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2008 AP Multiple Choice Exam
2008 APCalculus AB
MultipleChoice Exam
Section 1
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AP Calculus 2008 Multiple Choice Sotvts
t. l*fiffi* *-# =--t-(A) -3 @9 G) 2 (D) 3 (E) nonexistent
, Ii*= J x"J" = +' = -*r'(A) lnr'?+C (B) -lnx2+C (C) -r r+C
@-"' *, (E) lxa +C
AP Calculus 2008 Multiple Choice
3. If /(x)= (x-l)(x'z *2)' , then /'(x)=
(A) 6x(x'+z)'z
(B) 6r(x-1)(x'+z)'
(c) ('' +z)'(r' +3x-l)
r\()(l +z)'(tx'-6x+2)
(E) -3(x-r\(x'+z\'
(x.t) 3C x'+z)'.2( + (53-z)]' t
Cx+rrt[a$dx +*'ktl(xkd'( r*-6ntr')
+. J(sin(2r)+cos(zx))ar= - i coSaF+
tey |"o"(zr)+ lsin(z x)+ c
@i*'(zx)+jsin (zx)+c
(C) 2cos(2x)+ 2sin (2 x)+ C
(D) 2cos (2 x)- zsu(zx) + C
(E) -2cos(2x) + 2sin( 2x)+ C
t
2,9rw 2x{ c
AP Calculus 2008 Multiple Choice
5xo +8x2). llft----; lS
r+0 ]1' -l6X'
a'\ 1(v;Ls{kG)(g*-{r)
(B) o (c) 1 fot l+t (E) nonexistent"3
=+ r'\
f(x\={# if x*2
U if x=2
6. Let / be the function defined above. Which of the following statements about / are true?
I. ,f has a limit at x = 2. t'/
II. / is "ootiy'ous
at x=2.
lll. f is ditrerftiable at x=2.
f1e) oory
(B) II only
(C) trI only
(D) I and II only
(E) I, il, and III
(x\t(ttz)'-fr- 4' x+d= q h" *12'*.,
,// o Q,t>___r
7
AP Calculus 2008 Multiple Choice
7. A particle moves along the x-axis with velocity given by v(/) =3t2 + 6r for time r > 0. If the
particle is at position x = 2 at time f = 0, what is the position of the particle at t =l?
(A) 4 (c) e (D) 11 (E) 12
x(t) = L+PoC r"hcr^\-
X(r) = z+2-+
2
4
-
fttt'*, u )Jt
[1t* L+6t) Jb)O r
c+sf| '
+ l+3-fo+o)6
8. If /(x) = cos(3x), ** t'[;)= - 3 sih 3x luY^ 492.
-3 Srtn E --
il52
nlf, [.(A) "/t fB) f:22 (c) _Jt
2@) -;
AP Calculus 2008 Multiple Choice
9. The gaph of the piecewise linear firnction / is shoum in the figure above. Ifg(")=
I -t (r)Or, whichofthe following values is greatest? o* t*&rc
r$ g(-r) (B) g(-z) (c) s(o) @s(r) G) g(z)
crraphof/ 21'
AP Calculus 2008 Multiple Choice
v
R *rtr
10. The graph of function / is shown above for 0 <.r S 3. Of the following, which has the least
r,gt""?
( ) !:f V)e : S*aa( vlr^,6
(B) Left Riemann sum approximation of |.t 71r1a, with 4' 'C Jr- \ / subintervals of equal length
@}isn, Riemann sum approximation $ [:f @dx with 4 subintervals of equal length
(D) Midpoint Riemann sum approximation of J jf (r)ar with 4 subintervals of equal
length
(E) Trapezoidal sum approximation of Ii n.l* with 4 subintervals of equal length
craph ofl
AP Calculus 2008 Multiple Choice
('
Graph ofl 0- b
l/qp<-
11. The graph of a function ,f is shown above. Which of the following could be the graph "@the derivative of f ?
(A) v
AP Calculus 2008 Muftiple Choice
lz.lf f (x)= rl2r'l , then /'(x) =
(A) 2e{zt')lnx (B) e(2/4 (c) eF''}l ({$-2""'o (E) 1x2"(2t')\J x-
rl-- | = af' {' -zteY'0,r-- -2iz
rl.If ,f(x) =x'+2x, then ftVt^4)=
ffi'^'*' @) 2xlnx+2 (c) 2lnx+2 (D) 2tnx+? (E) 2'*2\-Zxxx
{tuo1 = (0^r)"*} hx
t'= z 0^,. (i) + ?
