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AP ® Calculus AB AP ® C a lc u lu s B C Free-Response Questions and Solutions 1989 – 1997 Copyrigh t © 2003 College E ntrance Examinat ion B oard. All rights reserved. College Boar d, Advanced Pla cem ent Pro gram , AP, AP Vertica l Team s, APCD, Pacese tter, Pr e-AP, SAT, Stude nt Searc h Ser vice, and t he acorn logo ar e regi stered trademarks of the College Entra nce Exa mination Board. AP Ce ntral is a tradema rk owned by the College Ent rance Examination Board. PSA T/NM SQT i s a reg istered trademark jointly owned by the Col lege Entrance E xaminati on B oard and the Nationa l Mer it Scholarship Corporation. Educational Testing Ser vice an d ETS are reg istered trademarks of Educ ational Testing Serv ice. Other products and se rvices may be tr ademarks of their res pective owners. For t he College B oard’s online home for A P prof essionals, visit AP Cen tral at a pcentral.collegeboard.com.

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Notes about AP Calculus Free-Respon se Questions • The solution to each free- response question is as it appeared on the scoring standard from the AP Reading. Other math em atically correct s olutions are possible. • Scientific calculators wer e permitted, but not required, on the AP Calculus Exams in 1983 and 1984. • Scientific (nongraphing) calculators were required on the AP Calculus Exams in 1993 and 1994. • Graphing calculators have been required on the AP Calculus Exams since 1995. From 1995-1999, the calculator could be used on all 6 fre e-response questions. Since the 2000 Exams, the free-response section has consiste d of two parts -- Part A (questions 1-3) requires a graphing calculator and Part B ( questions 4-6) does not allow the u se of a calculator. Copy right © 2 003 by College Ent rance Exa mination Boa rd. Al l rights reserve d. Available at apcentral.c ollege boa rd.c om

