MATH 1080 EXAM REVIEW
ReviewofFunctions
BoundedIntervals
min max nomin nomax
Lmin 41 1111
gnax
111111 1 L 1 1 1 11 WMDb
a b a b a a b
xlxc R.acxcb3ExlxelR.aexeb3Exlxc.IR a X Eb ExlxelR 2Excb
2 b Ca b 2b 2b
Unboundedintervals
a
11 11 11 L 6
a
1 11 4 11 17
a
4 11 11
a
7
xlxc.IR x 23 EXIXEIR.xza3Exlxc.IR xca EXIXER XE 23
a a to C a a C a
RightUnbounded LeftUnbounded
111111111111
Unboundedat
SetofallrealnumbersC 9 1 bothends
Functions Arealfunctionis a rule that associateseachnumberin asetcalledthedomainwith
exactlyonenumberin anothersetcalledtherange
Df thedomain off Rf therangeoff
BasicFunctions
Function Domain Range Eventodd
y c where C a to c Even
cisaconstant
y x l a a Ca a Odd
y x2 Ca a o a Even
3 Ca
y Ca a a Odd
y T C a 070 C0,010 Odd nointercepts
6 a O.to ratherhasasymptotes
3 C a to C a a Odd
y x
y r 0 a 0 a Neither powerfunction rx x 2
C a a Co a
4 1 1 Even symmetricalongthey axis
y xD C it 7 Seitfefgears ExDislargestintergerlessthaorequaltoxffouwnnd
PiecewiseFunction definedalgebraicallyorsymbolicallybytwoormorefunctionsdefinedforagivensetofx's
CompositeFunction
fogCxl fCgCx evaluatefwhen x gCx
two
Uses functions tomakeanewone
ExponentialandLogarithmicFunctions
ExponentialFunction y b whereb is aconstant boo btladxc.IR
bisthebase x istheexponent
Ocb 1 b I ifeveryinputgoesto auniqueoutput
ay ay thenthefunctionis 1 I
MY t whenahorizontallinecrossesthegraph
v v morethanonce thenthefunctionisnot 1 1
Inverseoftheexponentialfunctionis far logis x thelogarithmicfunction
Ocbc 1 b I blog.blbl I logblbu w
ay ay logbcxtyiffbY xblncxt.yiffet.is
logelxtIncx Ince w
u
a
f blog.cwi w Ince I
, ExponentialGrowthandDecayModels
t t Aoinitialamount 1 time
ACHAoe ACHAoe K growth1decayrate ACHamountpresentHeftatdesiredtime
HalfLife thetimeit takesforonehalfofthepresentsubstancetodecay
DiscreteModeling
DiscreteModel usedtodescribeapopulationwheredataaboutthepopulationisgivenonlyatdiscretetimesteps
Sequences collectionofnumbersthatfollowaparticularpatternEachpartofthesequenceis aterm
Infinitesequence an 3h Ea azas 3
Finitesequence E 2h37 Ea azas an
Arithmeticsequence eachterm is afixednumberlargerthanthetermbeforeit
an In d an a n nd
Geometricsequence multiplyeach termbysomefixednonzeronumberCa ar ar ar
successive
grit rgn gn r n g
ifthecommonratio r iszero thetermswillalternatebetweenpositiveandnegative
Fibonaccisequence nextnumberisthesumofthe twonumberspriorto I I 2 3,58,13
an bn an bn Ean Ebn3 Eanbn KEES EKan3 k is aconstant
Series
IEan FEbn FECan bn kEsIn FEKzn IEIn n man Feetan Ieuan FeiIn 1In
NNE k nk FE n N Ntl NNE n NINMgkNtl IEns N'CEIT
l Erm Crm
nnq.gr
g NEM
gr
ni
g in
FµCani an an am an an
µC an an nnqc.mn if N is
10 if N isodd
even
DifferenceEquations
Axn Xml Xn
ConstantGrowthCa D Xm Xn b xn Xo rib
ProportionalGrowth b 07 Xm aXn xn Xoan
Generalcase att xn i axn b Xn 41a xo b1 a an
SteadyStateeventualsize
if l c a e l orXo Kia O
iia an nine exo a
DNE if a E l Xo Ki a O
if b O
1h17 Xo rib if b co
Xo if b O
Equilibriumstate Replaceall xn'swithXethensolveforXe
Cob
webbing Stability
whenthecobwebheads in thedirectionof apointonthegraphXE is stable
whenthecobwebheadsawayfrom apointon thegraph XEisunstable
LimitsandContinuity
finfa Xx L lyinga fan L therefore 7 fan L
v u
Lefthandlimit Righthandlimit
approachesfrom approachesfrom
thenegativeside thepositiveside
Determinelimitsusingthefactoringtechniquerationalizenumeratortechnique orthecommondenominator
