Connecting AB to BC: Fundamental Theorem of Calculus Oneofthemostimportantconceptsincalculusistheconnectionbetweentheareabounded

byacurveandthedefiniteintegral.Thisconnectionleadsuseventuallytothe

𝐅𝐮𝐧𝐝𝐚𝐦𝐞𝐧𝐭𝐚𝐥𝐓𝐡𝐞𝐨𝐫𝐞𝐦𝐨𝐟𝐂𝐚𝐥𝐜𝐮𝐥𝐮𝐬!

Let!ssaywewanttofindtheareabetweenthegraphof𝑓!(𝑥)andthe𝑥axisfrom𝑥 = 𝑎to

𝑥 = 𝑏.Today,wewillreviewafewdifferentapproachesthatallconnectbacktothe

FundamentalTheoremofCalculus.

Connecting AB to BC: Fundamental Theorem of Calculus

Topic: 10.1 Defining Convergent and Divergent Infinite Series

Date: March 30, 2020 AP CALCULUS BC YouTube Live Virtual Lessons Mr. Bryan Passwater Mr. Anthony Record

Topic: Unit 6 Connecting AB to BC

Fundamental Theorem of Calculus

Date: April 22, 2020

𝐓𝐨𝐝𝐚𝐲!𝐬𝐁𝐢𝐠𝐈𝐝𝐞𝐚𝐬(𝐓𝐨𝐩𝐢𝐜𝐬)

Topics Approach

Areaapproximations Riemannsums, trapezoidalapproximations

Findingexactareas Usinggeometricareas

Findingexactareas FundamentalTheoremofCalculus

Findingexactareas Usingpropertiesofintegrals

Connectingintegralsandderivatives FundamentalTheoremofCalculus

Accumulation ModifiedFTC

Quick Check 𝐓𝐨𝐩𝐢𝐜:𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐒𝐮𝐦𝐬𝐚𝐧𝐝𝐀𝐫𝐞𝐚𝐀𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐢𝐨𝐧𝐬

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟏:Let𝑓beafunctionthatistwicedifferentiableforallrealnumbers.Thetableabovegives

selectedvaluesfor𝑓!intheclosedinterval2 ≤ 𝑥 ≤ 12.UsearightRiemannsumwiththefour

subintervalsindicatedbythedatainthetabletoapproximate f 𝑓!(𝑥)𝑑𝑥."#

#Showtheworkthatleadsto

youranswer.

𝐓𝐨𝐩𝐢𝐜:𝐔𝐬𝐢𝐧𝐠𝐆𝐞𝐨𝐦𝐞𝐭𝐫𝐢𝐜𝐀𝐫𝐞𝐚𝐬

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟐:Let𝑔!bethefunctiongivenaboveontheclosedinterval[−2, 16], consistingoffourline

segmentsandasemicircle.

(𝐚)Evaluate# 𝑔′(𝑥)𝑑𝑥5

−2(𝐛)Evaluate# 𝑔′(𝑥)𝑑𝑥

16

3

𝐓𝐨𝐩𝐢𝐜:𝐓𝐡𝐞𝐅𝐮𝐧𝐝𝐚𝐦𝐞𝐧𝐭𝐚𝐥𝐓𝐡𝐞𝐨𝐫𝐞𝐦𝐨𝐟𝐂𝐚𝐥𝐜𝐮𝐥𝐮𝐬𝐏𝐚𝐫𝐭𝐈…𝐨𝐫𝐢𝐬𝐢𝐭𝐏𝐚𝐫𝐭𝐈𝐈?

𝑥 2 4 8 9 12

𝑓′(𝑥) 4 1 −2 0 3

𝒙 −𝟏 𝟐 𝟓 𝟔 𝟏𝟏

𝐺(𝑥) 1 −3 7 −2 4

𝑔(𝑥) 5 9 −11 2 0

𝑔′(𝑥) 3 −1 −8 4 15

𝑔′′(𝑥) 0 −3 4 8 2

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟑:Thefunction𝑔iscontinuousandhasderivativesofallordersforallrealvalues𝑥.Selected

valuesof𝑔, itsfirstandsecondderivativesand𝐺, theantiderivativeof𝑔, aregiveninthetableabove.

