtion Alternativel-% we ma- refer to x

as t0e independent variable and -

as t0e dependent variable


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&B&B

FunctionFunction

or example% if - * f(x) * x2% weobtain - b- s=uarin+ t0e value of x>

!n speif-in+ a funtion% we s0ouldin priniple speif- its domain (/)

Example: -*x2%x1 -*x2%x1D

One to one funtion :f(x&)f(x2) x&x2% x&%x2/


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&&&&

One to one Function:One to one Function:

exampleexample

1atna

!ra

Endan+

An+$i

5a+d6a

Ari

un+si erni$a0anf(xi)*"(xi%f(xi):f(xi)*xi

2%&x;%xiD

f(x&)f(x2) x&x2% x&%x2/

&23.4;

xi

&.C

&;24

3;

f(xi

)


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&2&2

SummarySummary

&to& Forrespondene&to& Forrespondene

untionuntion

1elation1elation

1elation untion

&to& orrespondene


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&3&3

Inerse !unctionInerse !unction

!f a funtion - * f(x) isone9to9one% wit0 domainA and ran+e 5% we anonstrut its inverse f9&%

wit0 domain 5 and ran+eA% su0 t0at if (xi% f(xi)) isan ordered pair of t0efuntion f t0en t0ereexists (-i%f9&(-i)) su0 t0at

-i* f(xi) and f9&(-i) * xi f(x)*2xf(x)*2x ff9&9&(-)*&G2 f(x)(-)*&G2 f(x) -*&Gx x*&G- f(t)*etfH(t)*lo+e-*ln -

5A

5 A


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&.&.

Inerse !unction"Inerse !unction"

graphicallygraphically

.4B

t*ln -

-

t

.4B

-*et

t

-


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&4&4

#omposite !unctions#omposite !unctions

Fomposite funtions: !f - is afuntion of x and x is a funtion of

t% t0en - is a funtion of t Ienerall-% if - * f(x) and x * +(t)%

t0en we write - * f(+(t))

or example% if - * 2x and x * t2%t0en - * f(+(t)) * 2t2


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&;&;

$eneral and Speci%c$eneral and Speci%c

FunctionFunction Ieneral funtion: t0e funtion is writtenIeneral funtion: t0e funtion is written

onl- in s-mbols of independent variablesonl- in s-mbols of independent variablesand ommas wit0 no numerialand ommas wit0 no numerialoeJientsoeJients

Example :Example : F*F(K)F*F(K)

//fxfx*/*/fxfx(!#%O%(!#%O%%%L%i%iL%eML%i%iL%eM imim%)%) UU&&* U* U&&(P(P&&&&%P%P&&22%%P%%P&&nn%U%U22%U%U33%%U%%Unn))

F*FF*FBBD#F>KD#F>Kdd /epends on t0e level of +eneralit-/epends on t0e level of +eneralit-


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&Kdd ##dd*93BBi D &BBBK*93BBi D &BBBK

QQdxdx*93*93xxD.D.--9B>49B>477D&BKD&BK


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&@&@

Explicit s implicitExplicit s implicit

!unctions!unctions ExpliitExpliit w0en we an determine w0i0w0en we an determine w0i0

variable is t0e dependent variable% and w0i0variable is t0e dependent variable% and w0i0variable(s) are t0e independent variable(s)variable(s) are t0e independent variable(s)

/ibatasi ole0 tanda * (atau ine=ualit- si+ns)/ibatasi ole0 tanda * (atau ine=ualit- si+ns) !mpliit!mpliit w0en we an not determine (arew0en we an not determine (are

not sure) w0i0 variable is t0e dependent Gnot sure) w0i0 variable is t0e dependent Gindependent variableindependent variable All variables are on t0e left 0and side of t0eAll variables are on t0e left 0and side of t0e

e=ualit- (ine=ualit-) si+ne=ualit- (ine=ualit-) si+n

Example of impliit funtion:Example of impliit funtion:

f(x%-%7)*BR .xD4-92*Bf(x%-%7)*BR .xD4-92*B


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&C&C

&ypes o! !unctions&ypes o! !unctions

un+si

un+si Al6abar un+si non9all6abar

un+si irasional un+si rasional un+si e$sponensial

un+si lo+aritma

un+si tri+onometri

un+si linear

un+si $uadrat

un+si ubi$

un+si pan+$at (tin++i)

un+si pea0


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2B2B

'inear Function'inear Function

-*aDbx% x-*aDbx% x!1!1 t0e power oft0e power ofindependent variable(s) is(are) * &independent variable(s) is(are) * &

