Math 2214 Solution Test 1 B Spring 2016
|
|
- Randolf Briggs
- 5 years ago
- Views:
Transcription
1 Mah 14 Soluion Te 1 B Spring 016 Problem 1: Ue eparaion of ariable o ole he Iniial alue DE Soluion (14p) e =, (0) = 0 d = e e d e d = o = ln e d uing u-du b leing u = e 1 e = + where C = for he iniial alue. e + 1 e =, (0) = 0 Problem : Gien ha = + 3 i a unique oluion for he iniial alue DE 3 e + = =. Find p(), g() and. 0 p ( ) g ( ) and (0) 0 Soluion (10p) conider he oluion : = e + 3 fir we hae = e = 1 3 = 3 3(0) 0 3 Now diide b e o ge ha e = 1+ e 3 e now conider he deriaie of boh ide e = + e + e e = e + = [ ] [3 1 9] e [3 8] From he proce of oling he fir order linear DE we know ha he 3 3d inegraing facor i u = e o p() = 3 and g() = 3 8 = e 1
2 Problem 3: Gien he iniial alue problem ealuae (-) when ep alue i h = 1. = 0, ( 4) =, ue Euler Mehod o Soluion (10p) = 0, ( 4) = uing he poin (-4, ) and he lope m = 1 = he equaion of he angen line i = (+) and ( 3) = 1 Nex uing he new poin ( 3,1) and lope m = 3 = he equaion of he angen line i = 3 8 and ( ) = ec Problem 4: Wha i he large -ineral for which ( 4) = ha a guaraneed + unique oluion for iniial alue (-1) = 3 according o he Exience Uniquene Theorem? Soluion (8p) ec ( 4) = + ec = ( 4)( + ) ( + ) 4, 0,, ( nπ ) and (-1) = 3 The large ineral for hi iniial alue i (- π, 0)
3 Problem 5: An objec weighing 19.6 Newon i dropped downward wih an iniial eloci of 0mec. The magniude of he force on he objec due o air reiance i R = S0. ( S i peed.) a) I reiance a poiie or negaie force in hi cenario? b) I eloci a poiie or negaie force in hi cenario? b) Se up he differenial equaion ha model hi cenario in erm of peed. (Do no ole) c) Se up he differenial equaion again in erm of eloci and gie iniial alue. (Do no ole) Soluion (1p) a) Reiance i poiie b) Veloci i negaie c m mg ) = + 0 = = = = 0 = d) 19.6 ince = ince eloci i negaie Problem 6: Conider he iniial alue non-linear DE = wih (0) = 0 ( 1) I a unique oluion guaraneed for hi iniial alue? Juif our concluion. = wih (0) = 0 ( 1) In order o deermine he open region ha would guaranee a unique oluion for hi iniial alue we will deermine where he following i coninuou. 1) ) f(, ) = = where 1, 1, 0 ( 1) f 1 = where 1, 1, 0, > 0 ( 1) Howeer he parial deriaie i no defined for he iniial alue o he oluion i no guaraneed. 3
4 Problem 7: Below i an example of he logiic equaion which decribe growh wih a naural populaion ceiling: Deermine he differenial equaion below ha i repreened b hi direcion field. Explain our reaoning for our choice. Explain wh ou rejeced he oher poibiliie. a) ( 15) = b) 15 ( ) = c) neiher a) or b) The equilibrium oluion occur a = 0 and = 15 for boh a) and b). Now e oher iocline: For a) le = 0 o ge lope = 0(0 15) = 100 which i poiie bu he acual lope i negaie. Now e b) a = 0 o ge = -100 which i he correc ign. Now we need o check b) a oher iocline o be ure ha i reall i he correc DE. Te = 10 o ge = 50 Which i poiie and mache. Now e = -10 o ge = -50 and ha he correc ign. Therefore b) i he correc. 4
5 Problem 8: Sole he following fir order linear DE uing he inegraing facor. + = + 1, (1) = 1 Soluion (14 p) + = + 1, (1) = and inegraing facor i u() = e e e + 1 o [ ] d = d = + + C = + + C uing iniial alue C = and = + ( ) d ln( ) ln( ) + = = = = Problem 9: A ank ha a capaci for 800 gallon and conain 300 gallon of waer wih 7 lb of al iniiall. A oluion conaining of 6 lbgal of al i pumped ino he ank a 10 galmin. A well irred mixure flow ou of he ank a he ame rae. a) Se up he differenial equaion ha model hi iuaion. b) Sole b uing he inegraing facor o find he equaion for he amoun of al in he ank a an ime. (how all work in deail) c) Find and explain he limiing alue of our oluion. d) Suppoe he rae a which he oluion flow ou change o 8 galmin, Se up he differenial equaion ha model hi new iuaion. (Do no ole hi DE.) e) Deermine how long i will ake o fill he ank uing he iuaion in par d). 5
Math 2214 Solution Test 1A Spring 2016
Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion
More informationMath 2214 Solution Test 1B Fall 2017
Mah 14 Soluion Tes 1B Fall 017 Problem 1: A ank has a capaci for 500 gallons and conains 0 gallons of waer wih lbs of sal iniiall. A soluion conaining of 8 lbsgal of sal is pumped ino he ank a 10 galsmin.
