AP Calculus AB 2008 Scoring Guidelines

Size: px
Start display at page:

Download "AP Calculus AB 2008 Scoring Guidelines"

Transcription

1 AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity. Foundd in 9, th association is composd of mor than 5,4 schools, collgs, univrsitis, and othr ducational organizations. Each yar, th Collg Board srvs svn million studnts and thir parnts,, high schools, and,5 collgs through major programs and srvics in collg admissions, guidanc, assssmnt, financial aid, nrollmnt, and taching and larning. Among its bst-known programs ar th SAT, th PSAT/NMSQT, and th Advancd Placmnt Program (AP ). Th Collg Board is committd to th principls of cllnc and quity, and that commitmnt is mbodid in all of its programs, srvics, activitis, and concrns. 8 Th Collg Board. All rights rsrvd. Collg Board, AP Cntral, Advancd Placmnt Program, AP, SAT, and th acorn logo ar rgistrd tradmarks of th Collg Board. PSAT/NMSQT is a rgistrd tradmark of th Collg Board and National Mrit Scholarship Corporation. All othr products and srvics may b tradmarks of thir rspctiv ownrs. Prmission to us copyrightd Collg Board matrials may b rqustd onlin at: AP Cntral is th onlin hom for AP tachrs: apcntral.collgboard.com.

2 8 SCORING GUIDELINES Qustion Lt R b th rgion boundd by th graphs of y = sin( π ) and y = 4, as shown in th figur abov. (a) Find th ara of R. (b) Th horizontal lin y = splits th rgion R into two parts. Writ, but do not valuat, an intgral prssion for th ara of th part of R that is blow this horizontal lin. (c) Th rgion R is th bas of a solid. For this solid, ach cross sction prpndicular to th -ais is a squar. Find th volum of this solid. (d) Th rgion R modls th surfac of a small pond. At all points in R at a distanc from th y-ais, th dpth of th watr is givn by h ( ) =. Find th volum of watr in th pond. (a) sin ( π ) = 4 at = and = Ara ( ( ) ( )) = sin π 4 d = 4 : : limits : intgrand : answr (b) 4 = at r = and s =.6759 s Th ara of th statd rgion is ( ( 4 )) r d : { : limits : intgrand (c) Volum = ( sin ( π ) ( 4 )) d = : { : intgrand : answr (d) Volum = ( )( sin ( π ) ( 4 )) d = 8.69 or 8.7 : { : intgrand : answr 8 Th Collg Board. All rights rsrvd.

3 8 SCORING GUIDELINES Qustion t (hours) Lt ()(popl) Concrt tickts wnt on sal at noon ( t = ) and wr sold out within 9 hours. Th numbr of popl waiting in lin to purchas tickts at tim t is modld by a twic-diffrntiabl function L for t 9. Valus of Lt () at various tims t ar shown in th tabl abov. (a) Us th data in th tabl to stimat th rat at which th numbr of popl waiting in lin was changing at 5: P.M. ( t = 5.5 ). Show th computations that lad to your answr. Indicat units of masur. (b) Us a trapzoidal sum with thr subintrvals to stimat th avrag numbr of popl waiting in lin during th first 4 hours that tickts wr on sal. (c) For t 9, what is th fwst numbr of tims at which L () t must qual? Giv a rason for your answr. (d) Th rat at which tickts wr sold for t 9 is modld by rt () = 55t t tickts pr hour. Basd on th modl, how many tickts wr sold by P.M. ( t =, ) to th narst whol numbr? L( 7) L( 4) 5 6 (a) L ( 5.5) = = 8 popl pr hour 7 4 (b) Th avrag numbr of popl waiting in lin during th first 4 hours is approimatly L( ) + L( ) L() ( ) ( ) ( ) ( ) + L L ( ) + L 4 ( 4 ) = 55.5 popl (c) L is diffrntiabl on [, 9 ] so th Man Valu Thorm implis L () t > for som t in (, ) and som t in ( 4, 7 ). Similarly, L () t < for som t in (, 4 ) and som t in ( 7, 8 ). Thn, sinc L is continuous on [, 9 ], th Intrmdiat Valu Thorm implis that L () t = for at last thr valus of t in [, 9 ]. OR Th continuity of L on [, 4 ] implis that L attains a maimum valu thr. Sinc L( ) > L( ) and L( ) > L( 4 ), this maimum occurs on (, 4 ). Similarly, L attains a minimum on (, 7 ) and a maimum on ( 4, 8 ). L is diffrntiabl, so L () t = at ach rlativ trm point on (, 9 ). Thrfor L () t = for at last thr valus of t in [, 9 ]. [Not: Thr is a function L that satisfis th givn conditions with L () t = for actly thr valus of t.] (d) rt () = Thr wr approimatly 97 tickts sold by P.M. : { : stimat : units : trapzoidal sum : { : answr : : : considrs chang in sign of L : analysis : conclusion OR : considrs rlativ trma of L on (, 9) : analysis : conclusion : { : intgrand : limits and answr 8 Th Collg Board. All rights rsrvd.