Irap0: strai+0t lineIrap0: strai+0t line a*interept (- axis)% t0e value of - w0ena*interept (- axis)% t0e value of - w0en

x*Bx*B T0e 0an+e in t0e dependent variableT0e 0an+e in t0e dependent variable

beause of one unit 0an+e in t0ebeause of one unit 0an+e in t0e

independent variable is onstant (t0eindependent variable is onstant (t0eslope% b% is onstant)slope% b% is onstant) b *b * -G-Gxx Example: a onstant opportunit- ostExample: a onstant opportunit- ost


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2&2&

Example: constant costExample: constant cost

((F((F -*&BB9(&G.)x%-*&BB9(&G.)x%

xxDD

,lope * 9&G.,lope * 9&G. !nterept * &BB!nterept * &BB

!f x is not!f x is notprodued% t0en aprodued% t0en a

ountr- anountr- anprodue aprodue a

maximum of &BB -maximum of &BB -

&BB

.BB

easible

re+ion

-

x


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2222

)uadratic !unction)uadratic !unction

-*ax-*ax22DbxD% aDbxD% aB% xB% x!1!1 T0e power ofT0e power of

independent variable(s)independent variable(s)is(are)is(are) 22

Often: domain is !1Often: domain is !1DD aSBaSB onvex (U9onvex (U9

s0aped)% 0as a minimums0aped)% 0as a minimumpointpoint

aBaB onaveonave(inverted U9s0aped)% 0as(inverted U9s0aped)% 0asa maximum pointa maximum point

#inimum or maximum#inimum or maximumours at t0e point x*9ours at t0e point x*9bG2abG2a

#aximumGminimum in a#aximumGminimum in apositive value of xpositive value of x ififbSB% in ne+ative valuebSB% in ne+ative valueof xof x if bBif bB

- axis interept * - axis interept * x axis interept(s)x axis interept(s)

-*B-*B =uadrati=uadratie=uation% an use t0ee=uation% an use t0eformula:formula:

=

2

1,2

4

2

b b a cx

a


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2323

)uadratic !unction)uadratic !unction

/isriminant : / * b/isriminant : / * b22 .a .a !f /SB : bot0 of t0e roots are real!f /SB : bot0 of t0e roots are real

numbers and 0ave dierent valuesnumbers and 0ave dierent values !f /*B : bot0 of t0e roots are real and!f /*B : bot0 of t0e roots are real and

0ave e=ual values0ave e=ual values !f /B : bot0 of t0e roots are not real!f /B : bot0 of t0e roots are not real

/G9.a*t0e maximum value of t0e/G9.a*t0e maximum value of t0edependent variable (if aB) ordependent variable (if aB) orminimum value (if aSB)minimum value (if aSB)


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2.2.

ExampleExample

QQdd*9*922D .D .

aB% b*B% SB%aB% b*B% SB%

/*B9(.>(9&)>(.))/*B9(.>(9&)>(.))*&;SB*&;SB An inverted UAn inverted U

s0aped urves0aped urve

wit0 a maximumwit0 a maximumpoint at *B andpoint at *B and

0as real roots0as real roots

Qd

92 2

.

