AP Calculus AB 2009 Scoring Guidelines

Save this PDF as:

Size: px
Start display at page:

Transcription

1 AP Calculus AB 9 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 19, the association is composed of more than,6 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents,, high schools and,8 colleges through major programs and services in college readiness, college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT, the PSAT/NMSQT and the Advanced Placement Program (AP ). 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. 9 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: AP Central is the official online home for AP teachers: apcentral.collegeboard.com.

2 9 SCORING GUIDELINES Question 1 Caren rides her bicycle along a straight road from home to school, starting at home at time t = minutes and arriving at school at time t = 1 minutes. During the time interval t 1 minutes, her velocity vt (), in miles per minute, is modeled by the piecewise-linear function whose graph is shown above. (a) Find the acceleration of Caren s bicycle at time t = 7. minutes. Indicate units of measure. (b) Using correct units, explain the meaning of vt () dt in terms of Caren s trip. Find the value 1 of vt () dt. 1 (c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. (d) Larry also rides his bicycle along a straight road from home to school in 1 minutes. His velocity is modeled by the function w given by wt () = sin ( t), where wt () is in miles per minute for 1 1 t 1 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. v( 8) v( 7) (a) a( 7.) = v ( 7.) = =.1 miles minute (b) vt () dt is the total distance, in miles, that Caren rode during the 1 minutes from t = to t = v() t dt = v() t dt v() t dt + v() t dt = = 1.8 miles (c) Caren turns around to go back home at time t = minutes. This is the time at which her velocity changes from positive to negative. 1 (d) wt () dt= 1.6; Larry lives 1.6 miles from school. 1 vt () dt= 1.; Caren lives 1. miles from school. Therefore, Caren lives closer to school. : { 1 : units 1 : meaning of integral : { 1 : value of integral : { 1 : reason : : Larry s distance from school 1 : integral 1 : value 1 : Caren s distance from school and conclusion 9 The College Board. All rights reserved.

3 9 SCORING GUIDELINES Question The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R() t = 18t 67t for t hours; R() t is measured in people per hour. No one is in the auditorium at time t =, when the doors open. The doors close and the concert begins at time t =. (a) How many people are in the auditorium when the concert begins? (b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. (c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by w () t = ( t) R() t. Find w( ) w( 1 ), the total wait time for those who enter the auditorium after time t = 1. (d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c). people : { 1 : integral (a) Rt () dt= 98 (b) R () t = when t = and t = The maximum rate may occur at, a = 1.696, or. R ( ) = Ra ( ) = 8.7 R ( ) = 1 : 1 : considers R () t = 1 : interior critical point and justification The maximum rate occurs when t = 1.6 or 1.6. (c) w( ) w() 1 = w () t dt = ( t) R() t dt = The total wait time for those who enter the auditorium after time t = 1 is 87. hours. : { 1 : integral 1 1 (d) w( ) = ( t) R( t) dt = On average, a person waits.77 or.776 hour. : { 1 : integral 9 The College Board. All rights reserved.

4 9 SCORING GUIDELINES Question Mighty Cable Company manufactures cable that sells for \$1 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is x meters from the beginning of the cable is 6 x dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the company s cost of producing the cable.) (a) Find Mighty s profit on the sale of a -meter cable. (b) Using correct units, explain the meaning of 6 x dx in the context of this problem. (c) Write an expression, involving an integral, that represents Mighty s profit on the sale of a cable that is k meters long. (d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer. (a) Profit = = dollars : { 1 : integral 1 6 xdx (b) 6 x dx is the difference in cost, in dollars, of producing a cable of length meters and a cable of length meters. with units (c) Profit 1 k k 6 x dx = dollars : { 1 : integral 1 : expression (d) Let P( k ) be the profit for a cable of length k. P ( k) = 1 6 k = when k =. This is the only critical point for P, and P changes from positive to negative at k =. Therefore, the maximum profit is P ( ) = 16, dollars. : 1 : P ( k) = 1 : k = 1 : justification 9 The College Board. All rights reserved.

