MATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)

Transcription

1 MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide ddiionl prcice for he meril h will be covered on he finl em When solving hese problems keep he following in mind: Full credi for correc nswers will only be wrded if ll work is shown Ec vlues mus be given unless n pproimion is required Credi will no be given for n pproimion when n ec vlue cn be found by echniques covered in he course The nswers, long wih commens, re posed s sepre file on hp://mhrizonedu/~clc 1 A funcion f () is coninuous nd differenible, nd hs vlues given in he ble below f () Fill in he ble wih pproime vlues for he funcion f () f () Arrnge he following numbers from smlles (1) o lrges (5) using he grph of f shown below: f( + h) f() lim h h The slope of f = 1 f (16) The verge re of chnge of f from = 1 o = 4 dy d = 8 y Suppose g () = 3 nd g () = 1 Find g( ) nd g ( ) ssuming ) g ( ) is n even funcion b) g ( ) is n odd funcion 4 For priculr pin medicion, he size of he dose, D, depends on he weigh of he pien, W We cn wrie D = fw ( ) where D is mesured in milligrms nd W is mesured in pounds ) Inerpre f (15) = 15 nd f (15) = 3in erms of his pin medicion b) Use he informion in pr ) o esime f (155)

2 5 Use he grph of f ( ) given below o skech grph of f ( ) y 6 Deermine if he semen is rue (T) or flse (F) No need o mke correcions ) If g ( ) is coninuous =, hen g ( ) mus be differenible = b) If r ( ) is posiive hen r ( ) mus be incresing c) If ( ) is concve down, hen ( ) mus be negive d) If h ( ) hs locl mimum or minimum = hen h ( ) mus be zero 7 Skech grph of f ( ) h sisfies ll of he following condiions: i) f ( ) is coninuous nd differenible everywhere ii) he only soluions of f( ) = re =,, nd 4 iii) he only soluions of f ( ) = re = 1 nd 3 iv) he only soluion of f ( ) = is = Find he following limis for f( ) = e ) lim f ( ) b) lim f ( ) c) lim f ( ) d) 1 + lim f ( ) e) lim f ( ) e e 9 Find lim h h nd some vlue (3 + h) (3) by recognizing he limi s he definiion of f ( ) for some funcion f 1 A priculr cr ws purchsed for $5, in 4 Suppose i loses 15% of is vlue ech yer Le V () represen he vlue of he cr s funcion of he yers since i ws purchsed Find V () nd use i o find he ec vlue of V (3) 11 Use he grph of f ( ) he righ o find he vlue(s) of so h ) f( ) = b) f ( ) = c) f ( ) = f ( )

3 1 Use he grph of f ( ) he righ o find inervls where ) f ( ) is decresing b) f ( ) is concve down f ( ) Le be posiive consn Find dy for ech of he following: d 3 ) y= rcn( +) b) y = c) y = cos ( ) d) y= + e) y = sinh 14 Le f ( ) be coninuous funcion wih f (4) = 3 nd f (4) = 5 ) Find he equion of he ngen line o h ( ) = f( ) + 7 = 4 b) Is g ( ) = incresing or decresing = 4? f ( ) c) Find k () where k ( ) = f( ) ( ) d) Find m (4) where m ( ) = e f 3 15 If g ( ) = nd g ( ) = 3, find 16 Find he indiced derivives: ) dm m m = o 1 v c dv for ( ) b) g ( ) for g ( ) = < < 17 Le f( ) = 4 = 4 4 > ) Is f ( ) coninuous = 1? Differenible = 1? b) Is f ( ) coninuous =? Differenible =? c) Find f ( ) Epress your nswer s piecewise funcion

