AP Calculus BC AP Exam Problems Chapters 1 3

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1 AP Eam Problms Captrs Prcalculus Rviw. If 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 onl I and II onl I, II, and III II onl II and III onl. If f g( ) ln, f ( ) ln, and g ( ) 0 for all ral, tn g ( ) +. Wat is t domain of t function f givn b f( )? : : : and : : and. If ln ln, tn = 5. If f( ), tn t invrs function, f, is givn b f ( ) 6. Wic of t following dos NOT av a priod of? f ( ) sin f ( ) sin f ( ) sin f ( ) tan f ( ) tan

2 AP Eam Problms Captrs 7. T grap of wic of t following quations as = as an asmptot? ln sin 8. Wic of t following is continuous for all ral numbrs? I. II. III. tan Non I onl II onl I and II I and III 9. If f( ) is continuous for all ral numbrs and if f( ) wn, tn f ( ) 0 0. If ( ) f g( ), wr f and g( ), tn ( ) ( ). T fundamntal priod of cos is 6. If t grap of tn a + c = a b c as a orizontal asmptot = and a vrtical asmptot, f() k. T function f is continuous on t closd intrval [0, ] and as valus givn in t tabl abov. T quation f( ) must av at last two solutions in t intrval [0, ] if k = 0

3 AP Eam Problms Captrs. Lt f b t function dfind b t following. sin, 0, 0 f( ) For wat valus of is f NOT continuous? 0 onl onl onl 0 and onl 0,, and 5. Wic of t following functions is continuous at =? I. ln II. III. ln I onl II onl I and II onl II and III onl I, II, and III 6. Lt f b t function givn b f( ) continuous for all ral numbrs? a. For wat positiv valus of a is f Non onl onl onl and onl Limits and Continuit 7. If f f ( ) f (0), tn 0 ( ) is 0 nonistnt ln for 0 8. If f( ) ln for tn f( ) is ln ln 8 ln 6 nonistnt 9. Find n n n 0,000n 0,500 nonistnt

4 AP Eam Problms Captrs 0. If f ( ) L, wr L is a ral numbr, wic of t following must b tru? a f ( a) ists. f ( a) L f( ) is continuous at = a. Non of t abov f( ) is dfind at = a.. Find n 5n n n n 5 nonistnt cos. Find 0 sin 0 8 nonistnt. If a 0, tn a a a is a a 6a 0 nonistnt Drivativs. If f ( ), wic of t following is qual to f ( )? T tan tan 0 is 0 sc ( ) sc ( ) cot( ) nonistnt

5 AP Eam Problms Captrs 6. If f is a diffrntiabl function, tn f ( a) is givn b wic of t following? I. 0 f ( a ) f ( a) II. a f ( ) f ( a) a III. a f ( ) f ( ) I onl II onl I and II onl I and III onl I, II, and III 7. Find sin( ) sin 0 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 II onl III onl I and III onl I, II, and III 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 b if f( ) 6 9 if is undfind nitr continuous nor diffrntiabl continuous but not diffrntiabl bot continuous and diffrntiabl not continuous and diffrntiabl. If f ( ), tn f (5) Find d d at =

6 AP Eam Problms Captrs. If f ( ), tn f (). If f ( ), tn f () If cos sin, tn sin( ) cos sin 0 cos sin 6. If f ( ) sin, tn f 7. If tan cot, tn d d sc csc sc csc sc csc sc csc sc csc 8. If tan, tn d d 6

7 AP Eam Problms Captrs 9. If tan cos, tn d d sin cos sc (cos ) sc (cos ) sin cos cos 0. If, tn d d 6 6. T valu of t drivativ of 8 at = 0 is 0. Wat is t instantanous rat of cang at = of t function f givn b f( )? 6 6. If f( ) tan, tn f. If, tn d d 7

8 5. If f ( ) sin, tn f (0) AP Calculus BC AP Eam Problms Captrs 0 6. If f and g ar diffrntiabl functions suc tat g( ) f( ) and ( ) ( ) ( ) f, tn ( ) g 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 uvw uvw w uvw uvw uvw w uvw uvw uvw w 8. If f ( ), tn t fourt drivativ of f( ) at = 0 is If cos, tn d d 8cos cos sin cos cos 50. If f ( ), tn (0) f is 0 5. If f ( ) sin, tn f( ) cos cos cos cos cos 8

