A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).

Transcription

1 A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically, first dtrmin if f a) xists and if so, f a) is lim x "a. If as a fraction, factor both numrator and dnominator if possibl, do any cancllations and xists and if so, f a) is lim. If f a) dos not xist, thn lim may How to find it: To find lim x "a not, xprss again, first dtrmin if f a x "a x "a sin x vry wll not xist. Thr ar cass in which that limit vry wll may xist; for xampl lim x "0 x. Although not in th AB curriculum, it is rcommndd that studnts b familiar with L Hopital s rul for such limits. as a fraction. If both numrator and dnominator ar To find lim x "# or lim x "$#, xprss polynomials, a) If th highr powr o is in th dnominator, th H.A. is y = 0. b) If both th numrator and dnominator hav th sam highst powr, th H.A. is th ratio of th cofficints of th highst powr trm in th numrator and th cofficint of th highst powr trm in th dnominator. c) If th highr powr o is in th numrator, thr is no H.A. Whn th numrator or dnominator dos not contain polynomials, studnts should s th ffct of plugging in vry larg or vry small numbrs. Not: L Hopital s rul can b usd to find ths limits if studnts hav larnd it. L Hopital s rul is not in th AB curriculum. I rcommnd taching it at th nd of th yar if thr is tim.. Find lim x "0 6x 5 # 8x 3 9x 3 # 6x 5 A. 3 B. "8 9 C. 4 3 D. "8 3 E. nonxistnt = #8 9 6x 5 # 8x 3 B. lim x "0 9x 3 # 6x = lim x 3 3x # 4 5 x "0 3x 3 3 # x x # 4 x #) 3x #. Find lim x "#$ x + A. " 3 B. 3 C. 3 4 D. E. " x # 4 x #) = lim x "#$ 3x # C. lim x "#$ x + 3x 3 # x #x + 4 4x 3 # 3x # = Dmystifying th MC AB Calculus Exam

2 3. Th figur to th right shows th graph of f x). Which of th following statmnts ar tru? I. lim x " # II. lim x " + f x) xists f x) xists III. lim x " xists A. I only B. II only C. I and II only D. I, II and III E. non ar tru D. Sinc lim x " # = 3 and lim x " + = 3, it follows that lim x " = 3. = ax Th function f is givn by. Th figur to th x 4 + b right shows a portion of th graph of f. Which of th following could b th valus of th constants a and b? A. a = -3, b = - B. a = 3, b = C. a = 3, b = - D. a = -3, b = E. a = 6, b = - A. Sinc f has a vrtical asymptot at x = ", b must qual ". Sinc f has a horizontal asymptot at y = "3, a must qual " 3. x n $ a n 5. If a " 0 and n is a positiv intgr, thn lim is x #a x n n $ a A. a B. n a C. D. 0 E. nonxistnt n n a x n # a n x n # a n B. lim = lim x "a x n n # a x "a x n # a n ) x n + a n = lim x "a x n + a = n a n 6. What ar all th horizontal asymptots of f x) = 6 + 3x in th xy-plan? 3 " x A. y = 3 only B. y = "3 only C. y = only D. y = "3 and y = 0 E. y = "3 and y = x E. lim = #3 lim x "! 3# x x "#! 3# 0 = Dmystifying th MC AB Calculus Exam

