FlashCardConstructionInstructionsTheleftcolumnisthequestionandtherightcolumnistheanswers.Cutouttheflashcardsandpastethequestiontoonesideofanotecardandtheanswertotheotherside.Becarefultopastethecorrectanswertoitscorrespondingquestion!

Arclength(betweenaandb)6.4

𝑠 = 1+ [𝑓! 𝑥 ]!𝑑𝑥!

!

Arclength(betweencandd)

6.4

𝑠 = 1+ [𝑔! 𝑦 ]!𝑑𝑦!

!

𝑢𝑑𝑣

7.2

𝑢𝑣 − 𝑣𝑑𝑢

WhatdoesLIATEstandfor?7.2

L–logarithmicpartI–inversetrigonometricpartA–algebraicpartT–trigonometricpartE–exponentialpart

L’Hopital’sRule7.7

lim!→!

𝑓(𝑥)𝑔(𝑥) = lim

!→!

𝑓′(𝑥)𝑔′(𝑥)

ImproperIntegrals/InfiniteLimits

𝑓 𝑥 𝑑𝑥!

!

7.8

lim!→!

𝑓 𝑥 𝑑𝑥!

!

ImproperIntegrals/InfiniteLimits

𝑓 𝑥 𝑑𝑥!

!!

7.8

lim!→!!

𝑓 𝑥 𝑑𝑥!

!

ImproperIntegrals/InfiniteLimits

𝑓 𝑥 𝑑𝑥!

!!

7.8

𝑓 𝑥 𝑑𝑥 + 𝑓 𝑥 𝑑𝑥!

!

!

!!

ImproperIntegrals/Infinite

Discontinuities

𝑓 𝑥 𝑑𝑥!

!

7.8

lim!→!!

𝑓 𝑥 𝑑𝑥!

!

ImproperIntegrals/Infinite

Discontinuities

𝑓 𝑥 𝑑𝑥!

!

7.8

lim!→!!

𝑓 𝑥 𝑑𝑥!

!

ImproperIntegrals/Infinite

Discontinuities

𝑓 𝑥 𝑑𝑥!

!

7.8

𝑓 𝑥 𝑑𝑥 + 𝑓 𝑥 𝑑𝑥!

!

!

!

𝑑𝑥𝑥!

!

!

7.8

!

!!! 𝑝 > 1

diverges𝑝 ≤ 1

Whatarethe6convergence/divergencetestswestudy

8.6

• nthtermtest• intregraltest• ratiotest• directcomparisontest• limitcomparisontest• alternatingseriestest

Whatarethe4infiniteserieswestudy8.6

• geometric• telescoping• p-series• alternatingseries

Whatistheconditionofconvergenceforthegeometricseries?

8.6

|r|<1

Whatisthesumofageometricseries?8.6

𝑆 =𝑎

1− 𝑟

Whatistheformatofageometricseries?8.6

𝑎𝑟!!

!!!

Whatistheformatofap-series?8.6

1𝑛!

!

!!!

Whendoesap-seriesdiverge?8.6

0 < 𝑝 ≤ 1

Whendoesap-seriesconverge?8.6

𝑝 > 1

Whatistheformatofanalternatingseries?

8.6

(−1)!!!𝑎!

!

!!!

Whataretheconditionsforanalternatingseriestoconverge?

8.6

• lim!→! 𝑎! = 0• 𝑎!!! ≤ 𝑎!

Whatisthecondition(s)requiredfortheratiotesttoindicateconvergence?

8.6

lim!→!

𝑎!!!𝑎!

< 1

Whatisthecondition(s)requiredforthedirectcomparisontesttoindicate

convergence?8.6

• 0 < 𝑎! ≤ 𝑏!

• 𝑏! 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠!!!!

Whatisthecondition(s)requiredforthelimitcomparisontesttoindicate

convergence?8.6

• lim!→!!!!!= 𝐿 > 0

• 𝑏! 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠!

!!!

WhatistheformatofaTaylor

polynomial

8.7

𝑃!(𝑥) = 𝑓 𝑐 + 𝑓!𝑐 𝑥 − 𝑐 +⋯ 𝑓 ! 𝑐𝑛!

(𝑥 − 𝑐)!

WhatistheformatofaMaclaurin

polynomial

8.7

𝑃!(𝑥) = 𝑓 0 + 𝑓! 0 𝑥 +⋯ 𝑓 ! (0)𝑛!

(𝑥)!

WhatistheformatoftheLagrangeform

ofthereminder

8.7

𝑅! 𝑥 = 𝑓 !!! (𝑧)𝑛 + 1 !

(𝑥 − 𝑐)!!!

Whatistheformatofapowerseries

8.8

𝑎!(𝑥 − 𝑐)!!

!!!

Whatarethethreetypesofpowerseries

convergence

8.8

• singlepoint• radius• absolute(everywhere)

PowerSeriesfor!!

8.10

1− 𝑥 − 1 + 𝑥 − 1 ! − 𝑥 − 1 ! +⋯Intervalofconvergence0 < 𝑥 < 2

PowerSeriesfor !!!!

8.10

1− 𝑥 + 𝑥! − 𝑥! + 𝑥! −⋯

Intervalofconvergence−1 < 𝑥 < 1

PowerSeriesfor𝑙𝑛𝑥8.10

𝑥 − 1 −𝑥 − 1 !

2 +𝑥 − 1 !

3 −⋯Intervalofconvergence0 < 𝑥 < 2

PowerSeriesfor𝑒!8.10

1+ 𝑥 + !!

!!+ !!

!!+⋯

Intervalofconvergence−∞ < 𝑥 < ∞

PowerSeriesfor𝑠𝑖𝑛𝑥8.10

𝑥 −𝑥!

3! +𝑥!

5! −𝑥!

7! +⋯Intervalofconvergence−∞ < 𝑥 < ∞

PowerSeriesfor𝑐𝑜𝑠𝑥8.10

1−𝑥!

2! +𝑥!

4! −𝑥!

6! +⋯Intervalofconvergence−∞ < 𝑥 < ∞

PowerSeriesfor𝑎𝑟𝑐𝑡𝑎𝑛𝑥8.10

𝑥 −𝑥!

3 +𝑥!

5 −𝑥!

7 +⋯Intervalofconvergence−1 < 𝑥 < 1

PowerSeriesfor𝑎𝑟𝑐𝑠𝑖𝑛𝑥8.10

𝑥 +𝑥!

2 ∗ 3+1 ∗ 3𝑥!

2 ∗ 4 ∗ 5+1 ∗ 3 ∗ 5𝑥!

2 ∗ 4 ∗ 6 ∗ 7+⋯Intervalofconvergence−1 < 𝑥 < 1

PowerSeriesfor(1+ 𝑥)! 8.10

1+ 𝑘𝑥 +𝑘(𝑘 − 1)𝑥!

2! +⋯Intervalofconvergence−1 < 𝑥 < 1 ∗

Standardformofaparabola

(Verticalaxis)9.1

(𝑥 − ℎ)! = 4𝑝(𝑦 − 𝑘)

Standardformofaparabola

(horizontalaxis)9.1

(𝑦 − 𝑘)! = 4𝑝(𝑥 − ℎ)

Standardformofanellipse(Majoraxisishorizontal)

9.1

(𝑥 − ℎ)!

𝑎! +(𝑦 − 𝑘)!

𝑏! = 1

Standardformofanellipse(Majoraxisisvertical)

9.1

(𝑥 − ℎ)!

𝑏! +(𝑦 − 𝑘)!

𝑎! = 1

Standardformofanhyperbola(Transverseaxisishorizontal)

9.1

(𝑥 − ℎ)!

𝑎! −(𝑦 − 𝑘)!

𝑏! = 1

Standardformofanhyperbola(Transverseaxisisvertical)

9.1

(𝑦 − 𝑘)!

𝑎! −(𝑥 − ℎ)!

𝑏! = 1

Parametricformofthederivative

9.3

𝑑𝑦𝑑𝑥 =

𝑑𝑦𝑑𝑡𝑑𝑥𝑑𝑡

Arclengthinparametricform

9.3

𝑠 = [𝑓! 𝑡 ]! + [𝑔! 𝑡 ]!!

!𝑑𝑡

SlopeinPolarForm

9.4

𝑑𝑦𝑑𝑥 =

𝑑𝑦𝑑𝜃𝑑𝑥𝑑𝜃

=𝑓 𝜃 𝑐𝑜𝑠𝜃 + 𝑓! 𝜃 𝑠𝑖𝑛𝜃−𝑓 𝜃 𝑠𝑖𝑛𝜃 + 𝑓! 𝜃 𝑐𝑜𝑠𝜃

AreainPolarCoordinates

9.5

𝐴 =12 𝑟!𝑑𝜃, 𝑟 = 𝑓(𝜃)

!

!

Velocity(vectors)

9.8

𝑣 𝑡 = 𝑟! 𝑡 = 𝑥! 𝑡 ,𝑦′(𝑡)

Acceleration(vectors)

9.8

𝑎 𝑡 = 𝑟!! 𝑡 = 𝑥!! 𝑡 ,𝑦′′(𝑡)

Speed(vectors)

9.8

[𝑥! 𝑡 ]! + [𝑦! 𝑡 ]!

Displacement(vectors)

𝑣 𝑡 𝑑𝑡!

!

9.8

𝑥! 𝑡 𝑑𝑡, 𝑦! 𝑡 𝑑𝑡!

!

!

!

TotalDistance(vectors)

𝑣 𝑡 𝑑𝑡!

!

9.8

[𝑥! 𝑡 ]! + [𝑦! 𝑡 ]!!

!

𝑑𝑡