|
|
|
- Lee Watts
- 7 years ago
- Views:
Transcription
1
2
3
4
5
6
7 969 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 ).. Which of th following dfins a function f for which f ( ) f( )? f ( ) f ( ) sin f ( ) cos f ( ) log f ( ). ln if and only if. If 5 7 f( ), for, f() k and if f is continuous at, thn k d 6 5. If y y, thn th valu of dy d at is not dfind AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
8 969 AP Calculus AB: Sction I 6. What is h h 8 lim? h Th limit dos not ist. It cannot b dtrmind from th information givn. 7. For what valu of k will k hav a rlativ maimum at? Non of ths and if th rmaindr is whn p( ) is dividd by, thn k 8. If p( ) k 6 9. Whn th ara in squar units of an panding circl is incrasing twic as fast as its radius in linar units, th radius is t. Th st of all points (, t ), whr t is a ral numbr, is th graph of y ln ln. Th point on th curv y that is narst th point, occurs whr y is non of th abov AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
9 969 AP Calculus AB: Sction I. If f( ) and g ( ), thn th solution st of f g ( ) g f( ) is,,. Th rgion boundd by th -ais and th part of th graph of y cos btwn and is sparatd into two rgions by th lin k. If th ara of th rgion for k is thr tims th ara of th rgion for k, thn k = arcsin arcsin 6. If th function f is dfind by f ( ) 5 f ( ), thn f, th invrs function of f, is dfind by If f ( ) and g ( ) ist and f ( ) g( ) for all ral, thn th graph of y f( ) and th graph of y g( ) intrsct actly onc. intrsct no mor than onc. do not intrsct. could intrsct mor than onc. hav a common tangnt at ach point of intrsction. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
10 969 AP Calculus AB: Sction I 6. If y is a function of such that y for all and y for all, which of th following could b part of th graph of y f( )? 7. Th graph of 5 y 5 has a point of inflction at (,) only (,6) only (,56) only (,) and (,6) (,) and (, 56) 8. If f( ) for all, thn th valu of th drivativ f ( ) at is nonistnt 9. A point movs on th -ais in such a way that its vlocity at tim t t is givn by At what valu of t dos v attain its maimum? v ln t. t Thr is no maimum valu for v. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
11 . An quation for a tangnt to th graph of arcsin y at th origin is 969 AP Calculus AB: Sction I y y y y. At, which of th following is tru of th function f dfind by f ( )? f is incrasing. f is dcrasing. f is discontinuous. f has a rlativ minimum. f has a rlativ maimum. d. ln d. Th ara of th rgion boundd by th curv qual to y, th -ais, th y-ais, and th lin is y. If sin,, what is dy d in trms of? tan cot cot tan csc AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
12 969 AP Calculus AB: Sction I 5. A rgion in th plan is boundd by th graph of m, m. Th ara of this rgion y, th -ais, th lin m, and th lin is indpndnt of m. incrass as m incrass. dcrass as m incrass. dcrass as m incrass whn m ; incrass as m incrass whn m. incrass as m incrass whn m ; dcrass as m incrass whn m. 6. d is non of th abov dy 7. If tan, thn y d tan C sc C ln sc C ln cos C sc tan C 8. Th function dfind by f ( ) cos sin has an amplitud of AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
13 969 AP Calculus AB: Sction I 9. cos d sin ln ln ln ln ln. If a function f is continuous for all and if f has a rlativ maimum at (, ) and a rlativ minimum at (, ), which of th following statmnts must b tru? Th graph of f has a point of inflction somwhr btwn and. f ( ) Th graph of f has a horizontal asymptot. Th graph of f has a horizontal tangnt lin at. Th graph of f intrscts both as.. If f ( ) f( ) and f (), thn f ( ). If abcd,,,, and ar ral numbrs and a, thn th polynomial quation 7 5 a b c d has only on ral root. at last on ral root. an odd numbr of nonral roots. no ral roots. no positiv ral roots.. What is th avrag (man) valu of t t ovr th intrval t? AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
14 969 AP Calculus AB: Sction I. Which of th following is an quation of a curv that intrscts at right angls vry curv of th family y k (whr k taks all ral valus)? y y y y y ln 5. At t a particl starts at rst and movs along a lin in such a way that at tim t its acclration is t ft pr scond pr scond. Through how many ft dos th particl mov during th first sconds? Th approimat valu of y sin at., obtaind from th tangnt to th graph at, is Which is th bst of th following polynomial approimations to cos nar? 8. d ln C ln C C C C 9. If y tan u, u v, and v ln, what is th valu of dy v d at? sc AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
15 n n. If n is a non-ngativ intgr, thn 969 AP Calculus AB: Sction I d d for no n n vn, only n odd, only nonzro n, only all n. If f ( ) 8 for, f( ) lswhr, thn f ( d ) is a numbr btwn and 8 8 and 6 6 and and and. What ar all valus of k for which th graph of -intrcpts? y k will hav thr distinct All k All k k, k All k. sin d cos C cos C cos cos cos C 5 C C. Th fundamntal priod of th function dfind by f( ) cos is 5 6 d d d g ( ) f ( ), d 5. If f ( ) g( ) and d thn f( ) 6 f g g 9 f 6 6g f 6 g d AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
16 969 AP Calculus BC: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. t Th asymptots of th graph of th paramtric quations, y t t, y only, y only, y. What ar th coordinats of th inflction point on th graph of y ( )arctan?,,,,,. Th Man Valu Thorm guarants th istnc of a spcial point on th graph of y btwn, and,. What ar th coordinats of this point?,,,, Non of th abov. 8 d 6 5. If y y, thn th valu of dy d at is not dfind AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
17 6. What is h h 8 lim? h It cannot b dtrmind from th information givn. 969 AP Calculus BC: Sction I Th limit dos not ist. 7. For what valu of k will k hav a rlativ maimum at? Non of ths 8. If h ( ) f ( ) g ( ), f( ) g ( ), and g( ) f( ), thn h( ) f ( g ) ( ) g ( ) f( ) g ( ) f( ) 9. Th ara of th closd rgion boundd by th polar graph of r cos is givn by th intgral cosd cosd cosd cos d cosd. d ln ln AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