3 B. Dfinition of Drivativ What you ar finding: Th drivativ of a function is a formula for th slop of th tangnt lin to th graph of that function. Thr ar two dfinitions that ar commonly usd that studnts should know. f x + h) $ f x) How to find it: f " x) = lim h #0 f x) $ f a) or f " x) = lim x #a. Studnts nd to know that h x $ a diffrntiabl functions at a point drivativ xists at th point) ar ncssarily continuous but continuous functions ar not ncssarily diffrntiabl. Ths typs of problms usually ncssitat that studnts rcogniz ths limits as a drivativ and us drivativ ruls to calculat it. sin # + h) $ sin# 7. lim h "0 = h A. 0 B. cos x C. - D. π E. C. This is asking for th drivativ of y = sin x at x = ".!!!!! d dx sin x ) = cos x and cos" = #. # f 4) 8. Lt f b a function such that lim x "4 x # 4 I. f is continuous at x = 4. II. f is diffrntiabl at x = 4. III. Th drivativ of f " is continuous at x = 4. =. Which of th following must b tru? A. I only B. II only C. I and II only D. I and III only E. I, II and III C. Studnts should rcogniz that th statmnt tlls thm that th drivativ of f xists at x = 4 and thus it must b tru that f is continuous at x = 4 as wll. Th qustion is, what about f "? It is asy to construct a picwis function for f " whos lft and right limits ar at x = 4. = f " x x # 6, x $ 4 &, x > 4 and thus f " x = &,x $ 4 0,x > 4 so f " is not continuous at x = 4. And it is asy to gnrat constants so that f is continuous at x = 4 : = x # 6x,x $ 4 & x #6,x > Dmystifying th MC AB Calculus Exam

4 f x) # f ) 9. If f x) = tan " x), find lim x " x #. A. 7 B. 7 C. " 5 D. " 5 E. nonxistnt A. Studnts must rcogniz that thy ar bing askd to find th drivativ of tan " x = f # x + 4x = at x =. + 4x $ f # = 7 0. Th graph of f x) consists of a curv that is symmtric to th y- axis on [-, ] and a lin sgmnt as shown to th right. Which of th following statmnts about f is fals? A. lim x "0 [ # f 0) ] = 0 B. lim x "0 C. lim h "0 f 0 + h # f 0 # h) h # f 0) = 0 D. lim x " x # x # f ) = 0 = # # f ) f + h E. lim h "0 h dos not xist B. A is tru. Th function is continous at x = 0. B is fals. f " x dos not xist at x = 0. C is tru. This is th slop of th scant lin btwn h, f h D is tru. Th slop of th lin at x = is #. E is tru. f " x dos not xist at x = 0. ) and #h, f #h) which is Dmystifying th MC AB Calculus Exam

5 C. Taking Drivativs with Basic Functions What you ar finding: Th drivativ of a function is a formula for th slop of th tangnt lin to th graph of that function. Studnts ar rquird to know how to tak drivativ of basic functions, trig functions, logarithmic and xponntials, and invrs trig functions. Thy also nd to b abl to tak drivativ of invrs functions. Finally, ruls such as powr, product, quotint, chain rul, and taking drivativs implicitly must b a procss that studnts hav down prfctly. How to find it: Powr Rul : d dx x n Quotint Rul : d f x) * = g x dx & g x) ) d dx sin x = nx n" Product Rul : d [ dx f x) # g x) ] = # f $ x) " f x) # g $ x) Chain Rul : d [ g x) ] dx f g x) ) = cos x d = "sin x d = sc x d dx ln x = x dx cos x d dx x. If f x) = x +) x " 3 A. x " 3 dx tan x d = x = a x ln a dx ax 4, thn # = d dx csc x d dx sin" x [ ] = f $ [ g x) ] # $ # g $ x) + g x) # f $ x) g x = "csc x cot x d = sc x tan x d = "csc x = dx sc x d " x dx cos" x = dx cot x " d " x dx tan" x = 3 x + 4 x ") B. 4 x +) x " 3) C. 8x x +) x " 3) 3 3x + x " 3) E. x " 3) 9x + 4x " 3) D. x " 3 3 x E. Product/chain rul : f " x) = x +)4 x # 3. If A. "n x 3. If f " x) = x # 3) 3 9x + 4 x # 3) " = ln$ x n #, and n is a constant, thn f x & B. x n A. natural log ruls : f " x or = cos 3x) = = + x # 3 = ln ) ln x n =) nln x so f " x) = )n = 3, thn f " x 4 = x # 3) 3 [ 4 x x +) + x # 3] C. " x n D. x n E. 0 # $ x n &# 0 ) nx n) & # = x n &# n) )nx & = )nx ) = )n $ x n $ $ x n x x + x A. " B. 3 3 sin 3x " C. "sin3x D. 3 3 sin3x cos3x E. "sin3x 3 3 cos3x 3 cos 3x C. Chain rul. [ ] 3 " # = cos 3x) = 3 [ cos 3x ) 3 ]$ $sin 3x) [ ] 3 = $sin3x cos3x Dmystifying th MC AB Calculus Exam 3