B

!1!1DD


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2424

#u*ic !unction#u*ic !unction

-*ax-*ax33DbxDbx22DxDdDxDd T0e power of independent variable(s)T0e power of independent variable(s)

is(are)is(are) 33 To nd t0e roots (xTo nd t0e roots (xii) an use fatorin+) an use fatorin+ No +eneral rule of fatorin+ existsNo +eneral rule of fatorin+ exists trialtrial

and errorand error

To +rap0To +rap0 use urve train+ met0oduse urve train+ met0od Fan be found in Total Fost Furve orFan be found in Total Fost Furve or

Total rodut FurveTotal rodut Furve


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2;2;

#u*ic Function#u*ic Function

Example: - * xExample: - * x33 x x22 .x D . .x D . ind t0e rootsVind t0e rootsV

To nd t0e roots% -*BTo nd t0e roots% -*B

xx33 x x22 .x D . * B .x D . * B

(x &) (x D 2) (x 2) * B(x &) (x D 2) (x 2) * B

xx&&*&% x*&% x22*92 x*92 x33*2*2

Irap0Irap0mathcad


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2


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2@2@

(o,er" exponential"(o,er" exponential"

and logarithmicand logarithmic!unctions!unctions ower untion%ower untion%Ieneral form: -*axIeneral form: -*axbb

5asi rules (review)5asi rules (review) aannaamm*a*anDmnDm

(a(ann))mm*a*anmnm

aannGaGamm*a*an9mn9m

aaBB*&*&

aa&Gn&Gn**nnaa aa9n9n*&Ga*&Gann

aammbbmm*(ab)*(ab)mm

Exponential funtion%Exponential funtion%

Ieneral form: -*bIeneral form: -*bxx

bbbase of t0ebase of t0e

funtionfuntion !n man- ases b!n man- ases b

ta$es t0e numberta$es t0e numbere*2>


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3B3B

'ogarithmic !unction'ogarithmic !unction

Xo+arit0mi funtion is t0e inverse ofXo+arit0mi funtion is t0e inverse ofexponential funtionexponential funtion

Ieneral form: -*Ieneral form: -*bblo+ x or -*lo+lo+ x or -*lo+bbxx

T0e inverse: x * bT0e inverse: x * b--

Fommon lo+: b * &BFommon lo+: b * &B Natural lo+: b * eNatural lo+: b * e

Natural lo+arit0m% +eneral form: -*lo+Natural lo+arit0m% +eneral form: -*lo+ee- * ln x- * ln x T0e lo+ of t0e base * &T0e lo+ of t0e base * &

Example:Example:&BB * &B&BB * &B222 * lo+2 * lo+&B&B&BB or lo+ &BB * 2&BB or lo+ &BB * 2

ln e * &ln e * &

ln eln e33* 3* 3


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3&3&

'ogarithms Rules'ogarithms Rules

Y ln(uv) * ln u D ln vln(uv) * ln u D ln v(u%vSB)(u%vSB)Y ln(eln(e33>e>e22) * ln e) * ln e33D lnD ln

ee22* 3D2*4* 3D2*4Y ln(uGv) * ln u ln vln(uGv) * ln u ln v

(u%vSB)(u%vSB)Y ln(eln(e22G) *2 ln G) *2 ln

Y ln(uln(uaa)*a ln u)*a ln u

Y ln eln e&4&4

* &4* &4

Y ln(uvln(uvaa)* ln u D ln v)* ln u D ln vaa* ln u D a ln v* ln u D a ln vY ln(x-ln(x-22) * ln x D 2 ln -) * ln x D 2 ln -

Y

ln(uln(uv)v) ln uln u ln vln vY ln(eln(e44ee22)) ln (4D2)ln (4D2) ln (492)ln (492)

Y Xo+Xo+bbu*(u*(bblo+ e)(lo+ e)(eelo+ u)lo+ u)Y u * eu * eppp *p * eelo+ ulo+ ulo+lo+bbeepp*(*(bblo+ e) plo+ e) p

p lo+p lo+bb e * p (e * p (bblo+ e)lo+ e)pp bblo+ e * plo+ e * p bblo+ elo+ e

Y lo+lo+bb e *&G(lo+e *&G(lo+eeb)b)Y Xet u * bXet u * b


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3232

'ogarithms Rules'ogarithms Rules

Y lo+lo+bb e *&G(lo+e *&G(lo+eeb)b)Y Xet u * bXet u * b Y Xo+Xo+bb b*(lo+b*(lo+bbe)(lo+e)(lo+ee

b)b)& *(lo+& *(lo+bbe)(lo+e)(lo+eeb)b)

lo+lo+bbe* &G(lo+e* &G(lo+bbe)e)


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3333

Exponential e-uationExponential e-uation

,olve ab,olve abxx * B% lets tr- to solve * B% lets tr- to solvet0ist0is


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3.3.