More informationMA Study Guide #1
MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationAPPM 2360 Homework Solutions, Due June 10
2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing
More informationExam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp
Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationMA 366 Review - Test # 1
MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order
More informationChapter 12: Velocity, acceleration, and forces
To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationMath 1b. Calculus, Series, and Differential Equations. Final Exam Solutions
Mah b. Calculus, Series, and Differenial Equaions. Final Exam Soluions Spring 6. (9 poins) Evaluae he following inegrals. 5x + 7 (a) (x + )(x + ) dx. (b) (c) x arcan x dx x(ln x) dx Soluion. (a) Using
More informationPhys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole
Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationMath 36. Rumbos Spring Solutions to Assignment #6. 1. Suppose the growth of a population is governed by the differential equation.
Mah 36. Rumbos Spring 1 1 Soluions o Assignmen #6 1. Suppose he growh of a populaion is governed by he differenial equaion where k is a posiive consan. d d = k (a Explain why his model predics ha he populaion
More information1 st order ODE Initial Condition
Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 1 1 s order ODE Iniial Condiion f, sandard form LINEAR NON-LINEAR,, p g differenial form M x dx N x d differenial form is equivalen o a pair of differenial
More informationRectilinear Kinematics
Recilinear Kinemaic Coninuou Moion Sir Iaac Newon Leonard Euler Oeriew Kinemaic Coninuou Moion Erraic Moion Michael Schumacher. 7-ime Formula 1 World Champion Kinemaic The objecie of kinemaic i o characerize
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationPHYSICS 151 Notes for Online Lecture #4
PHYSICS 5 Noe for Online Lecure #4 Acceleraion The ga pedal in a car i alo called an acceleraor becaue preing i allow you o change your elociy. Acceleraion i how fa he elociy change. So if you ar fro re
More informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More informationAQA Maths M2. Topic Questions from Papers. Differential Equations. Answers
AQA Mahs M Topic Quesions from Papers Differenial Equaions Answers PhysicsAndMahsTuor.com Q Soluion Marks Toal Commens M 600 0 = A Applying Newonís second law wih 0 and. Correc equaion = 0 dm Separaing
More informationTP B.2 Rolling resistance, spin resistance, and "ball turn"
echnical proof TP B. olling reiance, pin reiance, and "ball urn" upporing: The Illuraed Principle of Pool and Billiard hp://billiard.coloae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") echnical proof originally
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informations in boxe wers ans Put
Pu answers in boxes Main Ideas in Class Toda Inroducion o Falling Appl Old Equaions Graphing Free Fall Sole Free Fall Problems Pracice:.45,.47,.53,.59,.61,.63,.69, Muliple Choice.1 Freel Falling Objecs
More informationMath 2214 Sol Test 2B Spring 2015
Mah 14 Sol Tes B Sring 015 roblem 1: An objec weighing ounds sreches a verical sring 8 fee beond i naural lengh before coming o res a equilibrium The objec is ushed u 6 fee from i s equilibrium osiion
More informationMain Ideas in Class Today
Main Ideas in Class Toda Inroducion o Falling Appl Consan a Equaions Graphing Free Fall Sole Free Fall Problems Pracice:.45,.47,.53,.59,.61,.63,.69, Muliple Choice.1 Freel Falling Objecs Refers o objecs
More informationMath 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm
Mah Spring 6 Differenial Equaions Final Exam Due Dae: Tuesday, June 6, 5pm Your name (please prin): Insrucions: This is an open book, open noes exam. You are free o use a calculaor or compuer o check your
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationENV 6015 Solution to Mixing Problem Set
EN 65 Soluion o ixing Problem Se. A slug of dye ( ) is injeced ino a single ank wih coninuous mixing. The flow in and ou of he ank is.5 gpm. The ank volume is 5 gallons. When will he dye concenraion equal
More informationy = (y 1)*(y 3) t
MATH 66 SPR REVIEW DEFINITION OF SOLUTION A funcion = () is a soluion of he differenial equaion d=d = f(; ) on he inerval ff < < fi if (d=d)() =f(; ()) for each so ha ff
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationReview - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y
Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More informationEquations of motion for constant acceleration
