Review - 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
|
|
- Gloria Fox
- 5 years ago
- Views:
Transcription
1 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, ou ma ge addiional soluions in he case of Separable Equaions. Check for division b zero.) II. Firs Order Linear Equaions (FOL). d + p() = q() Mehod of Soluion : = 1 [ µ() ] µ()q() + C, where µ() =e p() III. Homogeneous Equaions (HE). d = f(, ), where f(, ) =f(, ) OR, M(, ) +N(, ) d =, where M(, ) = n M(, ) andn(, ) = n N(, ) Mehod of Soluion : Le Subsiue hese ino d (Someimes ou ma wan o use he subsiuion = u, so ha = u and d = du + u. = f(, ) o obain he Separable Equaion du + u = F (u). = v o simplif calculaions.) IV. Eac Equaion (EE). M(, ) + N(, ) d = where M = N Mehod of Soluion : Implici soluion given b f(, ) = C where : f = M(, ) = f = f f = N(, ) == = ( M(, ) + g() ) M(, ) + g()
2 d () Slope Fields. A soluion = φ() o he d.e. = f(, ) has slope f(, ) ahepoin (, ). The slope field of he d.e. indicaes he slope of soluions a various poins (, ). The slope field is used o give qualiaive informaion abou he behavior of soluions as (or,or, ec). Slope fields ma also be used o approimae he inerval of definiion of a soluion hrough a cerain poin. (3) Auonomous Equaions. Equaions of he form d d = F () ( ) are said o be auonomous since d does no depend on he independen variable. Such equaions ma have soluions ha are consan (i.e., K) hese are called Equilibrium Soluions d and are found b solving F () =. You should be able o find all equilibrium soluions o differenial equaions of he form ( ) and classif he sabili of hese soluions as follows : (a) Asmpoicall Sable - Soluions which sar near = K will alwas approach = K as : = K (b) Unsable - Soluions which sar near = K does no alwas approach = K as : = K (c) Semisable - This is a special pe of unsable soluion. In his case soluions on one side of = K will approach = K as, while soluions on he oher side of = K will no approach = K : = K
3 (4) Eisence and Uniqueness of Soluions. (a) THEOREM (Firs Order LINEAR). If p() andq() are coninuous on an inerval α<<βconaining, hen he IVP + p() = q() { has a unique soluion ( )= = φ() on he inerval α<<β, for an. (The larges such inerval is he inerval of definiion of he soluion.) (b) THEOREM (Firs Order NONLINEAR). If f(, ) and f are coninuous in some recangle { R: a<<b, and c<<dand (, ) lies inside he recangle R, hen he IVP = f(, ) has a unique soluion on he inerval ( )= h<< +h, for some number h>andan : d a c ( ) h +h b (To find he inerval of definiion of hese nonlinear equaions, ou mus acuall solve he IVP.) (5) Numerical Mehods. You should be able o compue b hand he firs few approimaions o he IVP d = f(, ) ( )= using he Euler (Tangen Line) Mehod : n = n 1 + hf( n 1, n 1 ) For he Euler Mehod, ou can ell if he Euler approimaion is smaller or larger han he rue soluion φ() near b looking a he sign of d a : d > a = () concave up a = EULER approimaion <φ() near d < a = () concave down a = EULER approimaion >φ() near
4 Pracice Problems 1. Deermine he order of each of hese differenial equaions; also sae wheher he equaion is linear or nonlinear: (a) + =1 (b) + =1 (c) ( ) 3 + =1 (d) + =1. (a) Which of he funcions 1 () = and () = are soluions of he IVP =, () =? (b) Which of he funcions 1 () = and () = are soluions of he IVP =, (1)=1? 3. Forwhavalue(s)ofr is = e r asoluionof 5 +6 =? 4. (a) Show ha = 3 is a soluion of he iniial value problem =3 /3,()=. (b) Find a differen soluion of he iniial value problem. 5. Find an eplici soluion of he iniial value problem =, (1) = 1. Indicae he inerval in which he soluion is valid. 6. (a) Find an implici soluion of he iniial value problem = +1, ()=. (b) Find an eplici soluion of he iniial value problem =, () = Forwhavalue(s)ofa is he soluion of he IVP +e =, () = a bounded on he inerval? 8. Deermine wheher each of he following differenial equaions is separable, homogeneous, linear, and/or eac or none of hese. (a) + +( +3) d = (c) ( +3 +1) +( + +1)d = (e) ( +1)d +( +1) = (b) +3 +( + )d = (d) +1+( +1) d = 9. Find implici soluions o hese differenial equaions : (a) + = (b) 1+ = (c) + + = 1. Find an implici form of he general soluion of he differenial equaion d = Find an implici soluion of he IVP +1+( +) d =, (1) = Consider he IVP : = +, (3) = 1. (a) Is he soluion increasing or decreasing near =3? (b) Is he soluion concave up or concave down near =3?