= }/-lx+z
AP Calculus 2008 Multiple Choice
table
iva
givenderivative
t2\t\f\t,t
in the
de^
14. The rylw$function "f has selected values of its second
above. Which ofthe following statements ry:t be tru,e
r(4 @) f isf€asneonthe interval (0, z). J 6t,, '
ct /lt, ,", f ,;ufrsingontheinterval (0,2). tS' qC//\\ (C) f $s a local maximum at x=1. ) ,r
(D) The graph of / has a point ofinflection at x = l({(n.Itel \n" graph of / changes concavity in the interval (0,
F*'r-j- *rqt
x 0 t 2 3
f"(r) 5 0 *7 4
,'I;rd.={ dJ" l^l*eu [+ c$< i'- q
u'; )xI)
fu*@i^,,-+l+c
@) 2hlx'z-+l+c
te) jarctan(;). t
AP Calculus 2008 Multiple Choice
16. Ir sin(rr )= *, th* *= bt(f?) (t Xt*5 ' t\ > t
v#2";fu
t-"or(r,y)cos(ry)
-ycos(.rry)xcos(r/)
tey Y(t-cos(tD
X J'*9 =
xJ=
!bs xJ
)-1bJry
(c) t- 9qrx1=) { firs:i
4
3
7
I
+ t-t*-,5uLS
17.The graphofthefunction / shownabovehashorizontaltangents at x= 2 and.r=5. Let Ibe the function defined bV s (r) = [ , t Oar. For what values of x does the graph of g
have a point of inflection?
r\) 4 only V and 5 only (D) 2,4, and 5 (E) 0,4, and 6(A) 2 only (B
AP Calculus 2008 Multiple Choice
18. In the .rry-plane, the line .r+ ! = k, where & is a constant, is tangent to the graph ofy=xz +3x+1. Whatisthevalue of k?
D, (B)-2 (c)-1
X+)z K --t
Xr Xt+3lc+t
(D) 0 (E) I
-X +\4 -'. SL"W = -l
,('t3 a -l /"
J
+ xra* X= -L
a* (-2,-t1
X+g.K + K:-2'\2?
{z (+lL+t(+) +l
5t -[
19. What are all horizontal asymptotes of the eraph of y = +1+ in the rry-plane?l- 2'
(A) y--1 only
(B) y-0 only
(c) y=5 only
(D) y=-1 and y=0
(r)r= -r and y = 5
horilo-Vv(
\,* jxtq)
\' 3,x?d)
l*"+|4)
5+)fruZl- d
-@(+)e-2
L- )*
-l
a
ry-b\- Y"
AP Calculus 2008 Multiple Choice
20.Let f be afunction with a second derivative given by f"(r)= t'(t-3)(r-6). What are
the .r-coordinates of the points of inflection of the graph of f ?
k./\.l,n4 Q" Ll\*"^'aa-l' srA'-'(A) 0 only
(B) 3 only
(C)0and6only
()3 and 6 only
(E) 0, 3, and 6
{u=r:* (p$ (r*l
+ . r tr- tr {"o3L,x€-\ I x-t \** 1 x- i
4,,, +-- \*-- l.*-[ +**' d '7rL
o
2l- Aparticle moves along a straight line. The graph of the particle's position x(t) attime t isshownabovefor 0<t<6. Thegraphhashorizontaltangents at tl and l=5 andapointofinflection at t = 2. For what values of t is the yglocity of the patirG i"cteffi'gZ
(),D <t <2
(B) lcr<5
(c) 2<t <6
(D) 3</<5 only
(E)l<t<2and5<r<6
\ 'tnC(€..o.t* uAl'^^ q > O Ccr"' ca^t" VP
\ ={/ S, C@ '?)