4 8 SCORING GUIDELINES Qustion Oil is laking from a piplin on th surfac of a lak and forms an oil slick whos volum incrass at a constant rat of cubic cntimtrs pr minut. Th oil slick taks th form of a right circular cylindr with both its radius and hight changing with tim. (Not: Th volum V of a right circular cylindr with radius r and hight h is givn by V = π r h. ) (a) At th instant whn th radius of th oil slick is cntimtrs and th hight is.5 cntimtr, th radius is incrasing at th rat of.5 cntimtrs pr minut. At this instant, what is th rat of chang of th hight of th oil slick with rspct to tim, in cntimtrs pr minut? (b) A rcovry dvic arrivs on th scn and bgins rmoving oil. Th rat at which oil is rmovd is R() t = 4 t cubic cntimtrs pr minut, whr t is th tim in minuts sinc th dvic bgan working. Oil continus to lak at th rat of cubic cntimtrs pr minut. Find th tim t whn th oil slick rachs its maimum volum. Justify your answr. (c) By th tim th rcovry dvic bgan rmoving oil, 6, cubic cntimtrs of oil had alrady lakd. Writ, but do not valuat, an prssion involving an intgral that givs th volum of oil at th tim found in part (b). (a) Whn r = cm and h =.5 cm, and dr =.5 cm min. dr dh = πr h + πr = π( )(.5)(.5) + π( ) dh =.8 or.9 cm min dh = cm min dr : = and =.5 4 : : prssion for : answr (b) = R() t, so = whn Rt () =. This occurs whn t = 5 minuts. Sinc > for < t < 5 and < for t > 5, th oil slick rachs its maimum volum 5 minuts aftr th dvic bgins working. : : Rt () = : answr : justification (c) Th volum of oil, in cm, in th slick at tim t = 5 minuts 5 is givn by 6, + ( R() t ). : limits and initial condition : { : intgrand 8 Th Collg Board. All rights rsrvd.

5 8 SCORING GUIDELINES Qustion 4 A particl movs along th -ais so that its vlocity at tim t, for t 6, is givn by a diffrntiabl function v whos graph is shown abov. Th vlocity is at t =, t =, and t = 5, and th graph has horizontal tangnts at t = and t = 4. Th aras of th rgions boundd by th t-ais and th graph of v on 5, 6 ar 8,, and, rspctivly. At tim t =, th particl is at =. th intrvals [, ], [, 5 ], and [ ] (a) For t 6, find both th tim and th position of th particl whn th particl is farthst to th lft. Justify your answr. (b) For how many valus of t, whr t 6, is th particl at = 8? Eplain your rasoning. (c) On th intrval < t <, is th spd of th particl incrasing or dcrasing? Giv a rason for your answr. (d) During what tim intrvals, if any, is th acclration of th particl ngativ? Justify your answr. (a) Sinc vt () < for < t < and 5 < t < 6, and vt () > for < t < 5, w considr t = and t = 6. ( ) = + v( t) = 8 = 6 ( 6) = + v( t) = 8 + = 9 Thrfor, th particl is farthst lft at tim t = whn its position is ( ) =. (b) Th particl movs continuously and monotonically from ( ) = to ( ) =. Similarly, th particl movs continuously and monotonically from ( ) = to ( 5) = 7 and also from ( 5) = 7 to ( 6) = 9. By th Intrmdiat Valu Thorm, thr ar thr valus of t for which th particl is at t () = 8. (c) Th spd is dcrasing on th intrval < t < sinc on this intrval v < and v is incrasing. (d) Th acclration is ngativ on th intrvals < t < and 4 < t < 6 sinc vlocity is dcrasing on ths intrvals. : idntifis t = as a candidat 6 : : considrs vt () : conclusion : : positions at t =, t = 5, and t = 6 : dscription of motion : conclusion : answr with rason : { : answr : justification 8 Th Collg Board. All rights rsrvd.

