AP Calculus BC 2010 Scoring Guidelines


 Colleen Bridges
 1 years ago
 Views:
Transcription
1 AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board is composed of more han 5,7 schools, colleges, universiies and oher educaional organizaions. Each year, he College Board serves seven million sudens and heir parens,, high schools, and,8 colleges hrough major programs and services in college readiness, college admission, guidance, assessmen, financial aid and enrollmen. Among is widely recognized programs are he SAT, he PSAT/NMSQT, he Advanced Placemen Program (AP ), SpringBoard and ACCUPLACER. The College Board is commied o he principles of excellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies and concerns. The College Board. College Board, ACCUPLACER, Advanced Placemen Program, AP, AP Cenral, SAT, SpringBoard and he acorn logo are regisered rademarks of he College Board. Admied Class Evaluaion Service is a rademark owned by he College Board. PSAT/NMSQT is a regisered rademark of he College Board and Naional Meri Scholarship Corporaion. All oher producs and services may be rademarks of heir respecive owners. Permission o use copyrighed College Board maerials may be requesed online a: Visi he College Board on he Web: AP Cenral is he official online home for he AP Program: apcenral.collegeboard.com.
2 SCORING GUIDELINES Quesion There is no snow on Jane s driveway when snow begins o fall a midnigh. From midnigh o A.M., snow cos accumulaes on he driveway a a rae modeled by f() = 7e cubic fee per hour, where is measured in hours since midnigh. Jane sars removing snow a 6 A.M. ( = 6. ) The rae g (), in cubic fee per hour, a which Jane removes snow from he driveway a ime hours afer midnigh is modeled by for < 6 g () = 5 for 6 < 7 8 for 7. (a) How many cubic fee of snow have accumulaed on he driveway by 6 A.M.? (b) Find he rae of change of he volume of snow on he driveway a 8 A.M. (c) Le h () represen he oal amoun of snow, in cubic fee, ha Jane has removed from he driveway a ime hours afer midnigh. Express h as a piecewisedefined funcion wih domain. (d) How many cubic fee of snow are on he driveway a A.M.? 6 or.75 cubic fee : { : inegral (a) f() d =.7 : answer (b) Rae of change is f( 8) g( 8) = 5.58 or 5.58 cubic fee per hour. : answer (c) h ( ) = For < 6, h () = h( ) + gs ( ) ds= + ds=. For 6 7, < h () = h( 6) + gs ( ) ds= + 5 ds= 5( 6 ). For 7, 6 6 < h () = h( 7) + g( s) ds = ds = 5 + 8( 7 ). 7 7 for 6 Thus, h () = 5( 6) for 6 < ( 7) for 7 < : h () for 6 : h () for 6 < 7 : h () for 7 < : inegral (d) Amoun of snow is f() d h( ) = 6. or 6.5 cubic fee. : h( ) : answer The College Board. Visi he College Board on he Web:
3 SCORING GUIDELINES (hours) E() (hundreds of enries) Quesion A zoo sponsored a oneday cones o name a new baby elephan. Zoo visiors deposied enries in a special box beween noon ( = ) and 8 P.M. ( = 8. ) The number of enries in he box hours afer noon is modeled by a differeniable funcion E for 8. Values of E(), in hundreds of enries, a various imes are shown in he able above. (a) Use he daa in he able o approximae he rae, in hundreds of enries per hour, a which enries were being deposied a ime = 6. Show he compuaions ha lead o your answer. 8 (b) Use a rapezoidal sum wih he four subinervals given by he able o approximae he value of (). 