AP Calculus Multipl-Coic Qustion Collction 985 998
. f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b tru? I. T maimum valu of f ( ) is 5. II. T maimum valu of f ( ) is 7. III. T minimum valu of f ( ) is 0. I only II only I and II only II and III only I, II, and III. f g ( ) = ( +), f( ) = ln( ), and g ( )> 0 for all ral, tn g ()= ( ) ln + + + + +. Wat is t domain of t function f givn by {: } f ( ) =? {: } {: } { : and } { : an d }.. ln ln =, tn = 5. If f( ) = +, tn t invrs function, f, is givn by f ( )= + + 6. Wic of t following dos NOT av a priod of π? f ( )= sin f ( )= tan f ( )= tan f ( )= sin f ( )= sin
7. T grap of wic of t following quations as y = as an asymptot? y = ln y = sin y = + y = y = 8. Wic of t following functions ar continuous for all ral numbrs? I. y = II. y = III. y = tan Non I only II only I and II I and III 9. If t function f is continuous for all ral numbrs and if tn f ( ) = f( )= + wn, 0 0. If is t function givn by ) ( = f(g()), wr f( )= and g ( )=, tn ) ( =. T fundamntal priod of cos() is π π 6π. If t grap of y = a + b as a orizontal asymptot y = and a vrtical asymptot =, + c tn a+ c= 5 0 5
0 f ( ) k. T function f is continuous on t closd intrval [0, ] and as valus tat ar givn in t tabl abov. T quation f ( )= must av at last two solutions in t intrval [0, ] if k = 0 5. Lt f b t function dfind by t following. sin, < 0, 0 < f( ) =, <, For wat valus of is f NOT continuous? 0 only only only 0 and only 0,, and 5. Wic of t following functions ar continuous at =? I. ln II. III. ln( ) I only II only I and II only II and III only I, II, and III () ) 6. Lt f b t function givn by f( )=. For wat positiv valus of a is f a continuous for all ral numbrs? Non only only only and only (
7.. I f( ) = +, tn f ( ) f(0) 0 is 0 nonistnt 8. If ln for 0 < f( ) = ln for <, tn f () is ln ln 8 ln6 nonistnt 9. n is 0, 000n n n + 0,500 nonistnt 0. If f ( ) = L, a wr L is a ral numbr, wic of t following must b tru? f (a) ists. f ( ) is continuous at = a. f () is dfind at = a. f ( a) = L Non of t abov. n 5n n + is n n 5 nonistnt. cosθ θ 0 sin θ is 0 8 nonistnt
a. If a 0, tn a a is a a 6a 0 nonistnt. If f ( )=, wic of t following is qual to f ()? 0 + 0 + 0 + 0 + 0 + 5. T tan ( + ) tan 0 is 0 sc () sc () cot() nonistnt 6. If f is a diffrntiabl function, tn f ( a) is givn by wic of t following? I. II. III. f ( a+ ) f( a) f ( ) f(a) a a f ( + ) f( ) 0 a I only II only I and II only I and III only I, II, and III 7. sin ( + ) sin 0 is 0 sin cos nonistnt
8. If f( ) = 7, wic of t following must b tru? I. f is continuous at =. II. f is diffrntiabl at =. III. f () = 7 Non I and III only II only I, II, and III III only 9. Wic of t following functions sows tat t statmnt If a function is continuous at = 0, tn it is diffrntiabl at = 0 is fals? f ( )= f ( )= f ( )= f ( )= f ( )= 0. At =, t function givn by f ( ) undfind. continuous but not diffrntiabl. diffrntiabl but not continuous. nitr continuous nor diffrntiabl. bot continuous and diffrntiabl., < = 6 9, is. If f ( ) =, tn f (5) = 0 5 5 5. d + d at = is 6 0 6
. If f ( )=, tn f () =. If f( ) =, tn f () = 6 6 8 5. If y = cos sin, tn y = 0 sin() cos ( +sin) cos ( sin) π 6. If f ( ) = sin, tn f = 7. If y = tan cot, tn dy = d sc csc sc csc sc + csc sc csc sc + csc 8. If y = arctan( ), tn dy d = + + + 9. If y = arctan ( cos ), tn dy d = sin + cos ( arccos ) + ( arcsc cos ) + cos ( ) sin arcsc( cos ) ( )
dy 0. If y =, tn + d = 6 ( + ) ( + ) 6 ( + ) ( + ). T valu of t drivativ of y = + 8 + at = 0 is 0. Wat is t instantanous rat of cang at = of t function f givn by f( )=? 6 6. If f( ) =, tn f π tan = π + π π. If y =, tn dy d = ( + ) ( + ) + + 5. If f ( ) = ( ) sin, tn f (0) = 0