Rv vr..rr
AP Calculus 2008 Multiple Choice
22. A rumor spreads among a population of l[ people at rrafc proportional to the prodUglo.Jthe
number olfueop$who have heard the rumor and the numberglpeople who have not heard
the rumor. If p denotes the number ofpeople who have heard the ntmoq which of the
following differential equations could be used to model this situation with respect to time /,
where ft is a positive constant?
(A) ,(bd,r
@=b(N-p\G) #=:elb-N)
@) #=nt(x-t)
@) #=HU.N)
t(\,-f) A.,^Jh,r,rN|'rt v>1hncc*f,r'rr-
fl,orc uh, I^..**l lL ' w'r'"'
Yv w'ur
@=ndt
AP Calculus 2008 Multiple Choice
23. Which ofthe following
condition y(3)=-Zt
is the solution to the differential equation d! =x' with the initialdxv
[ ,^ trq = ft'4,JJ J )
j.\t- lt'+ cz l- 3
gti z{+ c
t\* (r,-z)
y=2e-*it3
,tr=-)"4+x3t3
/*E(D)y =E;
@,=-p-*
\,a
r .trP-
2,\
czqt L -(rl*f33'. z t' -vr. 3(-z) t='
,u * c
\'= if- rv + {?z c -
|' -F''v '?@o{ Y(:) =-a ?
24.The function / is twice differentiable with f (2)=I, f '(2) = 4, and f "(2)= 3. What is the
value ofthe upp-ffiffi using the line tangent to the graph of f at x = 2?
(A) 0.4 @ou (c) 0.7 (D) 1.3 (E) 1.4
f {x1= ('(x) (X-x) t-\,7
a4 (z , r) )x1z Y,, t ('b\= 4
f [r.$' q (l'ot-z)+ I
ro.ttl)tl= ,G
AP Calculus 2008 Multiple Choice
(cx+d for x<2f(*)=lr,_"* for x>2
25.Let / be the firnction defined above, where c and d are constants. If / is differentiable at
x=2, wltat is the value of c+ d? = 2 -{ = -%(A) -4 @-, (c) o (D) 2
t $, /, 1L: c (a) +,tl,jhn c--t I
4-20)=L6+dl tt x) 1 " lz-c(z)
024+d I + 4-zCzLc+adt'tl f ' o"I \zL whr'* Y'|L" {'= c . ht
4,'-.
(E) 4
f i s e"FC ir^^g|'la -c/rvhnd! (i" h'/^)
M lL fo*€tr(!('
X)2 i(t=LX-C --' 4+,'. C=q-C=} )C--1 , c2
26. What isthe sryOf the line tansent to the curve y = arctan(4x) at the point at which
t=L? - .-l'=j v
o/ .1 -l{@))t = a'=z(D) -;@ ."); (c)o (E) I
AP Calculus 2(X)8 Multiple Choice
tlllI lltI I l\t t t \-\ \ \\
vt | | | |
, | | | |
t t t | |
, t t | |
ttt,l-+x)
I I | / -llll| | tlffrlftt,
slope field for which ofthe following differential equations?
\\\llttt I I
rllllllllllltll
27. Shown above is a
@) 1='Ydx
Bl Q=xv-dx
@=ry+v
P) #=ry+x
@) X=(x+1)3
AP Calculus 2008 Multiple Choice
28. Let f b adifferentiable tunction such that,f(g) = 15, /(6) =?,
"f'(3) = -8, and
f '(6)= -2. The function g is differentiable and g(")= f-t (,) for all x ' What is the value of
s'(3)?4@;(B) -;
(c) *
(D) *(E) The value of g,(3) cannot be determined from the information given.
q'(b) = L
' f'(o)
1'G)= I -- l'
J . -. 4,(d _z
wl,rr^e IG)= IF.','Svt4vh
AP Calculus 2008 Multiple Choice
2008 APCalculus AB
MultipleChoice Exam
Section 2
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AP Calculus 2008 Multiple Choice
76-Thegfaph of .f', thederivative f , rsshownabove for -2<x55. Onwhatintervalsis /
Ttr C veox,zg b'Irr,^L F'> O
(-2, t) n (t r 3)
Gz-t)
(A) [-2,1] only
P1(nl-2, 11
(C) [3,5] onlY
(D) [0,1.5] and [3, 5]
(E) [-2, -l], [, 2], and [4, 5]
AP Calculus 2008 MultiPle Choice
\r". t D\r6
xiz
34
Graph of/
77. Thefigure above shows the graph ofa function / with domain 0< x < 4' Which ofthe
following statements are ffue?
