COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II

Transcription

1 COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios. Do ot us clcultor.. Sktch th grph of th fuctios y, us clcultor. y, 5 y, y o th sm systm of coordits. Do ot. Fid th domi of f( ).. ( Wht is o-to-o fuctio? ( How c you tll from th grph of fuctio whthr it is o-to-o? 5. A fuctio f( ) is giv by th tbl blow. Dtrmi if it is o-to-o. 5 6 f( ) ( Wht is fuctio? Wht r its domi d rg? ( Wht is th grph of fuctio? (c ) How c you tll whthr giv curv is th grph of fuctio? 7. Fid formul for th ivrs of ch of th followig fuctios. Also idict th domi d rg of f f ( ) d ( ) :

2 ( f ( ) ( f( ) (c) f ( ) ( f( ) 8. ( Wht is th turl logrithm? ( Sktch th grphs of th turl logrithms fuctio d th turl potil fuctio with commo st of is. Do ot us clcultor. 9. Usig proprtis of logrithms, clcult: ( log6 6 ( log8 (c) log log 5. Solv ch qutio for : ( l ( 5 (c) l l 7 (. Solv ch iqulity for : ( ( l Crfully justify your swrs/mthod. (c).. Dtrmi whthr th sttmt is tru or fls. If it is tru, pli why. If it is fls, pli why or giv mpl tht disprovs th sttmt: ( If f is fuctio with f ( s) f ( t) th s t. ( If f is fuctio, th f ( s t) f ( s) f ( t). (c ) If d f is dcrsig fuctio, th f f. ( Evry fuctio f hs ivrs fuctio f. () Th idtity fuctio f :, f ( ) is its ow ivrs (f) If f is o-to-o fuctio, th f ( ). f( ) (g) If 6, th l 6l. (h) If b, th l l b.

3 (i) (j) If f( ) is icrsig d f( ) o I, th g ( ) f( ) is dcrsig o I.. Clcult: (. Justify your mthod. ( '( ) A f for f ( ) B (c ) f '( ) for f ( ) ( f '( ) for f ( ) cos () f '( ) for f ( ) kt (f) f '( ) for f ( ) (g) f '( ) for f( ) (h) f '( ) for 5 f ( ) cos (i) f '( ) for (k) f '( ) for ( ) f (j) f '( ) for f ( ) log f ( ) l 5 (l) '( ) f for f ( ) l (m) f '( ) d fid th domi of f( ) () f '( ) for f ( ) l l (o) y y ' if y (p) y ' for y / (q ) Th grl tidrivtiv F ( ) of PROBLEMS PLUS: f( ). O of th lgths of right trigl hs lgth cm. Eprss th lgth of th ltitud prpdiculr to th hypotus s fuctio of th lgth of th hypotus. 5. Th ltitud prpdiculr to th hypotus of right trigl is cm. Eprss th lgth of th hypotus s fuctio of th primtr of th trigl. 6. Solv th iqulity 5. Show ll stps. 7. Evlut log log log 5... log.

4 8. ( Show tht th fuctio f ( ) l is odd fuctio. ( Us prt ( to show tht f( ) is o-to-o ovr its domi d th fid th ivrs of f( ). 9. Solv th iqulity l.. Us proof by cotrdictio to show tht log 5is irrtiol umbr.. Show tht si th t sih.. Evlut si. DO NOT us l Hospitl s Rul. B. OTHER PROBLEMS:. Clcult th its:. Clcult th its of typ : 5. Clcult f '( ) for: c) si f ( ) sih( ) cosh( ) f ( ) c) f ( ) l f ( ) ) f) f( ) l t f ( ) g) f( ) h) f( ) l i) f( ) f ' for f '( ) rcsi j) k) f ' for f ( ) rct l) f ' for f ( ) l l

5 6. Clcult th followig itgrls: rct( ) d d d d rct( ) d d d rct( t) c) d dt d ) cos l( ) d ) t( )l cos( ) d f) d g) d h) sc( ) t( ) d sc( ) / t( ) sc ( t) dt j) i) t d k) d l) d m) log ( ) d 5

6 II. CHAPTER Tchiqus of Itgrtio A. FROM CALCULUS BY J. STEWART:. Clcult th followig itgrls. Show ll stps: cos m d l d c) g) 6 cos d ) t d h) si d f) cos t d i) l d t d d cos j) d k) d l) d m) d ) d o) 6 d p) d r) l d s) / d t) d u) csc cot d v) d ) A) u du z) u u 8 d B) 7 t sc d w) csc si d C) cos d d D) d E) / si d cos F) t t d d H) G) cos d 6

7 B. PROBLEMS PLUS:. d Hit: Try to void th lgthy prtil frctios pproch by usig substitutio. 7. If is turl umbr, show tht. Show tht I I d!, for vry.! l d!. Hit: Strt by showig tht if I dots th itgrl, th. Clcult d. 5. Th circl with rdius show i th figur touchs th curv y twic. Fid th r of th rgio tht lis btw th two curvs. 7

8 C. OTHER PROBLEMS:. Clcult d.. Clcult si si cos d. d.. Clcult. Clcult si cos t t dt. t t 5. Clcult d. 6. Clcult cos si si cos si d. 7. Clcult d 8*. Cosidr th fuctios f : R R dfid by: ( ) f d f ( ) f( t) dt for N d for R. Show tht ( ) f, R. Clcult f ( ), R. c) Usig mthmticl iductio, show tht f ( )..., N d for R.!!! 8