6 4. Th tabl blow givs valu of th diffrntiabl function f and g at x = ". If = f x ) " g x), thn h #") = f x) h x A. " " 3 4 B. + 3 x g x) f " x) g " x) # # 4 #3 C. " 6 8 D. Quotint rul : h " x) = f x) f " x) # g " x) " h "#) = f #!!!!!!or h x h " # $ = & # g x) f x) = # + # #3 -, D. " 3 4 [ ] # [ f x) # g x) ][ f " x) ] 4[ f x) ] [ f #) # g "#) ] # [ f #) # g #)][ f "#) ] = # [ f #)] # 4 4 ) * h " x = #. 0 = # 3 / 4 + -, - g " x) # g x) f " x) f x) [ ] 5. Th functions f and g ar diffrntiabl and f g x f 4) = 8, g 4) = 8, f " 8) = #, what is th valu of g " 4)?. 0 / 0 x for all x. If + 6 ) = 4 #) E. "4 " # 6 = # 3 4 A. " 8 B. " C. " D. "4 E. Insufficint data D. f " g x ) g " x) = x # g " x) = x f " g x ) = 8 f " 8) = 8 $ = $4 g " x ln tan x 6. f x) =, thn f " x = ln tan x ) A. ln tan x ) B. C. ln tan x ) sin x D. x tan x cos 3 x E. 4x sin x ) cos 3 x ) E. log ruls : " # $ = & ln tan x f x) = tan x )sc x ) x = [ tan x )] = 4x sin x $ Dmystifying th MC AB Calculus Exam " # $ cos x " $ & # $ cos x )& 4x sin x = cos 3 x

7 D. Tangnt Lins and Local Linar Approximations What you ar finding: You typically hav a function f and you ar givn a point on th function. You want to find th quation of th tangnt lin to th curv at that point. whr m is th slop and x,y ) is th How to find it: You us your point-slop quation: y " y = m x " x point. Typically, to find th slop, you tak th drivativ of th function at th spcifid point: f " x ). What you ar finding: You typically hav a function f givn as a st of points as wll as th drivativ of th function at thos x-valus. You want to find th quation of th tangnt lin to th curv at a valu c clos to on of th givn x-valus. You will us that quation to approximat th y-valu at c. This uss th concpt of local linarity th closr you gt to a point on a curv, th mor th curv looks lik a lin. How to find it: You us your point-slop quation: y " y = m x " x ) whr m is th slop and x, y ) is th point closst to c. Typically, to find th slop, you tak th drivativ f " x ) of th function at th closst x-valu givn. You thn plug c into th quation of th lin. Raliz that it is an approximation of th corrsponding y-valu. If nd drivativ valus ar givn as wll, it is possibl to dtrmin whthr th approximatd y-valu is abov or blow th actual y-valu by looking at concavity. For instanc, if w wantd to approximat f.) for th curv in th graph to th right, w could find f " x), us it to dtrmin th quation of th tangnt lin to th curv at x = and thn plug. into that linar quation. If information wr givn that th curv was concav down, w would know that th stimation ovr-approximatd th actual y-valu. 7. If f x) = sin 3 4x $ #, find f "& ). 3 A. " 3 B. " 9 C. "3 3 D. 3 3 E. " 9 8 B. f " x) = 3sin 4x)cos 4x) # 4 =sin 4x)cos 4x) $ f " * =sin 4$ & 3) & 3 ) * cos 4$ & 3 * = ) & 8. If f x) = x "x", find th quation of th tangnt lin to f at x = ". 3 * + * = + 9 ) & ) A. y = x + 4 B. y = "6x " 4 C. y = " ln x + D. y = " 6ln x + E. y = " 3ln x +) x # D. f " x) = x #x# f # ln $ f "#) = ) #3)ln = #6ln = $ y # = #6ln x +) $ y = # 6ln x +) Dmystifying th MC AB Calculus Exam

