Consider the differential equation (a) Find in terms of x an. AP CALCULUS BC 06 SCORING GUIDELINES x y. = Question 4 (b) Let y = f ( x) be the particular solution to the given differential equation whose graph passes through the point (, 8 ). Does the graph of f have a relative minimum, a relative maximum, or neither at the point (, 8 )? Justify your answer. (c) Let y = g( x) be the particular solution to the given differential equation with g( ) =. Find g( x) lim ( ). Show the work that leads to your answer. x x + (d) Let y = h( x) be the particular solution to the given differential equation with h ( 0) =. Use Euler s method, starting at x = 0 with two steps of equal size, to approximate h (. ) (a) = x = x ( x y) : in terms of x and y (b) ( x, y) = (, 8) ( x, y) = (, 8) ( ) = 0 8 = ( ) = ( ) ( ) 8 = 4 < 0 Thus, the graph of f has a relative maximum at the point (, 8 ). : conclusion with justification (c) lim ( g( x) ) = 0 and lim ( x + ) = 0 x x Using L Hospital s Rule, g( x) ( ) lim lim g x x = ( x ) x 6 ( x + ) + { : L Hospital s Rule : : answer lim g ( x) = 0 and lim 6( x + ) = 0 x x Using L Hospital s Rule, ( ) ( ) lim g x lim g x x 6( x ) = = = x 6 + 6 = + = + + = 4 (d) h( ) h( 0) + h ( 0) ( ) h( ) h( ) h ( ) ( ) { : Euler s method : : approximation 06 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Consider the differential equation x y. = AP CALUCLUS AB/CALCULUS BC 0 SCORING GUIDELINES Question 4 (a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find in terms of x an. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer. (c) Let y = f( x) be the particular solution to the differential equation with the initial condition f ( ) =. Does f have a relative minimum, a relative maximum, or neither at x =? Justify your answer. (d) Find the values of the constants m and b for which y = mx + b is a solution to the differential equation. (a) : { : slopes where x = 0 : slopes whe re x = (b) (c) = = ( x y) = x + y In Quadrant II, x < 0 an > 0, so x + y > 0. Therefore, all solution curves are concave up in Quadrant II. = = ( ) = =/ 0 ( x, y) (, ) Therefore, f has neither a relative minimum nor a relative maximum at x =. : : : concave up with reason : : considers ( x, y) = (, ) : conclusion with justification (d) y = mx + b d = ( mx + b ) = m x y = m x ( mx + b) = m ( m) x ( m + b) = 0 m = 0 m = b = m b = d : ( mx + b ) = m : : x y = m : answer Therefore, m = and b =. 0 The College Board. Visit the College Board on the Web: www.collegeboard.org.
0 SCORING GUIDELINES Consider the differential equation y ( x. ) = + Let y = f( x) be the particular solution to the differential equation with initial condition f ( 0) =. (a) Find f( x) + lim. Show the work that leads to your answer. x 0 sin x (b) Use Euler s method, starting at 0. x = with two steps of equal size, to approximate f ( ) (c) Fin = f( x), the particular solution to the differential equation with initial condition f ( 0) =. (a) lim ( f( x) + ) = + = 0 and lim sin x = 0 x 0 x 0 Using L Hospital s Rule, f( x) + f ( x) f ( 0 ) ( ) lim = lim = = = x 0 sin x x 0 cos x cos0 : L Hospital s Rule : { : answer (b) f( ) f( 0) + f ( 0 4 )( 4) = + ( )( ) = 4 : Euler s method : { : answer ( ) ( ) + 4 ( 4 )( 4) ( ) ( )( ) f f f = + + = 4 4 (c) = y ( x + ) = ( x + ) y = ( x + ) y = x + x + C y = 0 + 0 + C C = = x + x + y y = = x + x + ( x + ) : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web: www.collegeboard.org.