1989 AB1 Let f be the function given by . f(x)= x3− 7x+6 (a) Find the zeros of f . (b) W rite an equation of the line tangent to the g raph of f at x =− 1. (c) Find the number c that satisfies the conclus ion o f the Mean Value Theorem for f on the clo sed in terval [1 . ,3] Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB1 Solution (a) () () ( )( 3 76 12 1, 2, 3 fx x x xx x xx x =− + =− − + == =− )3 2 (b) () () () () 2 37 14 , 1 1 12 4 1 or 48 or 48 fx x ff yx xy yx ′ =− ′−= − − = −= − + += =− + (c) () () () 2 31 12 0 6 31 2 37 13 3 ff cf c c − − == − ′ −= = = 6 Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB2 Let R be the region in the first qua drant enclosed by the graph of y= 6x+4, the line , and the y-axis. y=2x (a) Find the area of R . (b) Set up, but do not integrate, an integral expression in term s of a single variable for the volum e of the solid generated when R is revolved about the x-axi s. (c) Set up, but do not integrate, an integral expression in term s of a single variable for the volum e of the solid generated when R is revolved about the y-axi s. Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB2 Solution (a) () 2 0 2 3/2 2 0 Ar ea 6 4 2 12 64 63 64 8 20 4 99 9 x xd x x x =+ − =⋅ + − =− − = \b ∫ (b) Volume about x-ax is V = ()2 2 0 64 4 x xd x π +− ∫ or V = ()2 0 32 64 3 xd x π π + − ∫ (c) Volum e about y-axis V = () 2 0 4 2 26 x xx π +− ∫ dx or V = 4 2 4 2 2 0 2 4 26 yy dy dy ππ  −  −  \b \b ⌠ ⌠  ⌡ ⌡ Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB3 A particle moves along the x-ax is in such a way that its acceleration at tim e t for t is given by . At tim e , the veloc ity of the partic le is v and its position is ≥0 a(t)=4cos( 2t) t=0 (0) =1 x(0 )= 0. (a) W rite an equation for th e veloc ity of the particle. v(t) (b) W rite an equation for th e position x(t) of the particle. (c) For what values of t , 0≤ t≤π, is the partic le at res t? Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB3 Solution (a) vt () () () () 4c os2 2s in2 01 1 2s in2 1 tdt vt t C vC vt t = =+ =⇒ = =+ ∫ (b) () () () () 2s in2 1 cos 2 00 1 cos 2 1 x tt dt x tt t xC xt t t =+ =− + + =⇒ = =− + + ∫ C (c) 2s in2 1 0 1 sin 2 2 71 1 2, 66 71 1 , 12 12 t t t t ππ ππ += =− = = Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB4 Let f be the function given by 2 () 4 x fx x = − . (a) Find the dom ain of f . (b) W rite an equation for each vertical asym ptote to the graph of f . (c) W rite an equation for each horiz ontal asym ptote to the graph of f . (d) Find () . f x′ Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB4 Solution 2 o r 2 or 2 x x x > (a) (b) xx 2, 2 == − (c) 2 2 m 1 4 lim 1 4 1, 1 x x x x x x yy →∞ →− ∞ = − =− − == − li (d) () () () 1/2 22 2 2 2 2 2 3/2 2 1 44 2 4 4 4 4 4 4 x xx x fx x x x x x x −  −− −  ′ = − −− − = − − = − 2 Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB5 The figure above shows the graph of ′f , the derivative of a function f . The do main of f is the s et of all real num bers x such that − . 10 ≤ x≤10 (a) For what values of x does the graph of f have a horizontal tangent? (b) For what values of x in t he inte rval does f have a relative maxim um? Justify your answer. (1 0,10) − (c) For value of x is the graph of f concave downward ? Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB5 Solution (a) () horizonta l tangent 0 7, 1,4,8 fx x ′ ⇔= =− − (b) Rela tive m axima at because 1, 8 x=− f′ cha nges from positiv e to ne gative at these points (c) () ( ) conc av e downw ard de crea sing 3,2 , 6,10 ff ′ ⇔ − Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB6 Oil is being pum ped continuously from a certa in oil well at a rate proportional to the am ount of oi l left in the well; that is, dy dt =ky , where y ,000 ,000 is the am ount of oil lef t in the well at any tim e t . Initially there w ere 1 gallons of oil in the well, and 6 years late r the re w ere 500 gallons rem aining. It will no longer be prof itable to pump oil when there are fewer than 50,000 gallons rem aining. ,000 (a) W rite an equation for y , the am ount of oil rem aining in the well at any tim e t . (b) At what rate is the am ount of oil in the well decreasing when there are 600,000 gallons of oil rem aining ? (c) In order no t to lose m oney, at what tim e t should oil no longer be pum ped from the well? Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 AB6 Solution (a) kt dy ky dt yC e = = or 1 1 ln ktC dy kd t y yk t C ye +  =   =+   =   66 1 6 6 ln2 66 66 01 0 , ln 10 1 6 2 ln 2 6 10 10 2 kt k tt tC C ye te k ye −− =⇒ = = ∴= =⇒ = ∴= − == ⋅ 10 (b) 5 5 5 ln 2 61 0 6 10 ln2 De cre asing at 10 ln2 gal /year dy ky dt == − ⋅ ⋅ =− 46 2 (c) 5 10 10 ln 20 ln 20 ln 2 6 ln 20 66 log 20 ln 2 ln 20 6 years aft er st art ing ln 2 kte kt t ⋅= ∴= − − − == ∴= Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC1 Let f be a function such that ′ ′ f (x)=6x+8. (a) Find f(x) if the graph of f is tangent to the line 3 . 