. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively? both re lwys discotiuous t (B) both c be cotiuous t (C) both re lwys cotiuous t (D) f g c be cotiuous t but f g is lwys discotiuous t (E) f g is lwys discotiuous t but f g c be cotiuous t.. Oil spilled from ruptured tker spreds i circle whose re icreses t costt rte of 6 mi / hr. How fst is the rdius of the spill icresig whe the re is 9 mi? mi/ hr (B) mi/ hr (C) mi/ hr (D) mi/ hr (E) mi/ hr. Regio R is eclosed by the -is, the verticl lie, d oe-to-oe fuctio y f ( ), stisfyig f ( ) d f ( ) b. Which of the followig defiite itegrl describes the volume of the solid S obtied by rottig the regio R bout the y-is? I = II= b III = IV = f( ) d { b [ f y[ f { b ( y)] ( y)] dy [ f ( )] } dy } d oly I (B) I d II (C) I d III (D) III d IV (E) I d IV 4. Where should the poit P be chose o the lie segmet AB so s to mimize the gle? 5 bove A (B) (D) 5 5 bove A (E) 5 5 5 bove A bove A (C) 5 5 bove A
5. Give f '( ) eists, which of the followig re lwys true? f ( h) f ( ) f ( ) f ( h) () f '( ) lim () f '( ) lim h h h h f ( h) f ( ) f ( h) f ( h) () f '( ) lim (4) f '( ) lim h h h h f ( h) f ( ) (5) f '( ) lim h h oly () is true (B) oly () d () re true (C) oly () d (4) re true (D) oly () d () re true (E) ll re true, if c 6. Let f ( ) b, if c for c >. If f '( c) eists the b (B) 8b (C) (8/9) b (D) (7 / 4) b (E) oe of the bove 7. Let,if,if f ( ) d g ( ) 4,if 4 f ( t) dt. The g '( ) = /,if 4 4 (B) (C) 4 (D) (E) does ot eist 4 4 8 4 4 4 4 9 9 9 9 8. The limit lim [ ] is best pproimted by (B) (C) (D) (E) 8 9 9. Cosider the equtio M X B, where X d B re colum vectors of legth, d M is mtri. Suppose the row reductio is used to solve equtio for X d the followig ugmeted mtri is obtied. 4 5 7 6 Which of the followig sttemet is correct? I MX B hs o solutio. II M X B hs ifiitely my solutios. III M does ot hve multiplictive iverse. I oly (B) II oly (C) III oly (D) both I d III (E) both II d III
. Evlute d. (B) (C) (D) (E) It is diverget improper itegrl t e cost. The legth of the curve described by C : t y e si t e (B) ( e ) (C) e, t is (D) ( e ) (E) e. The itervl of covergece of the series ( 4) is (, ) (B) [, ) (C) (, ] (D) [, ) (E) (, ]. Mr. Clculus hs $ d he wts to put i svig ccout to er some iterest for yers. Below re the optios offered by the TMTA bk: Optio I..8% APR (Aul Percetge Rte), iterest compouded every 4 moths Optio II..%APR, iterest compouded every 6 moths Optio III..7%APR, iterest compouded cotiuously. Reviewig these pls, Mr. Clculus rks them, ccordig to the iterest ered t the ed of yers, to be: Optio II > Optio I > Optio III (B) Optio I > Optio III > Optio II (C) Optio I > Optio II > Optio III (D) Optio III> Optio I > Optio II (E) Optio II > Optio III > Optio I 4. The sum of the series e e (B) e 4 e e e e is!! 4! (C) e e (D) e (E) It diverges 5. Let : for positive itegers. The = (B) (C) - (D) (E)
6. Cosider the followig sttemets: I. If f (t) d y g(t) re differetible, the dy dy dt, d / dt. d d dt d y d y dt II. If f (t) d y g(t) re twice differetible, the. d d dt III. The polr curves r si d r si hve the sme grph. 4 6 IV. The prmetric equtios t, y t hve the sme grph s t, y t. oly I is true (B) oly I d III re true (C) oly II is flse (D) oly IV is flse (E) they re ll flse. 7. A fuctio f () hs domi [,] d rge [,]. Wht re the domi d rge, respectively, of the fuctio g () defied by g ( ) f ( )? [-, ], [-, ] (B) [-, ], [, ] (C) [, ], [-, ] (D) [, ], [-, ] (E) [, ], [, ] 8. The prbol y b c hs verte ( p, p) d y-itercept (, p), where p. Wht is b? -p (B) (C) (D) 4 (E) p 9. A widow hs the shpe of rectgle surmouted by semicircle. If the perimeter of the widow is 5 feet, fid the legth i feet of the rdius of the semicircle tht will llow the mimum mout of light to eter through the widow. 5 (B) /( ) (C) 5/(4 ) (D) 5/(4 ) (E) /(4 ). If y t( y) dy, the = d sec ( y) (B) sec ( y) (E) sec ( y) y sec ( y) (C) ysec ( y) (D) sec ( y) sec ( y)