8 9. What is th slop of th lin tangnt to th graph of y = A. " B. " C. 0 D. D. y = x x " ) = "x x " y # = x " " x x " ) at x =? "x " "x $ y # ) = " x " ) E. " " " = 0. What is th quation of th lin normal to th graph of y = sin x cos x " sin x at x = 3#? A. y = x " 3# + B. y = "x + 3# + C. y = x " 3# 4 + D. y = " x + 3# 4 + E. y = " x + 3# 4 " [ + cos x cos x) ] # cos x = cos x # sin x C. y " = sin x #cos x 3$ y " * = 0 # & ) y 3$ & ) * = so y # = x # 3$ &. Calc) Lt f b th function givn by = 0 = # so m normal) = = 4 x * + y = x ) # 3$ 4 + # cos x. For what positiv valus) of c is " x "x " f c =? A.. B and C and 3.60 D E and A. f " c # 4x x + #x ) = x # #x ) = 8x x # #x or asir. Th function f is twic diffrntiabl with f "3 c is th approximation of f c = " and f #"3) = "4. For what valu of using th tangnt lin of f at x = c qual to c? A. "4.667 B. "0.4 C. ".8 D. ".5 E. "3.333 # y = "4 x "4 y c) = "4 c) "4 = c # 5c = "4 # c = ".8. C. Tangnt lin : y + = "4 x Dmystifying th MC AB Calculus Exam

9 3. Th lin x + y = k, whr k is a constant, is tangnt to th graph of y = x 3 " 9x " x +. What ar th only possibl valus of k? A. only B. 0 and - 9 C. and -9 D. 0 and 3 E. and -6 E. y = "x + k # y $ = " y = x 3 " 9x " x +# y $ = 6x "8x " = " # 6x x " 3) = 0 x = 0, y = so x + y = x = 3,y = "9 so x + y = "6 = 4x # 3 and f 4. What is th approximation for f.) found by 4. For th function f, f " x using th tangnt lin to th graph of f at x =. A. ".6 B. 4.5 C. 4.8 D. 5.4 E. 9.4 B. Tangnt lin : f " 4 ) # 3 = 5 $ y # 4 = 5 x # ) $ y.) 5.) # 6 = 4.5. B carful of th trap answr D). You arnt givn th tangnt lin quation, just th formula for th slop of th tangnt lin. 5. Calc) Th function f is dfind for x > 0 with f " x = " to approximat th valu of f 4 =, and f # x ) = sin & x and whthr it ovr or undr-approximats f Dmystifying th MC AB Calculus Exam $ ). Us th lin tangnt to f at A..69 undr-stimats B..69 ovr-stimats C..85 undr-stimats D..85 ovr-stimats E ovr-stimats B. Tangnt lin : f "! $ ) = 0.33 = sin & # y *= 0.33 x * #) + y = 0.33 x * # = & * f " x $ x ) cos $ & x ) f " x ++ y 4), * #) +=.69. < 0 on!,") so f is concav down and.69 is an ovrstimation. 6. Th numbr of vhicls waiting to gt past an accidnt scn is modld by a twic-diffrntiabl N of t, whr t is masurd in minuts. For 0 < t < 8, N " t rat of chang N " t)ovr th tim intrval 0 t 8. Th numbr of vhicls in lin at t = 4 is 45. By using th tangnt lin approximation at t = 4, which of th following is a possibl valu for th numbr of vhicls in lin at t = 4.5 minuts? t minuts) N " t > 0. Th tabl blow givs slctd valus of th vhicls pr minuts) I. 57 II. 59 III. 63 A. I only B. II only C. III only D. II and III only E. I, II and III B. Th slop of th tangnt lin incrass from t = 4 to t = 6. sinc N " t is gratr than N 4) + " is lss than N 6) + N " 4 N 4.5 N 4.5 N 4)#t = #t = > 0) = 57. = 6. So 57 < N 4.5) < 6