18 969 AP Calculus BC: Sction I. Th point on th curv non of th abov y that is narst th point, occurs whr y is. If t F( ) dt, thn F( ). Th rgion boundd by th -ais and th part of th graph of y cos btwn and is sparatd into two rgions by th lin k. If th ara of th rgion for k is thr tims th ara of th rgion for k, thn k arcsin arcsin 6. If and, thn dy y u du 6 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
19 969 AP Calculus BC: Sction I 5. If f ( ) and g ( ) ist and f ( ) g( ) for all ral, thn th graph of y f( ) and th graph of y g( ) intrsct actly onc. intrsct no mor than onc. do not intrsct. could intrsct mor than onc. hav a common tangnt at ach point of intrsction. 6. If y is a function such that y for all and y for all, which of th following could b part of th graph of y f( )? 7. Th graph of 5 y 5 has a point of inflction at, only,6 only, and,6, and, 56, 56 only 8. If f( ) for all, thn th valu of th drivativ f ( ) at is nonistnt AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
20 969 AP Calculus BC: Sction I 9. A point movs on th -ais in such a way that its vlocity at tim t t is givn by At what valu of t dos v attain its maimum? ln t v. t Thr is no maimum valu for v.. An quation for a tangnt to th graph of arcsin y at th origin is y y y y. At, which of th following is tru of th function f dfind by f ( )? f is incrasing. f is dcrasing. f is discontinuous. f has a rlativ minimum. f has a rlativ maimum.. If f ( ) dt, which of th following is FALSE? t f () f is continuous at for all. f () f () f ( ) AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
21 . If th graph of y f( ) + contains th point 969 AP Calculus BC: Sction I dy,, and f( ) for all, thn f ( ) d y + + y. If sin,, what is dy in trms of? d tan cot cot tan csc 5. A rgion in th plan is boundd by th graph of m, m. Th ara of this rgion y, th -ais, th lin m, and th lin is indpndnt of m. incrass as m incrass. dcrass as m incrass. dcrass as m incrass whn m ; incrass as m incrass whn m. incrass as m incrass whn m ; dcrass as m incrass whn m. 6. d is non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
22 969 AP Calculus BC: Sction I dy 7. If tan, thn y d tan C sc C ln sc C ln cos C sc tan C 8. What is lim tan? Th limit dos not ist. 9. d. ( ) is th Taylor sris about zro for which of th following functions? n! n n n sin cos ln( ). If f ( ) f( ) and f (), thn f ( ). For what valus of dos th sris n convrg? No valus of All valus of. What is th avrag (man) valu of t t ovr th intrval t? AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
23 969 AP Calculus BC: Sction I. Which of th following is an quation of a curv that intrscts at right angls vry curv of th family y k (whr k taks all ral valus)? y y y y y ln 5. At t a particl starts at rst and movs along a lin in such a way that at tim t its acclration is t ft pr scond pr scond. Through how many ft dos th particl mov during th first sconds? Th approimat valu of y sin at., obtaind from th tangnt to th graph at, is Of th following choics of, which is th largst that could b usd succssfully with an arbitrary in an psilon-dlta proof of lim 5? 5 8. If ( f ) ( ), thn f () ln(8 ) ln(8 ) ln() 8 9. If y tan u, u v, and v ln, what is th valu of dy v d at? sc AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
24 n n. If n is a non-ngativ intgr, thn 969 AP Calculus BC: Sction I d d for no n n vn, only n odd, only nonzro n, only all n. If f ( ) 8 for, f( ) lswhr, thn f ( d ) is a numbr btwn and 8 8 and 6 6 and and and. If cos d f( ) sin d, thn f ( ) sin cos C sin C cos sin C cos sin C cossin C. Which of th following intgrals givs th lngth of th graph of y tan btwn a and b, whr ab? b a b a b a b a b a tan tan d sc tan sc d d d d AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
25 . If f( ) f( ) f( ), f(), and f (), thn f () 969 AP Calculus BC: Sction I C) 5. Th complt intrval of convrgnc of th sris k is k k AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
26 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 ( ) 5 and ( ) 5, 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 5. 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 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
27 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 5 y y 6 8. 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 5. Th drivativ of f( ) attains its maimum valu at 5 5. If th lin y is tangnt in th first quadrant to th curv y k, thn k is 8. If f( ) A B5 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 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
28 97 AP Calculus AB: Sction I. Th acclration of a body moving in a straight lin is givn in trms of tim t by 8 6t. If th vlocity of th body is 5 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 5. 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 5 pr unit of tim t. At t, th numbr of bactria prsnt was 7,5. Find th numbr prsnt at t 5. t,, 7,5 5 7,5 5, What is th ara of th rgion compltly boundd by th curv y? y 6 and th lin d d 8. arcsin AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
29 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. 5 f ( ), find all valus of for which th graph of f is or or AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
30 97 AP Calculus AB: Sction I. h lim ln h is h nonistnt. Lt f ( ) cosarctan. What is th rang of f? 5. 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 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? 5 5 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
31 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 5arctan C 5 ln C 5ln C 5 5 C. Suppos that f is an odd function; i.., f ( ) f( ) for all. Suppos that f Which of th following must ncssarily b qual to f? f f ists. f f Non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