I. lim_/'(x) exists.x+T
II. lim /(r)
m hq/(x)
(A 'I only
,o"t"./
"*i"tr. /
exists. U0('
(B) II only @b".II
only
78' The frst derivative of the function / is defined by 'f '(") = sin("'-") ror 0 S 'r < 2' on
what interva(s) is / increasing? { w\C Uher. (' > O
(A) 1< x<1.445
(9/1< x s 1.6e1
(C)1.445<x<1.875
(D) 0.577 <x<1.445 and 1.875 <x32
(E) 0<x31 and 1.691<x<2
{,=o srw(a3-) =o
6o\ c.rta}'" C\aV+ (-t,l'tet )
(D) I and III only (E) I, n, and III
AP Calculus 2@8 MultiPle Choice
7s. rf I'-,f @)* = -r7 ^-i,: ilqdx = A,what is the value ot [' .t {4 a'z
(A) -21 6-t, (c) o (D) 13 (E) 2l\-/ t:t
-i ,li{ ..,,^}&\ \.*} \,u*d'unsl
v-w
80. The derivative of the function / is given Uy ,f '(t)= "' "os(tt ). How many points of
t"n"qr."A.es the graph of / have on the open interval (-2,2)?
(A) One (B) Two (C) Three @) Fotr Qfive
6-tpI {" 'x2 G sw*)' zx + us* '2{
t'v Cz,z\
huo uvrae v +' cl'v'"lh nW 4
AP Calculus 2008 Multiple Choice
81. If G(x) is an antiderivative for /(x) and G(2)=-7, then G(4) =
(A) ,r'(4)
(B) -?+ f'(4\
1c1 [iy@at
tol J,'(-z+ f (t))at
G(ql - GCD 3
4G{q -Ct)=\ f t,t? ("[{) =4 *
f1,., u
ltrru
.t *[irtt)a,
32.Aparticlemovesalongastraightlinewithvelocitygivenby v(r)=Z-(t.Ot1-' attime
t > 0. What is the qggglerajlgofthe particle at time r = l!/\
(A)-0.e14 (B) 0.055 (c) 5.486 (D) 6.086\-/
Cotc^z\4,1r., .o6ss
(E) 18.087
Lv(x)\ =d{ , tr3
AP Calculus 2008 Multiple Choice
83. What is the area enclosed by the curves lf xt -Bx2 +18'x-5 ^6 !.2= x+5?
(A) 10.667
@rr.s:r(c) 14.583
(D) 21.333
(E) 32
Yr-- tf, nU {{*- .\k/, sect\tut
l,= ftri'l.)L *
fr'l,r &A. Jt
-1,033
84. The graph of the derivative of a function / is shown in the figure above' The graph has
horizontal tangent lines at x = -1, x = lo and .r = 3. At which of the following values of x
does ;f have arytative maxg4ttr?
(A) -2 only
(B) I only-.\(y't ontv
(D) -l and 3 only
(E) -2,1, and 4
AP Calculus 2008 MultiPle Choice
g5. The table above gives values of a function / and its derivative at selected values of x. If f '
is continuous on the interval [-4, - U, what is the value of l-', t'Q)axt
(A) -4.5 ["[t.tt (c) 0 (D) 2.2s (E) 4.s
t.r
-l.s
-7"? ? -Z'lq'- -Q/,
x 4 -3 1 -lf (*) 0.75 -1.5 -2.25 -t.5
f '(r) -3 -1.5 0 1.5
-L -t
-l (r) bt +1(r) t's
AP Calculus 2008 MultiPle Choice
position. {(t), of the particle for 0 < t < 42
,'' .+ '' -a, -
-
z.t86. The table gives selected values ofthe velocity, v(r), of a particle moving along the x-axis.