5 13. Use he given slope field o skech he soluion of he corresponding iniial value problem = f(, ),( )= for he indicaed iniial value (, ): (a) (, ) (b) (, ) (c) ( 1, 3) (d) (, 4) (e) (, 3) 14. Skech (roughl) some soluions o he auonomous equaions below. Wha are he equilibrium soluions? Classif he sabili of each equilibrium soluion: d (a) d = ( +1) ( ) (b) d d = F (), where F () isasshownbelow: w w = F() For each of he iniial value problems deermine he larges inerval for which a unique soluion is guaraneed : (a) = 1, (1) = (b) + (an ) =sec, ()= (c) + 9 = 1,() = 1 (d) ( +4) = 1,( )=1
6 16. For each of he iniial value problems deermine all iniial poins (, ) for which a unique soluion is guaraneed in some inerval h<< + h: (a) = +, ( )= (b) = /, ( )= (c) = +, ( )= 1 (d) = 1/3 + 1/3,( )= (e) =,( )= 17. Find he eplici soluion of he iniial value problem = 1, () =. Where is his soluion defined? 18. Find he general soluion of hese differenial equaions : (a) =cos (b) e ( +) 1 d = 19. Esimae he soluion a =1.5 o he IVP = 5, (1) = using he Euler Mehod wih h =.5. Wha is he rue soluion a =1.5?. Is he Euler approimaion o he IVP d =, ( 1) = smaller or greaer han he rue soluion near = 1? Answers 1. (a) 1 s order nonlinear (b) 1 s order linear (c) 1 s order nonlinear (d) 3 rd order linear. (a) 1 and (b) 1 onl 3. r =,r=3 4. (a) =3 =3( 3 ) /3 =3 /3 ;=3() /3 (b) 5. = +1, > 1 6. (a) + = (b) = a =1 (8.) (a) HE and EE (b) HE (c) none of hese pes (d) FOL and EE (e) SE and EE 9. (a) (HE) ln 1 ( ) =ln + C and = and = (b) (SE) +1=C (c) (EE) + = C 1. 1 ( ) =ln + C =1 1. (a) (3) = < so is decreasing (b) = + +, so (3) = 3 < andso is concave down 13. (see las page for graph) 14. (a) Equilibrium soluions = (unsable); = (sable); = 1 (semisable) (b) Equilibrium soluions = (unsable); = (semisable); = 4 (sable) (See las page for graphs) 15. (a) >(b) π <<π (c) 3 <<(d) 4 << 16. (a) all (, ) (b) all (, )wih (c)all(, ) (, ) (d) all (, )wih (e) all (, )where 1 < < 1an 17. = 1+3e 1 3e, soluion defined for 1 ln 3 <<. sin 18. (a) = + C (b) = ln ( C 1 e) 19. =.375, rue soluion φ(1.5) = 1 5 (13 58 e.5 ).396. ( 1) > hus EULER approimaion, near = 1 (Graphs on ne page...)
7 13. (a) 14. (b) 4 1
MA 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 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 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 informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
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 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 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 informationIntermediate Differential Equations Review and Basic Ideas
Inermediae Differenial Equaions Review and Basic Ideas John A. Burns Cener for Opimal Design And Conrol Inerdisciplinary Cener forappliedmahemaics Virginia Polyechnic Insiue and Sae Universiy Blacksburg,
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 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 informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
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 informationThe fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation
Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or
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 informationMath 116 Second Midterm March 21, 2016
Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including
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 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 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 information(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)
Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion
More information(π 3)k. f(t) = 1 π 3 sin(t)
Mah 6 Fall 6 Dr. Lil Yen Tes Show all our work Name: Score: /6 No Calculaor permied in his par. Read he quesions carefull. Show all our work and clearl indicae our final answer. Use proper noaion. Problem
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 informationMath 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 informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationProblem set 6: Solutions Math 207A, Fall x 0 2 x
Problem se 6: Soluions Mah 7A, Fall 14 1 Skech phase planes of he following linear ssems: 4 a = ; 9 4 b = ; 9 1 c = ; 1 d = ; 4 e = ; f = 1 3 In each case, classif he equilibrium, =, as a saddle poin,
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 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 informationChapter 4. Truncation Errors
Chaper 4. Truncaion Errors and he Taylor Series Truncaion Errors and he Taylor Series Non-elemenary funcions such as rigonomeric, eponenial, and ohers are epressed in an approimae fashion using Taylor
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
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 information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationOutline of Topics. Analysis of ODE models with MATLAB. What will we learn from this lecture. Aim of analysis: Why such analysis matters?