10 E. Implicit Diffrntiation What you ar finding: You ar askd to ithr find dy dy or at som point x,y). You prform implicit dx dx diffrntiation whn th function is not givn xplicitly in th form y = or whn it is difficult or impossibl to put it in that form. How to find it: Tak th drivativ of vry trm in th givn xprssion, rmmbring that th chain rul is usd and th drivativ of y is not but dy dy. If you ar askd to find at a givn point, it is bst to plug dx dx th point in, onc you hav takn th drivativ and solv for dy dy. If th cancls out of th quation, th dx dx tangnt lin is vrtical at that point. Implicit xampls show up in rlatd rats problms whn you ar doing implicit diffrntiation with rspct for tim t. 7. If cos xy) = x + y, find dy dx. A. "" sin xy) B. + y sin xy D. " x sin xy "x sin xy) + y sin xy E. + y " x " sin xy) + y sin xy C. # D. " sin xy $ x dy dx + y & =+ dy dx dy dx [ " x sin xy) ] =+ y sin xy) ) dy dx = + y sin xy " x sin xy 8. Find th quation of th tangnt lin to x 3 + y 3 = 3xy + 5x " 3y at,). A. x + 9y = 0 B. 3x " 3y = 0 C. 9y " x =6 D. y = " 9 E. y = dy A. 3x + 3y dx = 3 " $ x dy # dx + y dy & dx + dy dx = 6 dy dx dy dx ) 9 dy dy = ) dx dx = 9 So th tangnt lin is y = 9 x ) ) x + 9y = 0 9. I + y = 34, find th bhavior of th curv at -4, 3). A. Incrasing, concav up B. Incrasing, concav down C. Dcrasing, concav up D. Dcrasing, concav down E. Dcrasing, inflction point B. x + 4y dy dx = 0 " dy dx = #x 4y = #x y d y dx = #y + x 4 y dy dx) d y dx #4,3 dy = 4 > 0 Curv is incrasing. dx #4,3) 6 = #6 # 4 ) 3 < 0 Curv is concav down Dmystifying th MC AB Calculus Exam

11 30. Find th y-intrcpt of th tangnt lin to 4 x + y = x + y + 3 at th point 9, 4). A. - B. 0 C. 5 D. 5 E. 4 9 B. x + y dy dy =+ dx dx " dy $ & dx y # ) =# x # dy dx = x y # " dy # = 3 = # dx 9,4) # 3!!!!!!Tangnt lin : y # 4 = # 3 x # 9 ) y # intrcpt : y # 4 = # 3 #9 ) " y =0 3. I + y = a, whr a is a constant, find d y dx. A. "y " x y B. x " y C. " "a D. y 3 y y E. 3 a D. x + y dy dx = 0 " dy dx = #x y dy d y #y + x #y # x dx = dx y = = #y # x = #a y y y 3 y 3 3. Lt x and y b functions of tim t that ar rlatd by th quation y = xy +. At tim t =, th valu of y is and dy dx =. Find whn t =. dt dt A. " B. + C. D. + E. " B. = x + " = x +) " x = # y dy dt = x dy dt + y dx dt " ) = #) + dx dt 4 = # + dx dt " dx dt = + or = x + " = x +) " x = # y dy dx = x dy dx + y " dy dx = y y # x dy dx = dy dx $ dx dt " = # #) $ dx dt " dx dt = Dmystifying th MC AB Calculus Exam

