bc mock exam 2020 solutions - afhs calculus · workers at a local office building are expected to...
TRANSCRIPT
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝑡 0 25 30 45 65 80
𝑊(𝑡) 2 10 13 30 60 81
𝐁𝐂𝟏:Workersatalocalofficebuildingareexpectedtobeatworkby9AM(𝑡 = 60)eachday.
Letthetwicedifferentiablefunction𝑦 = 𝑊(𝑡)betheamountofworkersinthebuildingat
time𝑡, inminutes, satisfyingthelogisticdifferentialequation𝑑𝑦𝑑𝑡
=115𝑦 S1 −
𝑦100
U
where𝑊(0) = 2.Selectedvaluesof𝑊(𝑡)aregiveninthetablewheret = 0is8AM.
(a)Find𝑊!(65)and𝑊!!(65).Usingcorrectunits, interpretthemeaningof𝑊!!(65)incontextof
thisproblem.
′W 65( ) = 115W 65( ) 1−
W 65( )100
⎛
⎝⎜
⎞
⎠⎟ =
115
60( ) 1−60( )
100
⎛
⎝⎜
⎞
⎠⎟ =
85
′′W t( ) = d2 ydt2
= 115
dydt
1− y100
⎛⎝⎜
⎞⎠⎟+ y − 1
100dydt
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
′′W 65( ) = 115
85
1− 60100
⎛⎝⎜
⎞⎠⎟+ 60( ) − 1
10085
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥= − 8
375
′′W 65( ) means the rate of the amount of workers in the
building is changing at a rate of − 8375
workers per minute
per minute at t = 65 or at 9:05 AM.
5:
1: ′W 65( )2 : ′′W t( )
0 / 2 if no product or chain rule
1: ′′W 65( )1: meaning in context
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝑡 0 25 30 45 65 80
𝑊(𝑡) 2 10 13 30 60 81
𝐁𝐂𝟏:Workersatalocalofficebuildingareexpectedtobeatworkby9AM(𝑡 = 60)eachday.
Letthetwicedifferentiablefunction𝑦 = 𝑊(𝑡)betheamountofworkersinthebuildingat
time𝑡, inminutes, satisfyingthelogisticdifferentialequation𝑑𝑦𝑑𝑡
=115𝑦 S1 −
𝑦100
U
where𝑊(0) = 2.Selectedvaluesof𝑊(𝑡)aregiveninthetablewheret = 0is8AM.
(b)Evaluate[ 𝑊!(3𝑡 + 25)𝑑𝑡"
#.Showtheworkthatleadstoyouranswer.
′W 3t + 25( )dt0
∞
⌠⌡ = lim
b→∞′W 3t + 25( )dt
0
b
⌠⌡
= limb→∞
13W 3t + 25( )⎡
⎣⎢
⎤
⎦⎥
0
b
= 13
limb→∞
W 3b+ 25( )−W 25( )⎡⎣ ⎤⎦
= 13
limb→∞W 3b+ 25( )−10⎡
⎣⎤⎦ =
13
100−10⎡⎣ ⎤⎦ = 30
W t( ) is a logistic curve with horizonal asymptote y = 100
so limb→∞W 3b+ 25( ) = 100.
3:1: improper integral1: use horizontal asymptote1: value
⎧
⎨⎪
⎩⎪
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝑡 0 25 30 45 65 80
𝑊(𝑡) 2 10 13 30 60 81
𝐁𝐂𝟏:Workersatalocalofficebuildingareexpectedtobeatworkby9AM(𝑡 = 60)eachday.
Letthetwicedifferentiablefunction𝑦 = 𝑊(𝑡)betheamountofworkersinthebuildingat
time𝑡, inminutes, satisfyingthelogisticdifferentialequation𝑑𝑦𝑑𝑡
=115𝑦 S1 −
𝑦100
U
where𝑊(0) = 2.Selectedvaluesof𝑊(𝑡)aregiveninthetablewheret = 0is8AM.
(c)UsealeftRiemannsumwiththreesubintervalsindicatedbythedatainthetabletoapproximatethe
valueof145[ 𝑊(𝑡)𝑑𝑡.$%
#Usingcorrectunits, explainthemeaningof
145[ 𝑊(𝑡)𝑑𝑡$%
#inthecontextofthe
problem.
(d)UseEuler!smethodwithtwostepsofequalsizestartingat𝑡 = 45minutestoapproximate
thenumberofworkersinsidetheofficebuildingby9: 25AM(𝑡 = 85minutes).