Lecure 3 Chaper 2 Physics I 01.29.2014 Equaions of moion for consan acceleraion Course websie: hp://faculy.uml.edu/andriy_danylo/teaching/physicsi Lecure Capure: hp://echo360.uml.edu/danylo2013/physics1spring.hml
More informationCourse II. Lesson 7 Applications to Physics. 7A Velocity and Acceleration of a Particle
Course II Lesson 7 Applicaions o Physics 7A Velociy and Acceleraion of a Paricle Moion in a Sraigh Line : Velociy O Aerage elociy Moion in he -ais + Δ + Δ 0 0 Δ Δ Insananeous elociy d d Δ Δ Δ 0 lim [ m/s
More informationINSTANTANEOUS VELOCITY
INSTANTANEOUS VELOCITY I claim ha ha if acceleraion is consan, hen he elociy is a linear funcion of ime and he posiion a quadraic funcion of ime. We wan o inesigae hose claims, and a he same ime, work
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationNote: For all questions, answer (E) NOTA means none of the above answers is correct.
Thea Logarihms & Eponens 0 ΜΑΘ Naional Convenion Noe: For all quesions, answer means none of he above answers is correc.. The elemen C 4 has a half life of 70 ears. There is grams of C 4 in a paricular
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationQ2.4 Average velocity equals instantaneous velocity when the speed is constant and motion is in a straight line.
CHAPTER MOTION ALONG A STRAIGHT LINE Discussion Quesions Q. The speedomeer measures he magniude of he insananeous eloci, he speed. I does no measure eloci because i does no measure direcion. Q. Graph (d).
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationElementary Differential Equations and Boundary Value Problems
Elemenar Differenial Equaions and Boundar Value Problems Boce. & DiPrima 9 h Ediion Chaper 1: Inroducion 1006003 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationPhysics for Scientists and Engineers. Chapter 2 Kinematics in One Dimension
Physics for Scieniss and Engineers Chaper Kinemaics in One Dimension Spring, 8 Ho Jung Paik Kinemaics Describes moion while ignoring he agens (forces) ha caused he moion For now, will consider moion in
More informationTheory of! Partial Differential Equations-I!
hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationDifferential Geometry: Numerical Integration and Surface Flow
Differenial Geomery: Numerical Inegraion and Surface Flow [Implici Fairing of Irregular Meshes using Diffusion and Curaure Flow. Desbrun e al., 1999] Energy Minimizaion Recall: We hae been considering
More informationAP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr
AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More information4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS
Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS Growing a a Consan Percen Rae Example 2 During he 2000 s, he populaion
More informationACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1
ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised 2019 vie.dang@humbleisd.ne 12/26/2018 7:27 PM 7.5A: Accumulaion 1 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationMath 115 Final Exam December 14, 2017
On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):
More informationOne-Dimensional Kinematics
One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion
More informationEF 151 Exam #2 - Spring, 2014 Page 1 of 6
EF 5 Exam # - Spring, 04 Page of 6 Name: Secion: Inrucion: Pu your name and ecion on he exam. Do no open he e unil you are old o do o. Wrie your final anwer in he box proided If you finih wih le han 5
More informationNuclear Decay kinetics : Transient and Secular Equilibrium. What can we say about the plot to the right?
uclear Decay kineics : Transien and Secular Equilibrium Wha can we say abou he plo o he righ? IV. Paren-Daugher Relaionships Key poin: The Rae-Deermining Sep. Case of Radioacive Daugher (Paren) 1/2 ()
More informationCHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
More informationComparison between the Discrete and Continuous Time Models
Comparison beween e Discree and Coninuous Time Models D. Sulsky June 21, 2012 1 Discree o Coninuous Recall e discree ime model Î = AIS Ŝ = S Î. Tese equaions ell us ow e populaion canges from one day o
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCING GUIDELINES (Form B) Quesion 4 A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime
More informationKinematics in two dimensions
Lecure 5 Phsics I 9.18.13 Kinemaics in wo dimensions Course websie: hp://facul.uml.edu/andri_danlo/teaching/phsicsi Lecure Capure: hp://echo36.uml.edu/danlo13/phsics1fall.hml 95.141, Fall 13, Lecure 5
More informationChapter 7: Inverse-Response Systems
Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More information6. Solve by applying the quadratic formula.