12 F. Drivativs of Invrs Functions What you ar finding: Thr is probably no topic that confuss studnts and tachrs) mor than invrss. Th invrs of a function f is anothr function f " that undos what f dos. So f " f x) ) = x. For instanc, th invrs of adding 5 is subtracting 5. Start with any numbr x, add 5, thn subtract 5, and you ar back to x. Do not confus th invrs f " with th rciprocal. x " = x but f x ) " # To find th invrs of a function, you rplac x with y and y with x. Th invrs to th function y = 4x " is x = 4y " or y = x +. In this sction, you ar concrnd with finding th drivativ of th invrs to a 4 [ f x )] ". function: d dx How to find it: Th formula usd is: dy dx =. But what I suggst, rathr than mmorizing this formula, f " y) is to switch x and y to find th invrs, and thn tak th drivativ, using implicit diffrntiation: x = f y) " = f # y) dy dx " dy dx =. Studnts should also know th formulas for drivativs of invrs f # y) trig functions and how thy ar found: d d " d dx sin" x) = " x dx cos" x) = " x dx tan" x) = + x 33. If f x) = x 3 + x + x + and g x) = f " x?, what is th valu of g " 4. A. 85 B. C. 57 D. 6 E. 4 D : Invrs : x = y 3 + y + y += 4 " y =. = 3y + y +) dy dx " dy dx = 3y + y +!!!!!!!!!!!!!!!!!!!! dy = dx y=) 3y + y + = Lt f b a diffrntiabl function such that f "4 function g is diffrntiabl and g x =, f 9) = "4, f # 4) = "6, f # 9) = 3. Th = f " x) for all x. What is th valu of g "#4)? A. " 6 B. " 4 C. 3 D. 9 E. Insufficint data C. g x) = f " x) # f g x) f $ g x = x. f g "4)) $ ) g $ x) =# $ = "4, thn g "4) = 9!!!!!Sinc f 9 f $ 9 = g "4 g $ "4) =# 3 g $ "4) =# g $ "4) = Dmystifying th MC AB Calculus Exam

13 35. Th function g is diffrntiabl for all ral numbrs. Th tabl blow givs valus of th function and its first drivativs at slctd valus o. If g " is th invrs function of g, what is th quation for th lin tangnt to th graph of y = g " x at x = 4? x g " x) g x # 4 4 #3 5 A. y + 3 = 5 x " 4 ) B. y + = 5 x " 4 ) C. y + = x " 4) D. y + 3 = x " 4 ) E. y + = x " 4 ) E. g ") = 4 # g " 4) = " g " ) $ 4) = g $ g " 4 g $ ") =!!!!!!Tangnt lin : y + = x " 4 ) 36. Calc) Find th drivativ of f " x) for f x) = x 3 " 3x " x + at x =. A B. "0.500 C D E A. Inv : x = y 3 " 3x " x += # y = 3.67 dy = dx y= 3.67) 3y " 6y " = If y = csc " x ), which of th following rprsnts dy dx? A. x sin x B. "x sin y tan y C. D. "x sin y E. "x sin x B. y = csc " x ) # csc y = x # sin y = x "cos y sin y dy dy = x # dx dx = "x sin y cos y # dy = "x sin y tan y dx x cos y Dmystifying th MC AB Calculus Exam