8 E d 8 Using correc unis, explain he meaning of () 8 E d in erms of he number of enries. (c) A 8 P.M., voluneers began o process he enries. They processed he enries a a rae modeled by he funcion P, where P () = hundreds of enries per hour for 8. According o he model, how many enries had no ye been processed by midnigh ( = )? (d) According o he model from par (c), a wha ime were he enries being processed mos quickly? Jusify your answer. E( 7) E( 5) (a) E ( 6) = hundred enries per hour (b) () 8 E d E( ) + E( ) E( ) + E( 5) E( 5) + E( 7) E( 7) + E( 8) =.687 or () 8 E d is he average number of hundreds of enries in he box beween noon and 8 P.M. (c) P () d= 6 = 7 : answer : rapezoidal sum : approximaion : meaning hundred enries : 8 { : inegral (d) P () = when =.85 and =.867. P() Enries are being processed mos quickly a ime =. : answer : considers P () = : idenifies candidaes : answer wih jusificaion The College Board. Visi he College Board on he Web:
4 SCORING GUIDELINES Quesion A paricle is moving along a curve so ha is posiion a ime is ( x(), y() ), where x () = + 8 and y () is no explicily given. Boh x and y are measured in meers, and is measured in seconds. I is known ha = e. d (a) Find he speed of he paricle a ime = seconds. (b) Find he oal disance raveled by he paricle for seconds. (c) Find he ime,, when he line angen o he pah of he paricle is horizonal. Is he direcion of moion of he paricle oward he lef or oward he righ a ha ime? Give a reason for your answer. (d) There is a poin wih xcoordinae 5 hrough which he paricle passes wice. Find each of he following. (i) The wo values of when ha occurs (ii) The slopes of he lines angen o he paricle s pah a ha poin (iii) The ycoordinae of ha poin, given y( ) = + e (a) Speed = ( x ( ) ) + ( y ( ) ) =.88 meers per second : answer (b) x () = = + e d =.587 or.588 meers Disance ( ) ( ) : { : inegral : answer d (c) = = when e = and d This occurs a =.7. Since x (.7) >, he paricle is moving oward he righ a ime =.7 or.8. : considers = : =.7 or.8 : direcion of moion wih reason (d) x () = 5 a = and = A ime =, he slope is A ime =, he slope is y() = y() = + d e + d = = d.. d = = = = d. d = = = : = and = : slopes : ycoordinae The College Board. Visi he College Board on he Web:
5 SCORING GUIDELINES Quesion Le R be he region in he firs quadran bounded by he graph of y = x, he horizonal line y = 6, and he yaxis, as shown in he figure above. (a) Find he area of R. (b) Wrie, bu do no evaluae, an inegral expression ha gives he volume of he solid generaed when R is roaed abou he horizonal line y = 7. (c) Region R is he base of a solid. For each y, where y 6, he cross secion of he solid aken perpendicular o he yaxis is a recangle whose heigh is imes he lengh of is base in region R. Wrie, bu do no evaluae, an inegral expression ha gives he volume of he solid. (a) Area ( x) ( x x ) x= = 6 = 6 = 8 x= : inegrand : aniderivaive : answer ( ) { : inegrand (b) Volume = π ( 7 x ) ( 7 6) : limis and consan y (c) Solving y = x for x yields x =. y y Each recangular cross secion has area y. = 6 Volume = 6 6 y { : inegrand : answer The College Board. Visi he College Board on he Web:
6 SCORING GUIDELINES Quesion 5 Consider he differenial equaion y. = Le y = f( x) be he paricular soluion o his differenial equaion wih he iniial condiion f () =. For his paricular soluion, f( x ) < for all values of x. (a) Use Euler s mehod, saring a x = wih wo seps of equal size, o approximae f (. ) Show he work ha leads o your answer. (b) Find lim f ( x). Show he work ha leads o your answer. x x (c) Find he paricular soluion y = f( x) o he differenial equaion = y wih he iniial condiion f () =. f f + x Δ ( ) ( ), = + = (a) ( ) () f ( ) f( ) + Δx (, ) 5 + ( ) = (b) Since f is differeniable a x =, f is coninuous a x =. So, ( ) ( ) lim f x = = lim x and we may apply L Hospial s x x Rule. f( x) f ( x) lim f ( x) x lim = lim = = x x x x lim x x : Euler s mehod wih wo seps : { : answer : use of L Hospial s Rule : { : answer (c) y = y = ln y = x + C ln = + C C = ln y = x y e x = 5 : : separaion of variables : aniderivaives : consan of inegraion : uses iniial condiion : solves for y Noe: max 5 [] if no consan of inegraion Noe: 5 if no separaion of variables f ( x) = e x The College Board. Visi he College Board on he Web:
