Empl 6: Lt rprsnt th mss, in ponds, of rdioctiv lmnt whos hlf-lif is 4000 rs. If thr r 00 ponds of th lmnt in n inctiv min, how mch will still rmin in 000 rs? Eprss or nswr to or mor dciml plc ccrc. 4 Assignmnt 5-6 For Problms -4, find gnrl soltion of ch diffrntil qtion. *Solv for in Problms nd 4.. d d *.. *4. ( ) d d For Problms 5-6, find prticlr soltion of th diffrntil qtion with th givn initil condition. (Rmmbr to writ or soltions in th form f.) 5. d d d nd () 4 6. nd () d 7. Find n qtion of fnction which contins th point (,) nd whos slop is for ch point on th grph of th fnction. 8. $000 is plcd into crtifict of dposit (CD) in which intrst is compondd continosl t rt of 5 % pr r (ctl rt of rtrn will b highr d to rt componding of intrst). Us or clcltor nd th forml A P to find:. th mont tht th CD wold b worth in r. 5 rs. 0 rs. b. th tim it wold tk th CD to b worth $,00. c. th tim it wold tk th CD to dobl in vl. 9. Sppos 00 bctri r introdcd into cltr to std thir rt of growth. Two ds ltr, th cltr is fond to contin 00 bctri. Assming th rt of growth is proportionl kt to th nmbr of bctri prsnt C, how mn bctri will b prsnt in mor ds (5 ds ftr th strt)? 0. Find th hlf-lif of rdioctiv isotop if 4.9 grms ot of n initil 5 grms of th isotop rmin ftr 0 rs.
. An isotop of crbon 44 4 ( C ) is sd for stimting how long go crtin living orgnisms wr 4 on rth. (Th mthod is clld crbon dting.) Th hlf-lif of C is pproimtl 570 rs. If th skll of n ncint primt contins 4 0% (.) of th C prsnt in th skll of modrn primt of similr spcis, stimt how long go th ncint primt livd (to th nrst thosnd rs).. A workr t hzrdos wst plnt ws ccidntll posd to toic chmicls which wr bsorbd into his bloodstrm. Upon fling ill, th workr wnt to hospitl nd hd som blood drwn for tsting. Th concntrtion of chmicl in th drwn blood ws fond to b.058 mg/ml. Epnsiv mdiction ws dministrd to contr th ffcts of th chmicl in th blood, bt th doctor on dt knw tht th concntrtion of th chmicl in th bloodstrm wold hv to dcrs grdll ovr tim ccording to th Bsic Lw of kt Eponntil Dc C. Mdiction wold hv to b dministrd vr hor ntil th concntrtion ws blow.0050 mg/ml. Two hors ltr, blood ws gin drwn, nd it ws fond to contin chmicl concntrtion of.06 mg/ml. Th doctor skd lb tchnicin to do th following. (Yo do th sm):. Writ th prticlr soltion for ponntil dc for th chmicl in th ptint s blood. (Lt t = 0 rprsnt th tim tht blood ws first drwn.) b. Sktch grph of th fnction from Prt. c. Find ot how long it will b bfor th ptint cn b tkn off mdiction. d. Whn th ptint hs onl ngligibl mont of chmicl in his bloodstrm (lss thn.000 mg/ml), h cn b rlsd from th hospitl. Find ot how long th ptint hs to b hospitlizd (from th tim h first cm to th hospitl nd hd his blood drwn).. Occsionll, ptints posd to this chmicl sffr dmg to thir cntrl nrvos sstms. A mimm concntrtion of.00 mg/ml rqirs follow-p mintion. Th doctor stimtd tht th mimm concntrtion of th chmicl in th workr s bloodstrm occrrd hor ftr posr. Th ptint stimtd hor ftr posr wold hv bn bot hors prior to his blood bing drwn for th first tim. Shold th ptint b skd to rtrn for follow-p m? Wh or wh not? f. Find th hlf-lif for th chmicl in th bloodstrm for th ptint. Simplif in Problms nd 4.. ln Diffrntit in Problms 5-7. 4. lnb 5. t t 6. f( ) 7. f( ) ln Antidiffrntit in Problms 8-0. 8. 9. 4 g( ) 0. (ln t) t
