Age (x) nx lx. Age (x) nx lx dx qx

Size: px
Start display at page:

Download "Age (x) nx lx. Age (x) nx lx dx qx"

Transcription

1 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 change age disribuion; assumes consan survivorship. Life Hisory Tables Time () = ime inerval used for separaing age caegories n = number alive a he beginning a age L = proporion of individuals alive a age Freq Age () n l Life Hisory Tables d = proporion of original populaion dying during he age inerval o +1 q = proporion of eising populaion dying during age inerval o +1; q = d /l Life epecancy E = T / l T = average life epecancy from curren ime: e.g. how much living will be done by cohor from beginning of period : T =Σ(L ); summed from o las Age () n l d q Age () n l d q L T e

2 aaliy f = oal naaliy; number of ferilized eggs produced in a given year by individuals surviving o age m = age specific naaliy; average number of ferilized eggs produced per individual surviving o beginning of age Loka proved ha any pair of unchanging l and m values will evenually give rise o a populaion wih a sable age disribuion Reproducive Rae R = rae of change in he populaion. If below 1., populaion is shrinking R = (l m ) Sum of he number of ferilized eggs produced per original individual during each age Age () n l d q L T e m f lm R = 8.5 ime age fecund Survive ,234 1, sum ,24 1,529 2, pc pc pc pc ime age fecund Survive ,234 1, sum ,24 1,529 2, pc pc pc pc R=1.5 ime age fecund Survive ,562 3, 5,785 11,141 21,457 41, ,735 3,342 6, sum ,839 3,535 6,814 13,125 25,279 48, pc pc pc pc R=1.83 2

3 Reproducive Raes R = ne reproducive rae; e number of offspring produced per individual. Also, a muliplier allowing us o deermine populaion size in he ne generaion. For simpliciy, assume discree generaions. R = (l m ) R rae of populaion increase as a funcion of ime. Uniless and dimensionless. sage a l d q Log1a Log1l k M f lm lm -r e -r lme -r L T e R o l m 8.75 l m e -r =1 lm Defining R R is a muliplier allowing us o deermine populaion size a fuure generaion. If discree generaions, ime in generaions hen =T and + 1 = * R General formula for populaion size a ime is: = * R Predics eponenial growh Year Reproducive Raes r = inrinsic rae of increase; per capia rae of increase; also Malhusian Parameer. When r is > populaions will increase, when i is < populaions will decrease; a populaion ha is no increasing or decreasing will have an r of. Unis number of new individuals per uni ime. sage a l d q Log1a Log1l R o lm 8.75 lm T c r.876 Double.787 number of individuals 1 Eponenial Growh ime r =.1 r =.2 r =.3 3

4 Deermining r If generaion ime (T) is known, can deermine r from Loka s Equaion: r 1 = e l using ieraion m This is difficul, and he equaion is no biologically meaningful r can be esimaed by: r R = ln( ) T c where T c is generaion ime. Cohor Generaion Time T c is he average birh-o-birh ime for a generaion. lm Tc = R Recall ha Thus T c R = = l m l m l m sage a l d q Log1a Log1l R o lm 8.75 lm T c r.876 Double.787 Loka s Soluion o r, solve for Tc firs, hen r T c = (19.25/8.75) = r = ln(2.3839)/3.68 =.876 How good is our esimae of r? True value of r difficul o ge: 1 = e r l m sage a l d q Log 1a Log 1l k M f l m l m -r e -r l m e -r L Ro l m 8.75 l m e -r =1 l m T c r.876 Double.787 sage a l d q Log 1a Log 1l k M f l m l m -r e -r l m e -r L Ro l m 8.75 l m e -r =1 l m T c r.876 Double.787 Value > 1 means esimae of r is oo small Value < 1 means esimae is oo large In his case, acual r slighly greaer han.876 4

