1973 AP Calculus AB: Section I

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1 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = and ( ), g= thn ( ( )) g f = Th slop of th lin tangnt to th graph of y ln ( ) = at = is. If f ( ) = + sin, thn f ( ) = + cos cos cos sin cos sin + cos. If f ( ) =, which of th following lins is an asymptot to th graph of f? y = = y = y = y = 6. If f( ) = for all, thn f () = + AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

2 97 AP Calculus AB: Sction I 7. Which of th following quations has a graph that is symmtric with rspct to th origin? + y = = + y = ( + ) y ( ) y = + y = A particl movs in a straight lin with vlocity tims t = and t =? vt () = t. How far dos th particl mov btwn If y = cos, thn dy d = 6sincos cos cos 6cos sincos. Th drivativ of f( ) = attains its maimum valu at =. If th lin y = is tangnt in th first quadrant to th curv y = + k, thn k is 8. If f( ) = + A + B and if f() = and f( ) = 7, what is th valu of A+ B? 6 It cannot b dtrmind from th information givn. AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

3 97 AP Calculus AB: Sction I. Th acclration α of a body moving in a straight lin is givn in trms of tim t by α= 8 6t. If th vlocity of th body is at t = and if st ( ) is th distanc of th body from th origin at tim t, what is s() s()? 8. If f( ) ( ) = for all, thn th domain of f is { } { > } { } { and } { is a ral numbr}. Th ara of th rgion boundd by th lins =, =, and y = and th curv y = is ( ) 6. Th numbr of bactria in a cultur is growing at a rat of pr unit of tim t. At t =, th numbr of bactria prsnt was 7,. Find th numbr prsnt at t =. t,, 7, 7,, What is th ara of th rgion compltly boundd by th curv y =? y = and th lin d d 8. ( arcsin ) = AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

4 97 AP Calculus AB: Sction I 9. Suppos that f is a function that is dfind for all ral numbrs. Which of th following conditions assurs that f has an invrs function? Th function f is priodic. Th graph of f is symmtric with rspct to th y-ais. Th graph of f is concav up. Th function f is a strictly incrasing function. Th function f is continuous.. If F and f ar continuous functions such that F ( ) = f( ) for all, thn f ( d ) is a F ( a) F ( b) F ( b) F ( a) Fa ( ) Fb ( ) Fb ( ) Fa ( ) non of th abov b. + ( + ) d=. Givn th function dfind by concav up. f ( ) =, find all valus of for which th graph of f is > < < or > < < or > > < < AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

5 97 AP Calculus AB: Sction I. + h lim ln h is h nonistnt. Lt f ( ) cos( arctan) =. What is th rang of f? < < { < < } { } { < } { }. tan d = + 6. Th radius r of a sphr is incrasing at th uniform rat of. inchs pr scond. At th instant whn th surfac ara S bcoms squar inchs, what is th rat of incras, in cubic inchs pr scond, in th volum V? S = r and V = r. 7. d = ln A point movs in a straight lin so that its distanc at tim t from a fid point of th lin is 8t t. What is th total distanc covrd by th point btwn t = and t =? AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

6 97 AP Calculus AB: Sction I 9. Lt f( ) = sin. Th maimum valu attaind by f is. d = ln ln ln + a log =, thn a = a. If a ( ) d = ( + ) + C arctan C ln ( ) + ( ) + + C ln + + C + C. Suppos that f is an odd function; i.., f ( ) f( ) Which of th following must ncssarily b qual to f ( )? f ( ) ( ) f = for all. Suppos that ( ) f ists. f ( ) f ( ) Non of th abov AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

7 . Th avrag valu of ovr th intrval is 97 AP Calculus AB: Sction I. Th rgion in th first quadrant boundd by th graph of y = sc, =, and th as is rotatd about th -ais. What is th volum of th solid gnratd? 8 n 6. If y =, thn n d y d n = n n n! n n n n n n n! dy 7. If = y and if y = whn =, thn y = d c 8. If f ( c) d= whr c is a constant, thn ( ) f d = c + c c c 9. Th point on th curv y = narst to ( ), is (, ) (, ) (, ) (, ) (,8 ). If tan( y) =, thn dy d = ytan( y)sc( y) tan( y)sc( y) sc ( y) y cos ( y ) cos ( y) cos ( y) y AP Calculus Multipl-Choic Qustion Collction 6 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

8 . Givn + for <, f( ) = cos for, f ( d ) 97 AP Calculus AB: Sction I = + +. Calculat th approimat ara of th shadd rgion in th figur by th trapzoidal rul, using divisions at = and = If th solutions of f( ) = ar and, thn th solutions of f = ar and and and and and. For small valus of h, th function 6 + h is bst approimatd by which of th following? h + h + h h h AP Calculus Multipl-Choic Qustion Collction 7 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

9 97 AP Calculus AB: Sction I. If f is a continuous function on [ ab, ], which of th following is ncssarily tru? f ists on ( ab., ) If f ( ) is a maimum of f, thn ( ) lim f ( ) = f lim for ( a, b) f ( ) = for som a, b [ ] f =. Th graph of f is a straight lin. AP Calculus Multipl-Choic Qustion Collction 8 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