(𝐚)Evaluate# 𝑔′(𝑥)𝑑𝑥5

2(𝐛)Evaluate# 𝑔(𝑥)𝑑𝑥

−1

6

Thefunction𝑓istwicedifferentiablehashorizontaltangentsat𝑥 = −3,−1, 2, and6.Aportion

ofthegraphof𝑓isgivenabovefor − 3 ≤ 𝑥 ≤ 6.Itisknownthat𝑓!(0) = 3.

(𝐜)Evaluate# 𝑓′(𝑥)𝑑𝑥6

−3(𝐝)Evaluate# 𝑓′′(𝑥)𝑑𝑥

0

−3

𝐓𝐨𝐩𝐢𝐜:𝐏𝐫𝐨𝐩𝐞𝐫𝐭𝐢𝐞𝐬𝐨𝐟𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐞𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬

#ℎ(𝑥)𝑑𝑥8

−1

= 16#ℎ(𝑥)8

4

𝑑𝑥 = 5# 2𝑘(𝑥)𝑑𝑥−1

4

= 6 #𝑘(𝑥)𝑑𝑥 = −28

−1

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟒:Thefunctionsℎand𝑘arecontinuousandsatisfytheequationsabove.Findthefollowing.

(𝐚) fℎ(𝑥).

/"

𝑑𝑥(𝐛)f[−2ℎ(𝑥) + 1]𝑑𝑥.

0

(𝐜)f𝑘(𝑥)𝑑𝑥.

0

(𝐝) f[2 − 3𝑘(𝑥)]𝑑𝑥0

/"

Thefunction𝑓istwicedifferentiablehashorizontaltangentsat𝑥 = −3,−1, 2, and6.Aportion

ofthegraphof𝑓isgivenabovefor − 3 ≤ 𝑥 ≤ 6.Itisknownthat𝑓!(0) = 3.

(𝐞)f [3 − 2𝑓(𝑥)]𝑑𝑥/#

1

(𝐟) f[𝑓!(𝑥) + 3𝑓′′(𝑥)]𝑑𝑥2

/3

𝐓𝐨𝐩𝐢𝐜:𝐂𝐨𝐧𝐧𝐞𝐜𝐭𝐢𝐧𝐠𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞𝐬𝐚𝐧𝐝𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬

𝐓𝐡𝐞𝐒𝐞𝐜𝐨𝐧𝐝(𝐨𝐫𝐅𝐢𝐫𝐬𝐭? )𝐅𝐮𝐧𝐝𝐚𝐦𝐞𝐧𝐭𝐚𝐥𝐓𝐡𝐞𝐨𝐫𝐞𝐦𝐨𝐟𝐂𝐚𝐥𝐜𝐮𝐥𝐮𝐬

𝐹(𝑥) = f 𝑓(𝑡)𝑑𝑡4

5𝐹!(𝑥) = 𝑓(𝑢) ∙ 𝑢′

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟓:Thefunctions𝑓, 𝑔andℎaretwicedifferentiable.Selectedvaluesfor𝑓areinthetable

aboveand𝑔andℎaredefinedbelow.

𝑔(𝑥) = 𝑥# −f 𝑓(𝑡)𝑑𝑡36

"ℎ(𝑥) = f cos(𝑡# − 1) 𝑑𝑡

3

78(;)

(𝐚)Find𝑔′(𝑥)anduseittoevaluate𝑔′(2).(𝐛)Findℎ′(𝑥)anduseittoevaluateℎ′%𝑒−1&.