5 How do we use his? R is he finie rae of populaion increase. I + 1 = * R represens he proporional change in a However, r is much more useful as a coninuous measure of 1 = R * populaion from o +1. Is always posiive. populaion growh. I is a raio, dimensionless and wih no unis. d d = r = e r Differenial equaion of logisic growh. Can only ell you he rae (d/d) of growh, can projec populaion size. Inegrae he firs equaion, and i is wrien in a form where you can projec populaion size given a ime period (). = R * By eension r R = e r Assuming infiniely small ime seps, swich from he discree o coninuous form. = ln( R) r is hus he naural log of R R = l m R is he ne reproducive rae, correced for moraliy. The unis of R is number of offspring per individual per lifespan. If R >1. populaion is increasing. 1 = e r l m True value of r, difficul o solve. The esimae from R is usually close enough. R rae of populaion increase as a funcion of absolue ime. Uniless, dimensionless. R rae of populaion increase as a funcion of generaion ime. Thus R=R only if T=1 Esimaing r: r R = ln( ) T However, his is only An esimaion of r. 5

6 r More abou r How long does i ake a populaion o double in size?? r = e =2 and o =1 =? = e ln 2=r r ime =.69/r Doubling Time for Populaion inrinsic rae of increase Human Populaion Growh year r doubling ime Given curren growh raes, wha will he world populaion be in 3 years?? = e r =6,426,11,45 e.125(3) 9,349,922,439 Variabiliy in r The maimum rae of increase obainable by a populaion is r ma Difference beween r ma and r is due o environmenal resisance r ma - qualiy of food, space, ec. are opimum, no compeiion or predaion. An eample sage a M Given hese daa answer he following Wha proporion of individuals survive o age 2? (l 2 ) Wha proporion of 2 year olds die before reaching age 3? (q 2 ) Wha is he epeced lifespan? (e ) How many oal offspring are produced by he hird year class? (f 3 ) Is he populaion growing or shrinking? (R ) Wha is he generaion ime? (T) How large will he populaion be in 243 years? (r) How long before he populaion doubles? (r) 6

7 An eample Wha proporion of individuals survive o age 2? (l 2 ) l = a /a = 2/15 =13.3% An eample Wha proporion of 2 year olds die before reaching age 3? (q 2 ) q = a +1 /a = 1/2 =5% sage a l sage a l d q An eample Wha is he epeced lifespan? (e ) L = (a +a +1 )/2 e =Sum(L )/a e =( )/15 = 145/15 =.94 An eample How many oal offspring are produced by he hird year class? (f 3 ) f = m *a f 3 =m 3 *a 3 = 6*1 = 6 sage a l d q L e sage a l d q M f

8 An eample Is he populaion growing or shrinking? (R ) R =Sum(l m ) R = = 1.28 An eample Wha is he generaion ime? (T) T=sum(l m )/sum(l m ) T=2.7/1.27 = 2.12 sage a l d q Log 1 a Log 1 l k M f l m l m sage a l d q Log 1 a Log 1 l k M f l m l m An eample How large will he populaion be in 243 years? (r) r=lnr /T r=ln(1.27)/2.12 =.114 = e r 243 =15 e 114* = 15 e sage a l d q Log 1 a Log 1 l k M f l m l m An eample How long before he populaion doubles? (r) Double ime =.69/r =.69/.114 = 6.5 years sage a l d q Log 1 a Log 1 l k M f l m l m

Math 36. Rumbos Spring Solutions to Assignment #6. 1. Suppose the growth of a population is governed by the differential equation.

Math 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 information

( ) ( ) ( ) + W aa. ( t) n t ( ) ( )

( ) ( ) ( ) + W aa. ( t) n t ( ) ( ) Supplemenary maerial o: Princeon Universiy Press, all righs reserved From: Chaper 3: Deriving Classic Models in Ecology and Evoluionary Biology A Biologis s Guide o Mahemaical Modeling in Ecology and Evoluion

More information

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon 3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of

More information

EXERCISES FOR SECTION 1.5

EXERCISES 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 information

5.1 - Logarithms and Their Properties

5.1 - 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 information

Math 2214 Solution Test 1A Spring 2016

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 information

Comparing Theoretical and Practical Solution of the First Order First Degree Ordinary Differential Equation of Population Model