14 G. Continuity and Diffrntiability What you ar finding: Typical problms ask studnts to dtrmin whthr a function is continuous and/or diffrntiabl at a point. Most functions that ar givn ar continuous in thir domain, and functions that ar not continuous ar not diffrntiabl. So functions givn usually tnd to b picwis and th qustion is whthr th function is continuous and also diffrntiabl at th x-valu whr th function changs from on pic to th othr. How to find it: Continuity: I lik to think of continuity as bing abl to draw th function without picking your pncil up from th papr. But to prov continuity at x = c, you hav to show that lim x "c = f c). Usually you will hav to show that lim f x) = lim f x). x "c # x "c + Diffrntiability: I lik to think of diffrntiability as smooth. At th valu c, whr th picwis function changs, th transition from on curv to anothr must b a smooth on. Sharp cornrs lik an absolut valu curv) or cusp points man th function is not diffrntiabl thr. Th tst for diffrntiability at x = c is to show that lim f x) = lim f x). So if you ar givn a picwis function, chck first for x "c # $ x "c + $ continuity at x = c, and if it is continuous, tak th drivativ of ach pic, and chck that th drivativ is continuous at x = c. Lins, polynomials, xponntials, sin and cosins curvs ar diffrntiabl vrywhr. 38. Lt f b th function dfind blow, whr c and d ar constants. If f is diffrntiabl at x = ", what is th valu of c d? $ & f x) = x + c +)x " d, x # " & x + + cx + 3d, x < " A. - B. 0 C. D. 3 E. 4 E. Continuity : lim x "# + = f x f x ) = lim x "# # x + c +,x & # ) x + + c, x < #!!!!!Diffrntiability : lim c # d = 3 += 4 x "# + f x ) $ # c +) # d =# c # 3d $ c + 4d = # = lim x "# # $ # + c += + c $ c = 3,d = # 39. Th graph of f x) = x " 0.0 is shown in th graph to th right. Which of th following statmnts ar tru? = 0. I. lim x "0 II. f is continuous at x = 0. III. f is diffrntiabl at x = 0. A. I only B. II only C. I and II only D. I, II, and III E. Non ar tru D. A trap problm. This looks lik x which is not diffrntiabl at x = 0. But th function is givn and f " x) = x x and f " 0 = Dmystifying th MC AB Calculus Exam

15 40. Lt diffrntiabl? b givn by th function blow. What valus of a, b, and c do NOT mak f x) = $ acos x,x " 0 & bsin x + c# ),x > 0 A. a = 0, b = 0, c = 0 B. a = 0, b = 0, c =00 C. a = 5, b = 5, c = 0.5 D. a = ", b =, c = E. a = "8, b = 8, c = ".5 D. For continuity, acos0 = bsinc" # a = bsinc" = f $ x asin x, x & 0 ) bcos x + c" ), x > 0 For diffrntiability, 0 = bcosc" If b = 0,a = 0 so choics A and B ar tru. If c = 0.5 or c =.5, b can tak on any valu. if c = 0.5 and b = 5, thn a = 5sin.5" = 5 so choic C is tru. if c =.5 and b = 8, thn a = 8sin.5" = 8 so choic E is tru. if c = and b =, thn f is not diffrntiabl so choic D is fals. 4. Lt f b th function dfind blow. Which of th following statmnts about f ar NOT tru? = $ x 3 " &,x # x " & 3,x = I. f has a limit at x =. II. f is continuous at x =. III. f is diffrntiabl at x =. IV. Th drivativ f is continuous at x =. A. IV only B. III and IV only C. II, III, and IV only D. I, II, III, and IV E. All statmnts ar tru A. lim x " = lim Sinc f = f # x = 3, so limit xists x " x + x + = 3, f is continuous x +,x $ & 0,x = f is diffrntiabl at x = as lim But sinc lim x " # $ # x " # = lim x " + # = 3 f 0), th drivativ of f is not continuous at x = Dmystifying th MC AB Calculus Exam