5 9 SCORING GUIDELINES Question Let R be the region in the first quadrant enclosed by the graphs of y = x and y = x, as shown in the figure above. (a) Find the area of R. (b) The region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x-axis has area Ax ( ) = sin ( x). Find the volume of the solid. (c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid. (a) Area = ( ) 1 x = x = x x dx : 1 : integrand 1 : antiderivative (b) Volume = sin ( ) = ( x) = cos x dx : 1 : integrand 1 : antiderivative y (c) Volume = ( y ) dy : { : integrand 1 : limits 9 The College Board. All rights reserved.

7 9 SCORING GUIDELINES Question 6 The derivative of a function f is defined by g( x) for x f ( x) = x. e for < x The graph of the continuous function f, shown in the figure above, has x-intercepts at x = and ( ) is a semicircle, and f ( ) =. x = ln. The graph of g on x (a) For < x <, find all values of x at which the graph of f has a point of inflection. Justify your answer. (b) Find f ( ) and f (. ) (c) For x, find the value of x at which f has an absolute maximum. Justify your answer. (a) f changes from decreasing to increasing at x = and from increasing to decreasing at x =. Therefore, the graph of f has points of inflection at x = and x =. 1 : identifies x = or x = : { with justification (b) f( ) = + g( x) dx = ( 8 ) = x ( ) = + ( ) x = + ( 1e x) f e dx = 8 1e : : f ( ) 1 : integral 1 : value : f ( ) 1 : integral 1 : antiderivative 1 : value (c) Since f ( x) > on the intervals < x < and ( ) ( ) < x < ln, f is increasing on the interval x ln. < on the interval ln( ) < x <, f is decreasing on the interval ln( ) x. Therefore, f has an absolute maximum at x = ( ) Since f ( x) ln. : { 1 : justification 9 The College Board. All rights reserved.

AP Calculus AB 2009 Free-Response Questions

AP Calculus AB 2009 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Calculus BC 2009 Free-Response Questions

AP Calculus BC 009 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Calculus AB 2011 Scoring Guidelines

AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the

AP Calculus AB 2012 Scoring Guidelines

AP Calculus AB Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

AP Calculus AB 2005 Free-Response Questions

AP Calculus AB 25 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 AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

AP Calculus AB 2007 Free-Response Questions

AP Calculus AB 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 to

AP Calculus AB 2008 Free-Response Questions

AP Calculus AB 2008 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 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

AP Calculus AB 2006 Scoring Guidelines

AP Calculus AB 006 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

AP Calculus BC 2007 Free-Response Questions

AP Calculus BC 7 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 college

AP Calculus BC 2008 Scoring Guidelines

AP Calculus BC 8 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

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

AP Calculus AB 2006 Free-Response Questions

AP Calculus AB 2006 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 AB 2004 Free-Response Questions

AP Calculus AB 2004 Free-Response 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

AP 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 not-for-profit membership association whose mission is to connect students to

AP Calculus AB 2010 Free-Response Questions

AP Calculus AB 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Calculus BC 2006 Free-Response Questions

AP Calculus BC 2006 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 2009 Free-Response Questions Form B

AP Calculus BC 009 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

AP 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

AP 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

AP Calculus BC 2012 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

AP Calculus AB 2012 Free-Response Questions

AP Calculus AB 1 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. 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

AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 29 SCORING GUIDELINES Question 1 Caren rides her bicycle along a straight road from home to school, starting at home at time t = minutes and arriving at school at time t = minutes. During

AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 2009 SCORING GUIDELINES Question 3 Mighty Cable Company manufactures cable that sells for \$120 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies

AP Calculus BC 2004 Free-Response Questions

AP Calculus BC 004 Free-Response 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

AP 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

AP 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 not-for-profit membership association whose mission is to connect students to college

AP Calculus BC 2003 Free-Response Questions

AP Calculus BC 2003 Free-Response 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