technique
ReviewofFunctions
BoundedIntervals
min max nomin nomax
Lmin 41 1111
gnax
111111 1 L 1 1 1 11 WMDb
a b a b a a b
xlxc R.acxcb3ExlxelR.aexeb3Exlxc.IR a X Eb ExlxelR 2Excb
2 b Ca b 2b 2b
Unboundedintervals
a
11 11 11 L 6
a
1 11 4 11 17
a
4 11 11
a
7
xlxc.IR x 23 EXIXEIR.xza3Exlxc.IR xca EXIXER XE 23
a a to C a a C a
RightUnbounded LeftUnbounded
111111111111
Unboundedat
SetofallrealnumbersC 9 1 bothends
Functions Arealfunctionis a rule that associateseachnumberin asetcalledthedomainwith
exactlyonenumberin anothersetcalledtherange
Df thedomain off Rf therangeoff
BasicFunctions
Function Domain Range Eventodd
y c where C a to c Even
cisaconstant
y x l a a Ca a Odd
y x2 Ca a o a Even
3 Ca
y Ca a a Odd
y T C a 070 C0,010 Odd nointercepts
6 a O.to ratherhasasymptotes
3 C a to C a a Odd
y x
y r 0 a 0 a Neither powerfunction rx x 2
C a a Co a
4 1 1 Even symmetricalongthey axis
y xD C it 7 Seitfefgears ExDislargestintergerlessthaorequaltoxffouwnnd
PiecewiseFunction definedalgebraicallyorsymbolicallybytwoormorefunctionsdefinedforagivensetofx's
CompositeFunction
fogCxl fCgCx evaluatefwhen x gCx
two
Uses functions tomakeanewone
ExponentialandLogarithmicFunctions
ExponentialFunction y b whereb is aconstant boo btladxc.IR
bisthebase x istheexponent
Ocb 1 b I ifeveryinputgoesto auniqueoutput
ay ay thenthefunctionis 1 I
MY t whenahorizontallinecrossesthegraph
v v morethanonce thenthefunctionisnot 1 1
Inverseoftheexponentialfunctionis far logis x thelogarithmicfunction
Ocbc 1 b I blog.blbl I logblbu w
ay ay logbcxtyiffbY xblncxt.yiffet.is
logelxtIncx Ince w
u
a
f blog.cwi w Ince I
, ExponentialGrowthandDecayModels
t t Aoinitialamount 1 time
ACHAoe ACHAoe K growth1decayrate ACHamountpresentHeftatdesiredtime
HalfLife thetimeit takesforonehalfofthepresentsubstancetodecay
DiscreteModeling
DiscreteModel usedtodescribeapopulationwheredataaboutthepopulationisgivenonlyatdiscretetimesteps
Sequences collectionofnumbersthatfollowaparticularpatternEachpartofthesequenceis aterm
Infinitesequence an 3h Ea azas 3
Finitesequence E 2h37 Ea azas an
Arithmeticsequence eachterm is afixednumberlargerthanthetermbeforeit
an In d an a n nd
Geometricsequence multiplyeach termbysomefixednonzeronumberCa ar ar ar
successive
grit rgn gn r n g
ifthecommonratio r iszero thetermswillalternatebetweenpositiveandnegative
Fibonaccisequence nextnumberisthesumofthe twonumberspriorto I I 2 3,58,13
an bn an bn Ean Ebn3 Eanbn KEES EKan3 k is aconstant
Series
IEan FEbn FECan bn kEsIn FEKzn IEIn n man Feetan Ieuan FeiIn 1In
NNE k nk FE n N Ntl NNE n NINMgkNtl IEns N'CEIT
l Erm Crm
nnq.gr
g NEM
gr
ni
g in
FµCani an an am an an
µC an an nnqc.mn if N is
10 if N isodd
even
DifferenceEquations
Axn Xml Xn
ConstantGrowthCa D Xm Xn b xn Xo rib
ProportionalGrowth b 07 Xm aXn xn Xoan
Generalcase att xn i axn b Xn 41a xo b1 a an
SteadyStateeventualsize
if l c a e l orXo Kia O
iia an nine exo a
DNE if a E l Xo Ki a O
if b O
1h17 Xo rib if b co
Xo if b O
Equilibriumstate Replaceall xn'swithXethensolveforXe
Cob
webbing Stability
whenthecobwebheads in thedirectionof apointonthegraphXE is stable
whenthecobwebheadsawayfrom apointon thegraph XEisunstable
LimitsandContinuity
finfa Xx L lyinga fan L therefore 7 fan L
v u
Lefthandlimit Righthandlimit
approachesfrom approachesfrom
thenegativeside thepositiveside
Determinelimitsusingthefactoringtechniquerationalizenumeratortechnique orthecommondenominator
technique
