BCCalculusFinal‘16-‘17
PartIA
NoCalculatorAllowed
1.
If f ( x ) = sin ln ( 2x ) ,then f ′ ( x ) = a)
(
(
)
sin ln ( 2x )
2x
1⎞
⎝ 2x ⎟⎠
)
b)
(
cos ln ( 2x )
x
)
c)
(
− cos ln ( 2x )
(
cos ln ( 2x )
2x
)
d)
cos ⎛⎜
e)
2.
If sin −1 x = ln y ,then
(c)
y
1+ x 2 (d)
y
1− x 2
(a)
e
(e)
esin x
1+ x 2 dy
= dx
xy
1− x 2
(b)
x
)
sin−1 x
−1
3.
Whichofthefollowingstatementsarefalse?
I.
⌠
⎮
⎮
⌡
((
II.
⌠
⎮
⌡
(x
III.
∫ csc x dx = ln csc x + cot x + c
a)
I only
x3 + x
5
)
4
)
x 4 + 2x 2 − 5 dx =
(
1 4
x + 2x 2 − 5
5
)
5
4
+c
)
1
sin x 6 dx = − cos x 6 + c
6
b)
d)
I and II only
II only
e)
III only
II and III only
0
4
x
f ( x) c)
25
6
30
8
50
12
4. Letfbeadifferentiablefunctionwhichhasvaluesshownonthetable
above.Usingthesub-intervalsdefinedbythetablevaluesandusingright
handRiemannsums, ∫0 f ( x ) dx ≈ 50
a)
b)
c)
d)
e)
290 360 380 390 430
5.
ThefunctionfisdefinedforallRealnumberssuchthat
⎧ x 2 − 7x +10
for x ≠ 2
⎪
.
f x = ⎨ b x−2
⎪
⎩ b for x = 2
(
()
)
Forwhichvalueofbwillthefunctionbecontinuousthroughoutitsdomain?
(a) −3 (b)
(d) 5 (e) Noneofthese
2 (c) 3 6.
a)
∫e
x
b)
e x sin e x +1 + c c)
e x sin e x + x + c cos ( ex +1) dx sin ( ex +1) + c (
)
(
)
d)
1
cos2 ( ex +1) + c
2
(e)
Noneofthese
7.
Shownaboveistheslopefieldforwhichofthefollowingdifferential
equations?
dy
dy
dy
a)
b)
c)
= xy − x = xy + x = y − x2 dx
dx
dx
dy
dy
3
d)
(e)
= ( y −1) x 2
= ( y −1)
dx
dx
8.
ThreegraphslabeledI,II,andIIIareshownabove.Oneisthegraphof
f ( x ) ,oneisthegraphof f ′ ( x ) ,andoneisthegraphof f ″ ( x ) .Whichofthe
followingcorrectlyidentitieseachofthethreegraphs?
f ( x ) = I , f ′ ( x ) = II , f ″ ( x ) = III a)
f ( x ) = II , f ′ ( x ) = I , f ″ ( x ) = III f ( x ) = II , f ′ ( x ) = III , f ″ ( x ) = I b)
c)
f ( x ) = III , f ′ ( x ) = I , f ″ ( x ) = II f ( x ) = III , f ′ ( x ) = II , f ″ ( x ) = I d)
e)
9.
Onwhichofthefollowingintervalisthegraphofthecurve
y = 2x 3 − 3x 2 − 12x + 15 bothdecreasingANDconcaveup?
1⎞
⎛
a)
b)
c)
( −∞, −1) ( −1, 2 ) ⎜⎝ −1, − ⎟⎠ 2
⎛1 ⎞
d)
(e) ( 2, ∞ )
⎜⎝ , 2 ⎟⎠
2
10. Whichofthefollowingisthesolutiontothedifferentialequation
dy
⎛π⎞
= ysec2 x withtheinitialcondition y ⎜ ⎟ = −1 ?