AP Calculus AB 2013 Free-Response Questions

AP Calculus AB 2013 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

AP Calculus AB 1998 Scoring Guidelines

AP Calculus AB 1998 Scoring Guidelines 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 BC 2010 Free-Response Questions

AP Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Calculus AB 2011 Free-Response Questions

AP Calculus AB 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 BC 2001 Free-Response Questions

AP Calculus BC 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

AP Macroeconomics 2009 Scoring Guidelines

AP Macroeconomics 2009 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Macroeconomics 2008 Scoring Guidelines

AP Macroeconomics 2008 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

AP Macroeconomics 2007 Scoring Guidelines

AP Macroeconomics 2007 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

AP 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 not-for-profit membership association whose mission is to connect students

Calculus 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

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,

Don'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

AP Calculus BC 2012 Free-Response Questions

AP Calculus BC 0 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 BC 1999 Scoring Guidelines

AP Calculus BC Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

AP Calculus AB 2001 Scoring Guidelines

P Calculus Scing Guidelines The materials included in these files are intended f non-commercial use by P teachers f course and eam preparation; permission f any other use must be sought from the dvanced

AP Microeconomics 2011 Scoring Guidelines

AP Microeconomics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Macroeconomics 2010 Scoring Guidelines

AP Macroeconomics 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Macroeconomics 2011 Scoring Guidelines Form B

AP Macroeconomics 2011 Scoring Guidelines Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

AP 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

2008 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

AP 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

AP Macroeconomics 2011 Scoring Guidelines

AP Macroeconomics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP Calculus BC 2013 Free-Response Questions

AP Calculus BC 013 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 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 not-for-profit membership association whose mission is to connect students to college success and

Practice 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

Student Performance Q&A:

Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief

AP 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 x-axis is modeled by a differentiable function

AP 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

Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

AP 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 twice-differentiable function f, on the interval [ 3, 4 ]. The graph of f has horizontal tangents

AP 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 not-for-profit membership association whose mission is to connect students to college success and opportunity.

x = 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

AP 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

Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.

CALCULUS WORKSHEET ON 7. Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.. f x x and g x x on x Find the area bounded by the given

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

AP 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

AP 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

correct-choice 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

AP 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

AP 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

AP Environmental Science 2008 Free-Response Questions

AP Environmental Science 2008 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

AP 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

AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

AP 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

AP Statistics 2007 Free-Response Questions Form B

AP Statistics 2007 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 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 not-for-profit membership association whose mission is to connect students to college success and

(b) Is the estimate in part (a) an over-estimate or underestimate of the actual area? Justify your conclusion.

Chapter. Define ( ) f x = x x + x. (a) Estimate the area between the graph of f and the x axis on the interval [, ] using a lefthand sum with four rectangles of equal width. (b) Is the estimate in part

AP Environmental Science 2009 Free-Response Questions

AP Environmental Science 2009 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

AP Statistics 2008 Scoring Guidelines Form B

AP Statistics 2008 Scoring Guidelines 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 Environmental Science 2006 Free-Response Questions

AP Environmental Science 2006 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

AP Physics B 2009 Scoring Guidelines

AP Physics B 009 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900,

PRACTICE 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

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,

AP 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

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

AP Statistics 2009 Scoring Guidelines Form B

AP Statistics 2009 Scoring Guidelines Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

AP 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

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

AP Computer Science A 2010 Free-Response Questions

AP Computer Science A 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

Definition: Let S be a solid with cross-sectional area A(x) perpendicular to the x-axis at each point x [a, b]. The volume of S is V = A(x)dx.

Section 7.: Volume Let S be a solid and suppose that the area of the cross-section of S in the plane P x perpendicular to the x-axis passing through x is A(x) for a x b. Consider slicing the solid into

Sample Activity: The Least Expensive Cable

Sample Activity: The Least Expensive Cable from the Pre-AP workshop Pre-AP : Topics for AP Vertical Teams in Mathematics For more information, see apcentral.collegeboard.com/preapworkshops. Copyright 2004

Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

AP Macroeconomics 2012 Scoring Guidelines

AP Macroeconomics 2012 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900,

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate