AP Calculus AB 2011 Scoring Guidelines


 Lee Long
 2 years ago
 Views:
Transcription
1 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 College Board is composed of more than 5,7 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 3, high schools, and 3,8 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT, the PSAT/NMSQT, the Advanced Placement Program (AP, SpringBoard and ACCUPLACER. The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: AP Central is the official online home for the AP Program: apcentral.collegeboard.com.
2 SCORING GUIDELINES Question For t 6, a particle is moving along the xaxis. The particle s position, x( t, is not explicitly given. t vt = sin e +. The acceleration of the particle is given by The velocity of the particle is given by ( ( t ( t cos( at = e e and x ( =. (a Is the speed of the particle increasing or decreasing at time t = 5.5? Give a reason for your answer. (b Find the average velocity of the particle for the time period t 6. (c Find the total distance traveled by the particle from time t = to t = 6. (d For t 6, the particle changes direction exactly once. Find the position of the particle at that time. (a v ( 5.5 =.5337, a ( 5.5 =.3585 : conclusion with reason The speed is increasing at time t = 5.5, because velocity and acceleration have the same sign. 6 (b Average velocity = vt ( =.99 6 : { : integral : answer 6 : { : integral (c Distance = vt ( =.573 : answer (d vt ( = when t = Let b = vt ( changes sign from positive to negative at time t = b. b xb ( = + vt ( =.3 or.35 : considers vt ( = : integral : answer The College Board.
3 SCORING GUIDELINES Question t (minutes H ( t (degrees Celsius As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t, where time t is measured in minutes and temperature H ( t is measured in degrees Celsius. Values of H ( t at selected values of time t are shown in the table above. (a Use the data in the table to approximate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer. (b Using correct units, explain the meaning of ( H t in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate (. H t (c Evaluate H ( t. Using correct units, explain the meaning of the expression in the context of this problem. (d At time t =, biscuits with temperature C were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that.73t B ( t = 3.8 e. Using the given models, at time t =, how much cooler are the biscuits than the tea? H( 5 H( (a H ( = =.666 or.667 degrees Celsius per minute 3 (b ( H t is the average temperature of the tea, in degrees Celsius, over the minutes H t ( ( = 5.95 (c H ( t = H( H( = 3 66 = 3 The temperature of the tea drops 3 degrees Celsius from time t = to time t = minutes. (d B( = + B ( t = 3.875; H( B( = 8.87 The biscuits are 8.87 degrees Celsius cooler than the tea. : answer : meaning of expression : trapezoidal sum : estimate : value of integral : { : meaning of expression : integrand : uses B( = : answer The College Board.
4 SCORING GUIDELINES Question 3 Let R be the region in the first quadrant enclosed by the graphs of f = 8x and g = sin ( π x, as shown in the figure above. (a Write an equation for the line tangent to the graph of f at x =. (b Find the area of R. (c Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y =. 3 (a f ( = f = x, so ( f = 6 An equation for the tangent line is y = + ( x 6. : f : ( : answer (b Area = ( g f dx 3 = ( sin( π x 8x dx = cos ( π x x π = + 8 π : : integrand : antiderivative : answer ( (( 8 3 ( sin( π (c π ( f ( g dx = π x x dx : limits and constant { : integrand The College Board.
5 SCORING GUIDELINES Question The continuous function f is defined on the interval x 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. x Let g = x + f( t. (a Find g( 3. Find g and evaluate g ( 3. (b Determine the xcoordinate of the point at which g has an absolute maximum on the interval x 3. Justify your answer. (c Find all values of x on the interval < x < 3 for which the graph of g has a point of inflection. Give a reason for your answer. (d Find the average rate of change of f on the interval x 3. There is no point c, < c < 3, for which f ( c is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. 3 9π (a g( 3 = ( 3 + f( t = 6 g = + f g ( 3 = + f( 3 = : g( 3 : g : g ( 3 (b g = when f( x =. This occurs at x = g 5 > for < x < and g < for 5 < x < 3. 5 Therefore g has an absolute maximum at x =. 5. : considers g = : identifies interior candidate : answer with justification (c g = f changes sign only at x =. Thus the graph of g has a point of inflection at x =. : answer with reason (d The average rate of change of f on the interval x 3 is f( 3 f( =. 3 ( 7 To apply the Mean Value Theorem, f must be differentiable at each point in the interval < x < 3. However, f is not differentiable at x = 3 and x =. : average rate of change : { : explanation The College Board.