𝐓𝐨𝐩𝐢𝐜:𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐢𝐨𝐧

𝐐𝐮𝐢𝐜𝐤𝐂𝐡𝐞𝐜𝐤𝟔:Thefunction𝑓iscontinuousanddifferentiablewithaportionof𝑓!, thederivativeof𝑓,

giveninthefigureabove.Itisknownthat𝑓(4) = −2

(𝐚)Find𝑓(−2)(𝐛)Find𝑓(6)

𝑥 1 2 3 6

𝑓(𝑥) 4 1 −2 0

𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐢𝐨𝐧𝐌𝐨𝐝𝐞𝐥(𝐌𝐨𝐝𝐢𝐟𝐢𝐞𝐝𝐅𝐓𝐂)

f 𝑓′(𝑥)𝑑𝑥=

5= 𝑓(𝑏) − 𝑓(𝑎) → 𝒇(𝒃) = 𝒇(𝒂) + f 𝒇′(𝒙)𝒅𝒙

𝒃

𝒂

Free Response Practice: Connecting AB to BC 2020 FRQ Practice Problem BC1

𝐁𝐂𝟏:Thefunction𝑓iscontinuousanddifferentiableforallvaluesof𝑥.Aportionofthegraph

of𝑓!, thederivativeof𝑓, isshownaboveontheclosedinterval[−3,6]. Itisknownthat𝑓(2) = −3.

(𝐚)Find𝑓(−3).

(𝐛)Evaluate# '2𝑓′(𝑥)+ 5(𝑑𝑥−1

3

(𝐜)Evaluate# 𝑓′′(4 − 3𝑥)𝑑𝑥2

1

(𝐝)Itisknownthat#12𝑓

′(𝑥)𝑑𝑥9

−1= −3.Find# 𝑓′(𝑥)𝑑𝑥

6

9.

The problem has been restated.

𝐁𝐂𝟏:Thefunction𝑓iscontinuousanddifferentiableforallvaluesof𝑥.Aportionofthegraph

of𝑓!, thederivativeof𝑓, isshownaboveontheclosedinterval[−3,6]. Itisknownthat𝑓(2) = −3.

(𝐞)Findall𝑥value(s)ontheopeninterval(−3, 6)where𝑓hasacriticalpoint.Foreach𝑥value, determine

if𝑓hasarelativeminimum, relativemaximum, orneither.Giveareasonforyouranswer.

(𝐟)Findthemaximumvalueof𝑓(𝑥)ontheclosedinterval[−3, 6].Justifyyouranswer.

(𝐠)Findanyopenintervalswherethegraphof𝑓isbothdecreasingandconcavedown.Giveareasonfor

youranswer.

(𝐡)Orderthevaluesof𝑓(2), 𝑓′(2), and𝑓′′(2)fromleastthegreatest.Explainyourreasoning.

The problem has been restated.

𝐁𝐂𝟏:Thefunction𝑓iscontinuousanddifferentiableforallvaluesof𝑥.Aportionofthegraph

of𝑓!, thederivativeof𝑓, isshownaboveontheclosedinterval[−3,6]. Itisknownthat𝑓(2) = −3.

(𝐢)Findallvaluesof𝑥where𝑓hasapointoninflectionontheopeninterval(−3, 6).Giveareasonfor

youranswer.

(𝐣)Findtheaveragevalueof𝑓′overtheclosedinterval[−3, 6]. Showtheworkthatleadstoyouranswer.

(𝐤)Theaveragerateofchangefor𝑓′(𝑥)overtheinterval[1, 6]isequalto25 butthereisnovalueof𝑐in

theopeninterval(1, 6)suchthat𝑓!!(𝑐) =25 .ExplainwhythisdoesnotcontradicttheMean

ValueTheorem.

The problem has been restated.

𝐁𝐂𝟏:Thefunction𝑓iscontinuousanddifferentiableforallvaluesof𝑥.Aportionofthegraph

of𝑓!, thederivativeof𝑓, isshownaboveontheclosedinterval[−3,6]. Itisknownthat𝑓(2) = −3.

(𝐥)Thefunction𝑦 = 𝑔(𝑥)satisfiesthedifferentialequation𝑑𝑦𝑑𝑥 =

𝑦𝑓′(𝑥 − 1)2 withinitialcondition

𝑔(2) = −3.UseEuler!smethodwithtwostepsofequalsizestartingat𝑥 = 2toapproximate𝑔(0).

(𝐦)Letℎbeatwicedifferntiablefunctiondefinedbyℎ(𝑥) = 2𝑥 + 3−# 𝑓′(𝑡 − 1)𝑑𝑡𝑒2𝑥

1.Findthe

seconddegreeMaclaurinpolynomialforℎ(𝑥).