Function ,ith 2 orFunction ,ith 2 or

more independentmore independentaria*lesaria*les !n eonomis% t0is t-pe of funtion is!n eonomis% t0is t-pe of funtion isused fre=uentl-used fre=uentl- Q*f(%X)Q*f(%X)

Fobb /ou+lass prodution funtion: Q*AFobb /ou+lass prodution funtion: Q*AXX

Example:Example: -*x-*x&&22>x>x2222

Q*Q*B>4B>4XXB>4B>4

U*xU*x&&xx2292x92x&& 3/ Irap0er (demo)3/ Irap0er (demo)
http://cobb-douglas.3dg/http://cobb-douglas.3dg/http://cobb-douglas.3dg/


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3434

#onexity and#onexity and

#oncaity o! a !unction#oncaity o! a !unction T0e funtion 7*f(xT0e funtion 7*f(x&&%x%x22) is onave) is onave

(onvex) if% for an- pair of distint(onvex) if% for an- pair of distint

points # and N on its +rap0 a surfaepoints # and N on its +rap0 a surfae line se+ment #N lies eit0er line se+ment #N lies eit0er on oron orbelowbelow(above) t0e surfae>(above) t0e surfae>

T0e funtion is stritl- onaveT0e funtion is stritl- onave

(onvex) i line se+ment #N lies(onvex) i line se+ment #N liesentirel- below (above) t0e surfae%entirel- below (above) t0e surfae%

exept at # and Nexept at # and N


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3;3;

#onexity and#onexity and

#oncaity o! a !unction#oncaity o! a !unction

#

N

#

N


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3


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3@3@

'eel Set'eel Set

Appl-in+ t0e denition of a levelAppl-in+ t0e denition of a levelset:set:

aa&&xx&&DaDa22xx22* * xx22*Ga*Ga22 (a (a&&GaGa22)x)x&&

xx&&aaxx22bb* * xx22*(x*(x&&9a9a))&Gb&Gb

Now an be +rap0ed in a 2/ spaeNow an be +rap0ed in a 2/ spae


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3C3C

'eel Set'eel Set

B&

2

x&

x2

B

&

2

x&

x2


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.B.B

Se-uences" Series" andSe-uences" Series" and'imits'imits


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.&.&

Se-uenceSe-uence

A suession of numbersR A funtion w0oseA suession of numbersR A funtion w0osedomain is t0e positive inte+ersR a set of numbersdomain is t0e positive inte+ersR a set of numberst0at are ordered aordin+ to a spei rulesRt0at are ordered aordin+ to a spei rulesR

T0e element of a se=uene is usuall- denoted b-T0e element of a se=uene is usuall- denoted b-aaiiaann Example:Example:

"&%2%3%.%a"&%2%3%.%ann aann*n*n

"3%;%C%&2%%a"3%;%C%&2%%ann aann*\*\

Use t0e formulae aUse t0e formulae ann**D(n9&)D(n9&)% w0ere% w0ere* t0e rst element of a se=uene* t0e rst element of a se=uene* t0e dierene between two onseutive elements* t0e dierene between two onseutive elementsaann* t0e n9t0 element of a se=uene* t0e n9t0 element of a se=uene

aa&B&B* 3 D (C)3 * 3B* 3 D (C)3 * 3B


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.2.2

'imit o! a se-uence'imit o! a se-uence

,ome se=uenes 0ave limits> or example:

f(n)*&Gn]B as n]^

f(n)*(nD&)G(nD2) ]& as n ]^

t0ese se=uenes are onver+ent Ot0er do not 0ave limits% e>+>% f(n)*2nD3

t0is se=uene is diver+ent A diver+ent se=uene is said to be denitel- diver+ent if:

or an- (arbitraril- lar+e) value of t0ere is an N suJientl-

lar+e t0at anS for all nSN% t0en t0e se=uene is denitel-diver+ent and limnan*

or an- (arbitraril- lar+e) value of t0ere is an N suJientl-lar+e t0at an9 for all nSN% t0en t0e se=uene is denitel-diver+ent and limnan*9