32 . Th avrag valu of ovr th intrval is 97 AP Calculus AB: Sction I 5. 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 8. If f cd5 whr c is a constant, thn c f d c 5 c 5 5 c c 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 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
33 . 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 5 divisions at and If th solutions of f( ) ar and, thn th solutions of f ar and 5 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 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
34 97 AP Calculus AB: Sction I 5. If f is a continuous function on ab,, which of th following is ncssarily tru? f ists on ab., f. If f is a maimum of f, thn lim f ( ) f lim for a, b f ( ) for som a, b Th graph of f is a straight lin. AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
35 97 AP Calculus BC: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. If f ( ), thn f( ) ( ). d 7 6. If f( ), thn th st of valus for which f incrass is,,,,,,,. For what non-ngativ valu of b is th lin givn by y b normal to th curv y? 5. d is nonistnt 6. If f( ) for all, thn f () AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
36 7. If y y ln, thn th valu of dy at th point (,) is d 97 AP Calculus BC: Sction I undfind 8. If y sin and intgr n for which ( n) y mans th nth drivativ of y with rspct to, thn th smallst positiv ( n) y y is If y cos, thn dy d 6sincos cos cos 6cos sincos. Th lngth of th curv y ln sc from = to = b, whr b, may b prssd by which of th following intgrals? b sc d b sc b d (sc tan d ) b ln sc d b sc tan d. Lt y. Whn = and d =, th valu of dy is AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
37 97 AP Calculus BC: Sction I. If n is a known positiv intgr, for what valu of k is n k n d? n n n n n n n. Th acclration of a body moving in a straight lin is givn in trms of tim t by 8 6t. If th vlocity of th body is 5 at t and if st ( ) is th distanc of th body from th origin at tim t, what is s() s()? 8. If t t and y, thn dy d t t t t t t t t t 5. Th ara of th rgion boundd by th lins,, and y and th curv y is 6. A sris pansion of sin t t is 6 t t t! 5! 7! 5 t t t t!! 6! 6 t t t! 5! 7! 5 t t t t!! 6! 5 7 t t t t! 5! 7! AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
38 97 AP Calculus BC: Sction I 5 7. Th numbr of bactria in a cultur is growing at a rat of, t pr unit of tim t. At t, th numbr of bactria prsnt was 7,5. Find th numbr prsnt at t 5.,, 7,5 5 7,5 5, Lt g b a continuous function on th closd intrval,. Lt g() and g(). Which of th following is NOT ncssarily tru? Thr ists a numbr h in, such that gh ( ) g ( ) For all a and b in,, if a b, thn ga ( ) gb ( ). Thr ists a numbr h in, such that Thr ists a numbr h in, such that For all h in th opn intrval 9. Which of th following sris convrg? gh ( ). gh ( ).,, lim g ( ) gh ( ). h for all in,. I. II. n n III. n n n ( ) n n I only III only I and II only I and III only I, II, and III. d C C C C C. ( ) d AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
39 97 AP Calculus BC: Sction I. A particl movs on th curv y ln so that th -componnt has vlocity ( t) t for t. At tim t, th particl is at th point,ln,,ln,. At tim t, th particl is at th point,ln 5 5,ln. h lim ln h is h nonistnt. Lt f ( ) for all ral and lt. For which of th following choics of is f( ) 7 whnvr? 5. tan d 6. Which of th following is tru about th graph of y ln in th intrval,? It is incrasing. It attains a rlativ minimum at,. It has a rang of all ral numbrs. It is concav down. It has an asymptot of. 7. If f ( ) 5 and th domain is th st of all such that 9, thn th absolut maimum valu of th function f occurs whn is 6 9 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
40 97 AP Calculus BC: Sction I 8. If th substitution sin y is mad in th intgrand of d, th rsulting intgral is sin ydy sin y dy cos y sin ydy sin ydy 6 sin ydy 9. If y y and if y y whn =, thn whn, y. d ln ln ln ( ) ln ln, thn f ( ). If f ln ln ln. If ln, y thn y is ln ln ln ln ln ln ln Non of th abov AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
41 . Suppos that f is an odd function; i.., f ( ) f( ) 97 AP Calculus BC: Sction I for all. Suppos that f Which of th following must ncssarily b qual to f? f f f f Non of th abov. Th avrag (man) valu of ovr th intrval is ists. 5. 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 6. d is ln ln ln ln divrgnt 7. cos ( ) lim 8. If f cd5 whr c is a constant, thn c f d c 5 c 5 5 c c 5 5 AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
42 9. Lt f and g b diffrntiabl functions such that f (), f (), f (), g(), g(), g() 5. ( ) ( ), thn h() If h fg 97 AP Calculus BC: Sction I 9 5. Th ara of th rgion nclosd by th polar curv r cos is. Givn for, f( ) cos for, f ( d ). Calculat th approimat ara of th shadd rgion in th figur by th trapzoidal rul, using 5 divisions at and AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
43 97 AP Calculus BC: Sction I. arcsin d sin d arcsin arcsin C d arccos arcsin d d. If f is th solution of f( ) f( ) such that ( ), f thn f C) 5. Suppos g( ) for all and statmnts is FALSE? F( ) tg( t) dt for all. Which of th following F taks on ngativ valus. F is continuous for all. F( ) g( ) g( t) dt F ( ) ists for all. F is an incrasing function. AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
44 985 AP Calculus AB: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () 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.. d If f( ), thn th th drivativ of f ( ) at is 8 8. If y, thn dy d 6 6 dy d. If cos, thn y cos cos sin C C sin sin C C C 5. n lim is n n, n,5 nonistnt AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
45 985 AP Calculus AB: Sction I 6. If f ( ), thn f (5) Which of th following is qual to ln? ln ln ln 8 ln t dt ln d dt t 8. Th slop of th lin tangnt to th graph of y ln at is 8 9. If d k, thn d k k k k k. If y, thn dy d ln ln ln. Th position of a particl moving along a straight lin at any tim t is givn by st () t t. What is th acclration of th particl whn t? 8. If f g f ( ) ln, ( ) ln, and g ( ) for all ral, thn g () AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