0 SCORING GUIDELINES The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t = 0, when the bird is first weighed, its weight is 0 grams. If B() t is the weight of the bird, in grams, at time t days after it is first weighed, then db = ( 00 B). Let y = B() t be the solution to the differential equation above with initial condition B ( 0) = 0. (a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning. d B d B (b) Find in terms of B. Use to explain why the graph of B cannot resemble the following graph. (c) Use separation of variables to fin = B(), t the particular solution to the differential equation with initial condition B ( 0) = 0. db (a) = ( 60 ) = B= 40 db B= 70 = ( 0 ) = 6 : db : uses : answer with reason db Because > db, the bird is gaining B= 40 B= 70 weight faster when it weighs 40 grams. d B db (b) ( 00 ) ( 00 ) = = B = B Therefore, the graph of B is concave down for 0 B < 00. A portion of the given graph is concave up. db (c) = ( 00 B) db = 00 B ln 00 B = t + C Because 0 B < 00, 00 B = 00 B. ln ( 00 0) = ( 0) + C ln ( 80) = C t 00 B = 80e t Bt () = 00 80 e, t 0 : : d B : in terms of B : explanation : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for B Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web: www.collegeboard.org.
0 SCORING GUIDELINES At the beginning of 00, a landfill contained 400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential dw equation = ( W 00) for the next 0 years. W is measured in tons, and t is measured in years from the start of 00. (a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first months of 00 (time t = ). 4 dw dw (b) Find in terms of W. Use to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time t =. 4 dw (c) Find the particular solution W = W( t) to the differential equation = ( W 00) with initial condition W ( 0) = 400. dw (a) = ( W ( 0) 00) = ( 400 00) = 44 t= 0 The tangent line is y = 400 + 44 t. ( ) ( ) W 400 + 44 = 4 tons 4 4 : at 0 : dw t = : answer dw dw (b) = = ( W 00) and W 400 6 dw Therefore > 0 on the interval 0 t. 4 The answer in part (a) is an underestimate. : dw : : answer with reason dw (c) = ( W 00) dw = W 00 ln W 00 = t + C ln ( 400 00) = ( 0) + C ln ( 00) = C W 00 = 00e t t W() t = 00 + 00 e, 0 t 0 : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for W Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web: www.collegeboard.org.
00 SCORING GUIDELINES Consider the differential equation y. = Let y = f( x) be the particular solution to this differential equation with the initial condition f () = 0. For this particular solution, f( x ) < for all values of x. (a) Use Euler s method, starting at x = with two steps of equal size, to approximate f ( 0. ) Show the work that leads to your answer. (b) Find lim f ( x). Show the work that leads to your answer. x x (c) Find the particular solution y = f( x) to the differential equation = y with the initial condition f () = 0. f f + x Δ ( ) ( ), 0 = 0 + = (a) ( ) () f ( 0) f( ) + Δx (, ) + ( ) = 4 (b) Since f is differentiable at x =, f is continuous at x =. So, ( ) ( ) lim f x = 0 = lim x and we may apply L Hospital s x x Rule. f( x) f ( x) lim f ( x) x lim = lim = = x x x x lim x x : Euler s method with two steps : { : answer : use of L Hospital s Rule : { : answer (c) y = y = ln y = x + C ln = + C C = ln y = x y e x = : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables f ( x) = e x 00 The College Board. Visit the College Board on the Web: www.collegeboard.com.
Consider the differential equation AP CALCULUS BC 009 SCORING GUIDELINES Question 4 6 x x y. = Let y f( x) differential equation with the initial condition f ( ) =. = be a particular solution to this (a) Use Euler s method with two steps of equal size, starting at x =, to approximate f ( 0. ) Show the work that leads to your answer. (b) At the point (, ), the value of f about x =. is. Find the second-degree Taylor polynomial for (c) Find the particular solution y = f( x) to the given differential equation with the initial condition f ( ) =. f f + x Δ (, ) = + 4 = 4 (a) ( ) ( ) : Euler s method with two steps : { : answer f ( 0) f( ) + Δx (,4 ) 7 4 + = 4 (b) P ( ) ( ) ( ) x = + 4 x + 6 x + : answer (c) x ( 6 y) = = x 6 y ln 6 y = x + C ln 4 = + C C = ln 4 ln 6 y = x ln 4 ( x ) 6 y = 4e + ( x ) y = 6 4e + ( ) 6 : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: max 6 [--0-0-0] if no constant of integration Note: 0 6if no separation of variables 009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.
008 SCORING GUIDELINES Question 6 y Consider the logistic differential equation = ( 6 y). Let y = f() t be the particular solution to the 8 differential equation with f ( 0) = 8. (a) A slope field for this differential equation is given below. Sketch possible solution curves through the points (, ) and ( 0, 8 ). (Note: Use the axes provided in the exam booklet.) (b) (c) (d) (a) Use Euler s method, starting at t = 0 with two steps of equal size, to approximate f (). Write the second-degree Taylor polynomial for f about t = 0, and use it to approximate f (). What is the range of f for t 0? : : solution curve through ( 0,8) : solution curve through (,) (b) ( ) ( )( ) () 7 ( )( ) f 8 + = 7 7 0 f + = 8 6 (c) d y = ( 6 y) + y ( ) 8 8 8 f( 0) = 8; f ( 0) = = ( 6 8) = ; and t= 0 8 8 f ( 0) = = ( )( ) + ( ) = 8 8 t= 0 The second-degree Taylor polynomial for f about t = 0 is P () t = 8 t + t. 4 9 f() P () = 4 (d) The range of f for t 0 is 6 < y 8. : answer : : Euler s method with two steps : approximation of f () : 4 : : second-degree Taylor polynomial : approximation of f () 008 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.
Consider the differential equation x y = + +. (a) Find in terms of x an. AP CALCULUS BC 007 SCORING GUIDELINES (Form B) rx (b) Find the values of the constants m, b, and r for which y = mx + b + e is a solution to the differential equation. (c) Let y = f ( x) be a particular solution to the differential equation with the initial condition f ( 0) =. Use Euler s method, starting at x = 0 with a step size of, to approximate f ( ). Show the work that leads to your answer. (d) Let y = g( x) be another solution to the differential equation with the initial condition g( 0 ) = k, where k is a constant. Euler s method, starting at x = 0 with a step size of, gives the approximation g () 0. Find the value of k. (a) = + = + ( x + y + ) = 6x + 4y + (b) If rx y = mx + b + e rx is a solution, then rx ( ) m + re = x + mx + b + e +. If r 0 : m = b +, r =, 0 = + m, : + : : answer rx : = m + re : : value for r : values for m and b so m =, r =, and OR b =. 4 If r = 0 : m = b +, r = 0, 0 = + m, so m =, r = 0, 9 b =. 4 7 + = + = (c) f( ) f( 0) f ( 0) ( ) 7 f ( ) ( ) + ( ) + = 9 f() f( ) f ( ) ( ) 7 9 + = + = 4 (d) g ( 0) = 0 + k + = k + g( ) g( 0) + g ( 0) = k + ( k + ) = k + = 0 k = : : Euler's method with steps : Euler's approximation for f () : g( 0) + g ( 0) : : value of k 007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).
006 SCORING GUIDELINES (Form B) Let f be a function with f ( 4) = such that all points ( x, y ) on the graph of f satisfy the differential equation y( x). = Let g be a function with g ( 4) = such that all points ( x, y ) on the graph of g satisfy the logistic differential equation y( y). = (a) Fin = f ( x). (b) Given that g ( 4) =, find lim g( x) and lim g ( x). (It is not necessary to solve for gx ( ) or to show how x x you arrived at your answers.) (c) For what value of y does the graph of g have a point of inflection? Find the slope of the graph of g at the point of inflection. (It is not necessary to solve for gx ( ). ) (a) y( x) = = ( x) y ln y = 6x x + C 0 = 4 6 + C C = 8 ln y = 6x x 8 6x x 8 y = e for < x < : : separates variables : antiderivatives : constant of integration : uses initial condition : solution Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables (b) lim g( x) = x lim g ( x) 0 = x : : lim g( x) = x : lim g ( x) = 0 x (c) = (6 4 y) Because 0 at any point on the graph of g, the concavity only changes sign at y =, half the carrying capacity. 9 = ( )( = ) = y : y = : : y = 006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6
006 SCORING GUIDELINES 6 Consider the differential equation = x for y. Let y = f ( x) be the particular solution to this y differential equation with the initial condition f ( ) = 4. (a) Evaluate and at (, 4 ). (b) Is it possible for the x-axis to be tangent to the graph of f at some point? Explain why or why not. (c) Find the second-degree Taylor polynomial for f about x =. (d) Use Euler s method, starting at x = with two steps of equal size, to approximate f ( 0. ) Show the work that leads to your answer. (a) = 6 (, 4) = 0x + 6( y ) = 0 + 6 6 = 9 (, 4) ( 6) : : (, 4) : : (, 4) (b) The x-axis will be tangent to the graph of f if = 0. ( k,0) The x-axis will never be tangent to the graph of f because = k + > 0 for all k. ( k,0) : = 0 an = 0 : : answer and explanation (c) Px ( ) = 4 + 6( x+ ) 9 ( x+ ) : quadratic and centered at x = : { : coefficients (d) f ( ) = 4 f ( ) 4 + ( 6) = f ( 0) + ( + 4 ) = 8 : : Euler's method with steps : Euler's approximation to f ( 0) 006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6
00 SCORING GUIDELINES Question 4 Consider the differential equation x y. = (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point ( 0, ). (Note: Use the axes provided in the pink test booklet.) 0, has a local minimum at ( ) (b) The solution curve that passes through the point ( ) x = ln. What is the y-coordinate of this local minimum? (c) Let y = f ( x) be the particular solution to the given differential equation with the initial condition f ( 0) =. Use Euler s method, starting at x = 0 with two steps of equal size, to approximate f ( 0.4 ). Show the work that leads to your answer. (d) Find in terms of x an. Determine whether the approximation found in part (c) is less than or greater than f ( 0.4 ). Explain your reasoning. (a) : : zero slopes : nonzero slopes : curve through ( 0, ) (b) = 0 when x y = The y-coordinate is ( ) ln. : sets = 0 : : answer (c) f ( 0.) f ( 0) + f ( 0)( 0.) = + ( )( 0.) =. f( 0.4) f( 0.) + f ( 0.)( 0.). + (.6)( 0.) =. : : Euler's method with two steps : Euler approximation to f ( 0.4) (d) = = x + y is positive in quadrant II because x < 0 an > 0.. < f ( 0.4) since all solution curves in quadrant II are concave up. : : : answer with reason Copyright 00 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).
004 SCORING GUIDELINES A population is modeled by a function P that satisfies the logistic differential equation dp = P ( P ). (a) If P ( 0) =, what is lim Pt ()? t If P ( 0) = 0, what is lim Pt ()? t (b) If P ( 0) =, for what value of P is the population growing the fastest? (c) A different population is modeled by a function Y that satisfies the separable differential equation dy = Y ( t ). Find Y() t if Y ( 0) =. (d) For the function Y found in part (c), what is lim Y() t? t (a) For this logistic differential equation, the carrying capacity is. If P ( 0) =, lim Pt () =. t If P ( 0) = 0, lim Pt () =. t : answer : : answer (b) The population is growing the fastest when P is half the carrying capacity. Therefore, P is growing the fastest when P = 6. : answer t t (c) = ( ) = ( ) dy Y 60 ln Y Y() t = K = Y() t = t t = + C 0 Ke t t 0 t t e 0 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for Y 0 if Y is not exponential Note: max [--0-0-0] if no constant of integration Note: 0 if no separation of variables (d) lim Y() t = 0 : answer t 0 if Y is not exponential Copyright 004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6
00 SCORING GUIDELINES A coffeepot has the shape of a cylinder with radius inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of h cubic inches per second. (The volume V of a cylinder with radius r and height h is V = r h. ) dh h (a) Show that. = dh h (b) Given that h = 7 at time t = 0, solve the differential equation = for h as a function of t. (c) At what time t is the coffeepot empty? (a) V = h dv dh = = h dh h h = = : dv : = h dv : computes : shows result (b) dh = h dh = h h = t + C 7 = 0 + C ( ) t 7 h = + 0 : separates variables : antiderivatives : constant of integration : : uses initial condition h = 7 when t = 0 : solves for h Note: max / [--0-0-0] if no constant of integration Note: 0/ if no separation of variables (c) ( ) 7 0 0 t + = : answer t = 0 7 Copyright 00 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com. 6
00 SCORING GUIDELINES (Form B) @O! Г N +IE@AHJDA@EBBAHAJE=AGK=JE @N O = AJO BN >AJDAF=HJE?K=HIKJEJJDACELA@EBBAHAJE=AGK=JEBH N # IK?DJD=JJDAEA O Г EIJ=CAJJJDACH=FDBB.E@JDAN?H@E=JABJDAFEJB J=CA?O=@@AJAHEAMDAJDAHBD=I=?==NEK?=EEKHAEJDAH=JJDEI FEJKIJEBOOKH=IMAH > AJO CN >AJDAF=HJE?K=HIKJEJJDACELA@EBBAHAJE=AGK=JEBH Г N & MEJDJDAEEJE=?@EJE C$ Г".E@O CN = @O @N MDAN! @O ГO ГO=! ГN @N O Г! Г! IBD=I=?=EEK=JJDEIFEJ H *A?=KIABEI?JEKKIBH N # JDAHA EI=EJAHL=?J=EECN!MDE?D OJDEIEJAHL= @O @N EIAC=JELAJ JDAABJBN!=@ @O @N EIFIEJELAJJDA HECDJBN!6DAHABHABD=I=?= EEK=JN! > O@O! ГN @N O! N Г N + & & Г& + +& O $ N ГN $ O Г $ N ГN $ N!!?=EEK KIJEBE?=JE IAF=H=JAIL=HE=>AI =JE@AHEL=JELAB@OJAH =JE@AHEL=JELAB@N JAH $?IJ=JBEJACH=JE KIAIEEJE=?@EJEC$ Г" ILAIBHO JA=N!$EB?IJ=J BEJACH=JE JA$EBIAF=H=JEBL=HE=>AI Copyright 00 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 6
00 SCORING GUIDELINES @O +IE@AHJDA@EBBAHAJE=AGK=JE O " N @N Г = 6DAIFABEA@BHJDACELA@EBBAHAJE=AGK=JEEIFHLE@A@AJ?DJDAIKJE?KHLAJD=JF=IIAIJDHKCD JDAFEJ =@IAJ?DJDAIKJE?KHLAJD=JF=IIAIJDHKCDJDAFEJ Г > AJ B >AJDABK?JEJD=JI=JEIBEAIJDACELA@EBBAHAJE=AGK=JEMEJDJDAEEJE=?@EJE B 7IA -KAH\IAJD@IJ=HJEC=J N MEJD=IJAFIEABJ=FFHNE=JA B DMJDAMHJD=JA=@I JOKH=IMAH?.E@JDAL=KAB > BHMDE?D O N > EI=IKJEJJDACELA@EBBAHAJE=AGK=JEKIJEBOOKH =IMAH @ AJ C >AJDABK?JEJD=JI=JEIBEAIJDACELA@EBBAHAJE=AGK=JEMEJDJDAEEJE=?@EJE C,AI JDACH=FDB C D=LA=?=ANJHAK=JJDAFEJ BIEIJDAFEJ=?==NEKH=?= EEKKIJEBOOKH=IMAH = > B N B B= Г B N B B= N " Г " "? K>IJEJKJA O N > EJDA,- N > Г " N > I> 4 /KAII> O N @O 8AHEBO O Г " N " N Г " N @N @ CD=I?==NEK=J @O C= @N Г " =@ @O @O C == N " @N @N Г I C== C= Г " Г" IKJE?KHLAJDHKCD IKJE?KHLAJDHKCD Г +KHLAIKIJCJDHKCDJDAE@E?=JA@ FEJIBMJDACELAIFAEAI=@ ANJA@JJDA>K@=HOBJDAIFABEA@ -KAH\IAJD@AGK=JEIH AGKEL=AJJ=>A=FFEA@J=JA=IJ JMEJAH=JEI -KAH=FFHNE=JEJB JAECE>AMEJDKJBEHIJFEJ @ KIAI E,- N > @N > C=! IDMIC== Г"??KIE Copyright 00 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 6
AP CALCULUS BC 00 SCORING GUIDELINES Let f be the function satisfying f= () x Г xf() x, for all real numbers x, with f () 4 and lim fx ( ) 0. x @ (a) Evaluate @ Г xf( x). Show the work that leads to your answer. (b) Use Euler s method, starting at x = with a step size of 0., to approximate f (). (c) Write an expression for y f( x) by solving the differential equation Г xy with the initial condition f () 4. (a) = @ Г xf( x) b f= ( x) lim f= ( x) lim f( x) @ b b@ = lim fb ( ) Г f() 0 Г 4 Г 4 b@ b@ : : use of FTC : answer from limiting process (b) f(.) N f() f= ()(0.) = 4 Г ()(4)(0.) Г f() NГ f= (.)(0.) NГГ (.)( Г)(0.). : : Euler's method equations or equivalent table : Euler approximation to f () (not eligible without first point) (c) Г x y ln y Г x k y Ce Г x Г 4 Ce ; x y 4e e Г C 4e : separates variables : antiderivatives : : constant of integration : uses initial condition f () 4 : solves for y Note: max / [--0-0-0] if no constant of integration Note: 0/ if no separation of variables Copyright 00 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 6
998 Calculus BC Scoring Guidelines Copyright 998 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.