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.3.3

De%nition o! 'imit o! aDe%nition o! 'imit o! a

se-uencese-uence A se=uene 0as a limit X providedA se=uene 0as a limit X provided

all values of t0e se=ueneall values of t0e se=uene

Zbe-ond t0e term[ an be madeZbe-ond t0e term[ an be madeas lose to X as one wis0esas lose to X as one wis0es

aann*(&Gn)*(&Gn) 0as limit X*B0as limit X*B


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....

.um*er e and limit.um*er e and limit

/erivation of e/erivation of e Fonsider t0e funtion:Fonsider t0e funtion:

f(m) * (&D(&Gm))f(m) * (&D(&Gm))mm

!f lar+er and lar+er!f lar+er and lar+ervalues are assi+ned tovalues are assi+ned tom% t0en f(m) will alsom% t0en f(m) will alsoassume lar+er valuesassume lar+er values

f(&)*2R f(2)*2%24Rf(&)*2R f(2)*2%24R

f(.)*2%..&.&Rf(.)*2%..&.&Rf(&BB)*Rf(&BBB)*f(&BB)*Rf(&BBB)*

f(m) will onver+e tof(m) will onver+e tot0e number 2%


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.4.4

/ounded and/ounded and

un*ounded se-uenceun*ounded se-uence A se=uene is bounded if t0ere is someA se=uene is bounded if t0ere is some

nite value S B su0 t0atnite value S B su0 t0at

lim alim ann (bounded above)(bounded above)nn

lim alim annS 9S 9 (bounded below)(bounded below)

nn

A se=uene is bounded if and onl- if itA se=uene is bounded if and onl- if it0as a lower bound and an upper bound0as a lower bound and an upper bound


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.;.;

Monotonic Se-uenceMonotonic Se-uence

A se=uene is monotoniall-A se=uene is monotoniall-inreasin+ if ainreasin+ if a&&aa22 aa33

A se=uene is monotoniall-A se=uene is monotoniall-dereasin+ if adereasin+ if a&&aa22


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.


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.@.@

SeriesSeries

A speial t-pe of se=ueneA speial t-pe of se=uene Ea0 of its member is obtained b-Ea0 of its member is obtained b-

summin+ t0e members of t0esummin+ t0e members of t0ese=uenese=uene arit0mati seriesarit0mati series !f a!f att% t*&%2%3% is a se=uene% t0en s% t*&%2%3% is a se=uene% t0en snn

** nnt9&t9&aatt% n*&%2%3% is alled a% n*&%2%3% is alled a

(arit0mati) series(arit0mati) series /eret aritmati$a adala0 deret -an+/eret aritmati$a adala0 deret -an+diperole0 den+an men6umla0$an su$u9diperole0 den+an men6umla0$an su$u9su$u suatu barisan aritmati$asu$u suatu barisan aritmati$a


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.C.C

DeterminingDetermining

conergency o! aconergency o! a

seriesseries !f s!f snn**nnt9&t9&aatt is a series assoiatedis a series assoiatedwit0 se=uene awit0 se=uene attandand

XimXimnnWW _(a_(anD&nD&)Ga)Gann_ * X_ * X !t follows t0at!t follows t0at

!f X & t0en series s!f X & t0en series snn onver+esonver+es

!f X S & t0en series s!f X S & t0en series snn diver+esdiver+es !f X * & t0en series s!f X * & t0en series snn ma- onver+ema- onver+e

or diver+eor diver+e


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4B4B

$eometric Series$eometric Series

One of t0e most important series inOne of t0e most important series inmat0ematis and in eonomismat0ematis and in eonomis