Comparing Theoretical and Practical Solution of the First Order First Degree Ordinary Differential Equation of Population Model Open Access Journal of Mahemaical and Theoreical Physics Comparing Theoreical and Pracical Soluion of he Firs Order Firs Degree Ordinary Differenial Equaion of Populaion Model Absrac Populaion dynamics

More information

Sterilization D Values

Sterilization D Values Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

Solutions to Assignment 1

Solutions 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 information

4.1 - Logarithms and Their Properties

4.1 - Logarithms and Their Properties Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,

More information

Note: For all questions, answer (E) NOTA means none of the above answers is correct.

Note: 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

) were both constant and we brought them from under the integral.

) were both constant and we brought them from under the integral. YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha

More information

Problem Set 7-7. dv V ln V = kt + C. 20. Assume that df/dt still equals = F RF. df dr = =

Problem 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 information

N t = N o e rt. Plot N t+1 vs. N t. Recruits Four Desirable Properties of S-R Relationships

N t = N o e rt. Plot N t+1 vs. N t. Recruits Four Desirable Properties of S-R Relationships Populaion Models in Fisheries Models for fish populaions are similar as hose used for birds and mammals. Some differen applicaions have been developed specific o fisheries. Sock Recrui Relaionships Influence

More information

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

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 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

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-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 information

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates

Biol. 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 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

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

18 Biological models with discrete time

18 Biological models with discrete time 8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so

More information

Comparison between the Discrete and Continuous Time Models

Comparison 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 information

4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS

4.1 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 information

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions

MA 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 information

Exponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1

Exponential 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 information

CHAPTER 12 DIRECT CURRENT CIRCUITS

CHAPTER 12 DIRECT CURRENT CIRCUITS CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chaper 5 Eponenial and Logarihmic Funcions Chaper 5 Prerequisie Skills Chaper 5 Prerequisie Skills Quesion 1 Page 50 a) b) c) Answers may vary. For eample: The equaion of he inverse is y = log since log

More information

5.2. The Natural Logarithm. Solution

5.2. The Natural Logarithm. Solution 5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,

More information

Chapter 4. Truncation Errors

Chapter 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 information

Numerical Dispersion

Numerical Dispersion eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal

More information

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,

More information

Math 333 Problem Set #2 Solution 14 February 2003

Math 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 information

1 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.

1 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

EEEB113 CIRCUIT ANALYSIS I

EEEB113 CIRCUIT ANALYSIS I 9/14/29 1 EEEB113 CICUIT ANALYSIS I Chaper 7 Firs-Order Circuis Maerials from Fundamenals of Elecric Circuis 4e, Alexander Sadiku, McGraw-Hill Companies, Inc. 2 Firs-Order Circuis -Chaper 7 7.2 The Source-Free

More information

Echocardiography Project and Finite Fourier Series

Echocardiography Project and Finite Fourier Series Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every

More information

MAC1105-College Algebra

MAC1105-College Algebra . Inverse Funcions I. Inroducion MAC5-College Algera Chaper -Inverse, Eponenial & Logarihmic Funcions To refresh our undersnading of "Composiion of a funcion" or "Composie funcion" which refers o he comining

More information

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.

More information

The average rate of change between two points on a function is d t

The average rate of change between two points on a function is d t SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope

More information

Chapter 7 Response of First-order RL and RC Circuits

Chapter 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 information

Phys1112: DC and RC circuits

Phys1112: DC and RC circuits Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.