6 SCORING GUIDELINES Question 5 At the beginning of, a landfill contained 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 equation = ( W 3 for the next years. W is measured in tons, and t is measured in years from 5 the start of. (a Use the line tangent to the graph of W at t = to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of (time t =. (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 =. (c Find the particular solution W = W( t to the differential equation = ( W 3 with initial 5 condition W ( =. (a = ( W ( 3 = ( 3 = t= 5 5 The tangent line is y = + t. ( ( W + = tons : at : t = : answer (b = = ( W 3 and W 5 65 Therefore > on the interval t. The answer in part (a is an underestimate. : : : answer with reason (c = ( W 3 5 = W 3 5 ln W 3 = t + C 5 ln ( 3 = ( + C ln ( = C 5 W 3 = e t 5 t 5 W( t = 3 + e, t 5 : : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for W Note: max 5 [] if no constant of integration Note: 5 if no separation of variables The College Board.
7 SCORING GUIDELINES Question 6 sin x for x Let f be a function defined by f = x e for x >. (a Show that f is continuous at x =. (b For x, express f as a piecewisedefined function. Find the value of x for which f = 3. (c Find the average value of f on the interval [, ]. (a lim ( sin x = x + x x lim e = f ( = So, lim f = f(. x : analysis Therefore f is continuous at x =. cos x for x (b f = < x e for x > cos x 3 for all values of x <. e x 3 = 3 when x = ln ( >. Therefore f 3 3 = for x = ( ln. : f : value of x (c f dx = f dx + f dx x = ( sin x dx + e dx x = x + cos x + e x x = ( 3 cos( + ( e + = = : x : ( sin x dx and e dx : antiderivatives : answer Average value = ( f x dx 3 = cos( e 8 8 The College Board.
AP 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 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 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 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 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 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 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 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 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 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 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 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 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 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. 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Macroeconomics 2010 Scoring Guidelines
AP Macroeconomics 2010 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 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 Macroeconomics 2011 Scoring Guidelines
AP Macroeconomics 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 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 Macroeconomics 2011 Scoring Guidelines Form B
AP Macroeconomics 2011 Scoring Guidelines 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 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 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 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 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 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 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 Calculus BC 1999 Scoring Guidelines
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
More informationAP CALCULUS AB 2009 SCORING GUIDELINES (Form B)
AP CALCULUS AB 09 SCORING GUIDELINES (Form B) Question 6 t (seconds) vt () (meters per second) 0 8 25 32 3 5 10 8 4 7 The velocity of a particle moving along the xaxis is modeled by a differentiable function
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 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 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 informationAP Physics C: Electricity and Magnetism 2011 Scoring Guidelines
AP Physics C: Electricity and Magnetism 11 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and
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 Statistics 2011 Scoring Guidelines
AP Statistics 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 in
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 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 CALCULUS AB 2006 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 6 t (sec) vt () ( ft sec ) at () 2 ( ft sec ) 0 15 25 35 50 60 20 20 14 10 0 10 1 5 2 1 2 4 2 A car travels on a straight track. During the time
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 informationAP Physics C: Electricity and Magnetism 2010 Scoring Guidelines
AP Physics C: Electricity and Magnetism 010 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and
More informationAP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 4
AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 4 The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function 1 3 3 2 f. In the figure
More informationAP Environmental Science 2008 FreeResponse Questions
AP Environmental Science 2008 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect
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 informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationAP Statistics 2007 FreeResponse Questions Form B
AP Statistics 2007 FreeResponse Questions 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 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 informationAP Environmental Science 2010 FreeResponse Questions
AP Environmental Science 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.
More informationAP CALCULUS AB 2008 SCORING GUIDELINES
AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line
More informationAP Environmental Science 2009 FreeResponse Questions
AP Environmental Science 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.
More informationReview Sheet for Third Midterm Mathematics 1300, Calculus 1
Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,
More informationAP Human Geography 2010 Scoring Guidelines
AP Human Geography 2010 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 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 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 United States Government and Politics 2011 Scoring Guidelines
AP United States Government and Politics 2011 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationAP Environmental Science 2006 FreeResponse Questions
AP Environmental Science 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect
More informationAP Physics C: Electricity and Magnetism 2009 Scoring Guidelines
AP Physics C: Electricity and Magnetism 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.
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 informationx = y + 2, and the line
WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
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 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 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 Statistics 2009 Scoring Guidelines Form B
AP Statistics 2009 Scoring Guidelines Form B 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 Computer Science A 2010 FreeResponse Questions
AP Computer Science A 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.
More informationAP Psychology 2011 Scoring Guidelines
AP Psychology 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 in
More informationAP Statistics 2010 Scoring Guidelines
AP Statistics 2010 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
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 Chemistry 2010 Scoring Guidelines Form B
AP Chemistry 2010 Scoring Guidelines Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
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 informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
More informationAnalyzing Piecewise Functions
Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including
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 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 Statistics 2007 Scoring Guidelines
AP Statistics 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 college
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationAP Human Geography 2011 Scoring Guidelines
AP Human Geography 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 information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
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 Chemistry 2007 Scoring Guidelines Form B
AP Chemistry 2007 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 information