2008 name 2008 calculus ab -...

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


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


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@) -;


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'


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


('

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


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


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


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"


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


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


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


(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


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


2008 APCalculus AB

MultipleChoice Exam

Section 2

Calculator Active

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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


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


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


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


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) '(,)


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^"


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--"


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

[email protected] 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'

2008 name 2008 calculus ab -...

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