# AP CALCULUS BC 2016 SCORING GUIDELINES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 = + = + + = 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:

2 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:

3 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 = C C = = x + x + y y = = x + x + ( x + ) : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: max [ ] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web:

4 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 () = 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 [ ] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web:

5 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) = ( ) = 44 t= 0 The tangent line is y = t. ( ) ( ) W = 4 tons 4 4 : at 0 : dw t = : answer dw dw (b) = = ( W 00) and W 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 ( ) = ( 0) + C ln ( 00) = C W 00 = 00e t t W() t = e, 0 t 0 : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for W Note: max [ ] if no constant of integration Note: 0 if no separation of variables 0 The College Board. Visit the College Board on the Web:

6 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 [ ] 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:

7 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 ) = 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 [ ] 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:

8 008 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 + = f + = 8 6 (c) d y = ( 6 y) + y ( ) f( 0) = 8; f ( 0) = = ( 6 8) = ; and t= 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:

9 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 = = + = (c) f( ) f( 0) f ( 0) ( ) 7 f ( ) ( ) + ( ) + = 9 f() f( ) f ( ) ( ) = + = 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 (for students and parents).

10 006 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 = 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 [ ] 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 (for AP students and parents). 6

11 006 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 ) = = 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 (for AP students and parents). 6

13 004 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 [ ] 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 (for AP students and parents). 6

14 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 / [ ] if no constant of integration Note: 0/ if no separation of variables (c) ( ) t + = : answer t = 0 7 Copyright 00 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com. 6

15 00 SCORING GUIDELINES (Form Г O = AJO BN N # IK?DJD=JJDAEA O Г FEJKIJEBOOKH=IMAH > AJO CN Г N & C\$ ГO ГO=! O Г! Г! IBD=I=?=EEK=JJDEIFEJ H *A?=KIABEI?JEKKIBH N # JDAHA @N EIFIEJELAJJDA HECDJBN!6DAHABHABD=I=?= EEK=JN! > O! N Г N + & & Г& + +& O \$ N ГN \$ O Г \$ N ГN \$ N!!?=EEK KIJEBE?=JE IAF=H=JAIL=HE=>AI JAH \$?IJ=JBEJACH=JE 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

16 00 SCORING O " Г = JDAFEJ Г > AJ B B 7IA N MEJD=IJAFIEABJ=FFHNE=JA B > BHMDE?D O N > AJ C 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 8AHEBO O Г " N " N Г C == Г I C== C= Г " Г" IKJE?KHLAJDHKCD IKJE?KHLAJDHKCD Г JMEJAH=JEI -KAH=FFHNE=JEJB KIAI E,- 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

17 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. (a) 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( b = lim fb ( ) Г f() 0 Г 4 Г 4 : : 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 / [ ] 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

### AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

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

### AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

AP CALCULUS AB 004 SCORING GUIDELINES (Form B) Question 4 The figure above shows the graph of f, the derivative of the function f, on the closed interval 1 x 5. The graph of f has horizontal tangent lines

### AP Calculus BC 2011 Free-Response Questions

AP Calculus BC 11 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

### AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES

The figure above shows the graph of the piecewise-linear function f. For 4, the function g is defined by g( ) = f ( t) (a) Does g have a relative minimum, a relative maimum, or neither at =? Justify your

### AP Calculus AB. Scoring Guidelines

17 AP Calculus AB Scoring Guidelines 17 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the

### AP Calculus BC 2005 Free-Response Questions

AP Calculus BC 005 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

### AP Calculus BC Exam. The Calculus BC Exam. At a Glance. Section I. SECTION I: Multiple-Choice Questions. Instructions. About Guessing.