2 a t the point (0, 2) xy−= − (b) Find the average value o f f(x) on the c losed interva l [1 . ,1] − Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC1 Solution (a) () () () () 2 32 32 38 03 3 43 2 43 fx x x C f C 2 f xx x x d fx x x x ′ =+ + ′ = = =+ + + =− =+ + − d (b) () () 1 32 1 1 432 1 1 43 2 11 11 4 3 2 24 3 2 1 1 4 3 143 22 24 3 2 4 3 2 2 3 xx x dx xx x x − − ++ − −−  =+ + −    =+ + − − − + +   \b \b  =− ∫ Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC2 Let R be the region enclos ed by the graph of y= x2 x2+1, the line x=1, and the x-axis. (a) Find the area of R . (b) Find the volum e of the solid generated when R is rota ted abo ut the y-axis . Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC2 Solution (a) 1 2 2 0 1 2 0 1 0 Ar ea 1 1 1 1 arctan 1 4 x dx x dx x x x π = + =− + =− =− ⌠⌡ ⌠⌡ (b) () 1 2 2 0 1 2 0 1 2 2 0 Vol ume 2 1 2 1 1 2l n 22 1l n2 x xd x x x xd x x x x π π π π  =  + \b =− +  =− + \b =− ⌠ ⌡ ⌠⌡ 1 or () () 1/2 0 1/2 0 Vol ume 1 1 2l n 1 1l n2 y dy y yy π π π  =−  − \b =+ − =− ⌠ ⌡ Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC3 Consider the function f defined by f(x)=excos x with dom ain [0 . ,2 ]π (a) Find the abs olute m aximu m and m inimum values of f(x). (b) Find the in tervals on which f is incre asing. (c) Find the x-coordina te of each poin t of inflection of the graph of f . Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC3 Solution (a) () [] () sin cos cos sin 5 0 w hen sin cos , ,44 xx x fx e x e x ex x fx x x x ππ ′ =− + =− ′ == = 0 4 5 4 2 x π π π () /4 5/ 4 2 1 2 2 2 2 fx e e e π π π − Max: 25 2 ;M in: 2 ππ − /4 ee (b) 4 π 0 2π 5 4 π + − + () f x′ Increasing on 5 0, , ,2 44 ππ π         ] (c) () [] [ () sin cos cos sin 2s in 0 w hen 0, ,2 xx x f xe x x e x x ex fx x ππ ′′ =− − + − =− ′′ == Point of inf lection a t x π= Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC4 Consider th e curve giv en by the par ametric equations 32 3 2 3 and 12 x tt y t =− = − t (a) In term s of t , find dy dx. (b) W rite an equation f or th e line tangen t to th e curv e at th e poin t where t=− 1. (c) Find the x- and y-coord inates for each critical poin t on the cu rve and iden tify each point as having a vertical or horizontal tangent. Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC4 Solution (a) () ( () ) 2 2 22 22 31 2 66 22 31 2 4 66 2 2 2 1 dy t dt dx tt dt tt dy t t dx t t t t tt =− =− +− −− === −− − (b) xy () 5, 11 3 4 3 11 5 4 or 32 9 44 43 29 dy dx yx yx yx =− = =− −= − + =− + += (c) () () () () () ,t ype 2 28,16 hor izontal 00 ,0 vertical 11 , 11 vertica 2 4, 16 hor izontal tx y −− −− − l Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC5 At any tim e t≥0 , the velocity of a particle traveling along the x-axis is given by the differential equation dx dt−10 x=60 e4t. (a) Find the general solution x(t) for the position of the p article. (b) If the positio n of the particle at tim e t=0 is x=− 8, find the particular solution x(t) for the position of the particle. (c) Use the particular solution fr om par t (b) to find th e tim e at which the p article is at rest. Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC5 Solution (a) Integrating Factor: 10 10 dt t ee− − ∫ = () () 10 4 10 10 6 41 60 10 10 tt t tt tt d xe e e dt 0 xee C x te C −− −− = =− + =− + e or () () 10 4 44 10 4 41 0 60 10 10 t h t p tt tt xt Ce xA e Ae Ae e A xt Ce e = = −= =− =− 4t 2 (b) () 10 4 81 0; 21 0 tt CC x te e − = =− −= (c) 10 4 10 4 20 40 20 40 0 1ln 2 6 tt tt dx ee dt ee t = − −= = or () 41 0 41 0 4 10 10 2 60 0 100 20 60 1ln 2 6 tt tt t dx ee e dt ee e t −− + = +− = = 4t Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC6 Let f be a function that is everywhere differen tiable and that has the following properties. (i) f(x+h)= f(x)+ f(h) f(−x)+ f(−h) for all real num bers h and x . (ii) f(x)>0 for all real num bers x . (iii) ′ f (0 )=− 1. (a) Find the value of f(0) . (b) Show that f(−x)= 1 f(x) for all rea l nu mbers x . (c) Using part (b), show tha t f(x+h)= f(x)f(h) for all rea l num bers h and x . (d) Use the definition of the deriva tive to find () in terms of ( ) f xf′ x . Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1989 BC6 Solution (a) () ( ) () () () () Le t 0 00 00 0 00 xh ff ff ff == + =+ = = + 1 (b) () ( ) () ( ) () ( ) () ( ) () Le t 0 0 0 0 1 Use 0 1 and solv e for h fx f fx fx fx f ff x f x = + += = −+ − == − or Note that () (0 (0 ) () (0) ) f xf fx fx f −+ −+ = + is th e reciprocal of f(x). (c) () () () () () () () () () () () () () 11 fx f h fx h fx f h fx f h f xf h fh f x fx f h + += + + = + = (d) () () ( ) () () () () () () ( ) () 0 0 0 lim lim 1 lim 0 h h h f xh f x fx h f xf h f x h fh fx h f xf f x → → → +− ′ = − = − = ′ == − Copy right © 2003 by College Ent rance Exa mination Boa rd. Al l rights re serve d. Available at apcentral.c ollege boa rd.c om

