AP Calculus AB 2007 Scoring Guidelines Form B


 Bridget McCormick
 2 years ago
 Views:
Transcription
1 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 college success and opportunity. Founded in 9, the association is composed of more than 5, schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents,, high schools, and,5 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its bestknown programs are the SAT, the PSAT/NMSQT, and the Advanced Placement Program (AP ). The College Board is committed to the principles of ecellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. 7 The College Board. All rights reserved. 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: Visit the College Board on the Web: AP Central is the official online home for the AP Program: apcentral.collegeboard.com.
2 7 SCING GUIDELINES (Form B) Question Let R be the region bounded by the graph of y = e and the horizontal line y =, and let S be the region bounded by the graph of y = e and the horizontal lines y = and y =, as shown above. (a) Find the area of R. (b) Find the area of S. (c) Write, but do not evaluate, an integral epression that gives the volume of the solid generated when R is rotated about the horizontal line y =. e = when =.44657,.5594 Let P = and Q =.5594 R = d =.54 e P Q (a) Area of ( ) : integrand : limits : answer (b) e = when =, ( ) Area of S = e d Area of R =.66 Area of R =.546 : integrand : limits : answer P ( ) ( ) + + ( ) e d Q P e d = =.546 Q Q = d { : integrand P (c) Volume π ( e ) ( ) : constant and limits 7 The College Board. All rights reserved.
3 7 SCING GUIDELINES (Form B) Question A particle moves along the ais so that its velocity v at time t is given by vt () = sin ( t ). The graph of v is shown above for t 5 π. The position of the particle at time t is ( t ) and its position at time t = is ( ) = 5. (a) Find the acceleration of the particle at time t =. (b) Find the total distance traveled by the particle from time t = to t =. (c) Find the position of the particle at time t =. (d) For t 5 π, find the time t at which the particle is farthest to the right. Eplain your answer. (a) a( ) = v ( ) = 6cos9 = or : a ( ) (b) Distance = vt () dt=.7 For < t <, vt ( ) = when t = π =.7745 and t = π =.566 ( ) = 5 ( ) () π = 5 + v t dt = π π ( ) () π = 5 + v t dt = 5.44 ( ) = 5 + v( t) dt = ( π ) ( ) ( π ) ( π ) ( ) ( π ) + + =. 7 : { : setup : answer : integrand (c) ( ) = 5 + v( t) dt = 5.77 or : uses ( ) = 5 : answer (d) The particle s rightmost position occurs at time t = π =.77. The particle changes from moving right to moving left at those times t for which vt ( ) = with vt ( ) changing from positive to negative, namely at t = π, π, 5π ( t =.77,.7,.96 ). T Using ( T) = 5 + v( t) dt, the particle s positions at the times it changes from rightward to leftward movement are: T: π π 5π T ( ): The particle is farthest to the right when T = π. : sets vt () = : answer : reason 7 The College Board. All rights reserved.
4 7 SCING GUIDELINES (Form B) Question The wind chill is the temperature, in degrees Fahrenheit ( F, ) a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour ( mph ). If the air temperature is F, then the.6 wind chill is given by W( v) = 55.6.v and is valid for 5 v 6. (a) Find W ( ). Using correct units, eplain the meaning of W ( ) in terms of the wind chill. (b) Find the average rate of change of W over the interval 5 v 6. Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 v 6. (c) Over the time interval t 4 hours, the air temperature is a constant F. At time t =, the wind velocity is v = mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at t = hours? Indicate units of measure. (a).84 W ( ) =..6 =.85 or.86 When v = mph, the wind chill is decreasing at.86 F mph. : { : value : eplanation (b) The average rate of change of W over the interval W( 6) W( 5) 5 v 6 is =.5 or W( 6) W( 5) W ( v) = when v = : average rate of change : W ( v) = average rate of change : value of v dw dw dv = = 5 5 =.89 F hr dt dv dt (c) ( ) W ( ) t= t= W = 55.6.( + 5t).6 dw =.89 F hr dt t = dv : = 5 dt : uses v( ) = 5, or uses vt () = + 5t : answer Units of F mph in (a) and F hr in (c) : units in (a) and (c) 7 The College Board. All rights reserved.
5 7 SCING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 5 with f ( ) =. The graph of f, the derivative of f, consists of two semicircles and two line segments, as shown above. (a) For 5 < < 5, find all values at which f has a relative maimum. Justify your answer. (b) For 5 < < 5, find all values at which the graph of f has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of f is concave up and also has positive slope. Eplain your reasoning. (d) Find the absolute minimum value of f ( ) over the closed interval 5 5. Eplain your reasoning. (a) f ( ) = at =,, 4 f changes from positive to negative at and 4. Thus, f has a relative maimum at = and at = 4. : { : values : justification (b) f changes from increasing to decreasing, or vice versa, at = 4,, and. Thus, the graph of f has points of inflection when = 4,, and. : { : values : justification (c) The graph of f is concave up with positive slope where f is increasing and positive: 5 < < 4 and < <. : { : intervals : eplanation (d) Candidates for the absolute minimum are where f changes from negative to positive (at = ) and at the endpoints ( = 5, 5 ). 5 π f( 5) = + f ( ) d = + π > f ( ) = 5 f( 5) = + f ( ) d = + > The absolute minimum value of f on [ 5, 5] is f ( ) =. : identifies = as a candidate : considers endpoints : value and eplanation 7 The College Board. All rights reserved.