46 985 AP Calculus AB: Sction I. If y y, thn, in trms of and y, dy d y y y y y y y y. Th vlocity of a particl moving on a lin at tim t is mtrs did th particl travl from t to t? vt 5t mtrs pr scond. How many Th domain of th function dfind by f( ) ln is th st of all ral numbrs such that is a ral numbr 6. Th function dfind by f ( ) for all ral numbrs has a rlativ maimum at 7. d 8. If y cos sin, thn y sin cos sin cos sin 9. If f f f dfin f? for all ral numbrs and, which of th following could f ( ) f ( ) f( ) f ( ) f ( ) AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
47 985 AP Calculus AB: Sction I. If y arctan cos, thn dy d sin cos arccos arcsccos sin arcsccos cos. If th domain of th function f givn by f( ) is :, what is th rang of f? : : : : :. d 5 ln. d at is d 6 6. If 7 k d6, thn k 5. If f ( ), which of th following is qual to f ()? lim h h h lim h h h lim h h h lim h h h lim h h h AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
48 985 AP Calculus AB: Sction I 6. Th graph of y 9 is symmtric to which of th following? I. Th -ais II. Th y-ais III. Th origin I only II only III only I and II only I, II, and III 7. d If th position of a particl on th -ais at tim t is for t is 5t, thn th avrag vlocity of th particl Which of th following functions ar continuous for all ral numbrs? I. II. III. y y y tan Non I only II only I and II I and III. tan d ln cos( ) C ln cos( ) C ln cos( ) C ln cos( ) C sc( ) tan( ) C AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
49 985 AP Calculus AB: Sction I. Th volum of a con of radius r and hight h is givn by V r h. If th radius and th hight both incras at a constant rat of cntimtr pr scond, at what rat, in cubic cntimtrs pr scond, is th volum incrasing whn th hight is 9 cntimtrs and th radius is 6 cntimtrs? 5 8. sin d. Th graph of th drivativ of f is shown in th figur abov. Which of th following could b th graph of f? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
50 985 AP Calculus AB: Sction I. Th ara of th rgion in th first quadrant that is nclosd by th graphs of y 8 is y 8 and Th figur abov shows th graph of a sin function for on complt priod. Which of th following is an quation for th graph? y sin y sin y sin y sin y sin 6. If f is a continuous function dfind for all ral numbrs and if th maimum valu of f ( ) is 5 and th minimum valu of f ( ) is 7, thn which of th following must b tru? I. Th maimum valu of f ( ) is 5. II. Th maimum valu of f ( ) is 7. III. Th minimum valu of f ( ) is. I only II only I and II only II and III only I, II, and III 7. lim csc is AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
51 985 AP Calculus AB: Sction I 8. Lt f and g hav continuous first and scond drivativs vrywhr. If f ( ) g( ) for all ral, which of th following must b tru? I. f ( ) g( ) for all ral II. f ( ) g( ) for all ral III. f ( d ) gd ( ) Non I only III only I and II only I, II, and III 9. If ln f( ), for all, which of th following is tru? f is incrasing for all gratr than. f is incrasing for all gratr than. f is dcrasing for all btwn and. f is dcrasing for all btwn and. f is dcrasing for all gratr than.. Lt f b a continuous function on th closd intrval possibl valu of f ( d ) is,. If f( ), thn th gratst 8 6. If lim f ( ) L, whr L is a ral numbr, which of th following must b tru? a f ( a) ists. f ( ) is continuous at a. f ( ) is dfind at a. f ( a) L Non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
52 985 AP Calculus AB: Sction I. d d t dt An quation of th lin tangnt to y at its point of inflction is y 6 6 y y y y. Th avrag valu of on th closd intrval f( ), is Th rgion nclosd by th graph of y, th lin, and th -ais is rvolvd about th y -ais. Th volum of th solid gnratd is AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
53 985 AP Calculus BC: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () 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.. Th ara of th rgion btwn th graph of y and th -ais from to is At what valus of dos 5 f( ) 5 5 hav a rlativ maimum? only only only and only, and. d ln 8 ln ln 8 ln ln 8 ln ln. A particl movs in th y-plan so that at any tim t its coordinats ar At t, its acclration vctor is t and y t t.,,,,,8 AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
54 985 AP Calculus BC: Sction I 5. Th curvs y f( ) and y g( ) shown in th figur abov intrsct at th point, ara of th shadd rgion nclosd by ths curvs and th lin is givn by a ( ) ( ) ( ) ( ) f g d f g d ab. Th b gd ( ) f ( d ) b c c ( ) ( ) a f g d ( ) ( ) f g d a ( ) ( ) f g d 6. If f( ), thn tan f AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
55 985 AP Calculus BC: Sction I 7. Which of th following is qual to arcsin 5 5 d? C arcsin C arcsin C C 5 C 8. If f is a function such that f( ) f() lim, which of th following must b tru? Th limit of f ( ) as approachs dos not ist. f is not dfind at. Th drivativ of f at is. f is continuous at. f () 9. If y y 8, thn, at th point,, y is 5. For n n n ( ) if f( ), thn f ( ) n n n ( ) n n ( ) n n ( ) n n n ( ) n n n ( ) n n n AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
56 985 AP Calculus BC: Sction I. d ln d. d ( )( ) ln C ln ln C ln C ln ( )( ) C ln ( )( ) C. Lt f b th function givn by f ( ). What ar all valus of c that satisfy th conclusion of th Man Valu Thorm of diffrntial calculus on th closd intrval,? only only only and and. Which of th following sris ar convrgnt? I. II. III. n n n ( ) n I only III only I and III only II and III only I, II, and III 5. If th vlocity of a particl moving along th -ais is vt ( ) t and if at t its position is, thn at any tim t its position ( t ) is t t t t t t t t t t AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