7 SCORING GUIDELINES Quesion 6 cos x for x f( x) = x for x = The funcion f, defined above, has derivaives of all orders. Le g be he funcion defined by g( x) f( ) d. = + x (a) Wrie he firs hree nonzero erms and he general erm of he Taylor series for cos x abou x =. Use his series o wrie he firs hree nonzero erms and he general erm of he Taylor series for f abou x =. (b) Use he Taylor series for f abou x = found in par (a) o deermine wheher f has a relaive maximum, relaive minimum, or neiher a x =. Give a reason for your answer. (c) Wrie he fifhdegree Taylor polynomial for g abou x =. (d) The Taylor series for g abou x =, evaluaed a x =, is an alernaing series wih individual erms ha decrease in absolue value o. Use he hirddegree Taylor polynomial for g abou x = o esimae he value of g (). Explain why his esimae differs from he acual value of g () by less han. 6! n x x n x (a) cos( x) = + + ( ) +! ( n)! n x x n+ x f( x) = ( ) +! 6! ( n + )! (b) f ( ) is he coefficien of x in he Taylor series for f abou x =, so f ( ) =. f ( ) = is he coefficien of x in he Taylor series for f abou!! x =, so f ( ) =. Therefore, by he Second Derivaive Tes, f has a relaive minimum a x =. : erms for cos x : erms for f : firs hree erms : general erm : deermines f ( ) : : answer wih reason 5 x x x : wo correc erms (c) P5 ( x ) = + :! 5 6! { : remaining erms 7 (d) g() + =! 7 Since he Taylor series for g abou x = evaluaed a x = is alernaing and he erms decrease in absolue value o, we know g 7 (). 7 < 5 6! < 6! : { : esimae : explanaion The College Board. Visi he College Board on he Web:
AP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationAP Calculus AB 2010 Scoring Guidelines Form B
AP Calculus AB 1 Scoring Guidelines Form B The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 19, he
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationAP Calculus AB 2004 FreeResponse Questions Form B
AP Calculus AB 2004 FreeResponse Quesions Form B The maerials included in hese files are inended for noncommercial use by AP eachers for course and exam preparaion; permission for any oher use mus be
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationAP CALCULUS BC 2010 SCORING GUIDELINES. Question 1
AP CALCULUS BC 2010 SCORING GUIDELINES Quesion 1 There is no snow on Jane s driveway when snow begins o fall a midnigh. From midnigh o 9 A.M., snow cos accumulaes on he driveway a a rae modeled by f()
More informationAP * Calculus Review
AP * Calculus Review Posiion, Velociy, and Acceleraion Teacher Packe AP* is a rademark of he College Enrance Examinaion Board. The College Enrance Examinaion Board was no involved in he producion of his
More informationx x 1 1. F x 1 t dt 2. F x dt 3. F x 1 t dt t x x 2x x F x dt 8. F x dt 9. F x sin x 1/ x cos x 10 x
PreCalculus  Honors Secion 8. Uni 8 More on Inegraion Find F' Fundamenal Theorem of Calculus No Calculaor. F d. F d. F d. F d 5. F cos d. F d 7. F d 8. F d 9. F d sin / cos. F d. F sin d. F d. F d. F
More informationOne Dimensional Kinematics
Chaper B One Dimensional Kinemaics Blinn College  Physics 2425  Terry Honan Kinemaics is he sudy of moion. This chaper will inroduce he basic definiions of kinemaics. The definiions of he velociy and
More informationNet Change In 13, write a complete sentence including appropriate units of measure.
