AP Calculus BC 2010 Scoring Guidelines

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "AP Calculus BC 2010 Scoring Guidelines"

Transcription

1 AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi 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 piecewise-defined 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 one-day 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 x-coordinae 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 y-coordinae 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 : y-coordinae 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 y-axis, 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 y-axis 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 fifh-degree 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 hird-degree 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 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 2007 Scoring Guidelines AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

A Curriculum Module for AP Calculus BC Curriculum Module

A 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 no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

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

Week #9 - The Integral Section 5.1

Week #9 - The Integral Section 5.1 Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. 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

Chapter 2 Kinematics in One Dimension

Chapter 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 information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics 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 information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order 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 information

Section 7.1 Angles and Their Measure

Section 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 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

Chapter 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

Chapter 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 information

Relative velocity in one dimension

Relative 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 information

Fourier Series Solution of the Heat Equation

Fourier 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 information

The Transport Equation

The 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 information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 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 information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing 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 information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 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 information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

Chapter 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.

Chapter 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 information

4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F

4kq 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 information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: 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 information

Acceleration Lab Teacher s Guide

Acceleration 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

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking 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 information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. 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 information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 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 information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and 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 information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. 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 information

Random Walk in 1-D. 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 1-D. 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 information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE 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 information

Section 5.1 The Unit Circle

Section 5.1 The Unit Circle Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P

More information

CHAPTER FIVE. Solutions for Section 5.1

CHAPTER FIVE. Solutions for Section 5.1 CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5

More information

MOTION ALONG A STRAIGHT LINE

MOTION ALONG A STRAIGHT LINE Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Motion Along a Straight Line

Motion Along a Straight Line Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

More information

MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM MA6-A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A half-lane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t, Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

( ) in the following way. ( ) < 2

( ) in the following way. ( ) < 2 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

Section A: Forces and Motion

Section A: Forces and Motion I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and

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

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009.

Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009. Name: Teacher: DO NOT OPEN THE EXMINTION PPER UNTIL YOU RE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINL EXMINTION June 2009 Value: 100% General Insrucions This examinaion consiss of wo pars. Boh pars

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

AP1 Kinematics (A) (C) (B) (D) Answer: C

AP1 Kinematics (A) (C) (B) (D) Answer: C 1. A ball is hrown verically upward from he ground. Which pair of graphs bes describes he moion of he ball as a funcion of ime while i is in he air? Neglec air resisance. y a v a (A) (C) y a v a (B) (D)

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

2. Waves in Elastic Media, Mechanical Waves

2. Waves in Elastic Media, Mechanical Waves 2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in

More information

Basic Assumption: population dynamics of a group controlled by two functions of time

Basic Assumption: population dynamics of a group controlled by two functions of time opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of

More information

Discussion Examples Chapter 10: Rotational Kinematics and Energy

Discussion Examples Chapter 10: Rotational Kinematics and Energy Discussion Examples Chaper : Roaional Kinemaics and Energy 9. The Crab Nebula One o he mos sudied objecs in he nigh sky is he Crab nebula, he remains o a supernova explosion observed by he Chinese in 54.

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

More information

Lectures # 5 and 6: The Prime Number Theorem.

Lectures # 5 and 6: The Prime Number Theorem. Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

More information

Physics 111 Fall 2007 Electric Currents and DC Circuits

Physics 111 Fall 2007 Electric Currents and DC Circuits Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

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

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

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

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

The Torsion of Thin, Open Sections

The Torsion of Thin, Open Sections EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Rotational Inertia of a Point Mass

Rotational Inertia of a Point Mass Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Capacitors and inductors

Capacitors and inductors Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear

More information

Fourier Series Approximation of a Square Wave

Fourier Series Approximation of a Square Wave OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

More information

Renewal processes and Poisson process

Renewal processes and Poisson process CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

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

Chapter 2 Motion in One Dimension

Chapter 2 Motion in One Dimension Chaper Moion in One Dimension Concepual Problems 5 Sand in he cener of a large room. Call he direcion o your righ posiie, and he direcion o your lef negaie. Walk across he room along a sraigh line, using

More information

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

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

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

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Chapter 2: Principles of steady-state converter analysis

Chapter 2: Principles of steady-state converter analysis Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

More information

THE PRESSURE DERIVATIVE

THE PRESSURE DERIVATIVE Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

More information

What is a differential equation? y = f (t).

What is a differential equation? y = f (t). Wha is a differenial equaion? A differenial equaion is any equaion conaining one or more derivaives. The simples differenial equaion, herefore, is jus a usual inegraion problem y f (). Commen: The soluion

More information

Understanding Sequential Circuit Timing

Understanding Sequential Circuit Timing ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

More information

4.2 Trigonometric Functions; The Unit Circle

4.2 Trigonometric Functions; The Unit Circle 4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

More information

Kinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.

Kinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard. Chaper 2 Kinemaics in 1-D From Problems and Soluions in Inroducory Mechanics (Draf ersion, Augus 2014) Daid Morin, morin@physics.harard.edu As menioned in he preface, his book should no be hough of as

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

Using RCtime to Measure Resistance

Using RCtime to Measure Resistance Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

More information

Lecture III: Finish Discounted Value Formulation

Lecture III: Finish Discounted Value Formulation Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Suggested Reading. Signals and Systems 4-2

Suggested Reading. Signals and Systems 4-2 4 Convoluion In Lecure 3 we inroduced and defined a variey of sysem properies o which we will make frequen reference hroughou he course. Of paricular imporance are he properies of lineariy and ime invariance,

More information

Interference, Diffraction and Polarization

Interference, Diffraction and Polarization L.1 - Simple nerference Chaper L nerference, Diffracion and Polarizaion A sinusoidal wave raveling in one dimension has he form: Blinn College - Physics 2426 - Terry Honan A coshk x w L where in he case

More information

LAB 6: SIMPLE HARMONIC MOTION

LAB 6: SIMPLE HARMONIC MOTION 1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

More information

Chapter 12 PURE TORSION

Chapter 12 PURE TORSION Chaper 1 PURE TORSION 1.1 GENERALS A member is subjeced o pure orsion if in any cross secion of his member he single sress differen from zero is he momen of orsion or wising (shorer TORQUE). Pure orsion

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z

1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z o ffix uden abel ere uden ame chool ame isric ame/ ender emale ale onh ay ear ae of irh an eb ar pr ay un ul ug ep c ov ec as ame irs ame lace he uden abel ere ae uden denifier chool se nly rined in he

More information