57 985 AP Calculus BC: Sction I 6. Which of th following functions shows that th statmnt If a function is continuous at, thn it is diffrntiabl at is fals? f ( ) f ( ) f ( ) f ( ) f ( ) 7. If f ( ) ln, thn f ( ) ln ln ln 8. sin d cos C cos C cos C cos C cos C 9. If f and g ar twic diffrntiabl functions such that thn h ( ) f ( ) g ( ) and ( ) g( ) h( ) f, f ( ) f( ) f ( ) f( ) f ( ) f( ) f ( ) f( ) f ( ) f( ). Th graph of y f( ) on th closd intrval,7 is shown abov. How many points of inflction dos this graph hav on this intrval? On Two Thr Four Fiv AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
58 985 AP Calculus BC: Sction I. If f ( )sin d f( )cos cos d, thn ( ) f could b sin cos. Th ara of a circular rgion is incrasing at a rat of 96 squar mtrs pr scond. Whn th ara of th rgion is 6 squar mtrs, how fast, in mtrs pr scond, is th radius of th rgion incrasing? lim h h 5 h 8 d is nonistnt. Th ara of th rgion nclosd by th polar curv sin r for is 8 5. A particl movs along th -ais so that at any tim t its position is givn by valus of t is th particl at rst? () t t t. For what No valus only only only and 6. For, if y sin, thn dy d is ln sin sin cot sin cos sin cos sin sin cot ln sin AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
59 985 AP Calculus BC: Sction I 7. If f is th continuous, strictly incrasing function on th intrval a b as shown abov, which of th following must b tru? b I. f ( d ) fb ( )( b a) a b II. f ( d ) fa ( )( b a) a b III. f ( d ) fc ( )( b a) for som numbr c such that a c b a I only II only III only I and III only I, II, and III 8. An antidrivativ of f( ) is 9. sin lim is nonistnt. If t t and y t, thn dy d at t = is What ar all valus of for which th sris n convrgs? n n AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
60 985 AP Calculus BC: Sction I. An quation of th lin normal to th graph of y 7 at th point whr is y y y y 5 y 5 dy. If y dt and if y = whn t =, what is th valu of t for which y? ln ln ln. Which of th following givs th ara of th surfac gnratd by rvolving about th y-ais th arc of y from y = to y =? y 9y dy 6 y y dy y y dy y 9y dy 6 y y dy 5. Th rgion in th first quadrant btwn th -ais and th graph of y 6 is rotatd around th y-ais. Th volum of th rsulting solid of rvolution is givn by 6 6 d 6 6 d d 9 9 y dy 9 y dy AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
61 985 AP Calculus BC: Sction I 6. d is 6 6 nonistnt 7. Th gnral solution for th quation dy y is d y C y C y C y C y C C 8. lim 5 is 5 nonistnt 9. Th bas of a solid is th rgion nclosd by th graph of y, th coordinat as, and th lin. If all plan cross sctions prpndicular to th -ais ar squars, thn its volum is If th substitution u is mad, th intgral d u du u u du u u du u u du u u du u AP Calculus Multipl-Choic Qustion Collction 55 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
62 985 AP Calculus BC: Sction I. What is th lngth of th arc of y from = to =? Th cofficint of in th Taylor sris for about = is 6 9. Lt f b a function that is continuous on th closd intrval, such that f () dos not ist, f (), and f( ) for all cpt =. Which of th following could b th graph of f?. At ach point, y on a crtain curv, th slop of th curv is point,8, thn its quation is y. If th curv contains th y 8 y 8 y 7 y ln 8 y 8 5. If n is a positiv intgr, thn lim n n can b prssd as n n n n d d d d d AP Calculus Multipl-Choic Qustion Collction 56 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
63 988 AP Calculus AB: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () 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.. thn dy If y, d. What is th domain of th function f givn by : f( )? : : : and : and. A particl with vlocity at any tim t givn by vt () movs in a straight lin. How far dos th particl mov from t to t? t. Th graph of 5 y is concav downward for all valus of such that 5 5. sc d tan C csc C cos C sc C sc tan C AP Calculus Multipl-Choic Qustion Collction 57 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
64 988 AP Calculus AB: Sction I 6. If ln y, thn dy d ln ln ln 7. d 5 5 C 5 C C 5 C C 8. Th graph of y f( ) is shown in th figur abov. On which of th following intrvals ar dy d and d y d? I. a b II. b c III. c d I only II only III only I and II II and III AP Calculus Multipl-Choic Qustion Collction 58 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
65 988 AP Calculus AB: Sction I 9. If, thn at th point y y,, dy d is nonistnt k. If k d 8, thn k An quation of th lin tangnt to th graph of at th point, f ( ) ( ) is y 7 6 y 6 5 y y y 7 8. If f ( ) sin, thn f. If th function f has a continuous drivativ on,c, thn c f ( d ) f ( c) f() f() c f() f ( c ) f ( ) c f( c) f(). cos sin d AP Calculus Multipl-Choic Qustion Collction 59 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
66 988 AP Calculus AB: Sction I 5. If f ( ), thn f () 6. A particl movs along th -ais so that at any tim t its position is givn by () t t t 9t. For what valus of t is th particl at rst? No valus only only 5 only and 7. d If y cos, thn d y d 8cos cos sin cos cos 9. d ln ln ln ln ln 5. Lt f b a polynomial function with dgr gratr than. If a b and f( a) f( b), which of th following must b tru for at last on valu of btwn a and b? I. f( ) II. f( ) III. f( ) Non I only II only I and II only I, II, and III AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
67 . Th ara of th rgion nclosd by th graphs of y and 988 AP Calculus AB: Sction I y is. If ln ln, thn =. If f ( ) cos and g( ) for all, and if f() g(), thn f ( ) lim g ( ) is nonistnt d. ln d ln ln ln ln ln ln ln ln 5. For all, if f ( ) dt, thn f ( ) t ln ln 6. cos d AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
68 988 AP Calculus AB: Sction I 7. At, th function givn by f, 6 9, is undfind. continuous but not diffrntiabl. diffrntiabl but not continuous. nithr continuous nor diffrntiabl. both continuous and diffrntiabl. 8. d Th h tan h tan lim h is sc ( ) sc ( ) cot( ) nonistnt. A rgion in th first quadrant is nclosd by th graphs of y =, and th coordinat as. If th rgion is rotatd about th y -ais, th volum of th solid that is gnratd is rprsntd by which of th following intgrals?, y ln d d d ln ydy ydy AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
