du = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y),

Size: px
Start display at page:

Download "du = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y),"

Transcription

1 29. Differentil equtions. The conceptul bsis of llometr Did it occur to ou in Lecture 3 wh Fiboncci would even cre how rpidl rbbit popultion grows? Mbe he wnted to et the rbbits. If so, then he would be less concerned bout totl number of rbbit pirs thn totl weight of rbbit met, which vries continuousl (s opposed to discretel). Let W(t) be weight t time t. Then W is n ordinr function. But wht kind of function is W? In this lecture we investigte. Over ver short time intervl [t, t + h], weight increses b Diff(W, [t, t+h]) W(t+h) W(t), with verge rte of increse DQ(W, [t, t+h]) Diff(W, [t, t+h])/h. Although weight chnges on [t, t+h], it chnges onl little, so it is lws W(t) + O[h]. Therefore verge increse per rbbit on [t, t+h] is DQ(W, [t, t+h])/{w(t) + O[h]}. As h pproches zero, this expression pproches DQ(W,[t,t + h]) lim h 0 W(t) + O[h] W (t) W(t) r(t), (29.1) s, where r hs the sme domin s W nd W. We cll r the reltive growth rte. The preceding rgument pplies virtull without chnge to the growth of n mss of tissue. In prticulr, it pplies to the growth of single orgn or orgnism. Tht is, if W(t) denotes size t time t of n orgn or orgnism, then W (t) r(t)w(t) (29.2) where r is reltive growth rte. Eqution (2) embodies Julin Huxle's first "essentil fct" bout growth, nmel, "tht it is process of self-multipliction of living substnce i.e. tht the rte of growth of n orgnism growing equll in ll its prts is t n moment proportionl to the size of the orgnism" (Huxle, 1932, p. 6). Wht kind of function is r? Tking our cue from Lecture 5, where Fiboncci rbbit pirs reproduce t constnt rte (of 12 times per nnum), we conjecture tht r m be constnt, s r λ (> 0). Then (2) implies W (t) λw(t) or, if W(t), d λ. (29.3) This is ver simple exmple of n ordinr differentil eqution, or ODE. How do we "solve" this ODE? In other words, how do we find if we know d/? Hitherto we hve used the fundmentl theorem. But we cnnot ppl it to (3), becuse the right-hnd side is n unknown function of time. On the other hnd, if W is invertible with inverse U, so tht W(t) is exctl the sme thing s t U(), then from (20.39) we hve d d. (29.4) So we cn rewrite (3) s d 1 λ, (29.5) nd now it follows from the fundmentl theorem tht t C + where C is constnt. To obtin of C, we set ), 0 U() 1 du C λu λ u du, (29.6) the vlue W(t (29.7) 0 1

2 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 2 where t0 is initil time, nd so is initil weight. Then (6), with, implies U() C + C + 0. But W(t0) implies U() t0, functions. So (8) implies C t0, from which (6) implies 1 λu du (29.8) becuse U nd W re inverses t t λ 1 du u t λ ln(u) (29.9) t λ t λ {ln() ln()} ln( / ) on using Tble 22.6 nd Exercise So λ{t t0} ln(/), or exp(λ{t t0}) /. Tht is, In other words, with exp(λ{t t 0}) eλ(t t 0 ) e λt e λt 0. (29.10) W(t) Ae λt (29.11) A e λt 0. (29.12) When W(t) A e λt, it is trditionl to s tht W exhibits exponentil growth t rte λ (lthough wht is ctull ment is tht W hs reltive growth rte λ). There is evidence tht orgnisms do grow exponentill during erl development, lthough reltive growth rte ppers to decrese lter on. The crux of the evidence is tht (11) implies ln( ) ln( W(t) ) ln( Ae λt ) ln(a) + ln(e λt ) (29.13) ln(a) + λt. Tht is, if growth is exponentil, then ln() is liner function of t. So we cn test n exponentil growth hpothesis b plotting (t, ln()) dt pirs nd drwing the stright line tht fits them best. The closer the fit, the more confident we re in the hpothesis. TIME t (ds) WEIGHT (grms) ln( ) TIME t (ds) WEIGHT (grms) ln( ) Tble 29.1 Bckmn's dt on growth in weight of mize. Source: Exercise 5.10 For exmple, from Thompson (1942, p. 115) we hve Bckmn's dt on weight (grms) of mize t time t (ds); see Tble 1. From Figure 1(), where ln() is plotted ginst t, the dt points fll ver close to stright line for times between 18 nd 60