10 97 Answr Ky 97 AB 97 BC. E. E. B. A. A 6. D 7. B 8. B 9. A. C. B. C. D. D. C 6. C 7. C 8. D 9. D. D. B. B. C. B. B 6. E 7. E 8. C 9. C. B. D. D. A. C. C 6. A 7. A 8. B 9. B. E. D. D. E. B. C. A. D. A. C. B 6. D 7. D 8. B 9. A. A. E. D. D. A. C 6. A 7. C 8. D 9. D. E. B. C. C. A. B 6. D 7. E 8. C 9. A. B. E. C. A. C. C 6. E 7. E 8. B 9. D. C. D. D. E. A. E AP Calculus Multipl-Choic Qustion Collction Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

11 97 Calculus AB Solutions. E ( d ) = + C. E g g( f ) ( ) = ( ) =. B y = ln ; y = =. At =, y =.. A f ( ) = + sin ; f ( ) = cos. A lim = y = is a horizontal asymptot 6. D ()( + ) ( )() f ( ) =, f () = = ( + ) 7. B Rplac with ( ) and s if th rsult is th opposit of th original. This is tru for B. ( ) + ( ) = = ( + ). 8. B t d t dt t 7 Distanc = = = = ( ) = d d y = 6sincos 9. A y = cos ( cos) = cos ( sin ) ( ) = cos ( sin ) ( ) f > for < and f < for > f has its maimum at =.. C f ( ) = ; f ( ) = ; f ( ) = = ( ). B Curv and lin hav th sam slop whn = =. Using th lin, th point of tangncy is,. Sinc th point is also on th curv, = + k k = C Substitut th points into th quation and solv th rsulting linar systm. = 6 + A+ B and 7 = 6 + A B ; A=, B= A+ B=. d d AP Calculus Multipl-Choic Qustion Collction 7 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

12 97 Calculus AB Solutions. D vt ( ) = 8t t + C and v() = C= so () s() s() = v t dt = (t t + t) = vt () = 8t t +.. D f( ) = ( ) f = + = f is not dfind at = and at =. ( ) ( ) ( ) ( ) ( ). C Ara = d= = ( ) 6. C t t dn =, N = 7 + C and N() = 7 C = dt t ( ) N = 7, N = 7 7. C Dtrmin whr th curvs intrsct = = ( )( + ) = =, =. Btwn ths two valus th parabola lis abov th lin y =. 9 Ara = (( + + 6) ) d= + + = d d arcsin = ( ) = = d d 8. D ( ) ( ) ( ) 9. D If f is strictly incrasing thn it must b on to on and thrfor hav an invrs.. D By th Fundamntal Thorm of Calculus, f ( d ) = Fb ( ) Fa ( ) whr F ( ) = f( ).. B ( ) d (( ) d) ( ) ( ) b a = + = = =. B ( ) f( ) = ; f ( ) = 6 ; f ( ) = 6 = 6 Th graph of f is concav up whr f > : f > for > and for < <. AP Calculus Multipl-Choic Qustion Collction 7 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

13 97 Calculus AB Solutions. C h ( h) ln + ln lim whr ln h = = = f ( ) f ( ) = ; f ( ) f ( ) f( ) cos arctan ; = < arctan < and th cosin in this domain taks on all valus in th intrval (,].. B ( ). B tan (sc ) (tan ) d d = = = dv = dr = dr =. = dt dt dt 6. E r S ( ) 7. E d ( ) ( d) ( ) 8. C v() t 8 6 = = = = t changs sign at t =. Distanc = () + () =. Altrnativ Solution: Distanc = v() t dt = 8 6t dt = 9. C sin sin ; Th maimum for sin is.. B d d ( ) ( ) a. D a( ) = = ln + = ln+ ln+ = ln a log = loga = = a ; a= 6. D d = d = tan ( ) + C + +. A f ( ) = f( ) f ( ) ( ) = f ( ) f ( ) = f ( ) thus f ( ) = f ( ). AP Calculus Multipl-Choic Qustion Collction 7 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

14 . C. C Washrs: d= = = r whr r = y = sc. d Volum sc tan (tan = = = tan ) = 97 Calculus AB Solutions 6. A n n n ( n ) n n y =, y = n, y = n,, y = n dy 7. A y, y() d = =. This is ponntial growth. Th gnral solution is y = C. Sinc y () =, C = and so th solution is y =. 8. B Lt z c c c =. Thn = ( ) = ( ) f c d f z dz 9. B Us th distanc formula to dtrmin th distanc, L, from any point (, y) = (, ) on th curv to th point (,). Th distanc L satisfis th quation ( ) L = +. Dtrmin whr L is a maimum by amining critical points. Diffrntiating with rspct dl to, L = ( ) + = 8. changs sign from positiv to ngativ at d d = only. Th point on th curv has coordinats (, ).. E sc ( y) ( y + y) =, y sc ( y) + y sc ( y) =, ysc ( y) cos ( y) y y = = sc ( y). D f ( d ) = ( + ) d+ cos( d ) = ( + ) + sin( ) = + ( sin sin ) = AP Calculus Multipl-Choic Qustion Collction 7 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

15 97 Calculus AB Solutions. D 7 = ; T = =. E Solv = and = ; =,. B Us th linarization of f( ) = at = 6. f ( ) =, f (6) = h L ( ) = + ( 6); f(6 + h) L(6 + h) = +. C This uss th dfinition of continuity of f at =. AP Calculus Multipl-Choic Qustion Collction 76 Copyright by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6