AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

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1 004 SORING GUIDELINES (Form B) dy 4 onsider the differential equation ( y. ) d = On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the aes provided in the test booklet.) (b) While the slope field in part is drawn at only twelve points, it is defined at every point in the y-plane. Describe all points in the y-plane for which the slopes are negative. Find the particular solution y = f( ) to the given differential equation with the initial condition f ( 0) = 0. : : zero slope at each point (, where 0 or y = = positive slope at each point (, where 0 and y > : negative slope at each point (, where 0 and y < (b) Slopes are negative at points (, y ) where 0 and y <. : description 4 dy = d y 5 ln y = + 5 y = e e y = Ke, K = ± e = Ke = K y = e 6 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y 0 if y is not eponential Note: ma 3 6 [ ] if no constant of integration opyright 004 by ollege Entrance Eamination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 59 6

2 004 SORING GUIDELINES Question 6 dy onsider the differential equation ( y. ) d = On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the aes provided in the pink test booklet.) (b) While the slope field in part is drawn at only twelve points, it is defined at every point in the y-plane. Describe all points in the y-plane for which the slopes are positive. Find the particular solution y = f( ) to the given differential equation with the initial condition f ( 0) = 3. : : zero slope at each point (, where 0 or y = = positive slope at each point (, where 0 and y > : negative slope at each point (, where 0 and y < (b) Slopes are positive at points (, y ) where 0 and y >. : description dy = d y 3 ln y = + 3 y = e e y = Ke, K = ± e = Ke = K y = + e 6 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y 0 if y is not eponential Note: ma 3 6 [ ] if no constant of integration opyright 004 by ollege Entrance Eamination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 60 7

3 005 SORING GUIDELINES (Form B) Question 6 dy y onsider the differential equation =. Let d y = f( ) be the particular solution to this differential equation with the initial condition f ( ) =. On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the aes provided in the test booklet.) (b) Write an equation for the line tangent to the graph of f at =. Find the solution y = f( ) to the given differential equation with the initial condition f ( ) =. : zero slopes : : nonzero slopes ( 4 ) (b) Slope = = y = ( + ) : equation y dy = d = + y 4 = + ; = y = = : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: ma 3 6 [ ] if no constant of integration opyright 005 by ollege Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 6 7

4 onsider the differential equation dy d AP ALULUS AB 005 SORING GUIDELINES =. y Question 6 On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the aes provided in the pink test booklet.) (b) Let y = f ( ) be the particular solution to the differential equation with the initial condition (). f = Write an equation for the line tangent to the graph of f at (, ) and use it to approimate f (. ). Find the particular solution y = f ( ) to the given differential equation with the initial condition f () =. : zero slopes : { : nonzero slopes (b) The line tangent to f at (, ) is y + = ( ). Thus, f (.) is approimately 0.8. : : equation of the tangent line : approimation for f (.) dy = d y ydy= d y = + 3 = + ; = y = + 3 Since the particular solution goes through (, ), y must be negative. Thus the particular solution is y = 3. 5 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: ma 5 [ ] if no constant of integration Note: 0 5 if no separation of variables opyright 005 by ollege Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and 6 (for AP students and parents). 7

5 006 SORING GUIDELINES (Form B) dy onsider the differential equation = ( y ) cos ( π ). d On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the aes provided in the eam booklet.) (b) There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c. Find the particular solution y = f ( ) to the differential equation with the initial condition f () = 0. : zero slopes : { : all other slopes (b) The line y = satisfies the differential equation, so c =. : c = ( y ) dy = cos( π ) d ( y ) = sin( π ) + π = sin ( π ) + y π = sin( π ) + = π = sin ( π ) + y π π = sin( π) + π y π y = for < < sin ( π) + π 6 : : separates variables : antiderivatives : constant of integration : uses initial condition : answer Note: ma 3 6 [ ] if no constant of integration 006 The ollege Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 6 63

6 006 SORING GUIDELINES dy + y onsider the differential equation =, where 0. d On the aes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the aes provided in the pink eam booklet.) (b) Find the particular solution y = f ( ) to the differential equation with the initial condition f ( ) = and state its domain. : sign of slope at each point and relative steepness of slope lines in rows and columns (b) dy = d + y ln + y = ln + K ln + K + y = e + y = = + y = y = and < 0 or y = and < 0 7 : : separates variables : antiderivatives 6 : : constant of integration : uses initial condition : solves for y Note: ma 3 6 [ ] if no constant of integration : domain 006 The ollege Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 6 64

7 onsider the differential equation AP ALULUS AB 007 SORING GUIDELINES (Form B) dy y. d = + On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the aes provided in the eam booklet.) d y (b) Find in terms of and y. Describe the region in the y-plane in d which all solution curves to the differential equation are concave up. Let y = f( ) be a particular solution to the differential equation with the initial condition f ( 0) =. Does f have a relative minimum, a relative maimum, or neither at = 0? Justify your answer. (d) Find the values of the constants m and b, for which y = m + b is a solution to the differential equation. : Sign of slope at each point and relative steepness of slope lines in rows and columns. (b) d y dy = + = + y d d Solution curves will be concave up on the half-plane above the line y = +. dy d ( 0, ) d y = 0 + = 0 and d ( 0, ) Thus, f has a relative minimum at ( 0, ). = 0 + > 0 3 : d y : d : description : answer : { : justification (d) Substituting y = m + b into the differential equation: m = + ( m + b) = ( m + ) + ( b ) Then 0 = m + and m = b : m = and b =. : value for m : { : value for b 007 The ollege Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 65

8 008 SORING GUIDELINES dy y onsider the differential equation =, where 0. d On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the aes provided in the eam booklet.) (b) Find the particular solution y = f( ) to the differential equation with the initial condition f ( ) = 0. For the particular solution y = f( ) described in part (b), find lim f ( ). : zero slopes : { : all other slopes (b) dy = d y ln y = + y = e + y = e e y = ke, where k = ± e = ke k = e ( ) ( ) f = e, > 0 6 : : separates variables : antidifferentiates : includes constant of integration : uses initial condition : solves for y Note: ma 3 6 [ ] if no constant of integration ( ) lim e e = : limit 008 The ollege Board. All rights reserved. Visit the ollege Board on the Web: 66

9 998 AP alculus AB Scoring Guidelines 4. Let f be a function with f() = 4 such that for all points (, on the graph of f the slope is given by 3 +. y Find the slope of the graph of f at the point where =. (b) Write an equation for the line tangent to the graph of f at = and use it to approimate f(.). Find f() by solving the separable differential equation dy d = 3 + with the initial y condition f() = 4. (d) Use your solution from part to find f(.). dy d = 3 + y dy d = y = 4 = = 4 8 = : answer (b) y 4 = ( ) f(.) 4 (. ) { : equation of tangent line : uses equation to approimate f(.) f(.) = 4. y dy = (3 + ) d y dy = (3 + ) d y = = = y = y = is branch with point (, 4) f() = : separates variables : antiderivative of dy term : antiderivative of d term : uses y = 4 when = to pick one function out of a family of functions : solves for y 0/ if solving a linear equation in y 0/ if no constant of integration Note: ma 0/5 if no separation of variables Note: ma /5 [ ] if substitutes value(s) for, y, or dy/d before antidifferentiation (d) f(.) = : answer, from student s solution to the given differential equation in opyright 998 ollege Entrance Eamination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the ollege Entrance Eamination Board. 67

10 !0Å#ALULUSÅ!"n DY X #ONSIDERÅTHEÅDIFFERENTIALÅEQUATIONÅ Y DX E A &INDÅAÅSOLUTIONÅY F X ÅTOÅTHEÅDIFFERENTIALÅEQUATIONÅSATISFYINGÅ F B &INDÅTHEÅDOMAINÅANDÅRANGEÅOFÅTHEÅFUNTIONÅFÅFOUNDÅINÅPARTÅA Y ÅA E DY X DX Y E X # Y E X # LN Y X # LN # ÅÅ # E LN Y X E ÅSEPARATESÅVARIABLES ÅANTIDERIVATIVEÅOFÅDYÅTERM ÅANTIDERIVATIVEÅOFÅDX ÅTERM ÅONSTANTÅOFÅINTEGRATION Å ÅUSESÅINITIALÅONDITIONÅF ÅSOLVESÅFORÅY Å.OTEÅÅIFÅY ÅISÅNOTÅAÅLOGARITHMIÅFUNTIONÅOFÅX.OTEÅMAXÅÅ;=ÅIFÅNOÅONSTANTÅOF INTEGRATION.OTEÅÅIFÅNOÅSEPARATIONÅOFÅVARIABLES B $OMAINÅX E X E X E E ANGEÅd Y d ÅX E ÅDOMAIN Å ÅÅÅÅÅÅ.OTEÅÅIFÅÅISÅNOTÅINÅTHEÅDOMAIN ÅRANGE.OTEÅÅIFÅYÅISÅNOTÅAÅLOGARITHMIÅFUNTIONÅOFÅX opyright 000 by ollege Entrance Eamination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the ollege Entrance Eamination Board. 68

11 00 SORING GUIDELINES (Form Г N O = AJO BN >AJDAF=HJE?K=HIKJEJJDAELA@EBBAHAJE=AGK=JEBH N # IK?DJD=JJDAEA O Г EIJ=AJJJDAH=FDBB.E@JDAN?H@E=JABJDAFEJB J=A?O=@@AJAHEAMDAJDAHBD=I=?==NEK?=EEKHAEJDAH=JJDEI FEJKIJEBOOKH=IMAH > AJO N >AJDAF=HJE?K=HIKJEJJDAELA@EBBAHAJE=AGK=JEBH Г N & MEJDJDAEEJE=?@EJE $ Г".E@O ГO ГO=! O Г! Г! IBD=I=?=EEK=JJDEIFEJ H *A?=KIABEI?JEKKIBH N # JDAHA EIFIEJELAJJDA HEDJBN!6DAHABHABD=I=?= EEK=JN! > O@O! 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 $?IJ=JBEJAH=JE KIAIEEJE=?@EJE$ Г" ILAIBHO JA=N!$EB?IJ=J BEJAH=JE JA$EBIAF=H=JEBL=HE=>AI opyright 00 by ollege Entrance Eamination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the ollege Entrance Eamination Board. 69 6

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