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0% found this document useful (0 votes)
127 views36 pagesAnswer Key - I
Uploaded by
cicifoodchainCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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/ 36
ly 9 od 80 08 a 2
2_]| CALCULUS BC
, 245, x<-2
L fos aaa
a (ors x2-2
Let fbe the function defined above. The value of f”
(A) -2 (B) -4 (© 4 (D) nonexistent
Le Pore ) »- p nat cont of
Lig Poo DUE POP O anhabl
Xa-2”
xo-2
Xe-2 00
he Cex) = 10
X-a-27
Dis
2. The = 3x"
graph of y = 3x*— 2° has a relative maximum at
SAY. (0,0) only
BY (1,2) only
(©) (2,4) only
PF (4,16) only
2 6x-3x%0 yi 6-6x
3x(2-x) =O y'ror>o aun
X=0 X=2 "
4meo = max, Th SAMPLE EXAMINATION I 3
T
3, A particle moves in the xy-plane so that its velocity vector at time tis v(t) =(C, sin (1) and the
~ particle's position vector at time ¢ = 0 is (1, 0). What is the position vector of the particle when
t=3?
a (4) (©) (10,22) (D) (10,2)
sid=< OG, emevCy
-) = => Cig!
5jo) = C01, ~#r+C2> = <1, 0> ee
cide CHEM, ese +B
S0)= X10, 24S cal
4. For what values of x does the curve y?— x°— 15x? = 4 have horizontal tangent lines?
(A) x=0only ite -~3x7-30XK =O
(8) x=10only 2yy’= 3X?430%
-dy - 3x7430%
(D) x=-10,x=0, and x= 10 oe = ay
3x(Xx410) =O 2 _—
a “4. 1000 — =
X20" x2-10 i
t u
2 yr re do double checle
only He omen eg ued 210 (c] 4 | CALCULUS BC
ies Gr-5)"?
a
, seri
5, What isthe radius of convergence for the Power
) 3 ©1 ©) 3 4
Corker at a= %
Iax5I™ ff tax-s) = 13x3! a/x-%lel
lx-B]¢ Answe
: A
6. ly SS sinc) =
Non *° =x
“2 ®1 ©Oo
kL. Bx+csx 4
XO 4 X-l “a
ig ed 0K tra Boo pmognen —1?
SAMPLE EXAMINATION I | 5
1 ip foxy = V2 4sin x, then f=) =
w-2 Bo OF [> 2
foo: -4COSX _ _ 2 Cosx
Qna-asiny 2-a4sinx
Fim)> 2 =f
\2
8 [laoe+a- v]ae=
we fox] o«
(D) 55
Answer
DB
{lca : teat] 2 3ll6)+8 -2@+4)
~ 56-16 =40
or
a4
aot] = 3648+ — (30444) =40
Answer
&
nahn pig orming any of i pa apt
A 6 | CALCULUS BC
cin tat fr all positive integers n,
su
, ers
s of positive nue .nts must be true?
9. Let a,, by, and c, be sequences fone
= i following stat “uf
Gy, S by S Cy If Yb, converges, then which of the fo! Wi
il
1, }7a, converges (
ACY cy converges
a -
IIL. }\(a, +b,) converges
(A) only '
(B) Il only
(D) 1. Mand 111
Answer
D. Lo]
»
{ab 12.
a t
D
10. For an object moving along a .,
long a straight line, th
object as a function of time. At which of the meth ae Tepresents the velocity of the movil?
(A) A is the speed the greatest?
i ®B Oc Poo
i Loan
i
Speed = [Velocity |
1 Speed Ie
3
i
i Answel
‘ (2)
a —P
in in P+ Al ( (i)=
SAMPLE EXAMINATION I
T
(A) 13 (B) Be (D) 125
7
3
Z i 2+ 32. =243=5
5 5) °
_ = WF.
\edeegr'] sg ess)e Bos
Zz Answer
3
Por e
2?