⎝ 4⎠
dx
a)
y = −etan x d)
b)
y = −e(
y = − 2 tan x −1 −1+tan x )
y = −e(
c)
e)
y = e(
−1+tan x )
) sec 3 x−2 2 3
11.
3( −2)
The sum of the infinite series ∑
5n
n=3
a)
5
b)
∞
15
7
16
75
c)
n+1
is
48
175
d)
e) dne
⎧ x if x < 2
4
⎪
12. If f ( x ) isdefinedby f ( x ) = ⎨
,then ∫−1 f ( x ) dx = ⎪⎩ 3 if 2 ≤ x
a)
9
2
b)
13
2
c)
15
d)
2
17
e)
2
dne
7x − sin x
x 2 + sin ( 3x )
13.
lim
x→0
a)
6
14.
Let f and g be continuous functions such that
b)
2
c)
1
d)
0
e)
dne
∫0 f ( x ) dx = 9 ,
6
∫3 f ( x ) dx = 5 , and ∫3 g ( x ) dx = −7 . What is the value of
6
0
3
⌠ ⎛1
⎞
⎮ ⎜⎝ 2 f ( x ) − 3g ( x )⎟⎠ dx =
⌡0
a)
−23 b)
−19 c)
−
17
2
d)
19 e)
23 15. Thegraphsofthefunctionsfandgareshownabove.Thevalueof
lim
f g ( x ) is
x→1
(
a) 0
)
b)
1
c)
2
d)
3
e)
dne
16.
Statethestepthathasthefirstmistakeinthisprocess:
Step1:
Step2:
Step3:
Step4:
Step5:
a.
b.
c.
d.
e.
∫ sec
4
2x tan 3 2x dx = 1
sec2 2x tan 3 2x sec2 2x 2dx = ∫
2
1⌠
1− u 2 ) u 3du = (
⌡
2
1⌠ 3 5
(u − u ) du = 2⌡
1⎛ 1 4 1 6 ⎞
u − u + c⎟ = ⎠
2 ⎜⎝ 4
6
1 4
1
tan 2x − tan 6 2x + c 8
12
Step1
Step2
Step3
Step4
Thereisnomistake.
17. A particle moves along a straight line so that, at time t > 0, the position
of the particle is given by s(t), the velocity is given by v(t), and the
acceleration is given by a(t). Which of the following expressions gives the
average velocity of the particle on the interval [2, 8] ?
a)
1 8
a (t ) dt 6 ∫2
d)
b)
v(8) − v(2)
6
1 8
s (t ) dt 6 ∫2
e)
c)
s (8) − s (2)
6
v(8) − v(2) 18. Attimet,apopulationofbacteriagrowsattherateof 5e0.2t + 4t grams
perday,wheretismeasuredindays.Byhowmanygramshasthepopulation
grownfromtime t = 0 daysto t = 10 days?
5e2 + 40 5e2 +195 25e2 + 40 a)
b)
c)
25e2 +195 25e2 +175 d)
e)
19. Thefunctionfisdifferentiableandincreasingforallrealnumbersx,and
thegraphoffhasexactlyonepointofinflection.Ofthefollowing,whichcould
bethegraphof f ′ ( x ) ,thederivativeoff?
20. Functions w, x, and y are differentiable with respect to time and are
related by the equation w = x2y. If x is decreasing at a constant rate of 1 unit
per minute and y is increasing at a constant rate of 4 units per minute, at
what rate is w changing with respect to time when x = 6 and y = 20?
a)
−384 b)
−240 d)
276 e)
384 c)
−96 x
f ( x)
f ′( x)
0
2
4
8
3
4
9
13
0
1
1
2
21. Thetableabovegivesvaluesofadifferentiablefunctionfandits
derivativeatselectedvaluesofx.If h ( x ) = f ( 2x ) ,whichofthefollowing
statementsmustbetrue?