AP Calculus CHAPTER WORKSHEET INTEGRALS Name Sea # Ne Change In 13, wrie a complee senence including appropriae unis of measure. 1. If 1 w' is he rae of growh of a child in pounds per year, wha does w
More informationWeek #9  The Integral Section 5.1
Week #9  The Inegral Secion 5.1 From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,
More informationAP Calculus BC 2010 FreeResponse Questions
AP Calculus BC 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More information, where P is the number of bears at time t in years. dt
CALCULUS BC WORKSHEET ON LOGISTIC GROWTH Work he following on noebook paper Do no use your calculaor 1 Suppose he populaion of bears in a naional park grows according o he logisic differenial equaion =
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationCh. 2 1Dimensional Motion conceptual question #6, problems # 2, 9, 15, 17, 29, 31, 37, 46, 54
Ch. 2 1Dimensional Moion concepual quesion #6, problems # 2, 9, 15, 17, 29, 31, 37, 46, 54 Quaniies Scalars disance speed Vecors displacemen velociy acceleraion Defininion Disance = how far apar wo poins
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationPRINCIPLE OF ANGULAR IMPULSE & MOMENTUM (Sections ) Today s Objectives: Students will be able to: a) Determine the angular momentum of a
PRINCIPLE OF ANGULAR IMPULSE & MOMENTUM (Secions 5.55.7) Today s Objecives: Sudens will be able o: a) Deermine he angular momenum of a paricle and apply he principle of angular impulse & momenum. b) Use
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMath 308 Week 2 Solutions
Mah 308 Week Soluions Here are soluions o he evennumbered suggesed problems. The answers o he oddnumbered problems are in he back of your exbook, and he soluions are in he Soluion Manual, which you can
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationStraight Line Motion, Functions, average velocity and speed.
Sraigh Line Moion, Funcions, average velociy and speed. Moion occurs whenever an objec changes posiion. Since objecs canno insananeously change posiion, raher hey do so progressively over ime, ime mus
More informationCALCULUS I. Assignment Problems Derivatives. Paul Dawkins
CALCULUS I Assignmen Problems Derivaives Paul Dawkins Table of Conens Preface... 1 Derivaives... 1 Inroducion... 1 The Definiion of he Derivaive... Inerpreaions of he Derivaive... Differeniaion Formulas...
More informationπ π π For the angle t =, we can construct an equilateral triangle to show that the
Calculus, secion 8. Derivaive & Inegral of Sin and Cos noes prepared by Tim Pilachowski We begin by going back o he uni circle, wih r, and he definiion of sine and ine as coordinaes on he circle. Our firs
More informationExponential Growth and Decay S E C T I O N 6. 3
Exponenial Growh and Decay S E C T I O N 6. 3 The Grea Divide 10 minues o complee Follow up Quesions (Wih your parner be prepared o answer he following quesions abou his aciviy) Do he graphs represen a
More informationAP Calculus AB 2010 FreeResponse Questions
AP Calculus AB 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationDERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES. Math 225
DERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES Mah 225 Derivaives Along Vecors Suppose ha f is a funcion of wo variables, ha is, f : R 2 R, or, if we are hinking wihou coordinaes, f : E 2 R. The
More informationConcepTests and Answers and Comments for Section 2.5
90 CHAPTER TWO For Problems 9, le A = f() be he deph of read, in cenimeers, on a radial ire as a funcion of he ime elapsed, in monhs, since he purchase of he ire. 9. Inerpre he following in pracical erms,
More information1. Match each position vs. time graph with the corresponding velocity vs. time graph. Position vs. Time Graphs
Honors Physics Velociy, Acceleraion, s Workshee (Sec. 3.1 o 3.5) 1. Mach each posiion s. ime graph wih he corresponding elociy s. ime graph. Posiion s. Time s 1 2 3 4 5 Velociy s. Time s A B C D E (Answers:
More informationChapter 1 Limits, Derivatives, Integrals, and Integrals
Chaper Limis, Derivaives, Inegrals, and Inegrals Problem Se . a. 9 cm b. From o.: average rae 6. 4 cm/s From o.0: average rae 7. cm/s From o.00: average rae 7. 0 cm/s So he insananeous rae of change of
More informationChapter 2 OneDimensional Kinematics Description of motion in one dimension
Chaper 2 OneDimensional Kinemaics Descripion of moion in one dimension Unis of Chaper 2 Posiion, Disance, and Displacemen Average Speed and Velociy Insananeous Velociy Acceleraion Moion wih Consan Acceleraion
More informationREADING POSITION VERSUS TIME GRAPHS
Quesion 1. READING POSITION VERSUS TIME GRAPHS A person is iniially a poin C on he ais and says here for a lile while, hen srolls along he ais o poin A, says here for a momen and hen runs o poin B and
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationVelocity and Acceleration 4.6
Velociy and Acceleraion 4.6 To undersand he world around us, we mus undersand as much abou moion as possible. The sudy of moion is fundamenal o he principles of physics, and i is applied o a wide range
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationChapter 4 Logarithmic Functions
4.1 Logarihms and Their Properies Chaper 4 Logarihmic Funcions Wha is a Logarihm? We define he common logarihm funcion, or simply he log funcion, wrien log 10 x or log x, as follows: If x is a posiive
More information+ > for all x> 1, then f ( x) < 0 for the. (1 x) 0. = f between x= 0 and x= 0.5.