/eret +eometri$ adala0 deret -an+/eret +eometri$ adala0 deret -an+diperole0 den+an mem6umla0$an su$u9diperole0 den+an mem6umla0$an su$u9su$u barisan +eometrisu$u barisan +eometri 5arisan -an+ setiap su$u beri$utn-a5arisan -an+ setiap su$u beri$utn-a

diperole0 den+an men+ali$an su$udiperole0 den+an men+ali$an su$usebelumn-a den+an sebua0 bilan+an tetapsebelumn-a den+an sebua0 bilan+an tetap

tertentutertentu 5ilan+an tetap tersebut disebut rasio (r) atau5ilan+an tetap tersebut disebut rasio (r) atau


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4&4&

ExampleExample

5arisan +eometri$: u5arisan +eometri$: u&&%u%u22%u%u33%u%u..%u%unn'i$a : u'i$a : u22GuGu&&* u* u33GuGu22* u* unnGuGun9&n9&*r*r

'i$a u'i$a u&&*a% ma$a u*a% ma$a u22*ar% u*ar% u33*ar*ar22

,e0in++a u,e0in++a unn*u*u&&rrn9&n9&*ar*arn9&n9&

5entu$ umum barisan +eometri:5entu$ umum barisan +eometri:a%ar%ara%ar%ar22%ar%ar33%%ar%%arn9&n9&


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4242

ExampleExample

,,nn*aDarDar*aDarDar22DarDar33DDarDDarn9&n9&

uu&&*a*a

uunn*ar*arn9&n9&

,,nn*a(r*a(rnn&)G(r9&) if rS&&)G(r9&) if rS&

,,nn*a(&r*a(&rnn)G(&9r) if r&)G(&9r) if r&


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4343

Economic 0pplicationEconomic 0pplication

resent `alue (`) of a streamresent `alue (`) of a stream !nvestment!nvestment

Fost 5enet Anal-sis of a pro6etFost 5enet Anal-sis of a pro6et ` of a as0 stream` of a as0 stream r * ;G-earr * ;G-ear

e 0ave ue 0ave u&&*a*1p &BB%BBB%BBB>BB*a*1p &BB%BBB%BBB>BB

e 0ave n * &B -earse 0ave n * &B -ears

e 0ave a +eometri series of:e 0ave a +eometri series of:

` *` * `G(&Dr)`G(&Dr)nn


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4.4.

1.I0RI0&E #0'#1'1S1.I0RI0&E #0'#1'1Sand O(&IMIS0&IO.and O(&IMIS0&IO.


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4444

#ontinuity o! Functions#ontinuity o! Functions

!mportant: man- mat0ematial!mportant: man- mat0ematialte0ni=ues onl- appliable if t0ete0ni=ues onl- appliable if t0e

funtion is ontinuousfuntion is ontinuous Fontinuit- of a funtionFontinuit- of a funtion

explained easil- wit0 t0e aid of aexplained easil- wit0 t0e aid of a

+rap0+rap0 A untion is ontinuous if t0e +rap0 ofA untion is ontinuous if t0e +rap0 oft0e funtion 0as no brea$s or 6umpst0e funtion 0as no brea$s or 6umps


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4;4;

#ontinuity o! a#ontinuity o! a

!unction!unction

untion -*2x isuntion -*2x isontinuous at ever-ontinuous at ever-point xpoint x !1!1

x*3x*3 f(x)*;f(x)*; F0oose a smallF0oose a small

number%number%>0,>0, t0ere ist0ere issome valuesome value S BS B

su0 t0at all t0esu0 t0at all t0efuntion valuesfuntion valuesdened on t0e set ofdened on t0e set ofx valuesx values

-*2x

3

;

3D39

;D;9

x

f(x)