of Topics wih MATLAB Shan He School for Compuaional Science Universi of Birmingham Module 6-3836: Compuaional Modelling wih MATLAB Wha will we learn from his lecure Aim of analsis: Aim of analsis. Some
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 informationA First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
More informationMidterm Exam Review Questions Free Response Non Calculator
Name: Dae: Block: Miderm Eam Review Quesions Free Response Non Calculaor Direcions: Solve each of he following problems. Choose he BEST answer choice from hose given. A calculaor may no be used. Do no
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationBoyce/DiPrima/Meade 11 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors. A linear first order ODE has the general form
Boce/DiPrima/Meade h ed, Ch.: Linear Equaions; Mehod of Inegraing Facors Elemenar Differenial Equaions and Boundar Value Problems, h ediion, b William E. Boce and Richard C. DiPrima, 7 b John Wile & Sons,
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 informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More informationSecond-Order Differential Equations
WWW Problems and Soluions 3.1 Chaper 3 Second-Order Differenial Equaions Secion 3.1 Springs: Linear and Nonlinear Models www m Problem 3. (NonlinearSprings). A bod of mass m is aached o a wall b means
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
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 informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
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 informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
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 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 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 informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
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 informationProblem Set 7-7. dv V ln V = kt + C. 20. Assume that df/dt still equals = F RF. df dr = =
20. Assume ha df/d sill equals = F + 0.02RF. df dr df/ d F+ 0. 02RF = = 2 dr/ d R 0. 04RF 0. 01R 10 df 11. 2 R= 70 and F = 1 = = 0. 362K dr 31 21. 0 F (70, 30) (70, 1) R 100 Noe ha he slope a (70, 1) is
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationR.#W.#Erickson# Department#of#Electrical,#Computer,#and#Energy#Engineering# University#of#Colorado,#Boulder#
.#W.#Erickson# Deparmen#of#Elecrical,#Compuer,#and#Energy#Engineering# Universiy#of#Colorado,#Boulder# Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance,
More information10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
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 informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
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 informationLinear Dynamic Models
Linear Dnamic Models and Forecasing Reference aricle: Ineracions beween he muliplier analsis and he principle of acceleraion Ouline. The sae space ssem as an approach o working wih ssems of difference
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationMath 106: Review for Final Exam, Part II. (x x 0 ) 2 = !
Mah 6: Review for Final Exam, Par II. Use a second-degree Taylor polynomial o esimae 8. We choose f(x) x and x 7 because 7 is he perfec cube closes o 8. f(x) x / f(7) f (x) x / f (7) x / 7 / 7 f (x) 9
More informationChapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis
Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...
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 informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
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 Hughes-Halle, Gleason, McCallum e. al. Copyrigh 5 by John Wiley & Sons, Inc. This maerial is used by permission
More informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
More informationACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.
ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models
More informationMath 4600: Homework 11 Solutions
Mah 46: Homework Soluions Gregory Handy [.] One of he well-known phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
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 informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationTMA4329 Intro til vitensk. beregn. V2017
Norges eknisk naurvienskapelige universie Insiu for Maemaiske Fag TMA439 Inro il viensk. beregn. V7 ving 6 [S]=T. Sauer, Numerical Analsis, Second Inernaional Ediion, Pearson, 4 Teorioppgaver Oppgave 6..3,
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 informationSections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationKEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationAge (x) nx lx. Age (x) nx lx dx qx
Life Tables Dynamic (horizonal) cohor= cohor followed hrough ime unil all members have died Saic (verical or curren) = one census period (day, season, ec.); only equivalen o dynamic if populaion does no
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationTHE SINE INTEGRAL. x dt t
THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his-
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
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 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 informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationFishing limits and the Logistic Equation. 1
Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More information( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
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 informationUCLA: Math 3B Problem set 3 (solutions) Fall, 2018
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 This problem se concenraes on pracice wih aniderivaives. You will ge los of pracice finding simple aniderivaives as well as finding aniderivaives graphically
More informationChapter 2. Motion in One-Dimension I
Chaper 2. Moion in One-Dimension I Level : AP Physics Insrucor : Kim 1. Average Rae of Change and Insananeous Velociy To find he average velociy(v ) of a paricle, we need o find he paricle s displacemen
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationCheck in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
More informationMath 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas
Mah 527 Lecure 6: Hamilon-Jacobi Equaion: Explici Formulas Sep. 23, 2 Mehod of characerisics. We r o appl he mehod of characerisics o he Hamilon-Jacobi equaion: u +Hx, Du = in R n, u = g on R n =. 2 To
More informationNew Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations
9 J. Mah. Fund. Sci., Vol. 8, No.,, 9-5 New Seven-Sep Numerical Mehod for Direc Soluion of Fourh Order Ordinary Differenial Equaions Zurni Omar & John Olusola Kuboye Deparmen of Mahemaics, School of Quaniaive
More information