Chapter1–LinearEquationsandFunctions1.1 –SolutionsofLinearEquationsandInequalitiesinOneVariableSolveA=P+Prt

a) forr

b) forPBreak-EvenPoint:Thelevel(s)ofproductionwhereProfit=0.Inotherwords,R=C.Ex:#48,Page61

Break-evenDishSystemsmanufacturessatellitesystemsandhasitsmonthlyprofitPindollarsrelatedtothenumberofsatellitesystems,x,by4P=81x–29,970FindthenumberofsystemsthatDishSystemsneedstoproduceandsellinordertobreakeven.

A =P + Prt

⇒¥t¥¥fE¥¥±A =P + Prt

A =P ( I + rt )

+¥y =P (

H##Fia = P or P = tart

P@or R= C

4 ( o ) = 81 x - 29970

0 = 81 × - 29970

t 29,970+ 2 9 97 0

29,9 ftp.tems#f

Ex:#56,Page61

InvestmentYieldsOnesafeinvestmentpays10%peryear,andariskierinvestmentpays18%peryear.Awomanwhohas$145,600toinvestwantstohaveanincomeof$20,000peryearfromherinvestments.Howmuchshouldsheinvestateachrate?

EM.it?eEInyrtt#nx.ii-20,000O.IO#t8(Y5,60x)=20,000

0.10×+26,208-0.1811=20,00026,208-0.0811=20,000to .O8×+0.08×26,208=0,000+0.08×-20,000

- 20,000

6,208=0.08×62080.07=0.08×0.08

$77,600 = X

10% : $77,600

l8%:$±¥g%j"€.

1.2FunctionsIndependentvariable:Dependentvariable:Function:VerticalLineTest:Ex:Isthisgraphthegraphofafunction?Whyorwhynot?FunctionNotationEx:Evaluate,giventhatf(x)=2x3–3x2+1andg(x)=3x a)f(3) b)f(0) c)f(t) d)g(12) e)g(x+h)

x

y

x

first,

more choice

yComes second

Relation from x toywhere each × corresponds to just one y

stole,⇐iB÷¥e

.

T•Net a

FI

to:

+ of 3

= 2 (353-3 (3) 2+1.2( z 7) -3.9 + 1

= 54 - 27+1=28= 2 ( o )3-3192+1 = 1

= 2 t3 - 3+2+1

= 342 ) = 36

g ( x ) =3 X

= 3 ( x + h)= 3×+34

Ex:Givenf(x),find! "#$ %!(")

$ (h¹0)a)f(x)=x+3b)f(x)=2x2–4x+5

f ( xth ) = ( xth ) +3

= ×

th+3

=(

xthi.hn#.=fthtfh=DfCX+hn)-fCx=Cxth)(x+h)=x2txhthxthf(x+h)=2(x+h5-4(xth)+5=2(x2+2xh+hY.4x-4ht5fg.x+axntze4*

"h¥¥,×+±(2×2*+2424×-44.5)=(2×2

h

=

2#t4xh+2h#x4h##+4#=

= 4xhyzht.IM#.2h4)_=4.xty2hD

DomainandRangeDomain:Range:Ex:Findthedomainandrangeofeachofthefollowingfunctions. a)y=3x-2 b)y= ( c)f(x)= )"%*Ex:MortgagePaymentThistableshowstheapproximatenumberofyearsthatitwilltakeacoupletopayoffa$100,000mortgageatseveraldifferentinterestratesiftheypay$800permonth.Ifrdenotestherateandf(r)denotesthenumberofyears: r f(r) 2.6 12 5.2 15 6.3 17 7.4 20 9 30 a)Whatisf(6.3)andwhatdoesitmeaninEnglish? b)Iff(r)=30,whatisr? c)Does2*f(2.6)=f(2*2.6)?