1990 AB1 A particle, initially at rest, m oves along the x-ax is so that its acceleration at any tim e t≥0 is given by The position of the particle when a(t)=12 t2− 4. t=1 is x(1) =3. (a) Find the values of t for which the pa rticle is at re st. (b) W rite an expression for the position x(t) of the partic le at any tim e t≥0. (c) Find the to tal distance tra veled by th e particle f rom t=0 to t=2.

1990 AB1 Solution (a) vt () () () 3 3 2 44 44 0 41 Therefore 0 , 1 t t vt t t tt tt =− =− = =− = == 0 (b) () () () 42 4 42 2 31 1 2 1 31 4 24 xt t t C x C C C xt t t =− + == − ⋅ + =− = =− + (c) (0 ) 4 (1) 3 (2 ) 12 Dist ance 1 9 10 x x x = = = =+ =

1990 AB2 Let f be the function given by f(x)=ln x x−1 . (a) What is the dom ain of f ? (b) Find the value of the derivative of f at x=− 1. (c) W rite an expression for 11( ),where denotes the inv erse function of . f xf −− f

1990 AB2 Solution (a) 0 1 0a nd 1 0 1 0a nd 1 0 0 0o r 1 x x xx x xx x xx > − >− > ⇒ < () () () () () () 2 1 1 1 1 1 or 11 ln ln 1 1 1 1 2 xx x fx x x xx xx f x x x f −− − ′ =⋅ − − = − ′ −− ⇒ = − − ′−= − (b) (c) () ()1 ln 1 1 1 1 1 y yy y y x x x y x x e x x ee e x e e fx e −  = \b −  = − −= = − = −

1990 AB3 Let R be th e region enclosed by th e graphs of y=ex , y=( x−1)2, and the line x=1. (a) Find the area of R . (b) Find the volum e of the solid generated when R is revolved about the x-axis . (c) Set up, but do not integrate , an in tegral expressio n in te rm s of a single variable for the volum e of the solid generated when R is rev olved about the y-axis .

