AP Calculus BC 1999 Scoring Guidelines


 Basil Armstrong
 2 years ago
 Views:
Transcription
1 AP Calculus BC Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from the Advanced Placement Program. Teachers may reproduce them, in whole or in part, in limited quantities, for facetoface teaching purposes but may not mass distribute the materials, electronically or otherwise. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any thirdparty copyrights contained herein. These materials were produced by Educational Testing Service (ETS), which develops and administers the eaminations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in, the association is composed of more than, schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents,, high schools, and,5 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its bestknown programs are the SAT, the PSAT/NMSQT, the Advanced Placement Program (AP ), and Pacesetter. The College Board is committed to the principles of equity and ecellence, and that commitment is embodied in all of its programs, services, activities, and concerns. Copyright by College Entrance Eamination Board. All rights reserved. College Board, Advanced Placement Program, AP, and the acorn logo are registered trademarks of the College Entrance Eamination Board.
2 BC{. A particle moves in the y{plane so that its position at any time t, t, isgiven by (t) = t ; ln( + t) andy(t) = sin t. (a) Sketch the path of the particle in the y{plane below. Indicate the direction of motion along the path. (b) At what time t, t, does(t) attain its minimum value? What is the position ((t) y(t)) of the particle at this time? (c) At what time t, <t<, is the particle on the y{ais? Find the speed and the acceleration vector of the particle at this time. (a) y ( : graph : direction O (b) (t) =t ; +t = t + t ; = t = ;+p 5 or t =:6 in [ ] (:6) = ;: y(:6) = :7 (c) (t) = t ; ln( + t) = t =:5 or :6 (t) =t ; +t y (t) = cos t 4 : (t) = : solution for t : position : (t) = : solution for t : speed : acceleration vector speed = p ( (:6)) +(y (:6)) =:6 (t) =+ ( + t) y (t) =; sin t acceleration vector = < (:6) y (:6) > = <: ;:7 >
3 AB{ / BC{. The shaded region, R, is bounded by the graph of y = and the line y =4,asshown in the gure above. (a) Find the area of R. (b) Find the volume of the solid generated by revolving R about the {ais. (c) There eists a number k, k>4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral epression that can be used to nd the value of k. O y y y 4 (a) Area = Z ; = Z (4 ; ) d (4 ; ) d ( : integral : answer = = 4 ; =:666 or :667 (b) Volume = Z ;4 ; ( ) d = Z (6 ; 4 ) d > < : limits and constant : integrand : answer = 6 ; 5 5 = 56 5 = 6:4 or 6:5 (c) Z ; (k ; ) ; (k ; 4) d = > < : limits and constant : integrand <; > each error : equation
4 AB{ / BC{. The rate at which water ows out of a pipe, in gallons per hour, is given by a dierentiable function R of time t. The table above shows the rate as measured every hours for a 4{hour period. (a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approimate Z 4 R(t) dt. Using correct units, eplain the meaning of your answer in terms of water ow. (b) Is there some time t, <t<4, such that R (t) =? Justify your answer. (c) The rate of water ow R(t) can be approimated by Q(t) = ; 76 + t ; t. Use Q(t) toapproimate the 7 average rate of water ow during the 4{hour time period. Indicate units of measure. t R(t) (hours) (gallons per hour) (a) Z 4 R(t) dt 6[R() + R() + R(5) + R()] =6[:4+:+:+:] = 5.6 gallons This is an approimation to the total ow in gallons of water from the pipe in the 4{hour period. : R() + R() + R(5) + R() : answer : eplanation (b) Yes Since R() = R(4) = :6, the Mean Value Theorem guarantees that there is a t, <t<4, such thatr (t) =. ( : answer : MVT or equivalent (c) Average rate of ow average value of Q(t) = Z (76 + t ; t ) dt = :75 gal/hr or.74 gal/hr : limits and average value constant : Q(t) asintegrand : answer (units) Gallons in part (a) and gallons/hr in part (c), or equivalent. : units
5 BC{4 4. The function f has derivatives of all orders for all real numbers. Assume f() = ;, f () = 5, f () =, and f () = ;. (a) Write the third{degree Taylor polynomial for f about = and use it to approimate f(:5). (b) The fourth derivative off satises the inequality jf (4) ()j for all in the closed interval [:5 ]. Use the Lagrange error bound on the approimation to f(:5) found in part (a) to eplain why f(:5) 6= ;5. (c) Write the fourth{degree Taylor polynomial, P (), for g() = f( +) about =. Use P to eplain why g must have a relative minimum at =. (a) T (f )() =;+5( ; ) + ( ; ) ; ( ; ) 6 f(:5) T (f )(:5) = ;+5(;:5) + (;:5) ; 4 (;:5) = ;4:5=;4:5 4 : T (f )() < ; > each error : approimation of f(:5) (b) Lagrange Error Bound = 4! j:5 ; j4 =:75 f(:5) > ;4:5 ; :75 = ;4:66 > ;5 ( : value of Lagrange Error Bound : eplanation Therefore, f(:5) 6= ;5. (c) P () =T 4 (g )() = T (f )( +)=; The coecient of in P () isg (). This coecient is, so g ()=. : T 4 (g )() < ; > each incorrect, missing, or etra term : eplanation The coecient of in P () is. This coecient! is 5, so g () = which is greater than. g () Therefore, g has a relative minimum at =. Note: < ; > ma for improper use of + ::: or equality