(𝐧)Evaluate# sin(𝑥)𝑓′(𝑥)𝑑𝑥3

1.

The problem has been restated.

𝐁𝐂𝟏:Thefunction𝑓iscontinuousanddifferentiableforallvaluesof𝑥.Aportionofthegraph

of𝑓!, thederivativeof𝑓, isshownaboveontheclosedinterval[−3,6]. Itisknownthat𝑓(2) = −3.

(𝐨)Write, butdonotevaluate, anintegralexpressionintermsof𝑥thatwouldgivethearclengthof𝑓over

theinterval[−2, 1].

(𝐩)Let𝑝(𝑥) = 𝑒−𝑓(𝑥).IstheleftRiemannapproximationof# 𝑝(𝑥)𝑑𝑥6

2anoverorunderestimate?

Giveareasonforyouranswer.

2020 FRQ Practice Problem BC2

𝐁𝐂𝟐:Thefunctions𝑓(𝑥), 𝑎(𝑥)and𝑏(𝑥)aretwicedifferentiablewithselectedvaluesindicatedinthetable.

Let𝑔bethefunctiondefinedby𝑔(𝑥) = 𝑥# −f 𝑓(𝑡)𝑑𝑡6

#.

(𝐚)Approximatetheaveragevalueof𝑓(𝑥)overtheinterval[0,4]usingatrapezoidalsumwiththree

subintervalsindicatedinthetable.

(𝐛)Does𝑔havealocalminimum, alocalmaximum, orneitherat𝑥 = 1? Giveareasonforyour

answer.

The problem has been restated.

𝑥 0 1 2 4

𝑓(𝑥) 6 2 −1 0

𝑓′(𝑥) 5 3 3 −2

𝑎(𝑥) ? 36 24 16

𝑏(𝑥) 0 1 8 64

𝐁𝐂𝟐:Thefunctions𝑓(𝑥), 𝑎(𝑥)and𝑏(𝑥)aretwicedifferentiablewithselectedvaluesindicatedinthetable.

Let𝑔bethefunctiondefinedby𝑔(𝑥) = 𝑥# −f 𝑓(𝑡)𝑑𝑡6

#.

(𝐜)Find lim𝑥→2

𝑓(𝑥2)sin(𝜋𝑥).

(𝐝)Itisknownthat lim𝑥→1

𝑔(2𝑥)− 𝑥− 3𝑐𝑒𝑘(𝑥−1) + 2 = −

32 where𝑐and𝑘areconstants.Findthevalues

of𝑐and𝑘.

(𝐞)Evaluate# 5𝑥𝑓′%𝑥2&𝑑𝑥2

0.

The problem has been restated.

𝑥 0 1 2 4

𝑓(𝑥) 6 2 −1 0

𝑓′(𝑥) 5 3 3 −2

𝑎(𝑥) ? 36 24 16

𝑏(𝑥) 0 1 8 64

𝐁𝐂𝟐:Thefunctions𝑓(𝑥), 𝑎(𝑥)and𝑏(𝑥)aretwicedifferentiablewithselectedvaluesindicatedinthetable.

Let𝑔bethefunctiondefinedby𝑔(𝑥) = 𝑥# −f 𝑓(𝑡)𝑑𝑡6

#.

(𝐟)Itisknownthat# 𝑓′′(𝑥)𝑑𝑥5

0= 7.Find𝑓′(5).

(𝐠)Itisknownthat+1

𝑏(𝑛)

∞

𝑛=1isaconvergentpseries.Findallvaluesofαwhere+,

1𝑏(𝑛)-

2𝛼+1∞

𝑛=1converges.

(𝐡)Theseries+ 𝑎(𝑛)∞

𝑛=0isaconvergentgeometricseriesthatcanbewrittenintheform+ 𝐴𝑟𝑛

∞

𝑛=0where

𝐴and𝑟arebothpositiveconstants.Findthevaluesof𝐴and 𝑎(𝑛)K

LM2

𝑥 0 1 2 4

𝑓(𝑥) 6 2 −1 0

𝑓′(𝑥) 5 3 3 −2

𝑎(𝑥) ? 36 24 323

𝑏(𝑥) 0 1 8 64