Section I The Calculus BC Exam AP Calculus BC Exam SECTION I: Multiple-Choice Questions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Grade 50% Writing Instrument Pencil

### AP CALCULUS BC 2006 SCORING GUIDELINES (Form B) Question 2

AP CALCULUS BC 2006 SCORING GUIDELINES (Form B) Question 2 An object moving along a curve in the xy-plane is at position ( x() t, y() t ) at time t, where dx t tan( e ) for t 0. At time t =, the object

### AP Calculus AB 2015 Free-Response Questions

AP Calculus AB 015 Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online

### AP Calculus BC 2008 Free-Response Questions Form B

AP Calculus BC 008 Free-Response Questions Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### AP Calculus BC 2006 Free-Response Questions Form B

AP Calculus BC 2006 Free-Response Questions Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### AP Calculus BC 1998 Free-Response Questions

AP Calculus BC 1998 Free-Response Questions These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced

### AP Calculus AB AP Calculus BC

AP Calculus AB AP Calculus BC Free-Response Questions and Solutions 979 988 Copyright 4 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central,

### AP Calculus BC 2011 Free-Response Questions Form B

AP Calculus BC 11 Free-Response Questions Form B About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded

### AP Calculus AB 2008 Free-Response Questions Form B

AP Calculus AB 2008 Free-Response Questions Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### AP Calculus BC Multiple-Choice Answer Key!

Multiple-Choice Answer Key!!!!! "#\$\$%&'! "#\$\$%&'!!,#-! ()*+%\$,#-! ()*+%\$!!!!!! "!!!!! "!! 5!! 6! 7!! 8! 7! 9!!! 5:!!!!! 5! (!!!! 5! "! 5!!! 5!! 8! (!! 56! "! :!!! 59!!!!! 5! 7!!!! 5!!!!! 55! "! 6! "!!

### Student Study Session Topic: Interpreting Graphs

Student Study Session Topic: Interpreting Graphs Starting with the graph of a function or its derivative, you may be asked all kinds of questions without having (or needing) and equation to work with.

### AP* Physics B: Newton s Laws YOU MAY USE YOUR CALCULATOR

AP* Physics B: Newton s Laws Name: Period: YOU MAY USE YOUR CALCULATOR CLEARLY SHOW THE METHOD YOU USED AND STEPS INVOLVED IN ARRIVING AT YOUR ANSWERS. It is to your advantage to do this, because you may

### AP Calculus BC 2015 Free-Response Questions

AP Calculus BC 05 Free-Response Questions 05 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central

### Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..

### AP CALCULUS BC 2009 SCORING GUIDELINES

AP CALCULUS BC 009 SCORING GUIDELINES Question 6 The Maclaurin series for by f( x) = x e is 3 n x x x x e = 1 + x + + + + +. The continuous function f is defined 6 n! ( x 1) e 1 for x 1 and f () 1 = 1.

### Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

### Answer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2

Answer Key Calculus I Math 141 Fall 2003 Professor Ben Richert Exam 2 November 18, 2003 Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem

### AP Calculus AB 2001 Free-Response Questions

AP Calculus AB 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must

### Mark Howell Gonzaga High School, Washington, D.C.

Be Prepared for the Sylight Publishing Calculus Exam Mar Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita lbert Oa Ridge

### 1969 AP Calculus BC: Section I

969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric

### Graphical Relationships Among f, f,

Graphical Relationships Among f, f, and f The relationship between the graph of a function and its first and second derivatives frequently appears on the AP exams. It will appear on both multiple choice

### s-.,... PORTAr!TId.ntIflcatIoainformaUon;

AP Calculus BC Exam CiECTION II: Free Response 00 NOT OPEN THIS BOOKl.Er OR BEGIN PART B UNTIL YOU ARE TOLD TO DO SO. I I 0 At a Glance Total Tine 1 hour, 30 minutes Number of Que5tions 0 Percent of Total

### Math 116 Second Midterm November 14, 2012

Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

### IF (some things are true), then (some other thing is true).

Student Notes Student Study Session Topic: Important Theorems Facts, truth, ideas, etc. in mathematics are known as definitions, theorems, and postulates (also known as aioms or assumptions). Theorems

### 24. AB Calculus Step-by-Step Name. a. For what values of x does f on [-4,4] have a relative minimum and relative maximum? Justify your answers.