6. If f and g ar twic diffrntiabl functions suc tat g ( )= f ( ) and g ( )= ( ) f tn ) ( = ( ) f ( )+ f ( ) f ( ) + f ( ) f ( )+ f ( ) ( ) + f ( ) f ( ) f ( ) + f ( ) ( ), ( ) 7. If u, v, and w ar nonzro diffrntiabl functions, tn t drivativ of uv w is uv + u v w uvw uvw w uvw uv w u vw w u vw+ uvw +uvw w uv w + u vw uvw w ( ) 8. If f( ) = +, tn t t drivativ of f ( ) at = 0 is 0 8 0 8 9 If y = cos, tn d y d = 8cos cos sin cos cos 50. If f( ) = ( ), tn f (0) is 0
5. If f ( ) ( )= sin, tn f () = cos( ) cos( ) + cos( ) cos( ) ) cos( π 5. If f ( )= tan(), tn f 6 = 8 5. If f and g ar twic diffrntiabl and if ( )= f( g ( )), tn ( )= ( )[ g ( ) ] + f g f g ( ) ( ( ))g ( ) f ( g ( ))g ( )+ f ( g ( ))g ( ) ( )[ g ( ) ] f g ( ) f ( g ( ))g ( ) f ( g ( )) 5. d d ( ) = ( ) ( )ln ( )ln ln d 55. ln = d ( )
56. If f ( )= ln( ), tn f ( )= ln ( )+ ln ( )+ ln ( ) + 57. If f ( )= ln ( ), tn f ( )= 58. If f ( )=, tn ln ( f ()) = 0 59. d ln cos d π is π π cos tan π cos π π π tan π tan π 60 If f( )= ln( ), tn f () = 6. If f( )= tan, tn f ( )= tan tan sc tan tan tansc t a n tan tan
( ) 6. If y 0 =, tn dy d = ( ) ( ln0)0 ( ) ( ln ) 0 0 ( ) ( ) ( ) 0 ( ) ( ln0 0 ) 0 ( ) 6. If y = ln, tn dy d = ln ln + ln 6. d ln d ( ) = ln ( ln ) ( ln ) ( ln ) ( ln )( ln ) ln ( )( ln ) 65. If f( )= ln(), tn f () = ln( ) ln( ) 6(ln ) ln( ) 5 6 5 66. For 0< < π, if y = sin ( ), tn dy is d ln ( sin ) ( ) sin cot (C sin ( os ) ( ) c ) ( sin ) ( cos + sin ) ( sin ) ( cot + ln ( sin )) 67. If ( ) = +, f tn f ( )= + + (+ ) ( + ln(+ ) )
68. If + y+ y =0, tn, in trms of and y, dy d = + y + y + y + y + y + y + + y y 69. If + y y =, tn at t point (,), dy d is 0 nonistnt 70. If + y+ y =7, tn in trms of and y, dy d = + y + y + y + y + y + y + y y + y 7. If + y = 0, tn wn =, dy d = 7 7 7 7. If y + y = 8, tn, at t point (, ), y is 5 0
7. If y y = 6, tn dy d = y y y y y y y y y 7. If dy = y, tn d d y d = y y y y y 75. T slop of t lin tangnt to t curv y + ( y+ ) = 0 at (, ) is 0 76. T slop of t lin tangnt to t grap of ln( y) = at t point wr = is 0 77. T slop of t lin tangnt to t grap of y = ln at = is 8 78. T slop of t lin normal to t grap of y = ln(sc) at π = is nonistnt
79. An quation of t lin tangnt to t grap of f ( ) = ( ) at t point (,) is y =7+6 y = y = 6+5 y = 78 y = + 80. An quation of t lin tangnt to t grap of y = + at t point (, 5) is y =8 + y =66 + y =8 + y = y =6 8. An quation of t lin tangnt to t grap of y = + cos at t point ( 0,) is y = + y = + y = y = y = 0 8. Lt f b t function givn by f ( )= and lt g b t function givn by g ( )= 6. At wat valu of do t graps of f and g av paralll tangnt lins? 0.70 0.567 0.9 0.0 0.58 8. Wic of t following is an quation of t lin tangnt to t grap of f ( ) = + at t point wr f ( ) =? y = 85 y = +7 y = +0.76 y = 0. y =.6 8. An quation of t lin normal to t grap of y = + + 7 at t point wr = is + y = 0 y = y = + y = 5 + y =5
85. A particl movs along t -ais so tat at any tim t 0 its position is givn by ( t) = t t 9t+. For wat valus of t is t particl at rst? No valus only only 5 only and 86. T position of a particl moving along a straigt lin at any tim t is givn by st () = t +t+. Wat is t acclration of t particl wn t =? 0 8 87. A particl movs along a lin so tat at tim t, wr 0 t π, its position is givn by s(t) =cost t + 0. Wat is t vlocity of t particl wn its acclration is zro? 5.9 0.7..55 8. 88. A particl movs along t -ais so tat its position at tim t is givn by ( t) = t 6t+5. For wat valu of t is t vlocity of t particl zro? 5 89. T position of a particl moving along t -ais is ( t) = sin(t) cos( t) for tim t 0. Wn t =π, t acclration of t particl is 9 9 0 9 9