6 8 SCORING GUIDELINES Qustion 5 dy y Considr th diffrntial quation =, whr. d (a) On th as providd, sktch a slop fild for th givn diffrntial quation at th nin points indicatd. (Not: Us th as providd in th am booklt.) (b) Find th particular solution y = f( ) to th diffrntial quation with th initial condition f ( ) =. (c) For th particular solution y = f( ) dscribd in part (b), find lim f ( ). (a) : zro slops : { : all othr slops (b) dy = d y ln y = + C y = + C y = y = k, whr k = ± = k k = C ( ) ( ) f =, > C 6 : : sparats variabls : antidiffrntiats : includs constant of intgration : uss initial condition : solvs for y Not: ma 6 [----] if no constant of intgration Not: 6 if no sparation of variabls ( ) (c) lim = : limit 8 Th Collg Board. All rights rsrvd.

7 8 SCORING GUIDELINES Qustion 6 ln Lt f b th function givn by f( ) = for all >. Th drivativ of f is givn by ln f ( ) =. (a) Writ an quation for th lin tangnt to th graph of f at =. (b) Find th -coordinat of th critical point of f. Dtrmin whthr this point is a rlativ minimum, a rlativ maimum, or nithr for th function f. Justify your answr. (c) Th graph of th function f has actly on point of inflction. Find th -coordinat of this point. (d) Find lim f ( ). + ln (a) f( ), ln = = f ( ) = = 4 ( ) An quation for th tangnt lin is y = ( ) 4. ( ) ( ) : : f and f : answr (b) f ( ) = whn =. Th function f has a rlativ maimum at = bcaus f ( ) changs from positiv to ngativ at =. : : = : rlativ maimum : justification ( ln ) ln (c) f ( ) + = = for all > 4 f ( ) = whn + ln = : f ( ) : : answr = Th graph of f has a point of inflction at f ( ) changs sign at =. = bcaus (d) ln lim + = or Dos Not Eist : answr 8 Th Collg Board. All rights rsrvd.

AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com

AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos

More information

Question 3: How do you find the relative extrema of a function?

Question 3: How do you find the relative extrema of a function? ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

More information

AP Calculus AB 2008 Free-Response Questions

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

More information

AP Calculus BC 2008 Scoring Guidelines

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

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000 hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

More information

The Normal Distribution: A derivation from basic principles

The Normal Distribution: A derivation from basic principles Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

More information

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

More information

QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

More information

Section 7.4: Exponential Growth and Decay

Section 7.4: Exponential Growth and Decay 1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST: .4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

More information

New Basis Functions. Section 8. Complex Fourier Series

New Basis Functions. Section 8. Complex Fourier Series Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

More information

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13) con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

More information

Lecture 3: Diffusion: Fick s first law

Lecture 3: Diffusion: Fick s first law Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

More information

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

More information

Traffic Flow Analysis (2)

Traffic Flow Analysis (2) Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

(Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for the European Normal Black Scholes Formula) (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

AP Calculus AB 2004 Scoring Guidelines

AP Calculus AB 2004 Scoring Guidelines AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

More information

HOMEWORK FOR UNIT 5-1: FORCE AND MOTION

HOMEWORK FOR UNIT 5-1: FORCE AND MOTION Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.