More information

Direct Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1

Direct Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1 Direc Curren Circuis February 19, 2014 Physics for Scieniss & Engineers 2, Chaper 26 1 Ammeers and Volmeers! A device used o measure curren is called an ammeer! A device used o measure poenial difference

More information

USP. Surplus-Production Models

USP. Surplus-Production Models USP Surplus-Producion Models 2 Overview Purpose of slides: Inroducion o he producion model Overview of differen mehods of fiing Go over some criique of he mehod Source: Haddon 2001, Chaper 10 Hilborn and

More information

20. Applications of the Genetic-Drift Model

20. Applications of the Genetic-Drift Model 0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0

More information

Chapter 8 The Complete Response of RL and RC Circuits

Chapter 8 The Complete Response of RL and RC Circuits Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior

More information

APPM 2360 Homework Solutions, Due June 10

APPM 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 information

Chapter 10 INDUCTANCE Recommended Problems:

Chapter 10 INDUCTANCE Recommended Problems: Chaper 0 NDUCTANCE Recommended Problems: 3,5,7,9,5,6,7,8,9,,,3,6,7,9,3,35,47,48,5,5,69, 7,7. Self nducance Consider he circui shown in he Figure. When he swich is closed, he curren, and so he magneic field,

More information

Challenge 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 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 information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 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 information

R.#W.#Erickson# Department#of#Electrical,#Computer,#and#Energy#Engineering# University#of#Colorado,#Boulder#

R.#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 information

1 Differential Equation Investigations using Customizable

1 Differential Equation Investigations using Customizable Differenial Equaion Invesigaions using Cusomizable Mahles Rober Decker The Universiy of Harford Absrac. The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for

More information

Lab 10: RC, RL, and RLC Circuits

Lab 10: RC, RL, and RLC Circuits Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in

More information

Chapter 7: Solving Trig Equations

Chapter 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 information

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3 Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 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 information

non -negative cone Population dynamics motivates the study of linear models whose coefficient matrices are non-negative or positive.

non -negative cone Population dynamics motivates the study of linear models whose coefficient matrices are non-negative or positive. LECTURE 3 Linear/Nonnegaive Marix Models x ( = Px ( A= m m marix, x= m vecor Linear sysems of difference equaions arise in several difference conexs: Linear approximaions (linearizaion Perurbaion analysis

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

Module 4: Time Response of discrete time systems Lecture Note 2

Module 4: Time Response of discrete time systems Lecture Note 2 Module 4: Time Response of discree ime sysems Lecure Noe 2 1 Prooype second order sysem The sudy of a second order sysem is imporan because many higher order sysem can be approimaed by a second order model

More information

LAPLACE TRANSFORM AND TRANSFER FUNCTION

LAPLACE 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 information

10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e

10.1 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 information

Problem Set on Differential Equations

Problem Set on Differential Equations Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p()

More information

Summary of shear rate kinematics (part 1)

Summary of shear rate kinematics (part 1) InroToMaFuncions.pdf 4 CM465 To proceed o beer-designed consiuive equaions, we need o know more abou maerial behavior, i.e. we need more maerial funcions o predic, and we need measuremens of hese maerial

More information

Chapter 2. First Order Scalar Equations

Chapter 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 information

8. Basic RL and RC Circuits

8. Basic RL and RC Circuits 8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics

More information

INDEX. Transient analysis 1 Initial Conditions 1

INDEX. 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 information

MA Study Guide #1

MA 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 information

Zürich. ETH Master Course: L Autonomous Mobile Robots Localization II

Zürich. ETH Master Course: L Autonomous Mobile Robots Localization II Roland Siegwar Margaria Chli Paul Furgale Marco Huer Marin Rufli Davide Scaramuzza ETH Maser Course: 151-0854-00L Auonomous Mobile Robos Localizaion II ACT and SEE For all do, (predicion updae / ACT),

More information

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data

Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based

More information

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

KEY. 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 information

UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #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 information

Smoothing. Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T

Smoothing. Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T Smoohing Consan process Separae signal & noise Smooh he daa: Backward smooher: A an give, replace he observaion b a combinaion of observaions a & before Simple smooher : replace he curren observaion wih

More information

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance

More information

3.6 Derivatives as Rates of Change

3.6 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 information

Differential Equations

Differential 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 information

KINEMATICS IN ONE DIMENSION

KINEMATICS IN ONE DIMENSION KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec

More information

dy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page

dy 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 information

Chapter 14 Wiener Processes and Itô s Lemma. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 14 Wiener Processes and Itô s Lemma. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull Chaper 14 Wiener Processes and Iô s Lemma Copyrigh John C. Hull 014 1 Sochasic Processes! Describes he way in which a variable such as a sock price, exchange rae or ineres rae changes hrough ime! Incorporaes

More information

Math 2214 Solution Test 1B Fall 2017

Math 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 information

2.4 Cuk converter example

2.4 Cuk converter example 2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28 Analysis sraegy This converer

More information

Spatial Ecology: Lecture 4, Integrodifference equations. II Southern-Summer school on Mathematical Biology

Spatial Ecology: Lecture 4, Integrodifference equations. II Southern-Summer school on Mathematical Biology Spaial Ecology: Lecure 4, Inegrodifference equaions II Souhern-Summer school on Mahemaical Biology Inegrodifference equaions Diffusion models assume growh and dispersal occur a he same ime. When reproducion

More information

ln y t 2 t c where c is an arbitrary real constant

ln y t 2 t c where c is an arbitrary real constant SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies

More information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan

Ground 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 information

AP Calculus BC Chapter 10 Part 1 AP Exam Problems

AP 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 information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. 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 information

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs. Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers

More information

IB Physics Kinematics Worksheet

IB Physics Kinematics Worksheet IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6 Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is:

More information

ADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations

ADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations ADVANCED MATHEMATICS FOR ECONOMICS - /3 Shee 3: Di erenial equaions Check ha x() =± p ln(c( + )), where C is a posiive consan, is soluion of he ODE x () = Solve he following di erenial equaions: (a) x

More information

HOMEWORK # 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 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 information

Homework-8(1) P8.3-1, 3, 8, 10, 17, 21, 24, 28,29 P8.4-1, 2, 5

Homework-8(1) P8.3-1, 3, 8, 10, 17, 21, 24, 28,29 P8.4-1, 2, 5 Homework-8() P8.3-, 3, 8, 0, 7, 2, 24, 28,29 P8.4-, 2, 5 Secion 8.3: The Response of a Firs Order Circui o a Consan Inpu P 8.3- The circui shown in Figure P 8.3- is a seady sae before he swich closes a

More information

EE100 Lab 3 Experiment Guide: RC Circuits

EE100 Lab 3 Experiment Guide: RC Circuits I. Inroducion EE100 Lab 3 Experimen Guide: A. apaciors A capacior is a passive elecronic componen ha sores energy in he form of an elecrosaic field. The uni of capaciance is he farad (coulomb/vol). Pracical

More information

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive

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)

(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

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (

More information

Turbulence in Fluids. Plumes and Thermals. Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College

Turbulence in Fluids. Plumes and Thermals. Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College Turbulence in Fluids Plumes and Thermals enoi Cushman-Roisin Thayer School of Engineering Darmouh College Why do hese srucures behave he way hey do? How much mixing do hey accomplish? 1 Plumes Plumes are

More information

ASTR415: Problem Set #5

ASTR415: Problem Set #5 ASTR45: Problem Se #5 Curran D. Muhlberger Universi of Marland (Daed: April 25, 27) Three ssems of coupled differenial equaions were sudied using inegraors based on Euler s mehod, a fourh-order Runge-Kua

More information

Errata (1 st Edition)

Errata (1 st Edition) P Sandborn, os Analysis of Elecronic Sysems, s Ediion, orld Scienific, Singapore, 03 Erraa ( s Ediion) S K 05D Page 8 Equaion (7) should be, E 05D E Nu e S K he L appearing in he equaion in he book does

More information

Unit Root Time Series. Univariate random walk

Unit Root Time Series. Univariate random walk Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he

More information

Theory of! Partial Differential Equations-I!

Theory 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 information

Nuclear Decay kinetics : Transient and Secular Equilibrium. What can we say about the plot to the right?

Nuclear 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 information