. If f () is differetible d ever equl to o (, ), the the derivtive of t ( ) is equl to f ( ) the derivtive of t ( f ( )) (B) the reciprocl of the derivtive of t ( f ( )) (C) the squre of the derivtive of t ( f ( )) (D) the egtive of the derivtive of t ( f ( )) (E) oe of the bove. The fuctio f () is cotiuous for [,] d hs locl (reltive) miimum t = d =. Lbel sttemets I, II, d III s lwys true, sometimes true, or ever true. I. f '() II. f () hs iflectio poit betwee = d = III. f '() I. lwys true, II. lwys true, III. ever true (B) I. sometimes true, II. lwys true, III. ever true (C) I. sometimes true, II. lwys true, III. sometimes true (D) I. sometimes true, II. sometimes true, III. ever true (E) I. sometimes true, II. sometimes true, III. sometimes true. If f () is ivertible fuctio, d g ( ) f ( ) 5, wht is g ( )? 5 f ( ) 5 (B) f ( ) 5 (C) (D) f ( ) 5 (E) f ( ) f ( ) 5 4. Let u, for =,,. If u coverges, the (D) u coverges (B) u u diverges (C) diverges (E) oe of the bove u coverges 5. The ceter C (, b), i rectgulr coordites, of the coic sectio represeted i polr form r is t 4 8cos (,) (B) (,) (C) (,) (D) (, ) (E) (, ) 4 4 6. The solutio of the equtio Wht is b? 7 7 8 c be epressed i the form 7 7. log b 7/5 (B) 7/8 (C) 8/7 (D) 5/8 (E) 5/7
7. Evlute d C (B) C (C) C l (D) l C (E) l C 8. A cr is drivig log drk rod whose shpe is give by the grph of y. As the cr trvels log the rod from left to right, whe will its hedlights be poited directly t the poit (,8)? Whe the cr is t (4,6) (B) The cr s hedlights will ever poit t (,8) (C) Whe the cr is t (,4) d lso t (4,6) (D) Whe the cr is t (8/,64/9) (E) Whe the cr is t (,4) 9. A observer is stdig ectly mile south of est-west highwy. He wtches cr trvelig o the highwy d otes tht whe the gle betwee orth d his lie of sight to the cr is 45, the tht gle is icresig t rte of / sec. How fst is the cr trvelig t this time? 4 mile/hr (B) mile/hr (C) mile/hr (D) mile/hr (E) mile/hr. If ( ) ( ) f ( ) e e, fid '() ( e e ( e e ) (E) ( e e ) ) d f = f ( ) d (B) (C) f is ot differetible t (D) ( e e ) 4. lim = does ot eist (B) (C) l (D) (E). If f () is twice differetible d g( ) f ( ), wht is g "()? f "() 4 f '() f () (B) f "() f '() (C) f "() f '() f () (D) f f () f "() ( '()) (E) 4 f "() f '()
. If, d equls l( ) (B) l( ) (C) ( t ) (D) t (E) ( ) 4. Suppose tht f () is cotiuous. If f ( ) d, f ( ) d, f ( ) d 7, fid f ( 4) d /4 (B) 5/4 (C) (D) 9/4 (E) 7/4 5. Suppose, b, c, d. The (B) (C) (D) db c b lim equls c d (E) c 4 8 e 6. lim equls (B) (C) (D) the limit does ot eist (E) e 7. The regio bouded by the grphs of y ( ) d y is rotted bout the y- is to form solid. The volume of the solid is (B) /4 (C) / (D) /6 (E) / 8. The grph of f '( ) is s pictured. Whe is the grph of f () cocve up? (, ) (, ) (B) (,) (4, ) (C) (,) (, ) (D) (,) (4, ) (E) (, ) (,4)
9. I the digrm the rc of the circle hs legth 5 d the chord hs legth 4. The mesure of the gle t the ceter of the circle, i rdis, is best pproimted by.8 (B).6 (C).4 (D). (E). 4. I the epsio of coefficiet of 8? 7 4 ( )( ), wht is the 95 (B) 96 (C) 4 (D) 5 (E) 78
Key: E B D C B B D E B C E C 4 D 4 C 4 C 4 B 5 B 5 D 5 A 5 E 6 D 6 B 6 C 6 C 7 B 7 B 7 D 7 D 8 C 8 D 8 E 8 A 9 E 9 C 9 A 9 D A B B 4 D Prepred by: Mthemtics Fculty Teessee Techologicl Uiversity Cookeville, TN Coordited by: Yug-Wy Liu