4 18 Torricelli s Theorem ses h if here is hole in coniner of liquid h fee below he surfce of he liquid, hen he liquid will flow ou re given by R( h ) = gh where g = 3f sec Find liner funcion h cn be used o pproime his re for holes h re close o 5 fee below he surfce of he wer 19 For wh vlue(s) of k will 3 f ( ) k k k = + + hve n inflecion poin = 5? The funcion ( ) is defined implicily by he equion y cos( π ) y = ln y ) Find he vlue of he derivive of y wih respec o he poin (1, 1) b) Find he equion of he ngen line o he curve (1, 1) 3 1 A cble is mde of n insuling meril in he shpe of long, hin cylinder of rdius R I hs elecricl chrge disribued evenly hroughou i The elecricl field, E, disnce r from he cener of he cble is given below k is posiive consn kr r R E = kr r > R r ) Is E coninuous r = R? b) Is E differenible r = R? c) Skech E s funcion of r d) Find de dr 1 Le f() = + for Find 3 ) he criicl poin(s) nd deermine if i is locl mimum or minimum b) he inflecion poin(s) c) he globl mimum nd minimum on he given inervl 3 Le f ( ) 3 > 1 ) he coordines of he locl mim nd he locl minim b) he coordines of he inflecion poin(s) 3 4 = + wih consn Find (nswers will be in erms of ) 4 Find he ec vlue of he following limis: π sin( θ ) ) lim b) lim c) lim rcn π sin θ sin(7 θ )

5 B 5 Consider he fmily of funcions f() = Find he vlues of A nd B so h f () hs 1 + A criicl poin (4,1) 6 Consider he fmily of funcions y () = ln for > ) Find he -inercep Your nswer will be in erms of b) Find he criicl poin nd deermine if i is locl mimum or minimum (or neiher) 7 Find he vlues of, b, nd k so h he prmeric equions given below rce ou circle of rdius 3 cenered (,4) = + kcos, y= b+ ksin, π = Consider he lines prmeerized by y = 4 9 nd = 5+ 6 y = c+ 8 ) For wh vlue of c, if ny, will hese wo lines be prllel? b) For wh vlue of c, if ny, will hese wo lines inersec (5, 3)? 9 Suppose n objec moves in he y plne long ph given by prmeric equions 3 = 3 +1, y = 4 1, ) Deermine he ime when he objec sops Where will i sop? b) Deermine he ime when he objec his he -is 3 Wire wih ol lengh of L inches will be used o consruc he edges of recngulr bo nd hus provide frmework for he bo The boom of he bo mus be squre Find he mimum volume h such bo cn hve 31 Wh re he dimensions of he lrges recngle h cn be inscribed under he grph of y = 5 so h one side is on he -is? 3 A closed recngulr bo wih squre boom hs fied volume V I mus be consruced from hree differen ypes of merils The meril used for he four sides coss $18 per squre foo; he meril for he boom coss $339 per squre foo, nd he meril for he op coss $161 per squre foo Find he minimum cos for such bo in erms of V w c 33 The speed of wve rveling in deep wer is given by V( w) = k c + w where w is he wvelengh of he wve Assume c nd k re posiive consns Find he wvelengh h minimizes he speed of he wve

6 34 The grph of he funcion f ( ) nd is derivive f ( ) re given he righ ) Deermine which grph is f ( ) nd which grph is f ( ) b) Use he grphs o find he vlues of h mimize nd minimize he funcion g ( ) = f( e ) The grph below on he lef shows he number of gllons, G, of gsoline used on rip of M miles The grph below on he righ shows disnce rveled, M, s funcion of ime, in hours since he sr of he rip You cn ssume he segmens of he grphs re liner G (gllons) (7, 8) (1, 46) M (miles) ( 1, 7 ) (, 1 ) M (miles) (hours) ) Wh is he gs consumpion in miles per gllon during he firs 7 miles of he rip? During he ne 3 miles? b) If G= f( M) nd M = h (), wh does k () = f( h ()) represen? Find k(5) c) Find k (5) nd k (15) Wh do hese quniies ell us? 36 A cmer is focused on rin s he rin moves long rck owrds sion s shown he righ The rin rvels consn speed of 1 km hr How fs is he cmer roing (in rdins/min ) when he rin is km from he cmer? 37 Snd is poured ino pile from bove I forms righ circulr cone wih bse rdius h is lwys 3 imes he heigh of he cone If he snd is being poured re of 15 f 3 per minue, how fs is he heigh of he pile growing when he pile is 1 f high? h r