24. AB Calculus Step-by-Step Name The figure to the right shows the graph of f!, the derivative of the odd function f. This graph has horizontal tangents at x = 1 and x = 3. The domain of f is!4 " x "

### AP * Chemistry. Kinetics: Integrated Rate Law & Determining Ea. René McCormick

AP * Chemistry Kinetics: Integrated Rate Law & Determining Ea René McCormick *AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this

### AP Physics C: Mechanics 2005 Scoring Guidelines

AP Physics C: Mechanics 005 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### AP Calculus AB 2002 Free-Response Questions

AP Calculus AB 00 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be

### Taylor and Maclaurin Series. Approximating functions using Polynomials.

Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear

### AP Calculus Free-Response Questions 1969-present AB

AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions

### Calculus Graphical, Numerical, Algebraic 5e AP Edition, 2016

A Correlation of Graphical, Numerical, Algebraic 5e AP Edition, 2016 Finney, Demana, Waits, Kennedy, & Bressoud to the Florida Advanced Placement AB/BC Standards (#1202310 & #1202320) AP is a trademark

### Student Session Topic: Average and Instantaneous Rates of Change

Student Session Topic: Average and Instantaneous Rates of Change The concepts of average rates of change and instantaneous rates of change are the building blocks of differential calculus. The AP exams

### Taylor and Maclaurin Series. Approximating functions using Polynomials.

Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear

### Calculus with the Graphing Calculator

Calculus with the Graphing Calculator Using a graphing calculator on the AP Calculus exam Students are expected to know how to use their graphing calculators on the AP Calculus exams proficiently to accomplish

### AP Physics B 2007 Scoring Guidelines

AP Physics B 007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)

Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x

### AP Calculus AB AP Calculus BC

AP Calculus AB AP Calculus BC Free-Response Questions and Solutions 969 978 Copyright 4 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central,

### Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)

Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x

### Solutions to Math 41 Final Exam December 10, 2012

Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)

### AP Calculus AB. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 1. Scoring Guideline.

217 AP Calculus AB Sample Student Responses and Scoring Commentary Inside: RR Free Response Question 1 RR Scoring Guideline RR Student Samples RR Scoring Commentary 217 The College Board. College Board,

### AP Calculus BC. Functions, Graphs, and Limits

AP Calculus BC The Calculus courses are the Advanced Placement topical outlines and prepare students for a successful performance on both the Advanced Placement Calculus exam and their college calculus

### Review for the Final Exam

Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x

### AP* Kinetics Free Response Questions KEY page 1

AP* Kinetics Free Response Questions KEY page 1 Essay Questions 1983 a) three points Plot ln k or log k vs 1/T Eact = - R (slope) or - 2,303 R (slope) For partial credit, if the 2-point equation is given

### 5.1 Separable Differential Equations

5.1 Separable Differential Equations A differential equation is an equation that has one or more derivatives in it. The order of a differential equation is the highest derivative present in the equation.

### Slope Fields and Differential Equations

Student Stud Session Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on

### AP Physics C: Mechanics

2017 AP Physics C: Mechanics Scoring Guidelines 2017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. Visit

### AP Calculus Testbank (Chapter 9) (Mr. Surowski)

AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series

### Calculus Graphical, Numerical, Algebraic 2012

A Correlation of Graphical, Numerical, Algebraic 2012 To the Advanced Placement (AP)* AB/BC Standards Grades 9 12 *Advanced Placement, Advanced Placement Program, AP, and Pre-AP are registered trademarks

### AP* Electrochemistry Free Response Questions page 1

Galvanic (Voltaic) Cells 1988 Average score = 5.02 a) two points Sn ---> Sn 2+ + 2e Ag + + e ---> Ag AP* Electrochemistry Free Response Questions page 1 b) two points 2 Ag + + Sn ---> 2 Ag + Sn 2+ E =

### AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66

AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle

### AP CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 2. or meters 2 :

AP CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 2 A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the

### AP Physics C: Electricity and Magnetism

2017 AP Physics C: Electricity and Magnetism Scoring Guidelines 2017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College

### B L U E V A L L E Y D I S T R I C T C U R R I C U L U M & I N S T R U C T I O N Mathematics AP Calculus BC

B L U E V A L L E Y D I S T R I C T C U R R I C U L U M & I N S T R U C T I O N Mathematics AP Calculus BC Weeks ORGANIZING THEME/TOPIC CONTENT CHAPTER REFERENCE FOCUS STANDARDS & SKILLS Analysis of graphs.

### Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7

Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions /19/7 Question 1 Write the following as an integer: log 4 (9)+log (5) We have: log 4 (9)+log (5) = ( log 4 (9)) ( log (5)) = 5 ( log

### cos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =

MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)

### Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4

Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x

### AP Calculus AB 2nd Semester Homework List

AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this

### AP CALCULUS AB 2011 SCORING GUIDELINES

AP CALCULUS AB 2011 SCORING GUIDELINES Question 4 The continuous function f is defined on the interval 4 x 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure

### Math Honors Calculus I Final Examination, Fall Semester, 2013

Math 2 - Honors Calculus I Final Eamination, Fall Semester, 2 Time Allowed: 2.5 Hours Total Marks:. (2 Marks) Find the following: ( (a) 2 ) sin 2. (b) + (ln 2)/(+ln ). (c) The 2-th Taylor polynomial centered

### Calculus AP Edition, Briggs 2014

A Correlation of AP Edition, Briggs 2014 To the Advanced Placement AB/BC Standards AP is a trademark registered and/or owned by the College Board, which was not involved in the production of, and does

### AP Calculus BC: Syllabus 3

AP Calculus BC: Syllabus 3 Scoring Components SC1 SC2 SC3 SC4 The course teaches Functions, Graphs, and Limits as delineated in the Calculus BC Topic The course teaches Derivatives as delineated The course

### 2000 Advanced Placement Program Free-Response Questions

2000 Advanced Placement Program Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other

### AP CALCULUS AB 2007 SCORING GUIDELINES (Form B)

AP CALCULUS AB 27 SCORING GUIDELINES (Form B) Question 2 A particle moves along the x-axis so that its velocity v at time 2 t is given by vt () = sin ( t ). The graph of v is shown above for t 5 π. The

### 8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0

8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n

### Syllabus for AP Calculus BC

Syllabus for AP Calculus BC Underlying Focus: The emphasis in AP Calculus is on an intuitive understanding of all concepts and the interplay between the geometric and analytic information and on the use

### Mark Howell Gonzaga High School, Washington, D.C.

Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

### 2000 AP CHEMISTRY FREE-RESPONSE QUESTIONS

000 AP CHEMISTRY FREE-RESPONSE QUESTIONS CHEMISTRY SECTION II (Total time 90 minutes) Part A Time 40 minutes YOU MAY USE YOUR CALCULATOR FOR PART A. CLEARLY SHOW THE METHOD USED AND STEPS INVOLVED IN ARRIVING

### AP Calculus Testbank (Chapter 6) (Mr. Surowski)

AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The

### AP Physics C: Mechanics 2007 Free-Response Questions. The College Board: Connecting Students to College Success

AP Physics C: Mechanics 2007 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Solve the 4 multiple-choice problems below. A graphing calculator is required for some questions

### Math 221 Exam II Tuesday Mar 23 5:30-7:00 PM Answers

Math 221 Exam II Tuesday Mar 23 5:30-7:00 PM Answers I. (25 points.) Find. Note: The book sometimes writes D xy for. (a) y = (x 2 x + 1) 7 Answer: Let u = x 2 x + 1. Then y = (x 2 x + 1) 7 = u 7 so = d

### A.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3

A.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3 Each of the three questions is worth 9 points. The maximum possible points earned on this section

11.10 Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. We start by supposing that f is any function that can be represented by a power series f(x)= c 0 +c 1 (x a)+c 2 (x a)

### University of Connecticut Department of Mathematics

University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2015 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual

### Learning Target: I can sketch the graphs of rational functions without a calculator. a. Determine the equation(s) of the asymptotes.

Learning Target: I can sketch the graphs of rational functions without a calculator Consider the graph of y= f(x), where f(x) = 3x 3 (x+2) 2 a. Determine the equation(s) of the asymptotes. b. Find the

### PETERS TOWNSHIP HIGH SCHOOL

PETERS TOWNSHIP HIGH SCHOOL COURSE SYLLABUS: AP CALCULUS BC Course Overview and Essential Skills AP Calculus BC is a challenging class which will prepare students to take the AP Calculus BC Exam in May

### AP Calculus AB Free Response Notebook

AP Calculus AB Free Response Notebook Table of Contents Area and Volume... -5 Charts with Riemann Sums, MVT, Ave. Rates/Values... 4-5 Analyzing Graph of f... 54-59 Slope Fields with differential Equations...

### AP Calculus BC. Course Description:

AP Calculus BC Course Description: The two fundamental problems of Calculus include: 1) finding the slope of the tangent to a curve, determined by the derivative, and 2) finding the area of a region under

### Families of Functions, Taylor Polynomials, l Hopital s

Unit #6 : Rule Families of Functions, Taylor Polynomials, l Hopital s Goals: To use first and second derivative information to describe functions. To be able to find general properties of families of functions.

### MATH 108 FALL 2013 FINAL EXAM REVIEW

MATH 08 FALL 203 FINAL EXAM REVIEW Definitions and theorems. The following definitions and theorems are fair game for you to have to state on the exam. Definitions: Limit (precise δ-ɛ version; 2.4, Def.

### MATH 1241 Common Final Exam Fall 2010

MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the

### MIDLAND ISD ADVANCED PLACEMENT CURRICULUM STANDARDS AP CALCULUS BC

Curricular Requirement 1: The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC

### The AP exams will ask you to find derivatives using the various techniques and rules including

Student Notes Prep Session Topic: Computing Derivatives It goes without saying that derivatives are an important part of the calculus and you need to be able to compute them. You should know the derivatives

### AP Calculus Worksheet: Chapter 2 Review Part I

AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative

### AP Calculus BC Syllabus

Instructor: Jennifer Manzano-Tackett jennifer-manzano@scusd.edu (916) 395-5090 Ext. 506308 www.mt-jfk.com AP Calculus BC Syllabus Textbook: Calculus, 6th edition, by Larson, Hostetler and Edwards: Houghton

### UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test

UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following

### MATHEMATICS AP Calculus (BC) Standard: Number, Number Sense and Operations

Standard: Number, Number Sense and Operations Computation and A. Develop an understanding of limits and continuity. 1. Recognize the types of nonexistence of limits and why they Estimation are nonexistent.

### University of Connecticut Department of Mathematics

University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2014 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual

### Prentice Hall. Calculus: Graphical, Numerical, Algebraic National Advanced Placement Course Descriptions for Calculus BC.

Prentice Hall Grades 9-12 Calculus: Graphical, Numerical, Algebraic 2007 C O R R E L A T E D T O National Advanced Placement Course Descriptions for Calculus BC Grades 9-12 I Functions, Graphs, and Limits

### Math 116 Second Midterm November 12, 2014

Math 6 Second Midterm November 2, 204 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this eam until you are told to do so. 2. This eam has 3 pages including this cover. There are 9 problems. Note

### AP CALCULUS BC 2010 SCORING GUIDELINES. Question 2

AP CALCULUS BC 21 SCORING GUIDELINES t (hours) E() t (hundreds of entries) Question 2 2 5 7 8 4 13 21 23 A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in

### Test 2 Review Math 1111 College Algebra

Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.