9 III. CHAPTER Impropr itgrls d L Hospitl s Rul A. FROM TEXTBOOK:. Clcult th followig its (usig ithr mthods from Clculus I or th l Hospitl s Rul, whichvr is most fficit): l cos 7 t sc c) l l si l / / ) f) / f) g) (discuss ftr th vlus of ) h) l( ) (discuss ftr th vlus of ). I th followig problms, vlut th giv impropr itgrls or show tht it divrgs: t dt d c) d si tdt ) f) cos d d l( ) g) d h) 9 9 d i) d j) d k) 6 d 9

10 B. FROM CALCULUS BY J. STEWART:. Clcult th followig its (usig ithr mthods from Clculus I or th l Hospitl s Rul, whichvr is most fficit): 9 5 b (discuss ftr th vlus of d b ) c) t p q t rcsi ) f) h) t 5 g) b (discuss ftr th vlus of d b ). Epli why th followig itgrls r impropr: d / c) sc d 56 d 5. Which of th followig itgrls r impropr? Why? d d c) l d 6. Dtrmi whthr ch itgrl is covrgt or divrgt. Evlut thos tht r covrgt: d 5 d c) dz z d ) d f) rct d g) 9 9 d h) sc d i) t t dt j) d k) rct d

11 C. PROBLEMS PLUS: 7. Cosidr f :, giv by: Clcult f t f t dt, if f( ), if. 8. If f ' is cotiuous, f(), d f '() 7, vlut f f ( ) ( 5 ). 9. Clcult h h h d. Clcult d. Th figur blow shows sctor of circl with ctrl gl. Lt A( ) b th r of th sgmt btw th chord PR d th rc PR. Lt B( ) b th r of th trigl PQR. Fid A( ) B ( ).. Fid th rl umbrs d b such tht si( ) b

12 D. OTHER PROBLEMS:. Clcult th followig its (usig ithr mthods from Clculus I or th l Hospitl s Rul, whichvr is most fficit): 5 c) si l t ) f) cot g) cot h) l i) rct j) k) t t l) m) ) si o)

13 IV. CHAPTER Ifiit Sris A. FROM CALCULUS BY J. STEWART:. Dtrmi whthr th squc covrgs or divrgs. If it covrgs, fid its it (rmmbr th Clculus I d Clculus II mthods hr): 5 c) ) f) g) h) cos i)!! j) k) si l). Wht is th diffrc btw squc d sris? Wht is covrgt sris? Wht is divrgt sris?. Lt. Dtrmi whthr is covrgt. Dtrmi whthr is covrgt.. Dtrmi whthr th sris is covrgt or divrgt. If it is covrgt, fid its sum: c) f) g) ) h) rct( )

14 . Dtrmi whthr th sris is covrgt or divrgt. Us y of th tsts lrt, clrly idict th tst tht you usd, d show ll stps:... c) ) f) g) i) h) 5 j) l( )l l( ) k) l) 6 m)! )! o) q) si p) r) s). How my trms of th prtil sum for th sris do w d to dd i ordr for this prtil sum to pproimt th full sum of th sris to th idictd ccurcy? S S. S S.

15 5. Dtrmi whthr th sris is bsolutly covrgt, coditiolly covrgt or divrgt: c)! / 6. Fid th itrvl of covrgc for th powr sris: ) c) Fid powr sris rprsttio for th fuctio d dtrmi its itrvl of covrgc: f( ) ( ) f c) f( ) 5 f( ) (Hit: us prtil frctios dcompositio first) ) f ( ) l 5 f) g) f ( ) t g) f( ) f( ) l 8. Fid th McLuri sris for f( ) usig th formul for McLuri dirctly, or trsformtios of kow sris, whichvr is most covit: f ( ) si f ( ) 5 c) f ( ) cosh( ) f ( ) si (Hit: You c us tht si cos( ) ) ) f ( ) cos( ) 5

16 9. Fid th Tylor sris of f( ) roud giv. Us dirct clcultios (th Tylor psio thorm) or othr mthods, s covit: f ( ), f ( ), c) f ( ), f ( ) l( ), ) f ( ) cos( ), ) f ( ), 9. Evlut th idfiit itgrl s ifiit sris: si( ) d cos d c) d si d. Us th McLuri sris for Us th McLuri sris for si( ) to clcult to clcult si. corrct to fiv dciml plcs. corrct to fiv dciml plcs.. Fid th Tylor polyomils up to dgr 6 for f ( ) si( ) ctrd t. Grph f( ) d ths polyomils o commo scr. Evlut f( ) d ch of ths polyomils t, d. c) Commt o how th Tylor polyomils covrg to f( ).. For wht vlus of dos th sris l( ) covrg?. Us th sum of th first ight trms to pproimt th sum of th sris 5 md i this pproimtio.. Estimt th rror 6

17 B. PROBLEMS PLUS: 5. If f ( ) si, fid f ().. A fuctio f( ) is dfid by f( ) discotiuous? f( ). Whr is f( ) cotiuous d whr is. To costruct sowflk curv, strt with quiltrl trigl with sids of lgth. Stp i th costructio is to divid ch sid ito thr qul prts, costruct quiltrl trigl o th middl prt (o th outsid of th origil trigl), d th dlt th middl prt. Stp is to rpt Stp for ch sid of th rsultig polygo. This procss is th rptd t ch succdig stp. Th sowflk curv is th curv tht rsults from rptig this procss idfiitly (do Stp d lt ). Lt s, l d p rprst th umbr of sids, th lgth of sid d th totl lgth of th th pproimtig curv (th curv obtid ftr Stp of th costructio), rspctivly. Fid formuls for s, l d p. Show tht p. c) Sum ifiit sris to fid th r closd by th sowflk curv. Prts d c) show tht th sowflk curv is ifiitly log but closs oly fiit r. 7