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4


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4@4@

Formal de%nitionFormal de%nition

A funtion f(x) w0i0 is dened onA funtion f(x) w0i0 is dened onan open interval inludin+ t0ean open interval inludin+ t0e

point x*a is ontinuous at t0atpoint x*a is ontinuous at t0atpoint ifpoint if

XimXimxxaaf(x) exists% i>ef(x) exists% i>e

XimXimxxa9a9f(x) * Ximf(x) * XimxxaDaDf(x)f(x) XimXimxxaaf(x) * f(a)f(x) * f(a)


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4C4C

DERI0&IE 0.DDERI0&IE 0.DDIFFERE.&I0'DIFFERE.&I0'


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;B;B

DeriatieDeriatie

A onvenient wa- to express 0ow aA onvenient wa- to express 0ow a0an+e in t0e level of one variable (sa-0an+e in t0e level of one variable (sa-x) determines a 0an+e in t0e level ofx) determines a 0an+e in t0e level ofanot0er variable (sa- -)anot0er variable (sa- -)

Examples:Examples: 8ow a 0an+e in a tari rate determines a8ow a 0an+e in a tari rate determines a

0an+e in domesti (border) prie of0an+e in domesti (border) prie of

importablesimportables

8ow a 0an+e in export prie determines a8ow a 0an+e in export prie determines a0an+e in export suppl-0an+e in export suppl-


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;&;&

De%nition o! deriatieDe%nition o! deriatie

T0e derivative ofT0e derivative ofa funtion -*f(x)a funtion -*f(x)is simpl- t0eis simpl- t0eslope of t0eslope of t0etan+ent linetan+ent line A tan+ent line (toA tan+ent line (to

a urve) is a linea urve) is a linew0i0 6ustw0i0 6usttou0es t0e urvetou0es t0e urveat a +iven pointat a +iven point

x

-

-*f(x)


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;2;2


T0e derivative ofT0e derivative ofa funtion -*f(x)a funtion -*f(x)

at t0e pointat t0e point

*(x*(x&&%f(x%f(x&&)) is t0e)) is t0e

slope of t0eslope of t0etan+ent line attan+ent line at

t0at pointt0at point Notation:Notation: -G-Gxx

or d-Gdx or for d-Gdx or fHH(x)(x)

x

-

-*f(x)

tan+entline


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;3;3


x

-

-*f(x)

x xDx

x

- -

-D-

,eantline

D % i i !D % iti !


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;.;.

De%nition o!De%nition o!

Di3erentialDi3erential

!f f!f fH(xH(xBB) is t0e derivative of t0e) is t0e derivative of t0efuntion -*f(x) at t0e point xfuntion -*f(x) at t0e point xBB% t0en% t0en

t0e total dierential at a point xt0e total dierential at a point xBB

isis

d-*df(xd-*df(xBB%dx) * f%dx) * fH(xH(xBB) dx) dx

T0e dierential is a funtion of bot0T0e dierential is a funtion of bot0x and dxx and dx


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;4;4


T0e dierential provides us wit0 aT0e dierential provides us wit0 amet0od of estimatin+ t0e eet ofmet0od of estimatin+ t0e eet of

a 0an+e in x of amount dx *a 0an+e in x of amount dx * xxon -% w0ereon -% w0ere - is t0e exat- is t0e exat0an+e in - w0ile d- is t0e0an+e in - w0ile d- is t0e

approximate 0an+e in -approximate 0an+e in -


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;;;;


Y T0e dierential issometimes desribedas t0e linear (orstrai+0t9line)approximation to t0e

0an+e in - w0en x0an+es b- dx>

Y Overestimateddierential

Y or smaller 0an+esin x% t0e expressionof dierential oers abetter approximation

# diti !# diti !


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;


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;@;@

Rules o! di3erentiationRules o! di3erentiation

/erivative of a onstant funtio

/erivative of a linear funtio

/erivative of a power funti

/erivative of aonstant multiple of

a funtion


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;C;C

Rules o! di3erentiationRules o! di3erentiation


70/72


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Math#0D !or di3erentiMath#0D !or di3erentiationation
http://matek-met.mcd/http://matek-met.mcd/http://matek-met.mcd/http://matek-met.mcd/http://matek-met.mcd/http://matek-met.mcd/

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