12. oy six = aresin (%)+C
(B) 24-40
© Va-F4e
©) Fsin (Zs
Answer
raha congo eng ary part i ap daa 8 | CALCULUS BC
_-1, then the value of k is
jas a point of inflection 41 =
13. If the graph of f(x) = 2x" + >
©1
(a) -2 @ -1
'
Fon: 4x - kx
Fes de 2kxe 2 442k
x3
Fins 4-2k=0
k=2
Ase
D
14. If f(x) = cos" (2x), then f(x) =
set
hen oe evs hu
282k pence, — {p> SAMPLE EXAMINATION I | 9
T
15, Which of the following is an equation of the line tangent to the curve with parametric equations
y= 3e%y = 66" at the point where r= 0? Meteo > Wy) = (3.6)
Ss
jay 2e+y—12=0 eee ces
(B) 2x-y+12=0 da “3e*
(CQ. x-2y+9=0 dt
(D) 2x-y=0 m) = &=-2
=o
y-é 6>-2(x-3)
Yy- 6=-2x+6
Answer
2x Yas =O A
16 [=r = 2x74 2X+K41
2 +3x41 ae ee
2x0 HK)
(A) 2in aetlec (rad (2x41)
®) ner ec Aloxt) $BOH0 =|
. 2Ax+ A+ Bxt+ B=!
ZAtRh =0
ArB=!
pe eee +
At
tu 2_dx 22 i
- aa i
ee ck ea i
elnlxerteln exe tC ase |
é oo
10 | CALCULUS BC
>, then £2 at = equals
17, Ifx = 2sin rand y = cos*r, then 3
1
a -1 jo | © 0 (D)
dy. gt. Rogsnt « -sint
dx ak ost
Answer
18, The rate of change ofthe surface area ofa cube, A, with respect to time, tis directly
prpborggnal to the square root of one-sixth of the surface area. If at ¢ = 0 the surface area of th
cube is 24 and the rate of change of the surface area is 8, how fast is the surface area changing
when the side length of the cube is 4?
(B) 24 (© 32 (D) 48
2X = Ke
Az6x teat
dy
Ah « rand sKezd = key
dl
Aro): 24 ) dA 24x
oe dt
kod
age aa ad 3 Whur x=4 dha nsw
tie
aye 4 Br al pmgemin
Ql T SAMPLE EXAMINATION I |_11
Graph of f .
10, The graph of a piecewise linear function fis shown above. If g(x) = J £0 at, which of the
following is true?
av'ate4)=(-D gen | fu )dt=0
BS g(-3)=8@) geod: jnost
(2x)"
n=0
1 A SAMPLE EXAMINATION I |_17
T
29. The volume of the solid formed by revolving the region bounded by the graphs of y = 9 and
3)? about the line y = 9 is given by which of the followin;
“(92 -(x-3)4) de
mn
© af (x-3'-9) dr symnely
(0) 2 [0-37 a ‘
( (4-(x-3)4) ‘dx
°
Answer
30. Which of the following gives the total area of the region enclosed by the graph of the
polar curve r=1+cos 6?
A- 1 \r*do
2
(A) ff (1+ e050) 49 a
le j
) [+008 0240
[oreo A-1.2 J (406)? de
(© [0+ cos 9)240 23
(0) 30 +0080) 0
™%
fia SAMPLE EXAMINATION I | 19
T
31. Let fbe the function given by f(x) = tan.x and let g be the function given by g(x
value of xin the interval 0 SAMPLE EXAMINATION I | 21
T
35, Iffis differentiable and increasing on the interval [0,b] and c is the number guaranteed by the
Mean Value Theorem on this interval, then which statement must be true?
flb) F(b)>Flod
(A) PO="5 :
@) fo=0 Feey = Flb)-FIO) So
O) F>0 el
(D) f’(x) changes sign at x
36. Let R(x) be the radius of a round pipe that drains water from a dam, where x is measured in
30,000
feet from the dam. Which choice best explains the meaning of x ff (R(x)Pdx 2
hoo
(A) ‘The amount of water in squaré eet that the pipe can hold in the section from 10,000 to
30,000 feet from the dam.
(B) The amount of water in cubic feet that ghe pipe can hold in the section from 10,000 to
30,000 feet from the dam.
(©) The amount of water in cubic feet flowing through the pipe from 10,000 to 30,000 feet.