x valuesInput

y values

Output

.D :R

,all Real # s

' R :B

D :X > OR :y } 0

D :# 4 R :# O

FC 6.3=17 years At 6.3%,

it'lltake 17 years

r=9 topayottthe loan

f( 2. 6) = 12 f ( 2. 2. 6) =f( 5.4=15

2. f ( 2. 6) = 24

1.3LinearFunctionsThemathisreview,butthefinanceisnew.Slope:y=mx+bEx:Findanequationofthelinethatpassesthrough(2,3)and(4,-6)

Steepness

m=rri÷F To

sitopely . int

m=rnjnI= #Xz - X

,

m.IT#=3It=I=.Ey - y ,=m( * ×

, ) Point - Slope Formula

* mxtb Slope - Intercept Formula

m=-El2,3)°µ==Cz3)y . 3=

- Ek - 2) y=mx+by - 3=-9-2×+9

# 3= - £( 2) + §

y= - Ex +12 3=-9+6

Check : ( 4,

-6 ) +9 +9

- 6IzI( 4) +12µ-€8jy2

Y= -9-2×+12

RateofChange:Thismeasuretellsyouhowfastsomethingischanging,andinwhatdirectionitischanging.RateofChange=ROC= +,-./1$.234

56,72-,!+864+$.-9.:;.::4<Ex:#56GrossDomesticProductTheU.S.grossdomesticproduct(GDP),y,fortheyearsfrom2011andprojectedto2021isgivenby y=977.8x+13643.2wherexisthenumberofyearspast1990andyisinbillionsofdollars.a)Whatistheslopeofthegraphofthisfunction?b)Interprettheslopeasarateofchange.

977.8 or 977,800,000,000

Each year ,the GDP

increases by $977.8billion

1.6ApplicationsofFunctionsinBusinessandEconomicsMarginalCost:Thecosttoproduceonemoreunitatthecurrentlevelofproduction.Also:MarginalRevenue,MarginalProfitEx:AlinearcostfunctionisC(x)=27.55x+5180 a)WhataretheslopeandC-intercept?

b)Whatisthemarginalcost,andwhatdoesitmean?

SupplyandDemandSupply:Thenumberofunitsthatacompany/marketiswillingtoproduceatagivenprice.Demand:Thenumberofunitsthatconsumersarewillingtobuyatagivenprice.MarketEquilibrium:Apriceandquantitywheresupply=demand.Surplus:Thereisasurpluswhendemandislessthansupply(Thereareunitsleftover)Shortage:Thereisashortagewhendemandisgreaterthansupply(Therearen’tenoughunitstogoaround)

:= 27.55 Variable

C- int = 5180 fixed

Marginal Cost $27.55

To produce & sell one more

item,

it will cost anadditional

$27.55 .

Ex:IfthedemandfunctionforZ-brandphonesisp+2q=100,andthesupplyfunctionis35p–20q=350,whathappenswhenp=14?Aretheresurplusphones,oristhereaphoneshortage?

D: p+2g= 100

price giant't

5 : 35p - 205-350

f- 14 D: 14+4=100- 14 -14

¥q=43

← Demand

S : 3544 ) - 20q= 350

490-209=350- 490

- 490

-2=409=7 ← Supply

D > S Demand is greater thansupply -

D > S ⇒ The price is

to low

Shortage

To get to equilibrium ,we

Should raise the price

#40onpage112

Labeltheseas“Demand”and“Supply”andthenanswerthese:a)Ifthepriceis$30,whatquantityisdemanded?b)Ifthepriceis$30,whatquantityissupplied?c)Isthereasurplusorshortagewhenthepriceis$30?Byhowmanyunits?

•( 0,40 ) $40 ,

Oqugt.

•.

.

• ( 50,35)

: ; D(60,10)• ( 0,10) : •

$10,60:.

. quant

220=40

Surplus of 20 units

Ex:Findtheequilibriumpointforthefollowingsupplyanddemandfunctions: Demand:p=480–3q Supply:p–17q=80

.

p- 179=80 Substitution

( 480.3g ) -179=80480 -3g - hq=A

480-209=80-480-480-209=-400

9=20

Find p=480 -3g

= 480-3120 )= 480 - 60

$= 420

Market Equilibrium : $420,20 units

Check : p- 17g = 80

.