I.
hisincreasingon 2 < x < 4 II. Thereexistsc,where 0 < x < 4 ,suchthat h ( c ) = 12 III. Thereexistsc,where 0 < x < 2 ,suchthat h′ ( c ) = 3 a) Ionly
b) IIonly
c)
IandIIIonly
d) IIandIIIonly
e) I,II,andIII
22. Fortime t ≥ 0 ,thevelocityofaparticlemovingalongthex-axisisgiven
2
by v (t ) = (t − 5 )(t − 2 ) .Atwhatvaluesoftistheaccelerationoftheparticle
equalto0?
a) 2only
d) 2and4only
b)
4only
e)
2and5only
c)
5only
23. Thegraphofthefunctionfisshowninthefigureabove.Forhowmany
valuesofxintheopeninterval ( −4, 4 ) isfdiscontinuous?
a) One b) Two
c) Three
d)Four e)Five
24.
a)
Whichofthefollowingseriesconvergeabsolutely?
∞
∑ ( −1)
n+1
n=1
1
2n
b)
∞
∑ ( −1)
n+1
n=1
1
n
c)
∞
( −1)
∑
n=1
n+1
n
n +1
e)
Noneofthese
d)
( −1)
∑
n=1
lim
a)
0
x→2
x2 − 4
x
∫2
cos (π t ) dt
=
n
1⎞
⎜⎝ ⎟⎠ 2
25.
∞
n+1 ⎛
b)
1
c)
2
d)
4
e)
DNE
BCCalculusFinal‘16-‘17
PartIB
CalculatorAllowed
1.
Thefigureaboveshowsthegraphof f ′ ( x ) ,thederivativeoffunctionf,
for-6<x<8.Ofthefollowing,whichbestdescribesthegraphoffonthesame
interval?
a)
1relativeminimum,1relativemaximum,and3pointsofinflection
b)
1relativeminimum,1relativemaximum,and4pointsofinflection
c)
2relativeminima,1relativemaximum,and2pointsofinflection
d)
2relativeminima,1relativemaximum,and4pointsofinflection
e)
2relativeminima,2relativemaxima,and3pointsofinflection
2.
Thecost,indollars,toshredtheconfidentialdocumentsofacompanyis
modeledbyC,adifferentiablefunctionoftheweightofdocumentsinpounds.
Ofthefollowing,whichisthebestinterpretationofCʹ(500)=80?
a) Thecosttoshred500poundsofdocumentsis$80.
80
b) Theaveragecosttoshreddocumentsis
dollarperpound.
500
c) Increasingtheweightofdocumentsby500poundswillincreasethe
costtoshredthedocumentsbyapproximately$80.
d) Thecosttoshreddocumentsisincreasingatarateof$80perpound
whentheweightofthedocumentsis500pounds.
e) Noneofthese
x
3.
Thederivativeofthefunctionfisgivenby f ′ ( x ) = − + cos x 2 .Atwhat
3
valuesofxdoesfhavearelativeminimumontheinterval 0 < x < 3 ?
2.372 a)
b) 1.798 c)
1.094 and 2.608 2.493 d)
e)
1.798 and 2.372 ( )
4.
Whichofthefollowingseriescannotbeshowntoconvergeusingthe
∞
1
limitcomparisontestwiththeseries ∑ 2 ?
n=1 n
a)
∞
4
∑
2
n=1 3n − n
d)
b)
∞
⎛ lnn ⎞
∑
⎜ 2 ⎟
n=1 ⎝ n ⎠
∞
∑
n=1
15
n4 + 5
e)
c)
∞
sin ⎛⎜ 2 ⎞⎟ ∑
⎝n ⎠
n=1
1
Noneofthese
5.
Lethbethefunctiondefinedby h ( x ) =
ofhand g ( 2 ) = 3 ,whatisthevalueof g ( 4 ) ?
−0.020 0.152 a)
b)
3.152 d)
e)
1
x +1
5
c)
. Ifgisanantiderivative
3.031 4.798 6.