145
W t( )dt0
45
⌠⌡ ≈ 1
452( ) 25( )+ 10( ) 5( )+ 13( ) 15( )⎡⎣ ⎤⎦ =
29545
145
W t( )dt0
45
⌠⌡ means the average number of workers
in the building from t = 0 (8 AM) to t = 45 (8:45 AM)
is about 659
workers.
3:1: left Riemann sum1: approximation1: meaning in context
⎧
⎨⎪
⎩⎪
t W t( ) dy = 115y 1− y
100⎛⎝⎜
⎞⎠⎟dx
45 30 11530( ) 1− 30100
⎛⎝⎜
⎞⎠⎟20( )= 2( ) 7
10⎛⎝⎜
⎞⎠⎟20( )=28
65 58 11558( ) 1− 58100
⎛⎝⎜
⎞⎠⎟20( )= 58
15⎛⎝⎜
⎞⎠⎟42100
⎛⎝⎜
⎞⎠⎟20( )=32.48
85 90.48
2 :1: Euler's method1: approximation⎧⎨⎩
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝑡 0 25 30 45 65 80
𝑊(𝑡) 2 10 13 30 60 81
𝐁𝐂𝟏:Workersatalocalofficebuildingareexpectedtobeatworkby9AM(𝑡 = 60)eachday.
Letthetwicedifferentiablefunction𝑦 = 𝑊(𝑡)betheamountofworkersinthebuildingat
time𝑡, inminutes, satisfyingthelogisticdifferentialequation𝑑𝑦𝑑𝑡
=115𝑦 S1 −
𝑦100
U
where𝑊(0) = 2.Selectedvaluesof𝑊(𝑡)aregiveninthetablewheret = 0is8AM.
(e)Isthereatime𝑐, suchthat25 < 𝑐 < 45,when𝑊!(𝑐) = 1.Justifyyouranswer.
(f)Let𝑃&(𝑡)betheseconddegreeTaylorpolynomialfor𝑊(𝑡)centeredat𝑡 = 65.Find𝑃&(𝑡)and
useittoapproximatethenumberofworkersinsidetheofficebuildingat9: 25AM(𝑡 = 85).
′W c( ) =W 45( )−W 25( )45− 25
= 30−1020
= 1
W t( ) is twice differentiable which mean W t( ) is continuous
so by the Mean Value Theorem there is a time c between
25 and 45 where ′W c( ) = 1, the average rate of change
W 45( )−W 25( )45− 25
.
2 : 1:W 45( )−W 25( )
45− 251: justification using MVT
⎧⎨⎪
⎩⎪
P2 t( ) =W 65( )+ ′W 65( ) t − 65( )+ ′′W 65( )2!
t − 65( )2
P2 t( ) = 60+ 85 t − 65( )− 82! 375( ) t − 65( )2
W 85( ) ≈ P2 85( ) = 60+ 85 20( )− 82! 375( ) 20( )2
= 60+ 32− 6415
= 131615
= 87.7333…
3:2 : second degree
Taylor polynomial1: approximation
⎧
⎨⎪
⎩⎪
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝐁𝐂𝟐:Thefunction𝑓istwicedifferentiablefor𝑥 ≥ −6.Aportionofthegraphof𝑓isgiveninthe
figureabove.Theareasformedbytheboundedregionsbetweenthegraphof𝑓andthe𝑥axisare
labeledinthefigure.Letthetwicedifferentiablefunction𝑔bedefinedby𝑔(𝑥) = [ 𝑓(𝑡)𝑑𝑡'
&.
(a)Find lim'→)&
𝑔(2𝑥) − 4𝑥𝑒'!)$ + 𝑥 + 1
.
limx→−2
g 2x( )− 4x⎡⎣ ⎤⎦ = g −4( )− −8( ) = f t( )dt2
−4
⌠⌡ +8
= 8− f t( )dt−4
2
⌠⌡ = 8− 10+ −2( )( ) = 0
limx→−2
ex2−4 + x +1⎡
⎣⎤⎦ = e
0 + −2( )+1= 0
limx→−2
g 2x( )− 4x
ex2−4 + x +1
= limx→−2
2 ′g 2x( )− 4
ex2−4 2x( )+1
l'Hospital's Rule! "## $##
=2 ′g −4( )− 4e0 −4( )+1
=2 f −4( )− 4
−3( ) =2 0( )− 4
−3( ) = 43
3:
1: limx→−2
g 2x( )− 4x⎡⎣ ⎤⎦1: applies L'Hospital's rule1: answer
⎧
⎨⎪⎪
⎩⎪⎪
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝐁𝐂𝟐:Thefunction𝑓istwicedifferentiablefor𝑥 ≥ −6.Aportionofthegraphof𝑓isgiveninthe
figureabove.Theareasformedbytheboundedregionsbetweenthegraphof𝑓andthe𝑥axisare
labeledinthefigure.Letthetwicedifferentiablefunction𝑔bedefinedby𝑔(𝑥) = [ 𝑓(𝑡)𝑑𝑡'
&.
(b)Findtheabsolutemaximumof𝑔ontheinterval[−6,2].Justifyyouranswer.
′g x( ) = f x( ) = 0⇒ x = −4,0
x g x( )−6 f t( )dt
2
−6
∫ = − f t( )dt−6
2
∫ = − −5+10− 2( ) = −3
−4 f t( )dt2
−4
∫ = − f t( )dt−4
2
∫ = − 10− 2( ) = −8
0 f t( )dt2
0
∫ = − f t( )dt0
2
∫ = − −2( ) = 2
2 f t( )dt2
2
∫ = 0
The absolute maximum value of g is 2 when x = 0
3:
1: ′g x( ) = 0
1: candidates x = −4,01: answer with justification
⎧
⎨⎪⎪
⎩⎪⎪
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝐁𝐂𝟐:Thefunction𝑓istwicedifferentiablefor𝑥 ≥ −6.Aportionofthegraphof𝑓isgiveninthe
figureabove.Theareasformedbytheboundedregionsbetweenthegraphof𝑓andthe𝑥axisare
labeledinthefigure.Letthetwicedifferentiablefunction𝑔bedefinedby𝑔(𝑥) = [ 𝑓(𝑡)𝑑𝑡'
&.
(c)Evaluate[ 𝑥𝑓!(2𝑥)𝑑𝑥*
)+.
u = x⇒ du = dx dv = ′f 2x( )dx⇒ v = 12f 2x( )
x ′f 2x( )dx−3
1
⌠⌡ = 1
2x f 2x( )⎡
⎣⎢
⎤
⎦⎥−3
1
− 12
f 2x( )dx−3
1
⌠⌡
= 12f 2( )⎛
⎝⎜⎞⎠⎟− −3
2f −6( )⎛
⎝⎜⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥ −14
f 2x( )u!
2dx( )du!
−3
1
⌠
⌡⎮⎮
= 120( )⎛
⎝⎜⎞⎠⎟− −3
2−5( )⎛
⎝⎜⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥ −14
f u( )du−6
2
⌠⌡
= −152
⎡
⎣⎢
⎤
⎦⎥ −14
−5( )+ 10( )+ −2( )⎡⎣ ⎤⎦ = −152− 34= − 33
4
3:2 : integration by parts1: answer⎧⎨⎩
Written by Bryan Passwater Special Thanks to T. Gott, T. Record, and M. Kiraly Scoring Guide by Ted Gott
𝐁𝐂𝟐:Thefunction𝑓istwicedifferentiablefor𝑥 ≥ −6.Aportionofthegraphof𝑓isgiveninthe
figureabove.Theareasformedbytheboundedregionsbetweenthegraphof𝑓andthe𝑥axisare
labeledinthefigure.Letthetwicedifferentiablefunction𝑔bedefinedby𝑔(𝑥) = [ 𝑓(𝑡)𝑑𝑡'
&.
(d)TheregionRisthebaseofasolidwhosecrosssectionsperpediculartothe𝑥axisaresquares.
Write, butdonotevaluate, anintegralexpressionthatgivesthevolumeofthesolid.
(e)For𝑥 ≥ 5, thefunction𝑓isnotexplicitlygiven, but𝑓!(𝑥) = −4 m12n
')%.If𝑎, = 𝑓!(𝑛),
findq 𝑎,
"
,-%
orshowthattheseriesdiverges.
volume = f x( )⎡⎣ ⎤⎦2dx
2
5
⌠⌡⎮ 1: integral expression
−4 12
⎛⎝⎜
⎞⎠⎟
n−5
n=5
∞
∑ = −4 12
⎛⎝⎜
⎞⎠⎟
n−5
n=5
∞
∑ = −4 1
1− 12
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
= −4 22−1
⎡
⎣⎢
⎤
⎦⎥ = −8 2 :
1: recognizes infinite geometric series
1: sum
⎧
⎨⎪
⎩⎪