Connecting AB to BC: Fundamental Theorem of Calculus · AP CALCULUS BC YouTube Live Virtual Lessons...
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Connecting AB to BC: Fundamental Theorem of Calculus One of the most important concepts in calculus is the connection between the area bounded by a curve and the definite integral. This connection leads us eventually to the ! Let ! s say we want to find the area between the graph of ! () and the axis from = to = . Today, we will review a few different approaches that all connect back to the Fundamental Theorem of Calculus. Connecting AB to BC: Fundamental Theorem of Calculus Topic: 10.1 Defining Convergent and Divergent Infinite 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 ! () Topics Approach Area approximations Riemann sums, trapezoidal approximations Finding exact areas Using geometric areas Finding exact areas Fundamental Theorem of Calculus Finding exact areas Using properties of integrals Connecting integrals and derivatives Fundamental Theorem of Calculus Accumulation Modified FTC
Transcript of Connecting AB to BC: Fundamental Theorem of Calculus · AP CALCULUS BC YouTube Live Virtual Lessons...
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