More information

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog

More information

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among

More information

AP CALCULUS AB 2008 SCORING GUIDELINES

AP CALCULUS AB 2008 SCORING GUIDELINES AP CALCULUS AB 2008 SCORING GUIDELINES Question 3 Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per

More information

AP Calculus AB 2009 Scoring Guidelines

AP Calculus AB 2009 Scoring Guidelines 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,

More information

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book. Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

More information

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final. Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

ME 612 Metal Forming and Theory of Plasticity. 6. Strain

ME 612 Metal Forming and Theory of Plasticity. 6. Strain Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

More information

Current and Resistance

Current and Resistance Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9

More information

AP Calculus AB 2003 Scoring Guidelines Form B

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

More information

union scholars program APPLICATION DEADLINE: FEBRUARY 28 YOU CAN CHANGE THE WORLD... AND EARN MONEY FOR COLLEGE AT THE SAME TIME!

union scholars program APPLICATION DEADLINE: FEBRUARY 28 YOU CAN CHANGE THE WORLD... AND EARN MONEY FOR COLLEGE AT THE SAME TIME! union scholars YOU CAN CHANGE THE WORLD... program AND EARN MONEY FOR COLLEGE AT THE SAME TIME! AFSCME Unitd Ngro Collg Fund Harvard Univrsity Labor and Worklif Program APPLICATION DEADLINE: FEBRUARY 28

More information

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D 24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

More information

AP Calculus AB 2010 Free-Response Questions Form B

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.

More information

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS 25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

More information

Deer: Predation or Starvation

Deer: Predation or Starvation : Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

More information

A Note on Approximating. the Normal Distribution Function

A Note on Approximating. the Normal Distribution Function Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

AP Calculus AB 2011 Scoring Guidelines

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

More information

AP Calculus AB 2007 Scoring Guidelines Form B

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

More information

Financial Mathematics

Financial Mathematics Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

More information

Adverse Selection and Moral Hazard in a Model With 2 States of the World

Adverse Selection and Moral Hazard in a Model With 2 States of the World Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

Planning and Managing Copper Cable Maintenance through Cost- Benefit Modeling

Planning and Managing Copper Cable Maintenance through Cost- Benefit Modeling Planning and Managing Coppr Cabl Maintnanc through Cost- Bnfit Modling Jason W. Rup U S WEST Advancd Tchnologis Bouldr Ky Words: Maintnanc, Managmnt Stratgy, Rhabilitation, Cost-bnfit Analysis, Rliability

More information

AP Calculus BC 2007 Free-Response Questions

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

More information

Performance Evaluation

Performance Evaluation Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism

More information

Production Costing (Chapter 8 of W&W)

Production Costing (Chapter 8 of W&W) Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary

More information

AP Calculus AB 2012 Scoring Guidelines

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

More information

Installation Saving Space-efficient Panel Enhanced Physical Durability Enhanced Performance Warranty The IRR Comparison

Installation Saving Space-efficient Panel Enhanced Physical Durability Enhanced Performance Warranty The IRR Comparison Contnts Tchnology Nwly Dvlopd Cllo Tchnology Cllo Tchnology : Improvd Absorption of Light Doubl-sidd Cll Structur Cllo Tchnology : Lss Powr Gnration Loss Extrmly Low LID Clls 3 3 4 4 4 Advantag Installation

More information

AP Calculus AB 2003 Scoring Guidelines

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

More information

Parallel and Distributed Programming. Performance Metrics

Parallel and Distributed Programming. Performance Metrics Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

AP Calculus AB 2005 Free-Response Questions

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

More information

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman Cloud and Big Data Summr Scool, Stockolm, Aug., 2015 Jffry D. Ullman Givn a st of points, wit a notion of distanc btwn points, group t points into som numbr of clustrs, so tat mmbrs of a clustr ar clos

More information

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

AP Calculus AB 2009 Free-Response Questions

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

More information

SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

More information

Factorials! Stirling s formula

Factorials! Stirling s formula Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

More information

AP Calculus AB 2007 Free-Response Questions

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

More information

AP Calculus BC 2009 Free-Response Questions

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

More information

Architecture of the proposed standard

Architecture of the proposed standard Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th

More information

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

Category 7: Employee Commuting

Category 7: Employee Commuting 7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

Foreign Exchange Markets and Exchange Rates

Foreign Exchange Markets and Exchange Rates Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

AP Calculus AB 2010 Free-Response Questions

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

More information

A Theoretical Model of Public Response to the Homeland Security Advisory System

A Theoretical Model of Public Response to the Homeland Security Advisory System A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial

More information

Maintain Your F5 Solution with Fast, Reliable Support

Maintain Your F5 Solution with Fast, Reliable Support F5 SERVICES TECHNICAL SUPPORT SERVICES DATASHEET Maintain Your F5 Solution with Fast, Rliabl Support In a world whr chang is th only constant, you rly on your F5 tchnology to dlivr no mattr what turns

More information

Who uses our services? We have a growing customer base. with institutions all around the globe.