(D) The amount of water in cubic feet in any 20,000-foot section of the pipe.
Answer
B
nase conyngoreuing any pt oth page tgs 22_| CALCULUS BC
FO
7) =
|
s =4
3 3
ith f(x) < Oand f(x) <0,
ble ab Jected values of a function, f(x), with f
37. The table above gives selected values o}
If gc =F" (a), what isthe value of gD)?
1
awl 1
Is -4 (B) -5 Os 2
qo. sy,
j Fra) °
whore flad=15 a=3 mv
Fis)=-5 : [A
38. The first and second deri
If fay = 2, f(() =2, and f
°s of function f are positive for all Values of x in the interval (0,4)
1) = 2, which of the following could be the value of f(2)?
OF 3 @« fos
fr inceasin
» concave Up .
ew)
(1,2) fayzle2)
(2)%4
G,2) m=2 ft
tt fe mean oars Bak pnarmn SAMPLE EXAMINATION I | 23
——y=f's)
439, The figure above shows the graph of the derivative of a polynomial function f
How many points of inflection does the graph of fhave?
(A) One (B) Two (C) Three
The points of vel. exkroma. Por Pix) one POL Por fre D
Answer
5
f'(x)
0.998
0.980
0.999
0.995
1.000
1,000
1.001
0.995,
1,002 | 0.980.
\ Fix) inceesing > Feed concave vp
| Fen dee > feey concave down
40. The table above gives values of the derivative of a function f. Based on this information, it
appears that in the interval covered by the table
(A) fhas a
isincre:
it of inflection.
ing and concave down everywhere.
(C) fis increasing and concave up everywhere.
(D) fis decreasing and concave up everywhere.
Answer
A |
moran 6p eseng ay pt his age Hag oy ee av a E10 Boo peor
== ~“
uLUS BC —
24 | CALC in (x) intersects the gaph of y Si
ial for s!
» Maclaurin polyoma 7
41. The graph of the third-degree
at approximately tz=3
id
ee | {Th SAMPLE EXAMINATION I |
43. Irs) =e' Inx, then f"(e) =
(© e+e (Dy est
(a) tte (B)
Fox) = erlnx + C*
x
fle) = ene : e*('+S) = e*(%)
= 6% er!
44, A population increases according to the equation P(t) = 6000 — 5500e"'™ for 2 0,
25
Answer
8
measured in years. This population will approach a limiting value as time goes on. During
which year will the population reach half of this limiting value?
(A) Second PL) = 100 ( Ge iam)
(B) Third
Ve Ptb)= to0re0) = 6000
(D) Eighth toss ~ouset
3000 = 6000- S500 E-
pose
300a -
5500
(qas
t2in( 55) 38
0159
Answer
aur op sg any att ge eg ng sd i Ba 1 Bako pn
CALCULUS BC
a fi
fio) fo
=2
1 =1
3 I Eo.
jable function
—4, what is the value of f
i
45. For the twice diffe
f'eo.t [ foo de=
i
————T
jar (B) -3 © 3
ep ae?
(D) Ss
3
, 3
fle | - 2xFeo | +2%o| 3
1 1 ;
——<—__—J
f°)
-1
-2
ihe table above BiVeS select
5
ed values of (2), f"(x)ay
3
§ Fee dy =-4
(4Fi2) -Po) ) ~2( 3ft2)-fen) +2(Fe Fa)
4€8)-€2) -2(3¢-5)-C1)) + 2¢4)
— 2442+ 28-3
-S 28 | CALCULUS BC
.d for these problems.
Part Az Graphing calc
Section I
Person ka, (
/ s é 2} goott
wy’
y '
x?
L. As shown in the figure above a person whose eye level is 5 feet above the ground stands
oon the top of a hill overlooking a valley. The shape of the valley is modeled by the graph
of f(x) = 50 cos (95): ‘The person’s line of sight is tangent to the side of the hill at point 4
(2.50 0s( iba)}
(a) Write an equation of the tangent line in terms of the coordinates of point A.
(b) Find the value of a
(©) Can the person see the top of a 25-foot tall flagpole located at the lowest point of tht
valley? Justify your answer.
(a) Write an equation of the tangent line in terms of the coordinates of point A.
; .
F (xy= —50Sin( LY - 1 sin/x
Fro)=50 Foo= — 359 (ie) +25
+ (0,55) m=fla)=—1sin/ a
Fi = pail (3)
(4, 50cos/%,))
a
aes
(4 ,50005(%s0))
en dao 8 a Bad ensure, SAMPLE EXAMINATION L } 29
{7h
(b) Find the value of a.
(0/55) is also a point on the tangoot line
yGay+ 500s 760) ( (aloud
inf &
552-3 sin( 85
45.935
(c) Can the person see the top of a 25-foot tall flagpole located at the lowest point of the
valley? Justify your answer.
o_“ cos/. Ss) =+1
100.
5a
100
# xX=100T
y room) =-3 sin (Be ) (ot -4) + 50004
Sia aaa 643
Flioom)+25 = 804252 -25 <-4
) No He person com} see Hho
Flay pole <1 2a, the position of partic,
le ig =
is given by x(t) = si
(t) Sin tang iti
tH,
ime 0
2, Two parti move in the. plane, For time B
Dera id x = t,and the position of particle B
x(t) = cost and)
ided below, sketch the path of particles A
and Bla
direction of each particle along ;, *,
its
Ma
In the viewing
ths A and Bal indicate W"
Find the velocity Ve" ch particle.
5, which particle is moving faster to
which particle's upward speed is
(a)
(b)
i
{ ; @)
yw, sketch the path of particles A and
Bla
low provided belo’
e with arrows the directior
f ‘ion of each particle along it
Sat
the right? Justify your
pate ans
eater? Justify your ae
In the viewing wind
(a)
paths A and B and indicat
-1
A: xzcost
x mace
1 xX=sint
xX=siny
2
i
i
j
i
i
i
q {Ts SAMPLE EXAMINATION I | 31
T
(b) Find the velocity vector for each particle
Vy=<-sinty1>
Vag2
(©) Ats=5, which particle is moving faster to the right? Justify your answer.
L, Only look at He: honzondd
Component of Ha valcily.
—Sinl5)=, 0-954 | Ah tes parkcle Ais moving Paster i)
Cos(5)= O. 283
the nsht.
(4) Atr=5, which particle’s upward speed is greater? Justify your answer.
L. Only look at flo verbal
Cotpovent of He velocity }
For ary valve oft, Inde tes, fhe verbal comporuty
of We velocidy, [ror bole parkeles, 15 J a
bot. parkeles are maviy upward 4 sa
speed:
Unban eros ys i f' stds. sashen #2
f sey of te pale _
(a) Find epost nd vee Ne Pai the speed ereAsiNg OF decoy vu)
D isthe Pee wy
Gtr =3.what ee 7
La fry you sale changes eieation one Find the value of
© va os oat ies
ane pari is farthest ight and at Which,
‘Shey
cat tfor wil
@ he
locity of the particle when = 3.
(a) Find the position and
asi 154926 |
xeaye 1-364 SS2d8 2-1 15
6°
Vex: 7-8 +t?
V(s)e7-2449 216-942 EB ‘
@)
(b) Atr=3whatis
the speed of t
es the particle? Is the speed increasing or decreasing!“ XO)
Speed af Le is [F]
Alt). VG)=—grat
Al3)=-$46=-2 <0
“Te velociy epohve amo! dewey |
at t=3, Coamg are &
xt)
xl
——— ££ mwa SAMPLE EXAMINATION I _|_33
T
(c) Inthe interval (0,4) the particle changes direction once. Find the value of ¢ when this
change of direction occurs.