Dae: Chaper 7 Prerequisie Skills BLM 7.. Apply he Eponen Laws. Simplify. Idenify he eponen law ha you used. a) ( c) ( c) ( c) ( y)( y ) c) ( m)( n ). Simplify. Idenify he eponen law ha you used. 8 w a)
More informationPhysics 101: Lecture 03 Kinematics Today s lecture will cover Textbook Sections (and some Ch. 4)
Physics 101: Lecure 03 Kinemaics Today s lecure will coer Texbook Secions 3.1-3.3 (and some Ch. 4) Physics 101: Lecure 3, Pg 1 A Refresher: Deermine he force exered by he hand o suspend he 45 kg mass as
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationln 2 1 ln y x c y C x
Lecure 14 Appendi B: Some sample problems from Boas Here are some soluions o he sample problems assigned for Chaper 8 8: 6 Soluion: We wan o find he soluion o he following firs order equaion using separaion
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More informationEE202 Circuit Theory II
EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationAmit Mehra. Indian School of Business, Hyderabad, INDIA Vijay Mookerjee
RESEARCH ARTICLE HUMAN CAPITAL DEVELOPMENT FOR PROGRAMMERS USING OPEN SOURCE SOFTWARE Ami Mehra Indian Shool of Business, Hyderabad, INDIA {Ami_Mehra@isb.edu} Vijay Mookerjee Shool of Managemen, Uniersiy
More informationNEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationPHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections
PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx
More informationUNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Name: Par I Quesions UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS Dae: 1. The epression 1 is equivalen o 1 () () 6. The eponenial funcion y 16 could e rewrien as y () y 4 () y y. The epression
More informationSuggested Practice Problems (set #2) for the Physics Placement Test
Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are
More informationSolution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
More informationAlgorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov)
Algorihm and Daa Srucure 2011/ Week Soluion (Tue 15h - Fri 18h No) 1. Queion: e are gien 11/16 / 15/20 8/13 0/ 1/ / 11/1 / / To queion: (a) Find a pair of ube X, Y V uch ha f(x, Y) = f(v X, Y). (b) Find
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)
More informationExponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1
Eponenial and Logarihmic Funcions -- ANSWERS -- Logarihms racice Diploma ANSWERS www.puremah.com Logarihms Diploma Syle racice Eam Answers. C. D 9. A 7. C. A. C. B 8. D. D. C NR. 8 9. C 4. C NR. NR 6.
More informationMath Wednesday March 3, , 4.3: First order systems of Differential Equations Why you should expect existence and uniqueness for the IVP
Mah 2280 Wednesda March 3, 200 4., 4.3: Firs order ssems of Differenial Equaions Wh ou should epec eisence and uniqueness for he IVP Eample: Consider he iniial value problem relaed o page 4 of his eserda
More informationF.LE.A.4: Exponential Growth
Regens Exam Quesions F.LE.A.4: Exponenial Growh www.jmap.org Name: F.LE.A.4: Exponenial Growh 1 A populaion of rabbis doubles every days according o he formula P = 10(2), where P is he populaion of rabbis
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A ank conains 15 gallons of heaing oil a ime =. During he ime inerval 1 hours, heaing oil is pumped ino he ank a he rae 1 H ( ) = + ( 1 + ln( + 1) ) gallons per hour.
More informationSection 3.8, Mechanical and Electrical Vibrations
Secion 3.8, Mechanical and Elecrical Vibraions Mechanical Unis in he U.S. Cusomary and Meric Sysems Disance Mass Time Force g (Earh) Uni U.S. Cusomary MKS Sysem CGS Sysem fee f slugs seconds sec pounds
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More information3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate
1 5.1 and 5. Eponenial Funcions Form I: Y Pa, a 1, a > 0 P is he y-inercep. (0, P) When a > 1: a = growh facor = 1 + growh rae The equaion can be wrien as The larger a is, he seeper he graph is. Y P( 1
More informationRadioactive Decay BSEN-625 ADVANCES IN FOOD ENGINEERING
Radioacive Decay BSE-65 DVCES I FOOD EGIEERIG civiy The rae of decay of a radionuclide I is he number of aoms ha decay per uni ime Unis Bacquarel (Bq): one desinegaion/second Bq s - Curie-(Ci): aciviy
More informationDiscussion Session 2 Constant Acceleration/Relative Motion Week 03
PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll
More information