69 988 AP Calculus AB: Sction I. If f( ), thn th invrs function, f, is givn by f ( ). Which of th following dos NOT hav a priod of? f ( ) sin f ( ) sin f ( ) tan f ( ) tan f ( ) sin. Th absolut maimum valu of on th closd intrval, f( ) occurs at. Th ara of th shadd rgion in th figur abov is rprsntd by which of th following intgrals? c ( ) ( ) a c f g d f ( d ) gd ( ) b c ( ) ( ) a c c a g f d ( ) ( ) a b f g d ( ) ( ) ( ) ( ) a g f d f g d c b AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
70 988 AP Calculus AB: Sction I 5. cos cos sin cos sin cos sin cos sin cos 6. What is th avrag valu of y for th part of th curv y which is in th first quadrant? If f ( ) sin, thn th numbr of zros of f on th closd intrval, is 8. For, du d u C 8 C ln ln C ln C ln C 9. If f( ) d and f( ) d 7, thn f ( d ) AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
71 988 AP Calculus AB: Sction I y z dz d dy. Th sids of th rctangl abov incras in such a way that and. At th instant dt dt dt whn and y, what is th valu of d dt? 5 5. If lim f( ) 7, which of th following must b tru? I. f is continuous at. II. f is diffrntiabl at. III. f () 7 Non II only III only I and III only I, II, and III. Th graph of which of th following quations has y as an asymptot? y ln y sin y y y. Th volum of th solid obtaind by rvolving th rgion nclosd by th llips about th -ais is 9y AP Calculus Multipl-Choic Qustion Collction 65 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
72 988 AP Calculus AB: Sction I. Lt f and g b odd functions. If p, r, and s ar nonzro functions dfind as follows, which must b odd? I. p( ) f g( ) II. r ( ) f( ) g ( ) III. s ( ) f( g ) ( ) I only II only I and II only II and III only I, II, and III 5. Th volum of a cylindrical tin can with a top and a bottom is to b 6 cubic inchs. If a minimum amount of tin is to b usd to construct th can, what must b th hight, in inchs, of th can? 8 AP Calculus Multipl-Choic Qustion Collction 66 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
73 988 AP Calculus BC: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () 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.. Th ara of th rgion in th first quadrant nclosd by th graph of y and th -ais is d ( ) ln, thn f ( ). If f. If u, v, and w ar nonzro diffrntiabl functions, thn th drivativ of uv w is uv uv w uvw uvw w uvwuvw uvw w uvw uvw uvw w uvw uvw uvw w AP Calculus Multipl-Choic Qustion Collction 67 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
74 988 AP Calculus BC: Sction I 5. Lt f b th function dfind by th following. sin,, f( ),, For what valus of is f NOT continuous? only only only and only,, and 6. If y y 6, thn dy d y y y y y y y y y 7. d is ln nonistnt 8. If f ( ), thn ln f () AP Calculus Multipl-Choic Qustion Collction 68 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
75 988 AP Calculus BC: Sction I 9. Which of th following pairs of graphs could rprsnt th graph of a function and th graph of its drivativ? I only II only III only I and III II and III. sin lim h h sin h is sin cos nonistnt. If 7 y 9 is an quation of th lin normal to th graph of f at th point,, thn f () A particl travls in a straight lin with a constant acclration of mtrs pr scond pr scond. If th vlocity of th particl is mtrs pr scond at tim sconds, how far dos th particl travl during th tim intrval whn its vlocity incrass from mtrs pr scond to mtrs pr scond? m m 7 m 6 m m AP Calculus Multipl-Choic Qustion Collction 69 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
76 988 AP Calculus BC: Sction I. sin 5 n n ( )! 5! n! 5 n n ( ) ( ) ( ) ( )! 5! n! n n ( ) ( ) ( ) ( )!! n! 6 n!! 6! n! 5 n ( ) ( ) ( )! 5! n!. If t dt thn F( ) F( ), 6 6 t t dt 5. For any tim t, if th position of a particl in th y-plan is givn by y ln t, thn th acclration vctor is t and, t ( t ), t ( t ), (t ), (t ), (t ) AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
77 988 AP Calculus BC: Sction I 6. d C C C C C 7. ( )( ) d 9 5 ln 8 ln 5 ln 5 8. If thr qual subdivisions of, ar usd, what is th trapzoidal approimation of d? 9. A polynomial p( ) has a rlativ maimum at,, a rlativ minimum at,, a rlativ maimum at 5,7 and no othr critical points. How many zros dos p( ) hav? On Two Thr Four Fiv. Th statmnt lim f ( ) L mans that for ach, thr ists a such that a if a, thn f( ) L if f( ) L, thn a if f( ) L, thn a a and f( ) L if a, thn f( ) L AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
78 . Th avrag valu of on th closd intrval, is 988 AP Calculus BC: Sction I ln ln ln. If f ( ), thn f ( ) ln ln ln. Which of th following givs th ara of th rgion nclosd by th loop of th graph of th polar curv r cos( ) shown in th figur abov? 6 cos( ) d cos( )d 8 cos ( )d cos ( ) d cos ( )d AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
79 988 AP Calculus BC: Sction I. If c is th numbr that satisfis th conclusion of th Man Valu Thorm for th intrval, thn c = f ( ) on 5. Th bas of a solid is th rgion in th first quadrant nclosd by th parabola y, th lin, and th -ais. Each plan sction of th solid prpndicular to th -ais is a squar. Th volum of th solid is If f is a function such that f ( ) ists for all and f( ) for all, which of th following is NOT ncssarily tru? f( ) d f ( d ) f( d ) f ( d ) f( d ) f ( d ) f( d ) f ( d ) f( d ) f( d ) 7. If th graph of has a point of inflction at, 6 y a b, what is th valu of b? It cannot b dtrmind from th information givn. AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
80 988 AP Calculus BC: Sction I d 8. ln cos d is cos tan cos tan tan 9. Th rgion R in th first quadrant is nclosd by th lins and y 5 and th graph of y. Th volum of th solid gnratd whn R is rvolvd about th y -ais is in i n n n n n. d 8 6. Th gnral solution of th diffrntial quation y y is y = C C C C C AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