3 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 3 ds. This six-week period is n exponentil growth phse. The dotted line in Figure 1(), determined b the method of Appendix 2A, hs eqution ln() t. (29.14) Compring with (13), λ , ln(a) nd so A exp(0.2279) Thus W(t) 1.256e t (29.15) provides n excellent description of growth in weight of mize on [18, 60]; see Figure 1(b). On the other hnd, (15) provides ver poor description t lter times. It isn't difficult to see wh. B (2), constnt reltive growth rte implies tht W keeps incresing. But the dt in Figure 1 nd Tble 1 impl tht W eventull pproches zero, in which cse, r pproches zero, too. Intuitivel, growth per unit weight declines with weight becuse more cells compete for the sme resources. The simplest possibilit is tht r decreses linerl with W, s r(t) λ 1 W(t) (29.16) where is nother prmeter. We ssume tht W(t0) <, (29.17) where t0 is the initil time; i.e., W hs domin [t0, ). Then r is positive nd decreses towrd zero s W(t) increses towrd. Substituting (16) into (2) nd setting W(t), we obtin new ODE d λ 1 (29.18) in plce of (3). Assuming s before tht W is invertible with inverse U, we hve d λ( ), (29.19) in plce of (5), nd it follows from the fundmentl theorem tht t C + u) where C is constnt. s C t0. Thus U() λu( du, (29.20) Exctl bove, W(t0) implies t U() t λ Tble 22.6 nd Exercise 22.1 now impl tht t t λ ln u u t λ ln t λ So λ{t t0} ln({ }/{ }), impling { } { } ln { } du. (29.21) u( u) ln. (29.22) e λ(t t 0 ) (29.23) { } or, fter strightforwrd lgebric mnipultions (Exercise 1),

4 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 4 In other words, where Note tht s required. Correspondingl, W(t) e λ(t t 0 ) e λ(t t 0 ) +. (29.24) (29.25) 1 + Ae λt A { / 1}e λt 0. (29.26) W(t 0), (29.27) W (t) λw(t) 1 W(t) λae λt (29.28) (1 + Ae λt ) 2 from (18) nd (25). Growth ccording to (24) is trditionll clled logistic growth. From (17) nd (28), / > 1, impling A > 0. So W(t) is less thn but pproches smptoticll s t becuse e λt 0 s t. We interpret s mximum possible weight. In Figure 2, the functions W nd W re plotted ginst Bckmn's dt with A 389, 75.5 nd λ (vlues were obtined b method similr to tht of Appendix 2A). We find tht the logistic model provides much better overll fit, with mximum growth during d 51; see Exercise 2. We conclude this lecture b observing tht Huxle used pir of ODEs to secure conceptul foundtion for llometr. He resoned tht "the growth-rte of n prticulr orgn is proportionl simultneousl () to specific constnt chrcteristic of the orgn in question, (b) to the size of the orgn t n instnt, nd (c) to generl fctor dependent on ge nd environment which is the sme for ll prts of the bod" (Huxle, 1932, p. 6). If is the size of n llometric orgn nd x the size of its bod, nd if k2, k1 re the specific constnts for prt nd bod, respectivel, then becuse both orgn nd bod re exposed to the sme environment, () (c) impl nd d k 1 xg(t) (29.29) k 2 G(t) (29.30) where t denotes time nd G is some unknown function. Dividing (30) b (29), we hve where d 1 β x (29.31) β k 2. (29.32) k 1 Note tht the orgn is positivel or negtivel llometric ccording to whether k2 > k1 or k1 > k2, nd hence ccording to whether β > 1 or β < 1, s in Lecture 22. Assuming tht x is n invertible function of time, nd hence t n invertible function of x, 1 (29.33)

