MATH 122B AND 125 FINAL EXAM REVIEW PACKET (Fall 2014)

Transcription

1 MATH B AND FINAL EXAM REVIEW PACKET (Fll 4) The following questions cn be used s review for Mth B nd. These questions re not ctul smples of questions tht will pper on the finl em, but they will provide dditionl prctice for the mteril tht will be covered on the finl em. When solving these problems keep the following in mind: Full credit for correct nswers will only be wrded if ll work is shown. Simplified ect nswers must be given unless n pproimtion is required. Credit will not be given for n pproimtion when n ect nswer cn be found by techniques covered in the course. The nswers, long with comments, re posted s seprte file on (follow the course link).. A function f() t is continuous nd differentible, nd hs vlues given in the tble below. t f() t Fill in the tble with pproimte vlues for the function f () t. t f () t. Arrnge the following numbers from smllest () to lrgest () using the grph of f shown below: f( + h) f() lim h h The slope of f t = f (6) The verge rte of chnge of f from = to = 4 dy d = 8 y Suppose f ( ) is differentible decresing function for ll. In ech cse, determine which quntity is lrger: ) f ( ) or f ( + ) b) f( + ) or f ( ) + f( ) 4. Let g () = 3 nd g () =. Find g( ) nd g ( ) ssuming ) gis ( ) n even function. b) gis ( ) n odd function.

2 . For prticulr pin mediction, the size of the dose, D, depends on the weight of the ptient, W. We cn write D= f( W) where D is mesured in milligrms nd W is mesured in pounds. ) Interpret f () = nd f () = 3in terms of this pin mediction. b) Use the informtion in prt ) to estimte f (). 6. Use the grph of f( ) given below to sketch grph of f ( ). f y ( ) 7. Determine if the sttement is true (T) or flse (F). No need to mke corrections. ) If gis ( ) continuous t =, then gmust ( ) be differentible t =. b) If r ( ) is positive then r ( ) must be incresing. c) If tis ( ) concve down, then t ( ) must be negtive. d) If h ( ) hs locl mimum or minimum t = then h ( ) must be zero. e) f ( ) is the tngent line to f( ) t =. f) Instntneous velocity cn be positive, negtive, or zero. 8. Sketch grph of f( ) tht stisfies ll of the following conditions: i) f( ) is continuous nd differentible everywhere ii) the only solutions of f( ) = re =,, nd 4 iii) the only solutions of f ( ) = re = nd 3 iv) the only solution of f ( ) = is = 9. Find the following limits for f( ) =. + e ) lim f( ) b) lim f( ) c) lim f( ) d) + lim f( ) e) lim f( ) e e. Find lim h h nd some vlue. (3 + h) (3) by recognizing the limit s the definition of f ( ) for some function f. A prticulr cr ws purchsed for $, in 4. Suppose it loses % of its vlue ech yer. Let Vtrepresent () the vlue of the cr s function of the yers since it ws purchsed. Find Vtnd () use it to find the ect vlue of V (3).

3 . Use the grph of f( ) t the right to find the vlue(s) of so tht ) f( ) = b) f ( ) = c) f ( ) = f( ) Use the grph of f ( ) t the right to find intervls where ) f( ) is decresing b) f( ) is concve down f ( ) Let be positive constnt. Find dy for ech of the following: d 3 ) y = rctn( + ) b) y = c) y = sec ( ) + d) y = + e) y sinh = f) y = e ln( ). Let f( ) be continuous function with f (4) = 3, f (4) = nd f (4) = 9. ) Find the eqution of the tngent line to h ( ) = f( ) + 7 t = 4. b) Is g ( ) = incresing or decresing t = 4? f( ) c) Find k () where k ( ) = f( ). ( ) d) Find m (4) where m ( ) = e f. e) Is ( ) j ( ) = f( ) concve up or concve down t = 4? 6. If 3 g ( ) = 6 + nd g ( ) = 3, find. 7. Determine where the slope of y( θ) = θ + cos (3 θ) will equl on the intervl θ π. 8. Find the indicted derivtives: ) dm m m = o v c dv for ( ) d b) ( z 9 ) dz

4 9. Find A nd B so tht f( ) is continuous nd differentible on the intervl (, ). + < f( ) = + A ( ) + B < <. Torricelli s Theorem sttes tht if there is hole in continer of liquid h feet below the surfce of the liquid, then the liquid will flow out t rte given by R( h) = gh where g = 3ft sec. Find liner function tht cn be used to pproimte this rte for holes tht re close to feet below the surfce of the wter.. For wht vlue(s) of k will = + + hve n inflection point t =? 3 f ( ) k k k. The function ( ) y is defined implicitly by the eqution cos ( π ) y = ln y ) Find the vlue of the derivtive of y with respect to t the point (, ) b) Find the eqution of the tngent line to the curve t (, ) ) Find d for t 3 + t+ 9= 7. dt b) For wht vlue(s) of t will the tngent line to the curve be horizontl? c) For wht vlue(s) of will the tngent line to the curve be verticl? 4. Let y = ln( + 3) + 8 ) Find ll the intercepts. b) Find ll the criticl points nd clssify ech s locl mimum, minimum, or neither. c) Find ll the inflection points.. A cble is mde of n insulting mteril in the shpe of long, thin cylinder of rdius R. It hs electricl chrge distributed evenly throughout it. The electricl field, E, t distnce r from the center of the cble is given below. k is positive constnt. kr r R E = kr r > R r ) Is E continuous t r R =? b) Is E differentible t r R =? c) Sketch E s function of r. d) Find de dr

5 6. Let f() t = 3 t + t for t. Find ) the criticl point(s) nd determine if it is locl mimum or minimum. b) the inflection point(s). c) the globl mimum nd minimum on the given intervl Let f( ) = 3+ with constnt >. Find (nswers will be in terms of ) ) the coordintes of the locl mim nd the locl minim. b) the coordintes of the inflection point(s). 8. Find the ect vlue of the following limits: t π sin( θ ) ) lim b) lim c) lim rctn t π sin t θ sin(7 θ ) d) lim y + y e y Bt 9. Consider the fmily of functions f() t =. Find the vlues of A nd B so tht f() t hs + At criticl point t (4,). 3. Consider the fmily of functions y( t) = t t ln t for t >. ) Find the t-intercept. Your nswer will be in terms of. b) Find the criticl point nd determine if it is locl mimum or minimum (or neither). 3. Find the vlues of, b, nd k so tht the prmetric equtions given below trce out circle of rdius 3 centered t (,4) = + kcost, y = b+ ksin t, t π. = 3t 7 = t Consider the lines prmeterized by nd y = 4 9t y = ct + 8 ) For wht vlue of c, if ny, will these two lines be prllel? b) For wht vlue of c, if ny, will these two lines intersect t (, 3)? 33. Suppose n object moves in the y plne long pth given by prmetric equtions 3 = t 3t +, y = t 4t, t. ) Determine the time when the object stops. Where will it stop? b) Determine the time when the object hits the -is. c) Find the eqution of the tngent line to the curve t t =.

6 34. Wire with totl length of L inches will be used to construct the edges of rectngulr bo nd thus provide frmework for the bo. The bottom of the bo must be squre. Find the mimum volume tht such bo cn hve. 3. A closed rectngulr bo with squre bottom hs fied volume V. It must be constructed from three different types of mterils. The mteril used for the four sides costs $.8 per squre foot; the mteril for the bottom costs $3.39 per squre foot, nd the mteril for the top costs $.6 per squre foot. Find the minimum cost for such bo in terms of V. w c 36. The speed of wve trveling in deep wter is given by V( w) = k c + w where w is the wvelength of the wve. Assume c nd k re positive constnts. Find the wvelength tht minimizes the speed of the wve. 37. An electric current, I, in mps, is given by I = cos( ωt) + 3 sin( ωt) where ω is constnt. Find the mimum nd minimum vlues of I. For wht vlues of t will these occur if t π. 38. The hypotenuse of the right tringle shown t the right is the line segment from the origin to point on the grph of y = 4 ( ). Find the coordintes on the grph tht will mimize the re of the right tringle. 39. A stined glss window will be creted s shown t the right. The cost of the semi-circulr region will be $ per squre foot nd the cost of the rectngulr region will be $8 per squre foot. Due to construction constrints, the outside perimeter must be feet. Find the mimum totl cost of the window. Wht re the dimensions of the window? r w h 4. For bb+ y= b forms tringle in the first qudrnt with the -is nd the y-is. ) Find the vlue(s) of b so tht the re of the tringle is ectly. b) Find the vlue of b tht mimizes the re of the tringle. b >, the line ( )

7 4. The grph of the function f( ) nd its derivtive f ( ) re given t the right. ) Determine which grph is f( ) nd which grph is f ( ). b) Use the grphs to find the vlues of tht mimize nd minimize the function g ( ) = f( e ) The grph below on the left shows the number of gllons, G, of gsoline used on trip of M miles. The grph below on the right shows distnce trveled, M, s function of time t, in hours since the strt of the trip. You cn ssume the segments of the grphs re liner. G (gllons) (7,.8) (, 4.6) M (miles) (, 7 ) (, ) M (miles) t (hours) ) Wht is the gs consumption in miles per gllon during the first 7 miles of the trip? During the net 3 miles? b) If G = f( M) nd M= ht (), wht does kt () = f( ht ()) represent? Find k (.). c) Find k (.) nd k (.). Wht do these quntities tell us? 43. A cmer is focused on trin s the trin moves long trck towrds sttion s shown t the right. The trin trvels t constnt speed of km hr. How fst is the cmer rotting (in rdins/min ) when the trin is km from the cmer? 44. Snd is poured into pile from bove. It forms right circulr cone with bse rdius tht is lwys 3 times the height of the cone. If the snd is being poured t rte of ft 3 per minute, how fst is the height of the pile growing when the pile is ft high? h r 4. A voltge, V volts, pplied to resistor of R ohms produces n electricl current of I mps where V = I R. As the current flows, the resistor hets up nd its resistnce flls. If volts is pplied to resistor of ohms, the current is initilly. mps but increses by. mps per minute. At wht rte is the resistnce chnging if the voltge remins constnt?

8 46. The rte of chnge of popultion depends on the current popultion, P, nd is given by dp = kp ( L P ) for some positive constnts k nd L. dt ) For wht nonnegtive vlues of P is the popultion incresing? Decresing? For wht vlues of P does the popultion remin constnt? d P d P b) Find s function of P. For wht vlue of P will =? dt dt 47. The mss of circulr oil slick of rdius r is M K( r ln( r) ) = +, where K is positive constnt. Wht is the reltionship between the rte of chnge of the rdius with respect to time nd the rte of chnge of the mss with respect to time? 48. A sphericl snowbll is melting. Its rdius is decresing t. cm per hour when the rdius is cm. How fst is the volume decresing t tht time? How fst is the surfce re decresing t tht time? 49. A function f() t is continuous nd differentible, nd hs vlues given in the tble below. The vlues in the tble re representtive of the properties of the function. t f() t ) Find upper nd lower estimtes for b) Find.6 f () t dt...8 f () t dt using n = 4.. A function gt () is positive nd decresing everywhere. Arrnge the following numbers from smllest () to lrgest (3). 9 n gt ( k ) t gt ( k ) t lim gt ( k ) t k = k = n k =. Severl objects re moving in stright line from time t = to time t = seconds. The following re grphs of the velocities of these objects (in cm/sec ). ) Which object(s) is frthest from the originl position t the end of seconds? b) Which object(s) is closest to its originl position t the end of seconds? c) Which object(s) hs trveled the gretest totl distnce during these seconds?

9 d) Which object(s) hs trveled the lest distnce during these seconds? Velocity of Object A Velocity of Object B 3 t t Velocity of Object C Velocity of Object D 3 t t Illustrte the following on the grph of f( ) given below. Assume F ( ) = f( ). ) f( b) f( ) b) f f( ) f f( ) f( b) f( ) b b b Fb ( ) F ( ) c) Fb ( ) F ( ) d) b ff ( ) ff ( ) b b

10 3. Let b be positive constnt. Evlute the following: b+ ) ( b + ) d b) d c) d d) b+ + d ( b) 4. Find the ect res of the regions. Include sketch of the regions. ) The region bounded between y = (4 ) nd the -is. b) The region bounded between y = + nd y = It is predicted tht the popultion of prticulr city will grow t the rte of pt () = 3 t+ (mesured in hundreds of people per yer). How mny people will be dded to the city in the first four yers ccording to this model? 6. At time t = wter is pumped into tnk t constnt rte of 7 gllons per hour. After hours, the rte decreses until the flow of wter is zero ccording to rt ( ) = 3( t ) + 7, gllons per hour. Find the totl gllons of wter pumped into the tnk. 7. Use the grph of g ( ) given t the right to sketch grph of gso ( ) tht g () = 3. g ( ) g'() A cr going 8 ft sec brkes to stop in five seconds. Assume the decelertion is constnt. ) Find n eqution for vt (), the velocity function. Sketch the grph of vt (). b) Find the totl distnce trveled from the time the brkes were pplied until the cr cme to stop. Illustrte this quntity on the grph of vt () in prt ). c) Find n eqution for st (), the position function. Sketch the grph of st (). 9. Consider the function t =. F( ) e dt ) Find F () b) Find F ( ) c) Is F( ) incresing or decresing for? d) Is F( ) concve up or concve down for? 6. The verge vlue of f from to b is defined s 3 f( ) = cos over the intervl π. 4 b ( ) b f d. Find the verge vlue of

11 π 6. According to book of mthemticl tbles, ( ) ln + 4cos d = π ln. π π ) Find ln ( + 4cos ) d b) Find ln ( + 4cos( )) π d 6. Use the grph of f( ) below to nswer the following. Circle True or Flse... ) f ( ) d f ( ) d True Flse.4. b) f ( ) d f ( ) d True Flse.. c) f ( ) ( ( )) True Flse d) f ( ) d True Flse e) f ( ) d True Flse Assume ll functions below re continuous everywhere. ) If 6 f ( ) d = 7, find f ( ) d. 3 3 b) If gis ( ) n odd function nd g( ) d =, find g( ) d. c) If h ( ) is n even function nd ( h( ) 3) d =, find h( ) d. b d) If f () t dt = M b+, find ( ) f t dt Use the grph of g ( ) t the right to determine which sign is pproprite: ) gc ( ) < = > gd ( ) b) g ( B) < = > g ( C) c) g ( A) < = > g ( B) g ( ) A B C D 6. Let Ft () = e t ( t+ ) ) Find F () t. b) Find the ect vlue of t e (t + ) dt ln(3)

12 66. Let g( ) = f () t dt. In ech cse eplin wht grphicl feture of f( ) you used to determine the nswer. f( ) ) Wht is the sign of gb? ( ) A b) Wht is the sign of g? c) Wht is the sign of g ( A)? A B 67. The grph t the right shows the rte, rt (), in hundreds of lge per hour, t which popultion of lge is growing, where t is in hours. ) Estimte the verge vlue of the rte over the first. hours. b) Estimte the totl chnge in the popultion over the first 3 hours. rt () t 68. Find the vlue of K so tht the totl re bounded by f( ) = K nd the -is over the intervl [,9] is Evlute the following: ) sin(3 ) cos 4 θ (3 θ )d θ b) tn(4 v) dv c) te t dt 7. Suppose g( t) dt =. ) Find g d 3 b) Find 3 g ( 3 ) 3 t dt 7. Find the solution to the initil vlue problems. ds ) = 3 t +, () dt s = b) dθ = d +, θ() = π 7. A metl cylindricl cn with rdius 4 inches nd height 6 inches is coted with uniform lyer of ice on its side (but not the top or bottom). If the ice melts t cubic inches per minute, how fst is the thickness of the ice decresing when it is inch thick?

13 73. A compny hs produced new model of sktebord tht is mrketed towrd ineperienced skteborders who prefer not to spend lot of money on their first sktebord. Let D( p) represent the number of people who re willing to buy this type of sktebord when the price is p dollrs. For ech of the following epressions, give the units, the sign, nd prcticl interprettion. 4 ) D (3) b) lim D( p) c) D ( p) dp + p A 6 foot long snke is crwling long the corner of room. It is moving t constnt /3 foot per second while stying tucked snugly ginst the corner of the room. At the moment tht the snke s hed is 4 feet from the corner of the room, how fst is the distnce between the hed nd the til chnging? 7. For ech of the following descriptions, sketch grph of function stisfying the given conditions. ) A function f( ), whose slope is incresing for ll, such tht lim f( ) =. b) A function g ( ) tht hs globl minimum nd locl mimum but not globl mimum. 76. An internet provider modeled their rte of new subscribers to n old internet service pckge by rt () s shown t the right. ) Use the Left Hnd Sum with n = to estimte the totl number of new subscribers between nd. Include n illustrtion of this estimte. b) Rnk the following from smllest () to lrgest (3): r() t dt Left Hnd Sum with n =. Right hnd Sum with n =. c) Wht is the sign of r (3)? 8 d) Estimte r () t dt. 4 New Subscribers per yer rt () Yers Since 77. Find the locl lineriztion of π ner =. g

14 78. Determine the inflection points of ht () if h t = tt+ t e. ( ) ( 3) ( ) t 79. Consider the function f( ) = + 4. Find number c in the intervl (, ) so tht the instntneous rte of chnge of f is identicl to the verge rte of chnge of f over [, ]. Why re we gurnteed to find such c? 8. Set up the integrl(s) needed to find the re of the shded region. f( ) g ( ) 8. A. Find lim sin( ) t dt nd tn lim tdt. sin( t) dt B. Find lim. Does L Hopitl s Rule pply? tn( t) dt