81 988 AP Calculus BC: Sction I. Th lngth of th curv y from = to = is givn by 6 d d 9 d 9 d 9 d. A curv in th plan is dfind paramtrically by th quations An quation of th lin tangnt to th curv at t is t t and y t t. y y 8 y y 5 y 8 5. If k is a positiv intgr, thn lim k is k! nonistnt 6. Lt R b th rgion btwn th graphs of y and th solid obtaind by rvolving R about th -ais is givn by y sin from to. Th volum of sind cosd sin d sin d sin d 7. A prson mtrs tall walks dirctly away from a strtlight that is 8 mtrs abov th ground. If th prson is walking at a constant rat and th prson s shadow is lngthning at th rat of 9 mtr pr scond, at what rat, in mtrs pr scond, is th prson walking? AP Calculus Multipl-Choic Qustion Collction 75 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
82 988 AP Calculus BC: Sction I 8. What ar all valus of for which th sris n n convrgs? n All ral 9. If dy ysc and y = 5 whn =, thn y = d tan tan 5 5 tan tan 5 tan 5. Lt f and g b functions that ar diffrntiabl vrywhr. If g is th invrs function of f and if g( ) 5 and f (5), thn g( ) 5 5. lim n n n n n n d d d d d. If f( ) d 6, what is th valu of f (5 d )? 6 6 AP Calculus Multipl-Choic Qustion Collction 76 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
83 988 AP Calculus BC: Sction I. Bactria in a crtain cultur incras at a rat proportional to th numbr prsnt. If th numbr of bactria doubls in thr hours, in how many hours will th numbr of bactria tripl? ln ln ln ln ln ln ln 7 9 ln. Which of th following sris convrg? I. n ( ) n n II. n n n III. ln n n n I only II only III only I and III only I, II, and III 5. What is th ara of th largst rctangl that can b inscribd in th llips 9y 6? 6 6 AP Calculus Multipl-Choic Qustion Collction 77 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
84 99 AP Calculus AB: Sction I 9 Minuts Scintific Calculator Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () 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.. If f( ), thn f() Which of th following rprsnts th ara of th shadd rgion in th figur abov? d ( ) f ( ydy ) c ( ba) f( b) f( a) ( d c) f( b) f( a) b d f d f ( b) f( a) a. n 5n lim n n n is 5 nonistnt AP Calculus Multipl-Choic Qustion Collction 78 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
85 99 AP Calculus AB: Sction I. If yy 7, thn in trms of and y, dy d y y y y y y y y y 5. If th function f is continuous for all ral numbrs and if thn f ( ) f( ) whn, 6. Th ara of th rgion nclosd by th curv y, th -ais, and th lins and is 5 6 ln ln ln ln 6 7. An quation of th lin tangnt to th graph of y at th point, 5 is y 8 y 8 y 6 y 66 y AP Calculus Multipl-Choic Qustion Collction 79 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
86 99 AP Calculus AB: Sction I 8. If y tan cot, thn dy d sc csc sc csc sc csc sc csc sc csc 9. If h is th function givn by h ( ) f( g ( )), whr f( ) and g ( ), thn h ( ). If f ( ) ( ) sin, thn f (). Th acclration of a particl moving along th -ais at tim t is givn by at () 6t. If th vlocity is 5 whn t and th position is whn t, thn th position () t 9t t t t t t 6 t t 9t 6t t 77t 55. If f and g ar continuous functions, and if f( ) for all ral numbrs, which of th following must b tru? b b b I. f ( gd ) ( ) f( d ) gd ( ) a a a b b b II. f ( ) g( ) d f( ) d g( ) d a a a b III. f ( ) d f( ) d a b a I only II only III only II and III only I, II, and III AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
87 99 AP Calculus AB: Sction I. Th fundamntal priod of cos( ) is 6. d C C C ln C ln( ) C 5. For what valu of dos th function f( ) ( )( ) hav a rlativ maimum? Th slop of th lin normal to th graph of y ln(sc ) at is nonistnt AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
88 99 AP Calculus AB: Sction I 7. ( ) d ( ) ( ) 6 C C ( ) 5 C C C 5 8. If f( ) sin, thn thr ists a numbr c in th intrval that satisfis th conclusion of th Man Valu Thorm. Which of th following could b c? Lt f b th function dfind by about f is tru? f( ) for, for. Which of th following statmnts f is an odd function. f is discontinuous at. f has a rlativ maimum. f () f( ) for AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
89 99 AP Calculus AB: Sction I. Lt R b th rgion in th first quadrant nclosd by th graph of y ( ), th lin 7, th -ais, and th y-ais. Th volum of th solid gnratd whn R is rvolvd about th y -ais is givn by 7 ( ) d 7 ( ) d ( ) d ( ) d 7 ( y ) dy. At what valu of dos th graph of y hav a point of inflction? At no valu of. An antidrivativ for is ( ) ln( ) ln arcsc( ) arctan( ). How many critical points dos th function 5 f( ) ( ) ( ) hav? On Two Thr Fiv Nin. If f( ) ( ), thn f () is AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
90 99 AP Calculus AB: Sction I d 5. d ( ) ( )ln ( )ln ln 6. A particl movs along a lin so that at tim t, whr t, its position is givn by t st () cost. What is th vlocity of th particl whn its acclration is zro? Th function f givn by f( ) is incrasing for, dcrasing for, incrasing for dcrasing for, incrasing for incrasing for all dcrasing for all dcrasing for, incrasing for, dcrasing for d d cos lim sin is 8 nonistnt. Th rgion nclosd by th -ais, th lin, and th curv y is rotatd about th -ais. What is th volum of th solid gnratd? AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
91 99 AP Calculus AB: Sction I. If ln( ) f( ), thn f ( ) ln( ) ln( ) ln( ) 6(ln ) d 6 ln ln. If dy y and if y whn =, thn whn =, y = d. Th top of a 5-foot laddr is sliding down a vrtical wall at a constant rat of ft pr minut. Whn th top of th laddr is 7 ft from th ground, what is th rat of chang of th distanc btwn th bottom of th laddr and th wall? 7 ft pr minut 8 7 ft pr minut ft pr minut ft pr minut ft pr minut 5. If th graph of thn ac a b y c has a horizontal asymptot y and a vrtical asymptot, 5 5 AP Calculus Multipl-Choic Qustion Collction 85 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.
AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos
AP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
Question 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
New Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting
A Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
ME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
Traffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
QUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means
Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian
Financial Mathematics
Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution
Section 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
Lecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective
Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog
Basis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
Parallel and Distributed Programming. Performance Metrics
Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:
Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,
CHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil
CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
Chapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlx-valud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,
Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)
con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)
![[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)](/thumbs/40/20779684.jpg "[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)")
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
Current and Resistance
Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9
Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY
Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl
Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
Introduction to Finite Element Modeling
Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation
The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl
5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.
Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by
Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman
Cloud and Big Data Summr Scool, Stockolm, Aug., 2015 Jffry D. Ullman Givn a st of points, wit a notion of distanc btwn points, group t points into som numbr of clustrs, so tat mmbrs of a clustr ar clos
Answer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
AP Calculus AB 2004 Scoring Guidelines
AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from
AP Calculus AB 2010 Free-Response Questions Form B
AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.
Mark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Eam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice eam contributors: Benita Albert Oak Ridge High School,
AP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college
Core Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiple-choice questions in which you are asked to choose the
L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
Factorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
Foreign Exchange Markets and Exchange Rates
Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls
Intermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers)
Intrmdiat Macroconomic Thory / Macroconomic Analysis (ECON 3560/5040) Final Exam (Answrs) Part A (5 points) Stat whthr you think ach of th following qustions is tru (T), fals (F), or uncrtain (U) and brifly
Category 7: Employee Commuting
7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil
E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S
E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S Fair Valu 1 Valuation Variabls Tabl 1 blow shows th variabls us in th rspctiv valuation
Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental
Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il
LAB 3: VELOCITY AND ACCELERATION GRAPHS
Goas: LAB 3: ELOCITY AND ACCELERATION GRAPHS Invstigat accration vs. tim graphs Prdict accration graphs from vocity graphs Invstigat accration as sop of vocity vs. tim graph Part 1 - Making ocity- Graphs
Problem Solving Session 1: Electric Dipoles and Torque
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Dpatmnt of Physics 8.02 Poblm Solving Sssion 1: Elctic Dipols and Toqu Sction Tabl (if applicabl) Goup Mmbs Intoduction: In th fist poblm you will lan to apply Coulomb
Calculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
Fundamentals of Tensor Analysis
MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006 Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss;
correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
Physics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves
Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc
Calculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing
Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif
Adverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
UNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems
Business rules FATCA V. 02/11/2015
Elmnt Attribut Siz InputTyp Rquirmnt BUSINESS RULES TYPE ERROR ACK Xpath I.Mssag Hadr FATCA_OECD Vrsion xsd: string = Validation WrongVrsion ftc:fatca_oecd/vrsion SndingCompanyIN Unlimit d xsd: string
TOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
Intrnational Association of Scintific Innovation and Rsarch (IASIR) (An Association Unifing th Scincs, Enginring, and Applid Rsarch) ISSN (Print): 79-000 ISSN (Onlin): 79-009 Intrnational Journal of Enginring,
LIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data
FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among
Mathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
Performance Evaluation
Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism
HOMEWORK FOR UNIT 5-1: FORCE AND MOTION
Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.
MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
A Theoretical Model of Public Response to the Homeland Security Advisory System
A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial
(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM
RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs
Version 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This print-out should have 47 questions Multiple-choice questions may continue on the net column or page find all choices before answering
AP CALCULUS AB 2008 SCORING GUIDELINES
AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line
Section 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
Solutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
Gold versus stock investment: An econometric analysis
Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 268-8662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag -7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin
Mark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
Trigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
Visualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
An Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images
An Adaptiv Clustring MAP Algorithm to Filtr Spckl in Multilook SAR Imags FÁTIMA N. S. MEDEIROS 1,3 NELSON D. A. MASCARENHAS LUCIANO DA F. COSTA 1 1 Cybrntic Vision Group IFSC -Univrsity of São Paulo Caia
AP Calculus AB 2012 Free-Response Questions
AP Calculus AB 1 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
Job shop scheduling with unit processing times
Job shop schduling with unit procssing tims Nikhil Bansal Tracy Kimbrl Maxim Sviridnko Abstract W considr randomizd algorithms for th prmptiv job shop problm, or quivalntly, th cas in which all oprations
Establishing Wireless Conference Calls Under Delay Constraints
Establishing Wirlss Confrnc Calls Undr Dlay Constraints Aotz Bar-Noy aotz@sci.brooklyn.cuny.du Grzgorz Malwicz grg@cs.ua.du Novbr 17, 2003 Abstract A prvailing fatur of obil tlphony systs is that th cll
1. (from Stewart, page 586) Solve the initial value problem.
. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.
Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D
24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd
Constraint-Based Analysis of Gene Deletion in a Metabolic Network
Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany
Testing the gravitational properties of the quantum vacuum within the Solar System
Tsting th gravitational proprtis of th quantum vacuum within th Solar Systm Dragan Hajdukovic To cit this vrsion: Dragan Hajdukovic. Tsting th gravitational proprtis of th quantum vacuum within th Solar
AP Calculus BC 2010 Free-Response Questions
AP Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded
AP Calculus AB 2007 Scoring Guidelines Form B
AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
Vibrational Spectroscopy
Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3
Techniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration
AP Calculus BC 2006 Free-Response Questions
AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to