5 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 5 b nlog with (4). So (31) cn be rewritten s x. (29.34) If is function of t, however, t function of x, then is lso function of x. So the chin rule reduces (34) to x, (29.35) nother ODE. Compring with (3) we find tht the right-hnd sides of (3) nd (18) involve onl the dependent vrible, wheres the right-hnd side of (35) involves the independent vrible s well; nevertheless, (35) is no more difficult to solve. We proceed s follows. According to the chin rule, if is positive function of x then defines composition with derivtive on using Tble So (36) It now follows from the theorem tht d β nd is d β z ln() (29.36) dz dz d d { d d ln() } d d, (29.37) reduces (35) to dz β x. (29.38) x z C + β 1 du C + βln(u) u C + β{ln(x) ln()}, so tht (36) nd properties of the logrithm impl ln() C + β ln x Properties of the exponentil now impl tht x C + ln x ( ) exp(c)exp( ln({x / )} β ) exp C + ln({x / )} β exp(c) x β. β On setting β exp(c) α we hve αx β, greeing with (22.32). 1 β (29.39). (29.40) (29.41) References Huxle, Julin S. (1932). Problems of Reltive Growth. The Dil Press, New York Thompson, D'Arc W (1942). On Growth nd Form. Cmbridge Universit Press.

6 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 6 Exercises Verif (24)-(27) Show tht the logistic growth curve W(t) defined b (25) hs n inflection point t t t*, defined b t* ln(a)/λ. Verif tht W(t*) / Show tht (28) implies W (t) λ 3 W(t) Hence show tht the curve W (t) hs inflection points where t 1 λ ln ± 3 i.e., where t 39 nd t 61 in 1 W(t) 6W(t) 1 W(t) ({ 2 ± 3 }A), Figure 2(b)..

7 M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 7 Answers nd Hints for Selected Exercises 29.2 From (28), we hve So W (t) λw(t) 1 W(t) W (t) d. { W (t)} λ d λ W (t) 1 W(t) λ W (t) 1 W(t) W(t) 1 W(t) + W(t) d 1 W(t) + W(t) 0 1 W (t) λ 2 W(t), impling W 0 where W(t*) 0 or 1 W(t*)/ 0 or 1 2W(t*)/ 0. From (25), however, we hve W(t*) /{ 1 + Ae λt* }, so tht W(t*) 0 is impossible, nd 1 W(t*) so tht 1 W(t*)/ 0 is impossible (except in the limit s t* ). So the inflection point must be determined b 1 2W(t*)/ 0. But λ W (t) 1 W(t) 1 W(t) (t) λ W (t) 1 2W(t) 1 W(t) 2W(t) (t*) 1 1 Ae λt* + 1 Ae λt* 1 + Ae, λt* 1 2 W(t*) 1 2 Ae λt* + 1 Ae λt* Ae. λt* Thus 1 2W(t*)/ 0 implies Ae λt* 1 or e λt* 1/A. Hence λt* ln(1/a) ln(a), nd λt* ln(a) or t* ln(a)/λ. Note tht, becuse 0 < W(t) <, the sign of W (t) is completel determined b 1 2W(t)/, which is positive if t < t* but negtive if t > t*. Thus W hs mximum W (t*). You cn verif tht W (t*) 0 corresponds to minimum (s opposed to mximum) of W b checking the sign of W (t*): from Exercise 3, W (t*) λ 3 W(t*) which is 1 W(t*) negtive. 1 6W(t*) 1 W(t*) λ 3 8,