. Find th r of th rgion bondd b clcltor., 0,, nd 0 withot sing. Withot sing clcltor, find th volm of th solid formd b rvolving th rgion bondd b, 0, 0, nd, bot th -is.. Withot sing clcltor, find th volm of th solid formd b sqr cross sctions prpndiclr to th -is, whos bs is th rgion bondd b 4, 0,, nd 4. 45 4. Rgion R is shown bondd b th grph of sin nd th -is. Withot sing clcltor find th volm of th solid formd whn rgion R is rvolvd bot th -is. Us clcltor for Problms 5-. 5. Find th primtr of rgion R from Problm 4. R 6. Find th vrg vl of t on [,]. Show n intgrl st-p. For Problms 7-0, f ( ) ln. 7. Find f (4) 8. Find 7 f( ) d 9. Solv ln 0 0. f is discontinos whn 0. Is th discontinit hol, n smptot, or jmp?. For prticl moving long stright pth with vlocit v(t) = tln(.7 t), t 0, s or clcltor to find:. th tim whn th prticl is t rst. b. th spd of th prticl t tim t = 4. c. th cclrtion of th prticl t tim t = 5. d. th totl distnc trvld b th prticl on th intrvl [, 5]..t. Convrt 7 from ponntil form to logrithmic form. Thn solv for.. Us log proprtis to condns th qtion. Thn solv for. log log ( ) log 4. f nd g r invrs fnctions. Th grph of g psss throgh th points (,), nd (, ). f( ) nd f(). Find:. g( ) b. g ()
46 Slctd Answrs:.. 5. ln ln C. C or C C or ln ln C 4 6. 4. C 7. 8. A $056.54, A5 $6.5, 9. 55 bctri 0. 49.74 or 49.74 rs. 9,000 rs 4. b A 0 $7.5 c..60 or.60 rs 6. f 7. ln 8. ln 4 ln f C.. ln. 6..746 or.747 8. 49.6 9..488 or.489..7 5.67 d. TD 4.50 or 4.5. t c. ln 7. 4b. CALCULUS BC UNIT 5 SUMMARY Shll Mthod Volm Forml : V rhd or d r 0, h 0 b Arc Lngth Forml: Arc Lngth b f d Eponntil nd Logrithmic Grphs: Grph of f( ) Grph of g ( ) ln 0 (,) ln 0 ln (0,) (,0) (,) All bsic ponntil f( ) th grphs shown bov. nd logrithmic g ( ) log grphs with 0 r similr to
47 CALCULUS BC UNIT 5 SUMMARY (CONTINUED) f( ) nd g( ) ln r invrs fnctions, so ln ln. ln Chng of Form Dfinition: Eponntil form Logrithmic form log Proprtis of Logrithms: onl tr whn 0 nd b 0 n. ln( b) ln b. ln ln b. nln b ln Chng of Bs: log Drivtivs of Invrs Fnctions: If f nd g r invrs fnctions, thn f( ) whr ( b, ) is point on th grph of f nd g () b (, b ) is th img point on th grph of g. Procdr for Logrithmic Diffrntition: vribl ponnt vribl bs. Tk th ntrl log (ln) of both sids of th qtion.. Simplif th right hnd sid of th qtion (Log Proprt ).. Diffrntit both sids. 4. Solv for. 5. Sbstitt for if ncssr. Diffrntil Eqtions: (Eqtions involving drivtivs.) Solv b sprting vribls nd intgrting both sids of th qtion. Procdr for Solving Diffrntil Eqtions. Rwrit s d (if ncssr). d. Mltipl both sids of th qtion b d (if ncssr).. Sprt vribls. (This is th most crcil stp.) 4. Intgrt both sids of th qtion. (Rmmbr to dd C to on sid.) 5. Solv for (if ncssr). 6. Us n initil condition to solv for C (if n initil condition is givn). Stps 5 nd 6 r intrchngbl. Eponntil Growth nd Dc: C kt
48 CALCULUS BC UNIT 5 SUMMARY (CONTINUED) Drivtivs form form (chin rl) d d Eponntil Rl. d d d d. d d Log Rl. 4. d d ln ln d d d d log log d d Intgrls form form (rvrs chin rl) Eponntil Rl. d C. d C d C d C Log Rl. d ln C d ln C Trig Rls 4. tn d ln cos C tn d ln cos C 5. cot d ln sin C cot d ln sin C