16 H. Intrmdiat Valu and Man Valu Thorm MVT) What it says: IVT) If you hav a continuous function on [ a,b] and f b) " f a), th function must tak on vry valu btwn f a) and f b travling at 40 mph and a minut latr you ar travling at 50 mph, at som tim within that minut, you must hav bn travling at 4 mph, 4 mph, and vry possibl valu btwn 40 mph and 50 mph. What it says: MVT) If of c btwn a and b such that f " c at som point btwn x = a and x = b. For instanc, if you ar on a road is continuous on [ a,b] and diffrntiabl on a, b), thr must b som valu = f b) # f a). In words, this says that thr must b som valu b # a btwn a and b such that that th tangnt lin to th function at that valu is paralll to th scant lin btwn a and b. 4. Th function f is continuous and non-linar for "3 # x # 7 and f "3 valu c, whr "3 < c < 7, for which f " c A. For som k, whr " 3 < k < 7, f # k B. For som k, whr " 3 < k < 7, f # k C. For som k, whr " 3 < k < 7, f # k D. For " 3 < k < 7, f # k) xists. E. For som k, whr " 3 < k < 7, f # k = 5 and f 7) = "5. If thr is no = #, which of th following statmnts must b tru? < ". > ". = 0. dos not xist. E. This is th Man - Valu Thorm which stats that thr must b som valu k. " f "3) " 3 < k < 7 such that f 7 = "5 " 5 = ". Sinc thr is no such valu k, thn thr must b a valu k on " 3 < k < 7 whr f is not diffrntiabl. 43. A nw robotic dog calld th IPup wnt on sal at 9 AM) and sold out within 8 hours. Th numbr of customrs in lin to purchas th IPup at tim t is modld by a diffrntiabl function A whr 0 t 8. Valus of A t) ar shown in th tabl blow. For 0 t 8, what is th fwst numbr of tims at which A " t = 0? t hours) A t) popl A. 0 B. C. 3 D. 4 E. 5 C. A is diffrntiabl on 0,8 and for som t on 6,7). A " t [ ] so th MVT implis that A " t) > 0 for som t on 0,) < 0 for som t on,4) and for som t on 7,8). Sinc f is continuous, th IVT implis that thr must b at last on valu of t on 0,4 whr A " t) = 0 and th sam must b tru on 6,8). And thr must b at last on on valu of t on 4,5 whr A " t) = 0. So thr must b at last 3 valus of t on 0,8) whr A " t) = Dmystifying th MC AB Calculus Exam

17 44. A continuous function f is dfind on th closd intrval -4 x 4. Th graph of th function, shown in th figur to th right consists of a lin and two curvs. Thr is a valu a, -4 a < 4, for which th Man Valu Thorm, applid to th intrval [a, 4] guarants a valu c, a c < 4 at which f " c I. -4 II. 0 III. = 3. What ar possibl valus of a? A I only B. II only C. III only D. II and III only E. I, II, and III C. Th issu hr is diffrntiability. Th function is not diffrntiabl at x = 0 and x = so a cannot qual " 4 or 0. Th function is diffrntiabl on,4 So th MVT holds on,4) and a =. and f 4) " f ) 45. Lt f b a twic-diffrntiabl function such that f a and b, a < b. Lt g x) = f f x) ). Th Man Valu Thorm applid to g " on [a, b] guarants a valu k such that a < k < b such that A. g " k 4 " = 3. = b and f b) = a for two unknown constants a = 0 B. g " k) = 0 C. g " k) = D. g " k) = E. g " k) = b # a B. g " x = f " f x) ) # f " x) = f " f a) f a) = f " b) # f " a) g " a = " g " b # " f f a) )b # f " b = f " a) # f " b) so g " a) = g " b)!!!!!sinc f is twic diffrntiabl, g " is diffrntiabl so th MVT guarants that = g " b) $ g " a) g " k b $ a = Calc) Thr ar valus) of c that satisfy th Man-Valu Thorm for f x) = cos x " 4cosx on [ 0,# ]. Find th sum of ths valus. A..455 B C D..687 E C. f " c) = #sinc + 8sinc = f $ ) # f 0) = #6 + = #4 $ $ $!!!!!!By graphs, valus of c ar :.55 and 3.07: sum is Dmystifying th MC AB Calculus Exam