Who uses our services? We have a growing customer base. with institutions all around the globe. not taking xpr Srvic Guid 2013 / 2014 NTE i an affordabl option for audio to txt convrion. Our rvic includ not or dirct trancription rvic from prviouly rcordd audio fil. Our rvic appal pcially to tudnt

More information

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

More information

AP Calculus BC 2004 Scoring Guidelines

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

More information

GOAL SETTING AND PERSONAL MISSION STATEMENT

GOAL SETTING AND PERSONAL MISSION STATEMENT Prsonal Dvlopmnt Track Sction 4 GOAL SETTING AND PERSONAL MISSION STATEMENT Ky Points 1 Dfining a Vision 2 Writing a Prsonal Mission Statmnt 3 Writing SMART Goals to Support a Vision and Mission If you

More information

7 Timetable test 1 The Combing Chart

7 Timetable test 1 The Combing Chart 7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

More information

Intermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers)

Intermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers) Intrmdiat Macroconomic Thory / Macroconomic Analysis (ECON 3560/5040) Final Exam (Answrs) Part A (5 points) Stat whthr you think ach of th following qustions is tru (T), fals (F), or uncrtain (U) and brifly

More information

Student Performance Q&A:

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

More information

Renewable Energy Sources. Solar Cells SJSU-E10 S John Athanasiou

Renewable Energy Sources. Solar Cells SJSU-E10 S John Athanasiou Rnwabl Enrgy Sourcs. Solar Clls SJSU-E10 S-2008 John Athanasiou 1 Rnwabl Enrgy Sourcs Rnwabl: Thy can last indfinitly 1. Wind Turbin: Convrting th wind nrgy into lctricity Wind, Propllr, Elctric Gnrator,

More information

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List. Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

Magic Message Maker Amaze your customers with this Gift of Caring communication piece

Magic Message Maker Amaze your customers with this Gift of Caring communication piece Magic Mssag Makr maz your customrs with this Gift of aring communication pic Girls larn th powr and impact of crativ markting with this attntion grabbing communication pic that will hlp thm o a World of

More information

AP Calculus BC 2010 Free-Response Questions

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

More information

HSBC Bank International Expat Explorer Survey 08

HSBC Bank International Expat Explorer Survey 08 HSBC Bank Intrnational Expat Explorr Survy 08 Rport On: Expat Existnc Th Survy Th Expat Explorr survy qustiond 2,155 xpatriats across four continnts about th opportunitis and challngs thy fac. Th survy

More information

Presentation on Short-Term Certificates to the CAPSEE Conference. September 18, 2014

Presentation on Short-Term Certificates to the CAPSEE Conference. September 18, 2014 Prsntation on Short-Trm Crtificats to th CAPSEE Confrnc Sptmbr 18, 2014 Th first goal of City Collgs Rinvntion is focusd on nsuring studnts arn crdntials of conomic valu Rinvntion Goals Incras numbr of

More information

2008 AP Calculus AB Multiple Choice Exam

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

More information

AP Calculus BC 1999 Scoring Guidelines

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

More information

Continuity Cloud Virtual Firewall Guide

Continuity Cloud Virtual Firewall Guide Cloud Virtual Firwall Guid uh6 Vrsion 1.0 Octobr 2015 Foldr BDR Guid for Vam Pag 1 of 36 Cloud Virtual Firwall Guid CONTENTS INTRODUCTION... 3 ACCESSING THE VIRTUAL FIREWALL... 4 HYPER-V/VIRTUALBOX CONTINUITY

More information

Making and Using the Hertzsprung - Russell Diagram

Making and Using the Hertzsprung - Russell Diagram Making and Using th Hrtzsprung - Russll Diagram Nam In astronomy th Hrtzsprung-Russll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.