6 Consider the differential equation AP CALCULUS AB 7 SCING GUIDELINES (Form B) dy y. d = + Question 5 (a) On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the aes provided in the eam booklet.) d y (b) Find in terms of and y. Describe the region in the yplane in d which all solution curves to the differential equation are concave up. (c) Let y = f( ) be a particular solution to the differential equation with the initial condition f ( ) =. Does f have a relative minimum, a relative maimum, or neither at =? Justify your answer. (d) Find the values of the constants m and b, for which y = m + b is a solution to the differential equation. (a) : Sign of slope at each point and relative steepness of slope lines in rows and columns. (b) (c) d y dy = + = + y d d Solution curves will be concave up on the halfplane above the line y = +. dy d (, ) d y = + = and d (, ) Thus, f has a relative minimum at (, ). = + > d y : d : description : answer : { : justification (d) Substituting y = m + b into the differential equation: m = + ( m + b) = ( m + ) + ( b ) Then = m + and m = b : m = and b =. : value for m : { : value for b 7 The College Board. All rights reserved.
7 7 SCING GUIDELINES (Form B) Question 6 Let f be a twicedifferentiable function such that f ( ) = 5 and f ( 5) =. Let g be the function given by g( ) = f ( f ( ) ). (a) Eplain why there must be a value c for < c < 5 such that f ( c ) =. (b) Show that g ( ) = g ( 5 ). Use this result to eplain why there must be a value k for < k < 5 such that g ( k ) =. (c) Show that if f ( ) = for all, then the graph of g does not have a point of inflection. (d) Let h ( ) = f( ). Eplain why there must be a value r for < r < 5 such that hr ( ) =. (a) The Mean Value Theorem guarantees that there is a value c, with < c < 5, so that f( 5) f( ) 5 f ( c) = = =. 5 5 f( 5) f( ) : : 5 : conclusion, using MVT (b) g ( ) = f ( f( ) ) f ( ) g ( ) = f ( f ( ) ) f ( ) = f ( 5) f ( ) g ( 5) = f ( f( 5) ) f ( 5) = f ( ) f ( 5 ) Thus, g ( ) = g ( 5). Since f is twicedifferentiable, g is differentiable everywhere, so the Mean Value Theorem applied to g on [, 5] guarantees there is a value k, with < k < 5, such that g ( k) = g ( 5) g ( ) =. 5 : g ( ) : g ( ) = f ( 5) f ( ) = g ( 5) : uses MVT with g (c) g ( ) = f ( f( ) ) f ( ) f ( ) + f ( f( ) ) f ( ) If f ( ) = for all, then g ( ) = f ( ) f ( ) + f ( f( ) ) = for all. Thus, there is no value at which g ( ) changes sign, so the graph of g has no inflection points. If f ( ) = for all, then f is linear, so g = f f is linear and the graph of g has no inflection points. (d) Let h( ) = f( ). h( ) = f( ) = 5 = h( 5) = f( 5) 5 = 5 = Since h( ) > > h( 5), the Intermediate Value Theorem guarantees that there is a value r, with < r < 5, such that h( r ) =. : considers g : : g ( ) = for all : f is linear : { : g is linear : h( ) and h( 5 ) : : conclusion, using IVT 7 The College Board. All rights reserved.
AP 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 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 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 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 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 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 3 The wind chill is the temperature, in degrees Fahrenheit ( F, ) a human feels based on the air temperature, in degrees Fahrenheit, and the wind
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 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 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 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 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 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 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 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 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 informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
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 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 Statistics 2005 Scoring Guidelines
AP Statistics 2005 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 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 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 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 informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationAP 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
More informationAP Human Geography 2006 Scoring Guidelines
AP Human Geography 2006 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 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 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 informationActive Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122. Carroll College Mathematics Department
Active Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122 Carroll College Mathematics Department Last Update: June 2, 2015 2 To The Student This packet is NOT your
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
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 informationExam 1 Review Questions PHY 2425  Exam 1
Exam 1 Review Questions PHY 2425  Exam 1 Exam 1H Rev Ques.doc  1  Section: 1 7 Topic: General Properties of Vectors Type: Conceptual 1 Given vector A, the vector 3 A A) has a magnitude 3 times that
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 Computer Science A 2010 Scoring Guidelines
AP Computer Science A 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 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 informationVersion 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This printout should have 47 questions Multiplechoice questions may continue on the net column or page find all choices before answering
More informationSection 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
More informationTo define concepts such as distance, displacement, speed, velocity, and acceleration.