1990 AB3 Solution (a) () () () 1 2 0 1 2 0 1 1 3 0 0 1 21 1 1 3 14 1 33 x x x Ae x dx ex x d ex ee =− − =− + −   =− −   =− − = − ∫ ∫ x (b) () () 1 4 2 0 1 1 2 5 0 0 22 1 1 1 25 11 7 22 5 2 10 x x x dx e x ee π ππ ππ =− −   =− −      = −−= − \b     ∫ Ve  \b or () () 1 1 0 1 5/2 2 2 2 0 1 2 2 21 1 2 1 ln 21 1 1 22 ln 52 2 4 41 3 7 2 54 4 2 10 e e y dy y y yy y y e e ππ ππ ππ π  =− − + −  Vy dy y    =⋅ + − − \b        =+ − = − \b   ⌠⌡ ∫ (c) Vx () 1 2 0 21 xe x π  =− −  ∫ dx or () () 1 2 2 1 011 1 ln e dy ππ=− − + − ⌠ ⌡ ∫ Vy y dy

1990 AB4 The radius r of a sphere is increasing at a consta nt rate of 0.04 centim eters per second. (Note: The volum e of a sphere with radius r is V = 4 3πr3.) (a) At the tim e when the ra dius of the s phere is 10 c entim eters, what is th e r ate of increase of its volum e? (b) At the tim e when the volum e of the sphere is cubic cen tim eters, what is the rate of increase of the area of a cross s ection through the center of the sphere? 36 π (c) At the tim e when the volum e and the radius of the sphere are in creasing at the sam e num erical rate, what is th e radius ?

1990 AB4 Solution (a) () 2 23 4 3 3 Therefore when 10 , 0.04 41 0 0.04 16 cm /sec dV dr r dt dt dr r dt dV dt π ππ =⋅ == == (b) Vr r () 33 2 2 4 36 36 27 3 3 2 Therefore when 36 , 0.04 6 2 3 0.04 0.24 cm /sec 25 r Ar dA drr dt dt dr V dt dA dt π π π π π ππ =⇒ =⇒ = ⇒ = = = == =⋅ = = (c) 22 2 44 1 11 Therefore cm 4 2 dV dr dt dt dr dr rr dt dt rr ππ π π = =⇒ = =⇒ =

1990 AB5 Let f be th e function defined by f(x)=sin 2x−sin x for 0≤ x≤ 3π 2 . (a) Find the x-in tercepts of the graph of f . (b) Find the in tervals on which f is incre asing. (c) Find the absolute m aximum value an d the abso lute m inimum value of f . Justify your answer.

1990 AB5 Solution (a) 2 sin 0 Therefore si n 0 or si n 1 0, ,2 xx sin x x x π π −= == = (b) () () () 2s in cos cos cos 2sin 1 53 0 w hen , , , 62 6 2 fx x x x xx fx x ππ π π ′ =− =− ′ == 53 26 ππ 2 π 0 6 π − + + − f′ 53 and 62 6 xx ππ π ≤≤ ≤≤ 2 π (c) x ()f x 00 1 64 0 2 51 64 3 2 2 π π π π − − M aximu m v alue: 2 Minim um value: -1/4

1990 AB6 Let f be the function that is given by f(x)= ax +b x2−c and that h as the following properties. (i) The graph of f is symmetric with re spect to the y-axis. (ii) limx→2+f(x)=+ ∞ (iii) ′ f (1) =− 2 (a) Determ ine the values of a , b , and c . (b) W rite an equation for ea ch vertica l and each horizontal asymptote of the graph of f. (c) Sketch the g raph of f in the xy -plane provided below. −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 y x

1990 AB6 Solution (a) Graph symm etric to y-axi s f is even ⇒ () ( ) () () () () () 2 2 2 2 therefore 0 lim the refore 4 4 2 4 2 2 1 therefore 9 9 x fx f x a fx c b fx x bx fx x b fb + → −= = =+ ∞ = = − − ′ = − − ′ −= = = (b) () 2 9 4 Ve rti cal: 2, 2 Horiz ont al: 0 fx x xx y = − == − = (c) −5 −4 −3 −2 −1 1 2 3 4 5 5 −4 −3 −2 −1 1 2 3 4 5 x y

1990 BC1 A partic le s tarts a t tim e t=0 and m oves along the x-axis so that its position at any tim e t≥0 is given by x(t) 1)3(2t−3). =(t− (a) Find the ve locity of the particle at a ny tim e t≥0. (b) For what values of t is the veloc ity of the particle less than z ero? (c) Find the value of t when the partic le is m oving and the ac ce lera tion is ze ro.

1990 BC1 Solution (a) vt () () () ( ) () () ( ) 23 2 31 2 3 2 1 18 11 x t tt t tt ′ = =− − + − =− − (b) () ( ) ( ) 2 0 w hen 1 8 11 0 Therefore 8 11 0 and 1 11 or and 1 8 11 Since 0, ans wer i s 0 , except 1 8 t t tt tt tt

1990 BC2 Let R be the region in the xy -plane between the graphs of y=exand y= e−x from . 0 t o 2 xx== (a) Find the volu me of the solid generated when R is revolved about the x-axis . (b) Find the volu me of the solid generated when R is revolved about the y-axis .

1990 BC2 Solution (a) () 2 22 0 2 22 0 44 44 11 22 11 1 1 22 2 2 2 2 xx xx e dx ee ee ee π π π π − − − − =−  =+    =+ − + \b     =+ −  ∫ Ve (b) Vx () () ( ) () ( ) () 2 0 2 0 2 0 22 2 2 22 2 2 2 22 0 1 1 23 xx xx xx xx x x e e dx xe e e e dx xe e e e ee e e ee π π π π π − −− −− −− −  =−   =+ − +   =+ − −   =+ − − − − −       =+  ∫ ∫

1990 BC3 Let f(x)=12 − x2for x≥0and f(x)≥ 0. (a) The line tan gent to the g raph of f at the poin t (, intercep ts the x-axis a t ( )) kf k . W hat is th e value of k ? x=4 (b) An is osceles trian gle whose b ase is the in terval from (0 to has its vertex on ,0) (, 0) c the graph of f . For what value of c does the triangle have m aximum area? Justify your answer.

1990 BC3 Solution (a) () () () () () ( ) () () () 2 2 2 2 12 ; 2 slope of ta nge nt l ine at ,2 line thr ough 4,0 & , has sl ope 0 12 44 12 so 2 8 12 0 4 2or 6 but 6 24 so 6 is not in the dom ain. 2 fx x f x x kf k k kf k fk k kk k kk k k kk f k ′ =− =− =− − − = −− − −= ⇒ − + = − == =− = (b) 2 3 22 11 12 22 2 4 6 on 0 ,4 3 8 33 6; 6 0 w hen c 88 cc Ac f c c c dA c c dc   =⋅ = − \b \b   =−  =− − = =4. Candidate test First derivative 00 41 6 43 0 cA Max − A′ + 0 4 43 second derivative 2 2 4 3 0 so 4 gi ves a r elative m ax. c dA c dc = =− < = c = 4 is the only critical va lue in the dom ain inte rval, therefo re m aximum

1990 BC4 Let R be the region inside the graph of the polar curve r= 2 and outside the graph of the polar cu rve r=2(1 −sin θ). (a) Sketch the two polar curves in th e xy -plane provided below and shade the region R . −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 y x (b) Find the area of R .

1990 BC4 Solution (a) −5 5 −5 5 y x R O 5− 5− 5 5 y x () () () () () () [] 2 2 0 2 0 00 0 0 1 22 1 sin 2 22 sin sin 4s in 1 cos2 1 4c os sin2 2 41 4 1 0 8 Ad d dd π π ππ π π θθ θθ θ θθ θ θ θθ θ π π  =− −  =− =− −  =− − −  =− − + − −  =− ⌠ ⌡ ∫ ∫∫ (b)

1990 BC5 Let f be the function defined by f(x)= 1 x−1. (a) W rite the first four term s and the gen eral term of the Taylo r series expansion of f(x) about . x=2 (b) Use the resu lt f rom part (a) to f ind th e f irst four term s and the general term of the series expansion about for ln x=2 x−1. (c) Use the series in p art (b ) to com pute a num ber that differs fro m ln 3 2 by less than 0.05. Justify your answer.

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