Mah 3 CHAPTER 4 EXAM (PRACTICE PROBLEMS  SOLUTIONS) Fall 0 Problem Le f ( ) = + ln( + ) a Show ha he graph of f is always concave down b Find he linear approimaion for f ( ) = + ln( + ) near = 0 c Use
More informationTHE CATCH PROCESS. Deaths, both sources. M only F only Both sources. = N N_SMF 0 t. N_SM t. = N_SMF t. = N_SF t
THE CATCH PROCESS Usually we canno harves all he fish from a populaion all a he same ime. Insead, we cach fish over some period of ime and gradually diminish he size of he populaion. Now we will explore
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationChapter 2  Motion along a straight line
MECHANICS Kinemaics Chaper  Moion along a sraigh line I. Posiion and displacemen II. Velociy III. Acceleraion IV. Moion in one dimension wih consan acceleraion V. Free fall Paricle: poinlike objec ha
More informationLab 1: One Dimensional Kinematics
Lab 1: One Dimensional Kinemaics Lab Secion (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Name: Parners: PreLab You are required o finish his secion before coming o he lab, which will be checked
More informationExponential functions
Robero s Noes on Prerequisies for Calculus Chaper 3: Transcendenal funcions Secion 1 Eponenial funcions Wha ou need o know alread: Meaning, graph and basic properies of funcions. Power funcions. Wha ou
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationA : + B v 2 = v a c (s  s 0 )
91962_01_s12p00010176 6/8/09 8:05 M Page 1 12 1. car sars from res and wih consan acceleraion achieves a velociy of 15 m>s when i ravels a disance of 200 m. Deermine he acceleraion of he car and he ime
More informationOneDimensional Kinematics
OneDimensional Kinemaics Michael Fowler Physics 14E Lec Jan 19, 009 Reference Frame Mechanics sars wih kinemaics, which is jus a quaniaive descripion of moion. Then i goes on o dynamics, which aemps o
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More information1.Blowdown of a Pressurized Tank
.Blowdown of a ressurized Tank This experimen consiss of saring wih a ank of known iniial condiions (i.e. pressure, emperaure ec.) and exiing he gas hrough a choked nozzle. The objecive is o es he heory
More informationx is a square root function, not an exponential function. h(x) = 2 x h(5) = 2 5 h(5) = 32 h( 5) = 2 5 h(0) = 2 0
Chaper 7 Eponenial Funcions Secion 7. Characerisics of Eponenial Funcions Secion 7. Page 4 Quesion a) The funcion y = is a polynomial funcion, no an eponenial funcion. b) The funcion y = 6 is an eponenial
More informationPhysics 2001 Problem Set 2 Solutions
Physics 2001 Problem Se 2 Soluions Jeff Kissel Sepember 12, 2006 1. An objec moves from one poin in space o anoher. Afer i arrives a is desinaion, is displacemen is (a) greaer han or equal o he oal disance
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationImportant Instructions to the Examiners:
(ISO/IEC  75 Cerified) Summer Eaminaion Subjec & Code: Applied Mahs (7) Model Answer Page No: /7 Imporan Insrucions o he Eaminers: ) The Answers should be eamined by key words and no as wordoword as
More information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More information1. The reaction rate is defined as the change in concentration of a reactant or product per unit time. Consider the general reaction:
CHAPTER TWELVE CHEMICAL KINETICS For Review. The reacion rae is defined as he change in concenraion of a reacan or produc per uni ime. Consider he general reacion: A Producs where rae If we graph vs.,
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms)
More informationMultiple Choice  TEST II
Muliple Choice Tes IIClassical Mechanics Muliple Choice  TEST II The following informaion perains o Problems 1 hrough 5: Projeciles A, B, C, and D are fired a he same ime from a heigh h meers above
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More information1.6 Constant Acceleration: A Special Case
20 CHAPTER 1. MOTION ALONG A STRAIGHT LINE 1.6 Consan Acceleraion: A Special Case Velociy describes changing posiion and acceleraion describes changing velociy. A quaniy called jerk describes changing
More informationand Decay Functions f (t) = C(1± r) t / K, for t 0, where
MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae
More informationMath 6SL Vectors and Lines Practice Test Questions Name
Mah 6SL Vecors and Lines Pracice Tes Quesions Name. ABCD is a recangle and O is he midpoin of [AB]. D C A O B Epress each of he following vecors in erms of OC and OD (a) (b) CD OA AD (Toal 4 marks) 2.