9 AP Eam Problms Captrs 5. If f ( ) tan, tn f If f and g ar twic diffrntiabl functions and if ( ) f g( ) f g( ) g( ) f g( ) g ( ) f g( ) g( ) f g( ) g ( ) f g( ) f g( ) g ( ) d d 5. Find f g( ) g ( ), tn ( ) ln ln ln 55. Find d ln d 56. If f ( ) ln, tn f( ) ln ln ln 57. If f ( ) ln, tn f( ) 58. If f ( ), tn ln f () 0 9

10 AP Eam Problms Captrs d 59. Find ln cos d cos tan cos tan tan 60. If f ( ) ln, tn f( ) 6. If tan f ( ), tn f( ) tan tan tan tan tan sc tan tan tan sc 6. If 0, tn d d ln ln0 0 ln If ln, tn d d ln ln ln d d ln 6. Find ln ln ln ln ln ln ln ln 0

11 AP Eam Problms Captrs 65. If ln f ( ), tn f( ) ln 6ln ln 5 ln For 0, if sin, tn d d lnsin sin cos sin cot sin cos sin sin cot lnsin 67. If, tn f( ) f( ) ln 68. If 0, tn, in trms of and, d d 69. If, tn at t point (, ), d d 0 nonistnt

12 AP Eam Problms Captrs 70. If 7, tn, in trms of and, d d 7. If 0, tn wn =, d d If 7. If 8, tn at t point (, ), 5 6, tn d d 0 7. If d d d, tn d Tangnts and Normals 75. T slop of t lin tangnt to t curv 0 at, is T slop of t lin tangnt to t grap of ln at t point wr = is 0

13 AP Eam Problms Captrs 77. T slop of t lin tangnt to t grap of ln at = is T slop of t lin normal to t grap of lnsc at = is nonistnt 79. An quation of t lin tangnt to t grap of f ( ) at t point, is 80. An quation of t lin tangnt to t grap of at t point (, 5) is An quation of t lin tangnt to t grap of cos at t point (0, ) is = = 0 8. Lt f b t function givn b f ( ) and lt g b t function givn b valu of do t graps of f and g av paralll tangnt lins? (Calculator) g( ) 6. At wat Wic of t following is an quation of t lin tangnt to t grap of point wr f( )? (Calculator) f ( ) at t

14 AP Eam Problms Captrs 8. An quation of t lin normal to t grap of 7 at t point wr = is Position/Vlocit/Acclration 85. A particl movs along t -ais so tat at an tim t 0 its position is givn b ( t) t t 9t. For wat valus of t is t particl at rst? Non onl onl 5 onl and 86. T position of a particl moving along a straigt lin at an tim t is givn b Wat is t acclration of t particl wn t =? s( t) t t A particl movs along a lin so tat at tim t, wr 0 t, its position is givn b t s( t) cost 0. Wat is t vlocit of t particl wn its acclration is zro? (Calculator) A particl movs along t -ais so tat its position at tim t is givn b wat valu of t is t vlocit of t particl zro? ( t) t 6t 5. For T position of a particl moving along t -ais is ( t) sint cost Wn t, t acclration of t particl is for tim t

15 AP Eam Problms Captrs Fr Rspons Qustions 5. Lt f b t function givn b f( ). (a) Find t domain of f. (b) Writ an quation for ac vrtical and orizontal asmptot for t grap of f. (c) Find f ( ). (d) Writ an quation for t lin tangnt to t grap of f at t point 0, f (0).. Lt f b t function givn b f ( ) 6. (a) Find t domain of f. (b) Dscrib t smmtr, if an, of t grap of f. (c) Find f ( ). (d) Find t slop of t lin normal to t grap of f at = 5.. A particl movs on t -ais so tat its position at tim t 0 is givn b ( t) t t. (a) Find t acclration of t particl at t = 0. (b) Find t vlocit of t particl wn its acclration is 0. (c) Find t total distanc travld b t particl from t = 0 to t = 5. 5

16 AP Eam Problms Captrs 6