Vaye thebras (4-74= eo
+23 {t
E notin el
wt in inderval
a yO sued
Eagat 1 Leek ° xu)
For next part
(a) Whatis the value oft for which the particle is farthest right and at which i
Justify your answer.
ts4 teo tz
=H x=0 ¥210/, xl)
o
Xl0)= 70) =4(0)2+ is Sds-0 *
sore bo
X(1) = 71)- ayy § stds = 3+4 22
x(4) = 104) -4(4)*4 fod = 2g-G44 4647 -4Y,
The pa rhcle, is the fathest risht atta! and He
fuhot htt oh t24
on (4)
nmorze apg ating ya pane es i td
uh eno Bako penmanen,
44 Let R be the region 1°
the region to the Fi
Write an improt
which it converses:
Write an improp'
which it eae
(c) Ther
x-axis is @ aul
the value (if any)
(a)
)
(a) Write an improper integral that
the right
er integral
€8.
Ris the base oF
re, Write an impro]
) to which
which it converges.
ofx=t between the x-axis and the graph
fy.
cen horizontal line y= 1and the grapy
Oty.
right of x=! betw
per integral that gives the area of region R and find th,
aly
1 that gives the area of region S and find
1 Valu
4. For this solid every cross sectioy
per
f a solic
per integral that gives the volume of
oft
hi
it converges.
gives the area of region R and find th
1 Value
(b) {f~ SAMPLE EXAMINATION I_|_ 35.
T
(b) Write an improper integral that gives the area of region S and find the value (if any) to
which it converges.
peje 2) dx eh (rk sya (2+ 4], )
po (Hain) bebe
Dongs
(©) The region R is the base of a solid. For this solid every eross section perpendicular to the
x-axis is a square. Write an improper integral that gives the volume of this solid and find
the value (if any) to which it converges.
b
| V- ee he jae
S= fe
aed ‘Rt ed 0 0 ne aks panrounmn
aT
5. Consi
@
(b)
()
(d)
(@)
dy
ial equation gy iven differentia
tial eg field for the given differentia, Ua,
e differe®
der the dill i
ws provided, skete! :
on the axes PPO ;
points indicated.
X=
©
Ity(a)= 1, find lim 2—L. Show the work that leads to your answer,
yQ)= 1, find 1
Find the particular solution of the differential equation that satisfies the inital
y@=L
For what values of x will the particular solution, of the differential equation ing
have one or more
(i) horizontal asymptotes and/or
(ii) vertical asymptotes
Justify your answer,
On the axe i i
Siar inde sketch a slope field for the given differential equation at
(d
(ec abee)
7) <7 SAMPLE EXAMINATION I | 37
t
Ge
(b) Iya =I find lim 2 4 Show the work that leads to your answer.
i a
xy? , 2
(e_ Pol y=
ax xa 2
x-1
(©) Find the particular solution of the differential equation that satisfies the initial condition
zit => C2-3
G
S y2ey =(xdx
edb adres
(@) For what values of x will the particular solution of the differential equation in part (©)
have one or more
(i) horizontal asymptotes and/or sue
(ii) vertical asymptotes 4)}H.A: 4] =O oi i
Justify your answer. aaa
a) XPS: ey BUC 3) Vi)
LUnasharzes copying rng ay part of i page Hea. ‘The graph of f(x)
1
Gix)2 oo
s function defined on the closed interval [-2,3]. The graph of fens
6. Let f be a continuous
semicircle and a semi-ellipse, as shown above. Let G(x) = G (—2)+ f f(t) dt
(a) On what intervals, if any, is G concave down? Justify your answer.
(b) IF the equation of the line tangent to the graph of G(x) at the point where x =i:
y= mx+7, what is the value of m and the value of G(0)? Justify your answer.
(©) Ifthe average value of fon the interval 0 < x <3 is zero, find the value of G(3}
Show your work that leads to your answer,
(@) On what intervals, if any, is G concave down? Justify your answer.
Ecx) is concave down on (11.5), on this
inservel Gex)= (0x) fs dheveasing thare Fore
Ete =F lx) is nesahve. (7> SAMPLE EXAMINATION I | 39
T
ngent to the graph of G(x) at the point where x = Ois
(b) If the equation of the line ta
‘of mand the value of G(0) ? Justify your answer.
y = mx +7, what is the
Glo) tod 29 > [6cor=3|
m=Glo) = Ftoy= 2 => [MES
Ger) cond the tomngent line inderseet ak (oy
~m= 60) '
4) bG@)=7.
(c) Ifthe average value of fon the interval 0