More information

LAB 3: VELOCITY AND ACCELERATION GRAPHS

LAB 3: VELOCITY AND ACCELERATION GRAPHS Goas: LAB 3: ELOCITY AND ACCELERATION GRAPHS Invstigat accration vs. tim graphs Prdict accration graphs from vocity graphs Invstigat accration as sop of vocity vs. tim graph Part 1 - Making ocity- Graphs

More information

Teaching Computer Networking with the Help of Personal Computer Networks

Teaching Computer Networking with the Help of Personal Computer Networks Taching Computr Ntworking with th Hlp of Prsonal Computr Ntworks Rocky K. C. Chang Dpartmnt of Computing Th Hong Kong Polytchnic Univrsity Hung Hom, Kowloon, Hong Kong csrchang@comp.polyu.du.hk ABSTRACT

More information

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

Mark 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 information

[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)

[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2) MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris

More information

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 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 information

AP Calculus AB 2005 Scoring Guidelines Form B

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

More information

AP Calculus AB 2004 Free-Response Questions

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

More information

TIME MANAGEMENT. 1 The Process for Effective Time Management 2 Barriers to Time Management 3 SMART Goals 4 The POWER Model e. Section 1.

TIME MANAGEMENT. 1 The Process for Effective Time Management 2 Barriers to Time Management 3 SMART Goals 4 The POWER Model e. Section 1. Prsonal Dvlopmnt Track Sction 1 TIME MANAGEMENT Ky Points 1 Th Procss for Effctiv Tim Managmnt 2 Barrirs to Tim Managmnt 3 SMART Goals 4 Th POWER Modl In th Army, w spak of rsourcs in trms of th thr M

More information

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs

More information

Increasing Net Debt as a percentage of Average Equalized ValuaOon

Increasing Net Debt as a percentage of Average Equalized ValuaOon City of Orang Township Warning Trnd: Incrasing Nt Dbt as a prcntag of avrag qualizd valuation Nt Dbt 3 yr. Avg. qualizd Valuation Incrasing Nt Dbt as a prcntag of Avrag Equalizd ValuaOon rc 1.20% 1.00%

More information

AP CALCULUS AB 2008 SCORING GUIDELINES

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

More information

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scic March 15, 005 Srii Dvadas ad Eric Lhma Problm St 6 Solutios Du: Moday, March 8 at 9 PM Problm 1. Sammy th Shar is a fiacial srvic providr who offrs loas o th followig

More information

B-285141. April 21, 2000. The Honorable Charles B. Rangel Ranking Minority Member Committee on Ways and Means House of Representatives

B-285141. April 21, 2000. The Honorable Charles B. Rangel Ranking Minority Member Committee on Ways and Means House of Representatives Unit Stats Gnral Accounting Offic Washington, DC 20548 Halth, Eucation, an Human Srvics Division B-285141 April 21, 2000 Th Honorabl Charls B. Rangl Ranking Minority Mmbr Committ on Ways an Mans Hous of

More information

AP Calculus BC 2006 Free-Response Questions

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

More information

AP Calculus BC 2012 Scoring Guidelines

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

More information

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009 Volum 3, Issu 1, 29 Statistical Approach for Analyzing Cll Phon Handoff Bhavior Shalini Saxna, Florida Atlantic Univrsity, Boca Raton, FL, shalinisaxna1@gmail.com Sad A. Rajput, Farquhar Collg of Arts

More information

Dehumidifiers: A Major Consumer of Residential Electricity

Dehumidifiers: A Major Consumer of Residential Electricity Dhumidifirs: A Major Consumr of Rsidntial Elctricity Laurn Mattison and Dav Korn, Th Cadmus Group, Inc. ABSTRACT An stimatd 19% of U.S. homs hav dhumidifirs, and thy can account for a substantial portion

More information

C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Reports with SAS C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

More information

Lecture 20: Emitter Follower and Differential Amplifiers

Lecture 20: Emitter Follower and Differential Amplifiers Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

More information

CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER 4c. ROOTS OF EQUATIONS CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

More information