6 AB{5 / BC{5 5. The graph of the functionz f, consisting of three line segments, is given above. Let g() = f (t) dt. (a) Compute g(4) and g(;). (b) Find the instantaneous rate of change of g, with respect to, at =. (c) Find the absolute minimum value of g on the closed interval [; 4]. Justify your answer. (d) The second derivative ofg is not dened at = and =. How many of these values are {coordinates of points of inection of the graph of g? Justify your answer. 4 (, 4) (, ) O 4 (4, ) Z 4 (a) g(4) = f (t) dt = ++ ; = 5 g(;) = Z ; f (t) dt = ; () = ;6 ( : g(4) : g(;) (b) g () = f () = 4 : answer (c) g is increasing on [; ] and decreasing on [ 4]. Therefore, g has absolute minimum at an endpoint of[; 4]. : interior analysis : endpoint analysis : answer Since g(;) = ;6 and g(4) = 5, the absolute minimum value is ;6. (d) One = On (; ), g () =f () > On ( ), g () =f () < : choice of = only : show ( g()) is a point of inection : show ( g()) is not a point of inection On ( 4), g () =f () < Therefore ( g()) is a point of inection and ( g()) is not.
7 BC{6 6. Let f be the function whose graph goes through the point ( 6) and whose derivative is given by f () = +e. (a) Write an equation of the line tangent to the graph of f at = and use it to approimate f(:). (b) Use Euler's method, starting at =with a step size of.5, to approimate f(:). Use f to eplain why this approimation is less than f(:). (c) Use Z : f () d to evaluate f(:). (a) f () = +e y ; 6= +e ( ; ) =:4 or :4 : f () : equation : approimation of f(:) y =6+ +e ( ; ) f(:) 6+ +e (:) = 6:4 (b) f(:5) f() + f ()(:5) = 6+:74 = 6:74 f(:) 6:74 + f (:5)(:5) = 6:74 + (:775)(:5) = 6:6 f () = e ; ( + e ) 4 = ( ; )e ; 4 : Euler's method equations or equivalent table : Euler approimation to f(:) (not eligible without rst point) : f () : reason For, f () > e ; > and the graph of f is concave upward on ( :). Therefore, the Euler approimation lines at and.5 lie below the graph. (c) f(:) ; f() = Z : +e d f(:) = 6 + :7 = 6:7 or 6: Z : +e : d = f(:) ; f() : answer
AP Calculus AB 2004 Scoring Guidelines
AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from
More informationAP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAP Calculus AB 2003 Scoring Guidelines Form B
AP Calculus AB Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationAP Calculus AB 2001 Scoring Guidelines
P Calculus Scing Guidelines The materials included in these files are intended f noncommercial use by P teachers f course and eam preparation; permission f any other use must be sought from the dvanced
More informationAP Calculus BC 2001 FreeResponse Questions
AP Calculus BC 001 FreeResponse 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
More informationAP Calculus BC 2004 FreeResponse Questions
AP Calculus BC 004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationAP Calculus AB 2007 Scoring Guidelines Form B
AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus BC 2003 FreeResponse Questions
AP Calculus BC 2003 FreeResponse Questions The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationAP Calculus BC 2004 Scoring Guidelines
AP Calculus BC Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from
More informationAP Calculus AB 1998 Scoring Guidelines
AP Calculus AB 1998 Scoring Guidelines These materials are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced
More informationAP Calculus AB 2005 Scoring Guidelines Form B
AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAP Calculus AB 2004 FreeResponse Questions
AP Calculus AB 2004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationAP Calculus AB 2003 Scoring Guidelines
AP Calculus AB Scoring Guidelines The materials included in these files are intended for use y AP teachers for course and exam preparation; permission for any other use must e sought from the Advanced
More informationAP Calculus BC 2007 FreeResponse Questions
AP Calculus BC 7 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAP Calculus BC 2006 FreeResponse Questions
AP Calculus BC 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus AB 2007 FreeResponse Questions
AP Calculus AB 2007 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus AB 2010 FreeResponse Questions Form B
AP Calculus AB 2010 FreeResponse Questions Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity.