Math 31S. Rumbos Fall Solutions to Assignment #16

Math 31S. Rumbos Fall Solutions to Assignment #16 Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation 1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

More information

Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

MATH SS124 Sec 39 Concepts summary with examples

MATH SS124 Sec 39 Concepts summary with examples This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Conservation Law. Chapter Goal. 5.2 Theory

Conservation Law. Chapter Goal. 5.2 Theory Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100. Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

Module 6: LINEAR TRANSFORMATIONS

Module 6: LINEAR TRANSFORMATIONS Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

Math 116 Calculus II

Math 116 Calculus II Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

More information

Indefinite Integral. Chapter Integration - reverse of differentiation

Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

1 Probability Density Functions

1 Probability Density Functions Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

Math 153: Lecture Notes For Chapter 5

Math 153: Lecture Notes For Chapter 5 Mth 5: Lecture Notes For Chpter 5 Section 5.: Eponentil Function f()= Emple : grph f ) = ( if = f() 0 - - - - - - Emple : Grph ) f ( ) = b) g ( ) = c) h ( ) = ( ) f() g() h() 0 0 0 - - - - - - - - - -

More information

5 Accumulated Change: The Definite Integral

5 Accumulated Change: The Definite Integral 5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

More information

Calculus I-II Review Sheet

Calculus I-II Review Sheet Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

(4.1) D r v(t) ω(t, v(t))

(4.1) D r v(t) ω(t, v(t)) 1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

More information

INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

More information

Integral points on the rational curve

Integral points on the rational curve Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

f(a+h) f(a) x a h 0. This is the rate at which

f(a+h) f(a) x a h 0. This is the rate at which M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Calculus - Activity 1 Rate of change of a function at a point.

Calculus - Activity 1 Rate of change of a function at a point. Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Math 42 Chapter 7 Practice Problems Set B

Math 42 Chapter 7 Practice Problems Set B Mth 42 Chpter 7 Prctice Problems Set B 1. Which of the following functions is solution of the differentil eqution dy dx = 4xy? () y = e 4x (c) y = e 2x2 (e) y = e 2x (g) y = 4e2x2 (b) y = 4x (d) y = 4x

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

Review Exercises for Chapter 4

Review Exercises for Chapter 4 _R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

F (x) dx = F (x)+c = u + C = du,

F (x) dx = F (x)+c = u + C = du, 35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

APPROXIMATE INTEGRATION

APPROXIMATE INTEGRATION APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

5.5 The Substitution Rule

5.5 The Substitution Rule 5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

More information

MAT 168: Calculus II with Analytic Geometry. James V. Lambers

MAT 168: Calculus II with Analytic Geometry. James V. Lambers MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge

More information

Math Lecture 23

Math Lecture 23 Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

More information

Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model.

Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model. Popultion Dynmics Definition Model A model is defined s physicl representtion of ny nturl phenomen Exmple: 1. A miniture building model. 2. A children cycle prk depicting the trffic signls 3. Disply of

More information

Consequently, the temperature must be the same at each point in the cross section at x. Let:

Consequently, the temperature must be the same at each point in the cross section at x. Let: HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

Math 61CM - Solutions to homework 9

Math 61CM - Solutions to homework 9 Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Calculus II: Integrations and Series

Calculus II: Integrations and Series Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

More information

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones. Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description

More information

Chapter 5 : Continuous Random Variables

Chapter 5 : Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

More information

Theoretical foundations of Gaussian quadrature

Theoretical foundations of Gaussian quadrature Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Phse-plne Anlsis of Ordinr November, 7 Phse-plne Anlsis of Ordinr Lrr Cretto Mechnicl Engineering 5A Seminr in Engineering Anlsis November, 7 Outline Mierm exm two weeks from tonight covering ODEs nd Lplce

More information

1 Functions Defined in Terms of Integrals

1 Functions Defined in Terms of Integrals November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider

More information

MATH , Calculus 2, Fall 2018

MATH , Calculus 2, Fall 2018 MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information