Aparticle moves along the x-axis so that its velocity at time t ≥ 0 is
t 2 −1
given by v (t ) = 2 . What is the total distance traveled by the particle from
t +1
t = 0 to t = 2 ?
a)
0.214 0.927 d)
b)
0.320 1.600 e)
c)
0.600 2
⎛ 1 ⎞
istheanti-derivativeof f ( x ) ,then ∫1 f ( x ) dx = 2
⎟
⎝ x +1⎠
7. If sin ⎜
a)
−0.281
b) −0.102 c) 0.102 d)
0.260 e)
0.282 8.
Thetemperature,indegreesFahrenheit,ofwaterinapondismodeled
⎛ 2π
⎞
bythefunction H (t ) = 55 − 9cos ⎜
t +10 )⎟ ,wheretisthenumberofdays
(
⎝ 365
⎠
sinceJanuary1(t=0).Whatistheinstantaneousrateofchangeofthe
temperatureofthewaterattimet=90days?
a)
0.114°F / day b)
0.153°F / day c)
50.252°F / day d)
56.350°F / day 2π
e)
°F / day 365
x0 = 0 x1 = 2 x2 = 4 f ( x0 ) = 2 f ( x1 ) ≈ 6 f ( x2 ) ≈10 dy Ax 2 + 4
9.
Considerthedifferentialequation =
,whereAisaconstant.
dx
y
Let y = f ( x ) betheparticularsolutiontothedifferentialequationwiththe
initialcondition f ( 0 ) = 2 .Euler’smethod,startingat x = 0 withastepsizeof
2,isusedtoapproximate f ( 4 ) .Stepsfromthisapproximationareshownin
thetableabove.WhatisthevalueofA?
a)
1
2
b)
2
c)
5
d)
13
2
e) ThevalueofAcannotbedetermined
2
10. Thesecondderivativeofafunctiongisgivenby g″ ( x ) = 2 − x + cos x + x .
For −5 < x < 5, onwhatopenintervalsisthegraphofgconcaveup?
a)
−5 < x < −1.606 only b)
−1.606< x < 5only c)
0.463 < x < 2.100 only d)
−5 < x < 0.463 and 2.100 < x < 5 e)
−5 < x < −1.606 and 0.463 < x < 2.100 x
1
3
5
f ( x) 2.4 3.6 5.4 11. Thetableabovegivesselectedvaluesofafunctionf.Thefunctionis
twicedifferentiablewith f ″ ( x ) > 0. Whichofthefollowingcouldbethevalue
of f ′ ( 3) ?
0.6 b)
0.7 c)
0.9 d) 1.2 e)
1.5 a)
( 3x + 8 )( 5 − 4x ) . f x hasa
12. Letfbeafunctionsuchthat f ( x ) =
( )
2
( 2x +1)
horizontalasymptoteat
y = −6 a)
y = −3 b)
c)
d)
y = 0
e)
1
y=− 2
3
y= 2
Line l is tangent to the graph of y = ex at the point (k, ek ). What is the
1
positive value of k for which the y-intercept of l is ?
2
13.
a)
14.
0.405 1.560 d)
b)
0.768 c)
1.500 e)
Thereisnovalueofk
Thefunctionfiscontinuousandtwicedifferentiable.If f ′ ( x ) < 0 and
f "( x ) < 0 forallvaluesofx,whichofthefollowingcouldbeatableofvalues
for f ( x ) ?
A
x
1
2
3
4
f ( x ) B x f ( x ) 1 −3 −3 −4 2 −1 −6 3
3
−9 4 19 C x f ( x ) 1 −3
2
0
3
3
4
6
D
x
1
2
3
4
f ( x) −3 5
11 13 E
x
1
2
3
4
f ( x) −3 −4 −3 0
15.
The regions A, B, and C in the figure above are bounded by the graph
of the function f and the x-axis. The area of region A is 14, the area of region
B is 16, and the area of region C is 50. What is the average value of f on the
interval [0, 8] ?
a)
6
b)
10 c)
40
d)
3
80
e)
3
48