Chapter 7 Kinematics of a particle Overview In kinematics we are concerned with describing a particle s motion without analysing what causes or changes that motion (forces). In this chapter we look at
More informationResearch Report No. 20055 Nancy K. Scammacca and Barbara G. Dodd
Research Report No. 20055 An Investigation of Educational Outcomes for Students Who Earn College Credit Through the CollegeLevel Examination Program Nancy K. Scammacca and Barbara G. Dodd www.collegeboard.com
More information2013 MBA Jump Start Program
2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
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 informationMidterm 2 Review Problems (the first 7 pages) Math 1235116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 15116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationEight Components of College and Career Readiness Counseling
The College Board National Office for School Counselor Advocacy Eight Components of College and Career Readiness Counseling Equity Leadership Transformation The Need for College Readiness Counseling In
More information= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).
Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( )  f( x 1 ) = y
More informationLecture 3: Derivatives and extremes of functions
Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationSolutions 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
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationSolutions to Midterm #1 Practice Problems
MAT Fall 0 Solutions to Midterm # Practice Problems. Below is the graph of a function y = r(). y = r() Sketch graphs of the following functions: (a) y = r( 3) (b) y = r( ) 3 (c) y = r() + (d) y = r( +
More informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationA Closer Look at the
A Closer Look at the PrincipalCounselor Relationship A Survey of Principals and Counselors May 2009 Acknowledgments We would like to thank Crux Research for their contributions to the analyses and interpretation
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1.6 Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described by piecewise functions. LEARN ABOUT the Math A city parking lot
More informationSection 1.3: Transformations of Graphs
CHAPTER 1 A Review of Functions Section 1.3: Transformations of Graphs Vertical and Horizontal Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Combining Transformations Vertical and Horizontal
More informationScalar versus Vector Quantities. Speed. Speed: Example Two. Scalar Quantities. Average Speed = distance (in meters) time (in seconds) v =
Scalar versus Vector Quantities Scalar Quantities Magnitude (size) 55 mph Speed Average Speed = distance (in meters) time (in seconds) Vector Quantities Magnitude (size) Direction 55 mph, North v = Dx
More informationHomework #2 Solutions
MAT Fall 0 Homework # Solutions Problems Section.: 8, 0, 6, 0, 8, 0 Section.:, 0, 8,, 4, 8..8. Find the relative, or percent, change in W if it changes from 0. to 0.0. Solution: The percent change is R
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
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 informationThe Earlier the Better? Taking the AP in 10th Grade. By Awilda Rodriguez, Mary E. M. McKillip, and Sunny X. Niu
Research Report 201210 The Earlier the Better? Taking the AP in 10th Grade By Awilda Rodriguez, Mary E. M. McKillip, and Sunny X. Niu Awilda Rodriguez is a doctoral student in Higher Education at the
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationMATH ADVISEMENT GUIDE
MATH ADVISEMENT GUIDE Recommendations for math courses are based on your placement results, degree program and career interests. Placement score: MAT 001 or MAT 00 You must complete required mathematics
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationInvestigating Area Under a Curve
Mathematics Investigating Area Under a Curve About this Lesson This lesson is an introduction to areas bounded by functions and the xaxis on a given interval. Since the functions in the beginning of the
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More information2010 The College Board. Visit the College Board on the Web: www.collegeboard.com.
AP CALCULUS AB 21 SCORING GUIDELINES Question 2 A zoo sponsored a oneday contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ( t = ) and 8 P.M. ( t = 8. )
More informationMath 2280  Assignment 6
Math 2280  Assignment 6 Dylan Zwick Spring 2014 Section 3.81, 3, 5, 8, 13 Section 4.11, 2, 13, 15, 22 Section 4.21, 10, 19, 28 1 Section 3.8  Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationL 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:
More information2.2. Instantaneous Velocity
2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical
More information1 of 7 9/5/2009 6:12 PM
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationThe AP Program: Resources for Parents and Families
The AP Program: Resources for Parents and Families I chose to take AP courses because I love to challenge myself... I feel thoroughly prepared for college because of the rigor of my AP classes. Anica,
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationDerivatives as Rates of Change
Derivatives as Rates of Change OneDimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions
More informationSECTION 1.6 Other Types of Equations
BLITMC1B.111599_11174 12//2 1:58 AM Page 11 Section 1.6 Other Types of Equations 11 12. A person throws a rock upward from the edge of an 8foot cliff. The height, h, in feet, of the rock above the water
More informationSample Test Questions
mathematics Numerical Skills/PreAlgebra Algebra Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
More informationAnswer Key Building Polynomial Functions
Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationCollege Entrance Tests SRJC, PSAT, SAT, ACT, CLEP, AP
College Entrance Tests SRJC, PSAT, SAT, ACT, CLEP, AP College Entrance Tests SRJC : Math and English Placement tests which can be taken more than once. These indicate which level Math/English students
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationPolynomials Past Papers Unit 2 Outcome 1
PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More information25 Rational Functions
5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More informationPhysics of the Atmosphere I
Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uniheidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:
More informationM122 College Algebra Review for Final Exam
M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationHomework #1 Solutions
MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More information