MATH6 Fall 8 MIDTERM II PRACTICE QUESTIONS PART I. + if <.[.,.] Find f () for f( ) = ( ) if A. B. D. cos.[.] Finh limit: lim sin E. dos not ist A. B. D. E. dos not ist.[.] Finh limit: A. B. 6 lim + D. E. dos not ist.[.5] Finh limit: h + lim. h h A. B. D. + E. 5. [.,.5] Suppos f is a function with th proprty that f ( ) = cos( ). Find g (), whr g ( ) = f ( ). A. g ( ) = cos( ) B. g ( ) = cos( ) g ( ) = sin( ) D. undfind. E. non of ths. cosh 6. [.,.] Intrprt lim as a drivativ. h h d A. (cos) B. cos ( ) cos ( h) D. co s () E. co s () d 7. [.] On Earth, you can asily shoot a papr clip straight up into th air with a rubbr band. In t sc aftr firing, th papr clip is s= 6t 6t ft abov your hand. What is th maimum hight of th papr clip? A. 8 ft B. ft 96 ft D. 6 ft E. not nough information. 8. [.6] Th quation of th tangnt lin to th curv of function y = ( ) at (, ) is A. + y = B. y = + y + = D. y = E. + y + = 9. [.,.]Finh drivativ of f ( ) =
A. E. Non of ths. B. D. d + d A. ( + )( + ) B. ( + ) ( + ) ( + )( + ) D. ( + )( + ) E. Non of ths..[.5] Finh drivativ ( ). [.,.5] Finh drivativ of f ( ) = ( ) ( + ) at =. A. B. D. - E. non of ths.. [.,.5] Find A. = t y = t at t = π /. d y/ d for a paramtrizd curv (/ ) tan, (/ ) sc B.. [.8] Finh drivativ of y = ( ) sin ( ) A. 6( )sin ( ) + ( ) /( + ) 6( )sin ( ) + ( ) /( + ) E. ( )[sin ( ) + ( ) / ] D. π E. non of ths. B. ( ) cos ( ) + 6( )sin ( ) D. ( )[sin ( ) + ( ) / ]. [.]Th slop of th tangnt lin to th graph of y = + at = is A. B. D. dos not ist E. non of ths. y d y 5. [.7] Givn that = y, comput at th point (,). d A. ln B. D. E. non of ths. / ( + ) 6. [.,.5,.7] Finh drivativ of y = / ( + ) / / ( + ) ( + ) A. B. + / / ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) / / ( + ) ( + ) / D. + / ( + ) + + ( + ) + + E. Non of ths.
d 7. [.9] Find 9 A. B. 6 whr = d y + y + y and = whn = and y > D. E. 5 d y 8. [.9] Find whr 9 A. B. y + = 6 and 9 d = 9 whn = D. 8 E. 9 9. [.] A ston is falling vrtically from th top of a cliff with an zro initial vlocity. If th ston hits th ground with a spd of 9 m/s, how high is th cliff? You may assum acclration du to gravity is 9.8 m/s downwards, that th ston lands at th bas of th cliff anhat air rsistanc is ngligibl. A. 98 m B..5 m 7 m D. 55.5 m E. 96 m [.6] If cos y + y cos =, thn th valu of dy/d at (, ) is. A. B. cos / D. E. dos not ist [.,.5] On intrval [, ], thr is a horizontal tangnt lin to th graph of function ln f ( ) = at A. = B. = = D. = E. =. [.8] Suppos that w know that a function g has drivativ g ( ) = + 6 for all, anhat g ( ) =. Us a diffrntial approimation (tangnt lin approimation) to stimat th valu of g (.5). A.. B.. 75. 95 D.. 9 E. non of ths.. [.8] Find y whr y = sc(tan ). 8 + A. B. + + D. + E... [.,.5] Lt f ( ) = + +, finh quation of th normal to th graph of f at = A. y = B. + y+ 7= 8+ y 6= D. y = E. + y 9= 5. [.7] Giving f ( ) = ( ) and A. B. f () =, find ( f ) (). D. 6 E.
6. [.] If function f is diffrntiabl at = c, thn which of th following statmnt is NOT tru? A. lim f ( ) f( c) c f( + h) f( ) B. lim = f ( c) h h f is continuous at = c. D. lim f ( ) = f( c) c E. Non of th abov.. 7. [.,.] Lt f ( ) =, find f () A. B. D. E. DNE 8. [.] If st () = t t, thn th acclration at t = is A. B. D. E. 6 PART II.. [.] Without using a calculator, finh corrct valu of k that maks th function f() continuous on [, ], if f is dfind as follows:. [.,.5,.7]Two functions, f and g, ar continuous and diffrntiabl for all ral numbrs. Som valus of th functions anhir drivativs ar shown in this tabl: f () / / g () - -/ f () / 5/ / -/5 g () - / - -/ f" () -/ g" () Basd on that luscious tabl, finh following drivativs: d (a) f ( ) d, valuatd at = () Lt h ( ) = f ( g ( )), find h () (b) (f) Lt R ( ) = f ( g()), find R ()
(c) (g) Lt r ( ) = ln[ f ()], find r () (d) Lt H ( ) = f [ g( )], find H () (h) Lt g ( ) t ( ) = f ( ), find () t. [.9] Two popl start from th sam point. On walks ast at mil/hour anh othr walks northast at mil/hour. How fast is th distanc btwn th popl changing aftr 5 minuts?. [.]Find what valus of constants a and b dos y = a + bsin satisfy y + y = cos? 5. [.6,.7] Find y from th following implicit functions. (a) sin( y) y cos = ; (b) y = + y ; (c) ( ) y = ln 6. [.] Th position of a particl is givn by th quation s( t) = + 6t 6t, whr t is masurd in sconds and s in ft. (a) Finh vlocity v(t) and acclration a(t) at tim t. (b) What is th vlocity and acclration aftr sconds? (c) How far dos th particl travls aftr sconds? (d) What is th total distanc travld aftr sconds? 7. [.9] Gravl is bing dumpd from a convyor blt at a rat of ft /min and its coarsnss is such that it forms a pil in th shap of a con whos bas diamtr and hight ar always qual. (a) How fast is th hight of th pil incrasing whn th pil is ft high? (b) At what rat is gravl bing dumpd if th hight of th pil is incrasing at a rat of π ft/min whn it is ft high? (Th volum V of a con is V = π r h ) 8. [.9] An oil tankr slams into th Alaskan coastlin. Th oil sprads in a circl whos ara incrass at a constant rat of 6 mils /hour. How fast is th radius of th spill incrasing whn th ara is 9 mils? You should also look ovr your lctur nots, homwork assignmnt, problm-solving labs, and quizzs. Th Midtrm of Fall, Spring, and Fall can b found at http://www.uta.du/math/pags/main/oldams/calc/calc.htm Midtrm : Friday, Oct., 6: 8: pm, Room TBD 5
Answrs to PART I:.E. D.B..A 5.A 6.D 7..D 8.D 9.B.D.A.A.E.B 5.B 6.C 7.C 8.A 9.B.D.A.B.A.E 5.C 6.A 7.E 8.A Answrs to PART II:. k = /;. (a) /; (b) 7/9; (c) -5/8; (d) /9; () -/5; (f) 6; (g) -8/5; (h) -96ln...9 mph;. a =, b = /; ln sin y + y sin y ln 5.(a) (b) (c) cos cos y y 6.(a) v( t) = 6 t, a ( t) = (b) v( ) = ft/s, a ( ) = ft/s (c) = 58 ft (d) 8 ft 8 7. (a) (b) 5 ft /min 5π 8. Lt A = th ara of th oil sprad, r = th radius. Giving da dt = 6 mils /hour, find out dr whn A = 9 mils dt A= π r d d A ( r ) dt dt π da dr = π r dt dt Whn A = 9 mils, r = A/ π = 9/ π = π mil. Thrfor, dr da 6.56 dt π r dt 6 π π 6