7 38 A volge, V vols, pplied o resisor of R ohms produces n elecricl curren of I mps where V = I R As he curren flows, he resisor hes up nd is resisnce flls If 1 vols is pplied o resisor of 1 ohms, he curren is iniilly 1 mps bu increses by 1 mps per minue A wh re is he resisnce chnging if he volge remins consn? 39 A funcion f () is coninuous nd differenible, nd hs vlues given in he ble below The vlues in he ble re represenive of he properies of he funcion f () ) Find upper nd lower esimes for b) Find 16 f () d 1 18 f () dusing 4 1 n = 4 Severl objecs re moving in srigh line from ime = o ime = 1 seconds The following re grphs of he velociies of hese objecs (in cm/sec ) ) Which objec(s) is frhes from he originl posiion he end of 1 seconds? b) Which objec(s) is closes o is originl posiion he end of 1 seconds? c) Which objec(s) hs rveled he grees ol disnce during hese 1 seconds? d) Which objec(s) hs rveled he les disnce during hese 1 seconds? Velociy of Objec A Velociy of Objec B Velociy of Objec C Velociy of Objec D

8 41 Illusre he following on he grph of f ( ) given below Assume F ( ) = f( ) ) f () b f() b) f ( b) f( ) b fhl fhl b b c) Fb () F () d) Fb ( ) F ( ) b fhl fhl b b 4 A funcion g () is posiive nd decresing everywhere Arrnge he following numbers from smlles (1) o lrges (3) 1 g ( ) Δ k = 1 k 9 g ( k ) Δ k = n lim g ( k ) Δ n k = 1 43 Le b be posiive consn Evlue he following: b+ ) ( b + 1) d b) d c) d d) b+ 1+ d ( b) 44 Find he res of he regions Include skech of he regions ) The region bounded beween y= (4 ) nd he -is b) The region bounded beween y= + nd = 3 + y 45 I is prediced h he populion of priculr ciy will grow he re of p () = 3 + (mesured in hundreds of people per yer) How mny people will be dded o he ciy in he firs four yers ccording o his model?

9 46 A ime = wer is pumped ino nk consn re of 75 gllons per hour Afer hours, he re decreses unil he flow of wer is zero ccording o r ( ) = 3( ) + 75, gllons per hour Find he ol gllons of wer pumped ino he nk 47 Use he grph of g ( ) given he righ o skech grph of g ( ) so h g () = 3 g'() A cr going 8 f sec brkes o sop in five seconds Assume he decelerion is consn ) Find n equion for v (), he velociy funcion Skech he grph of v () b) Find he ol disnce rveled from he ime he brkes were pplied unil he cr cme o sop Illusre his quniy on he grph of v () in pr ) c) Find n equion for s (), he posiion funcion Skech he grph of s () 49 Consider he funcion F ( ) = e d ) Find F () b) Find F ( ) c) Is F ( ) incresing or decresing for? d) Is F ( ) concve up or concve down for? 5 The verge vlue of f from o b is defined s 3 π f( ) = over he inervl cos 4 1 b ( ) b f d Find he verge vlue of ( ) 51 According o book of mhemicl bles, ln 5 + 4cos d = π ln π b) Find ln ( 5 4cos( )) π + π ) Find ln ( cos ) d π d

10 5 Use he grph of f ( ) below o nswer he following Circle True or Flse ) b) 1 f ( d ) f( d ) True Flse 4 5 f ( d ) f( d ) True Flse 1 1 f d f d True Flse c) ( ) ( ( )) 1 d) f( ) d True Flse e) 1 f ( ) d 1 True Flse ) If 6 f( ) d= 17, find b) If g ( ) is n odd funcion nd c) If ( ) is n even funcion nd 5 f ( d ) h ( h ) 3 3 gd ( ) =, find gd ( ) ( ) 3 d= 5, find hd ( ) 54 Use he grph of g ( ) he righ o deermine which sign is pproprie: g ( ) ) gc ( ) < = > gd ( ) b) g ( B) < = > g ( C) A B C D c) g ( A) < = > g ( B)