More informationAP Calculus BC 2012 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in 9, the College
More informationAP Calculus AB 2009 Scoring Guidelines
AP Calculus AB 9 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 19,
More informationAP Calculus AB 2011 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the
More informationAP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAP Calculus AB 2005 FreeResponse Questions
AP Calculus AB 25 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus BC 2012 FreeResponse Questions
AP Calculus BC 0 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus AB 2006 FreeResponse Questions
AP Calculus AB 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus BC 2013 FreeResponse Questions
AP Calculus BC 013 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus BC 2009 FreeResponse Questions Form B
AP Calculus BC 009 FreeResponse Questions Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity.
More informationAP Calculus BC 2010 FreeResponse Questions
AP Calculus BC 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationAP Calculus AB. Practice Exam. Advanced Placement Program
Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this
More informationAP United States Government & Politics 2003 Scoring Commentary
AP United States Government & Politics 2003 Scoring Commentary The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must
More informationAP Calculus BC 2009 FreeResponse Questions
AP Calculus BC 009 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationAP Macroeconomics 2003 Scoring Guidelines Form B
AP Macroeconomics 2003 Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from
More informationAP Calculus AB 2008 FreeResponse Questions
AP Calculus AB 2008 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Microeconomics 2003 Scoring Guidelines
AP Microeconomics 2003 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationAP Calculus AB 2013 FreeResponse Questions
AP Calculus AB 2013 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded
More informationAP United States Government and Politics 2001 Scoring Guidelines
AP United States Government and Politics 2001 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for
More informationAP Calculus AB 2009 FreeResponse Questions
AP Calculus AB 2009 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationAP Calculus AB 2012 FreeResponse Questions
AP Calculus AB 1 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Microeconomics 2003 Scoring Guidelines Form B
AP Microeconomics 2003 Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from
More informationAP Calculus AB 2011 FreeResponse Questions
AP Calculus AB 11 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus AB 2012 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in 9, the College
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Eam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice eam contributors: Benita Albert Oak Ridge High School,
More informationAP Macroeconomics 2003 Scoring Guidelines
AP Macroeconomics 2003 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationAP Macroeconomics 2004 Scoring Guidelines
AP Macroeconomics 2004 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought
More informationStudent Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the 2008 freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More informationAP Calculus AB 2010 FreeResponse Questions
AP Calculus AB 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationSample Activity: The Least Expensive Cable
Sample Activity: The Least Expensive Cable from the PreAP workshop PreAP : Topics for AP Vertical Teams in Mathematics For more information, see apcentral.collegeboard.com/preapworkshops. Copyright 2004
More informationAP CALCULUS BC 2008 SCORING GUIDELINES
AP CALCULUS BC 2008 SCORING GUIDELINES Question 3 x hx ( ) h ( x) h ( x) h ( x) ( 4 h ) ( x ) 1 11 30 42 99 18 2 80 128 488 3 448 3 584 9 3 317 753 2 1383 4 3483 16 1125 16 Let h be a function having derivatives
More informationAP Microeconomics 2002 Scoring Guidelines
AP Microeconomics 2002 Scoring Guidelines 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
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 13 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1 Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationAP United States Government & Politics 2003 Scoring Guidelines
AP United States Government & Politics 2003 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must
More information2000 Advanced Placement Program FreeResponse Questions
2000 Advanced Placement Program FreeResponse 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
More informationAP Government & Politics: Comparative 2000 Scoring Guidelines
AP Government & Politics: Comparative 2000 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More informationAP Statistics 2001 Solutions and Scoring Guidelines
AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use
More informationAP Psychology 2003 Scoring Guidelines
AP Psychology 2003 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced
More informationAP CALCULUS AB 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and
More informationAP CALCULUS AB 2015 SCORING GUIDELINES
AP CALCULUS AB 2015 SCORING GUIDELINES Question 5 The figure above shows the graph of f, the derivative of a twicedifferentiable function f, on the interval [ 3, 4 ]. The graph of f has horizontal tangents
More informationAP Government and Politics: United States 1999 Scoring Guidelines
AP Government and Politics: United States 1999 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationAP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.
AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods
More informationAP English Language & Composition 2002 Scoring Guidelines Form B
AP English Language & Composition 2002 Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for
More informationAP Statistics 2002 Scoring Guidelines
AP Statistics 2002 Scoring Guidelines 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 sought
More informationAP CALCULUS AB 2006 SCORING GUIDELINES
The graph of the function f shown above consists of six line segments. Let g be the function given by x g( x) = f( t) dt. 0 (a) Find g ( 4, ) g ( 4, ) and g ( 4. ) (b) Does g have a relative minimum, a
More informationAP Macroeconomics 2007 Scoring Guidelines
AP Macroeconomics 2007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationDon't Forget the Differential Equations: Finishing 2005 BC4
connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College
More informationAP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in
More informationAP Macroeconomics 2008 Scoring Guidelines Form B
AP Macroeconomics 2008 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students
More informationAP Statistics 1998 Scoring Guidelines
AP Statistics 1998 Scoring Guidelines These materials are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement
More informationAP Psychology 2000 Scoring Guidelines
AP Psychology 2000 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationAP Computer Science A 2004 FreeResponse Questions
AP Computer Science A 2004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use
More informationFunctions Defined by Integrals Curriculum Module
AP Calculus: Functions Defined by Integrals 8 Curriculum Module AP Calculus Functions Defined by Integrals Scott Pass John H. Reagan High School Austin, TX Reasoning from the graph of the derivative function
More informationAP Statistics 2000 Scoring Guidelines
AP Statistics 000 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought
More informationAP Macroeconomics 2008 Scoring Guidelines
AP Macroeconomics 2008 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Microeconomics 2011 Scoring Guidelines
AP Microeconomics 2011 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationScoring and Practice Sections
Taking the SAT I: Reasoning Test Scoring and Practice Sections The materials in these files are intended for individual use by students getting ready to take an SAT Program test; permission for any other
More informationAP Statistics 2004 Scoring Guidelines
AP Statistics 2004 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought
More informationAP Macroeconomics 2009 Scoring Guidelines
AP Macroeconomics 2009 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationMark 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,
More informationSL Calculus Practice Problems
Alei  Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram
More information7.6 Approximation Errors and Simpson's Rule
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationAP Human Geography 2004 Scoring Guidelines
AP Human Geography 2004 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationWorksheet 1. What You Need to Know About Motion Along the xaxis (Part 1)
Worksheet 1. What You Need to Know About Motion Along the xaxis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the
More informationAP United States History 2003 FreeResponse Questions
AP United States History 2003 FreeResponse Questions The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought
More informationChapter 5 Applications of Integration
MA111 Application of Integration Asst.Prof.Dr.Supranee Lisawadi 1 Chapter 5 Applications of Integration Section 5.1 Area Between Two Curves In this section we use integrals to find areas of regions that
More informationPractice Exam. dollegeboard. Program
dollegeboard Advanced Program Placement Practice Exam The questions contained in this ApCRJ Calculus BC Practice Exam are written to the content specifications of AP Exams for this subject. Taking this
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More informationSample Activity: Treads and Risers. PreAP : Strategies in Mathematics Helping Students Learn Mathematics Through Problem Solving
Sample Activity: Treads and Risers from the PreAP workshop PreAP : Strategies in Mathematics Helping Students Learn Mathematics Through Problem Solving For more information, see apcentral.collegeboard.com/preapworkshops.
More informationAP CALCULUS AB 2006 SCORING GUIDELINES. Question 4
AP CALCULUS AB 2006 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) 0 10 20 30 40 50 60 70 80 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity vt () after being launched upward
More information1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate FormEliminate the Common Factor
Review for the BST MTHSC 8 Name : [] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions Evaluate cos 6 Indeterminate FormEliminate the Common Factor Find the
More informationAbout the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
More informationFree Response Questions Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom
Free Response Questions 1969005 Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom 1 AP Calculus FreeResponse Questions 1969 AB 1 Consider the following functions
More informationMark 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,
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationLesson 5.4 Exercises, pages
Lesson 5.4 Eercises, pages 8 85 A 4. Evaluate each logarithm. a) log 4 6 b) log 00 000 4 log 0 0 5 5 c) log 6 6 d) log log 6 6 4 4 5. Write each eponential epression as a logarithmic epression. a) 6 64
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationAP CALCULUS BC 2008 SCORING GUIDELINES
AP CALCULUS BC 008 SCORING GUIDELINES Question 6 dy 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)
More informationAP English Literature & Composition 2002 Scoring Guidelines
AP English Literature & Composition 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any
More information