More informationMethod of least squares J. M. Powers University of Notre Dame February 28, 2003
Mehod of leas squares J. M. Powers Universiy of Nore Dame February 28, 200 One imporan applicaion of daa analysis is he mehod of leas squares. This mehod is ofen used o fi daa o a given funcional form.
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationWhen v = e j is a standard basis vector, we write F
1. Differeniabiliy Recall he definiion of derivaive from one variable calculus Definiion 1.1. We say ha f : R R is differeniable a a poin a R if he quaniy f f(a + h f(a (a := h 0 h exiss. We hen call f
More informationWeek #13  Integration by Parts & Numerical Integration Section 7.2
Week #3  Inegraion by Pars & Numerical Inegraion Secion 7. From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 5 by John Wiley & Sons, Inc. This maerial is used by permission
More informationDaniel López Gaxiola 1 Student View Jason M. Keith
Supplemenal Maerial for Transpor Process and Separaion Process Principles Chaper 1 Liquid Liquid and Fluid Solid Separaion Processes This chaper includes examples of adsorpion processes where one or more
More informationUnit III: Acceleration & Kinematics
Name: Period: Table #: Uni III: Acceleraion & Kinemaics 2006 Bill Amend Quick Review Wha is Jason s disance and displacemen, in meers? If he ran i in 45 seconds, wha are his speed and velociy (in m/s)?
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More information/30/2009. Perhaps the most important of all the applications of calculus is to differential equations. Modeling with Differential Equations
10 DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Perhaps he mos imporan of all he applicaions of calculus is o differenial equaions. DIFFERENTIAL EQUATIONS When physical or social scieniss use calculus,
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More information19Kinematics UNCORRECTED PAGE PROOFS
19Kinemaics 9.1 Kick off wih CAS 9. Inroducion o kinemaics 9.3 Velociy ime graphs and acceleraion ime graphs 9.4 Consan acceleraion formulas 9.5 Insananeous raes of change 9.6 Review 9.1 Kick off wih CAS
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationChapter 29: Maxwell s Equations and Electromagnetic Waves
Chaper 9: Mawell s quaions and lecromagneic Waves Displacemen Curren & Mawell s quaions Displacemen curren Displacemen Curren & Mawell s quaions Displacemen curren con d Displacemen Curren & Mawell s quaions
More informationChapter 8: Production Decline Analysis
Chaper 8: Producion Decline Analysis 8. Inroducion Producion decline analysis is a radiional means of idenifying well producion problems and predicing well performance and life based on real producion
More informationLab 2 Position and Velocity
b Lab 2 Posiion and Velociy Wha You Need To Know: Working Wih Slope In las week s lab you deal wih many graphing ideas. You will coninue o use one of hese ideas in his week s lab, specifically slope. Howeer,
More informationEE Control Systems LECTURE 4
Copyrigh FL Lewis 999 All righs reserved EE 434  Conrol Sysems LECTURE 4 Updaed: Wednesday, February 0, 999 TRANSFER FUNCTION AND ODE SOLUTION TRANSFER FUNCTION, POLES, ZEROS, STEP REPONSE The Laplace
More informationIntroduction to Kinematics (Constant Velocity and Acceleration)
Inroducion o Kinemaics (Consan Velociy and Acceleraion) Inroducion To race moion of an objec, we have o know how i moves wih respec o ime. Namely, i is expeced o record he change of moion in erms of elapsed
More informationINTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER 2009.
INTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER 9. 5.1 Difference Equaions. Much of economics, paricularly macroeconomics
More informationChapter 12. Kinematics of Particle
Chaper 1 Kinemaics of Paricle Engineering Mechanics : Dynamics R.C. Hibbeler Kinemaics of paricle ha moing along a recilinear or sraigh line pah Posiion A paricle raels along a sraighline pah defined
More informationForce and Motion Student Test
Force and Moion Suden Tes 8 h Grade Par I: Muliple Choice Choose one correc answer for each of he following quesions. 1. You are waching he fligh of a small, remoeconrol model airplane as i flies in a
More informationUniform Accelerated Motion
5 h Year Applied Mahs Higher Level Kieran Mills Uniform Acceleraed Moion No par of his publicaion may be copied, reproduced or ransmied in any form or by any means, elecronic, mechanical, phoocopying,
More information1 Coordinates, Symmetry, and Conservation Laws in Classical Mechanics
Benjamin Good February 25, 20 Coordinaes, Symmery, and Conservaion Laws in Classical Mechanics This documen explores he relaionship beween coordinae changes, symmery, and quaniies ha are conserved during
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationVectors and Two Dimensional Kinematics
C.1  Vecor lgebra  I Chaper C Vecors and Two Dimensional Kinemaics linn College  Phsics 2425  Terr Honan Polar Coordinaes (, ) are he Caresian (or recangular) coordinaes of some poin on a plane. r
More informationFirst Name:Key. Physics 101 Fall2006: Test1 Free Response #1  Solution
Las Name: Key Firs Name:Key Physics 101 Fall2006: Tes1 Free Response #1  Soluion 1. (20 ps) A person is running a a maximum velociy of 10 f/s o cach a rain. When he person is a disance d from he neares
More informationKinematics in One Dimension
Kinemaics in One Dimension PHY 3  dkinemaics  J. Hedberg  7. Inroducion. Differen Types of Moion We'll look a:. Dimensionaliy in physics 3. One dimensional kinemaics 4. Paricle model. Displacemen Vecor.
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationFirst Order Partial Differential Equations
Firs Order Parial Differenial Equaions 1. The Mehod of Characerisics A parial differenial equaion of order one in is mos general form is an equaion of he form F x,u, u 0, 1.1 where he unknown is he funcion
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationKinematic Models of Mobile Robots
EE 5325/4315 Kinemaics of Mobile obos, Summer 24 Jose Mireles Jr., Seleced noes from 'obóica: Manipuladores y obos móviles' Aníbal Ollero. Kinemaic Models of Mobile obos Assumpions: a) The robo moves in
More informationMECHANICS. Motion in One Dimension Displacement, Velocity, and Speed Acceleration. Motion with Constant Acceleration Integration
TIPL2_2762hr 11/23/6 11:11 AM P A R T Page 27 I MECHANICS C H A P T E R 2 MOTION IN ONE DIMENSION IS MOTION ALONG A STRAIGHT LINE LIKE THAT OF A CAR ON A STRAIGHT ROAD. THIS DRIVER ENCOUNTERS STOPLIGHTS
More informationSolving Matrix Differential Equations
Solving Marix Differenial Equaions Seps for Solving a Marix Differenial Equaion Find he characerisic equaion ofa, de( A I) = Find he eigenvalues of A, which are he roos of he characerisic equaion 3 For
More informationChapter 17. Traffic Flow Modeling Analogies
Chaper 17 Traffic Flow Modeling Analogies Dr. Tom V. Mahew, IIT Bombay 17.1 February 19, 2014 Conens 17 Traffic Flow Modeling Analogies 1 17.1 Inroducion...................................... 2 17.2 Model
More informationPart I 1. An object goes from one point in space to another. After it arrives at its destination
Paerns Across Space and Time Moion Workshee Soluions Par I 1. An objec goes from one poin in space o anoher. Afer i arrives a is desinaion (a) is displacemen is he same as is disance raveled. (b) is displacemen
More informationExperiment 10 RC and RL circuits: Measuring the time constant.
Experimen 1 C and circuis: Measuring he ime consan. Objec: The objec of his lab is o measure he ime consan of an C circui and a circui. In addiion, one can observe he characerisics of hese wo circuis and
More information