At time 1= e the particle is at the origin. Which of the \owing
could be the graph of the
v
'(r)(B)
hu'it"-w6L fzr
t
+'(r)(E) v(r)=z
t 0 I 2 3 4
,(r) -l 2 3 0 4
(A) '(r)
(D) '(,)
AP Calculus 2008 MultiPle Choice
89.Thefunction / is_continuous.for -2<x<2 and f (-Z)=f (2)=0.ffthettitno t'-Y!g"
stu$r q.rf x=3 'ts )
a^^l {" <o CsY'Caloa
dI;"-'
('P
x f (')
2 2.5
3 5
4 6.5
\?"'(6t,.ts
I
e[r'nn".f
do'utll"\'
ts, D/ [ (,
x f (r)
2 3
3 5
4 7
E{--I
knd.'rL, Nb+ C, sr' 'c
* -?. ".2, frt dfoh f '(r)=0, which of the following statements must be true?
(A)For -2<k<2, f'(k)>0. _-_l_(B)For -2<k<2, f'(k)<O- -L L
R'llot $osw, sr(C) For -2<k <2, f '(k) exists. juavonVet t'=O . . .
(D)For -2<k<2, f'(klexists,but /'isnotcontinuous' Lt J'ltt"t is |}o wh']/l-
r\. r \*l
(DF*some k, where -2<k<2, f'(k\ doesnotexist. i-rol,lriho+ cs*+ u".h^'1 ^ -'"'l';7(utt*^tut' f til DIB
90. The function / is continuous on the closed interval [2, 4] andtwice differentiable on the
openinterval (2,4).If /'(3) =2 and,f"(r)<0 ontheopeninterval (2'4), whichofthe
9.5-2 "lsL
x f (*)
2 3
3 5
4 6.5
rn^ (2,u) a0-ylru l'fu'
.ft ,Ot*+
$rwa l't^^ sl'qF'@W daruV'aaY
o""l
\k tq-*l ctv (wttv'
f"z-z.{
Itst-fifu) =!'2-t-z url
t** Avillt^"
AP Calculus 2008 MultiPle Choice
g7. An object traveling in a straight line has position x(r) at time f. If the initial position is
x(0) = 2 andthe velocity of the object is v(r) - 1Jl+ i' , what is the position of the object at
tirne t =3?
(A) 0.43r (B) 2.1s4 (c) 4.sr2 @))6.tt, (E) 17.408
('\(a) '- L+ \ ittl &; }{-t.sl -- c'st z
gg. The radius of a sphere is decreasingat anteof 2 centimeters per second. At the instant when
the radius of the sphere is 3 centimeters, what is the rate of change, in square centimeters per
second, of the **ib"" area of the sphere? (The surface area S of a sphere with radius r is
S --4xr'\ fr
(A) -1082 (B) -72n s)+sz (D) -2at (E) -t6n
b - -, $= unrz&- A --wr halv=\ dk d&
z Brr(.;) Gz\
---49dfis .7
1--"
AP Calculus 2008 MultiPle Choice
91. What is the average value of y =;ffiuon the closed interval [-1, 3J?
(A) -0.08s (B) 0.0e0 @.t*, (D) 0.244 (E) 0.732
t. I e"'n, )* =! f EI- ]r= ,lB33--t ) x4+x+z-' +)rkx+z
C cul
<-a miles --+l
I7 miles
I92. A city located beside a river has a rectangular boundary as shown in the figure above. The
papulation densiqr of the city at anypoint along a strip x miles from the river's edge is
r@e.Whichofthefollowingexpressionsgivesthepopulationofthe city?
(A) I;/(x)dx
(rflt1irt'ta.
Q 281:f g)dx
@) !1 f V)dx
@ 4l:f @)dx
0q
fr,,l a ?e*nnt *- b^i?
Ar-.i \: h',ur^; m;
rw9.\ g-1*v l'
' \ rniS '/'o niL
hc;rrhf,4| -i lr, [wt^,-#
, 1w\ = rni2
f ?"o
J r,n'
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