# cooking trajectory boiling water B (t) microwave time t (mins)

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer (a a consan emperaure of 1 ), and he oher is o microwave hem a a LOW power. By coincidence, i urns ou ha he heaing ime is he same for boh mehods eacly 2 minues. Of course we can use eiher one bu suppose we can also use a combinaion of he wo one mehod for a cerain ime and hen he oher. Here s he problem can I lower my cooking ime by swiching? And if so how much? Of course we assume he swich is insananeous wih no loss of emperaure. The graph a he righ provides he T- (emperaure-ime) rajecories for boh mehods over he inerval 2. The difference beween he graphs come from he differen ways in which hea is ransferred o he egg. In he microwave oven, hea is absorbed by he egg a a consan rae, and as a consequence, is emperaure increases a a consan rae. However, in he boiling waer, he egg absorbs hea more quickly a he beginning when i is cool han laer when i is close o he emperaure of he waer. (a) (graphical) Find a consrucion on a copy of he graph which provides a soluion o he problem of how o shif beween he wo mehods in order o minimize he oal cooking ime. emperaure T (degs) cooking rajecory boiling waer B () microwave M ( ) ime (mins) Now here are differen ypes of argumen for an opimizaion problem like his. Some are local and aemp o argue ha your find your minimum simply by arranging o do he bes you can a every momen. Ohers are global and work by consrucing and comparing enire rajecories. Regardless of he ype of argumen you produce, make sure ha a he end you draw on he original se of aes he emperaure-ime rajecory of he egg if i is heaed in he opimal (shores ime) manner. Show clearly how your graph gives us he resuling minimum cooking ime. (b) (algebraic and numerical). The mahemaical form of he boiling waer curve follows from Newon s Law which saes ha he difference D in emperaure beween he waer and he egg is an equaion of eponenial decay. Use his o find an equaion for he boiling waer graph B(). Use his equaion o obain a rough numerical check of your resuls obained graphically, in (a). (c) (calculus). Use he ools of calculus o find he eac opimal swich poin and hereby check your answers o (a) and (b). 4.1 alligaor egg page 1

2 Soluion. This is a good problem because here are differen ways o make he argumen, graphical, numerical and analyic, and boh local and global. Wih luck, some sudens will use one approach and some he oher, and when hese are pu ogeher a he end, some imporan undersanding will be gained. (a) The local argumen. In is simples form, he local argumen assers ha we always wan he rae of increase of emperaure o be as big as possible, and ha s given by he slope, so we always wan o be on he graph ha has he bigges slope. Now if you look a he wo graphs, he microwave graph always has slope 4, bu he boiling-waer graph sars wih a high slope, greaer han 4, and ends wih a low slope, less han 4. So ha ells us o sar wih he high slope of he boiling waer graph, and sick wih i unil he slope has dropped o 4, and a ha poin swich o he microwave So he swich poin will be he poin on he boiling-waer graph wih slope 4. Tha means ha he angen a ha poin is parallel o he microwave graph. And ha seems o happen a around =8. Maybe a iny bi above. Say 8.2. Tha s when we ransfer i from he waer o he microwave. So wha is he resuling minimum cooking ime? Le s work i ou. We sar wih 8.2 minues in he waer, and ha brings i o a emperaure of abou 64 (read from he graph), so i has 26 o go o ge o 9, and a 4 /min ha ll ake 6.5 minues. So he oal is = We ge an overall cooking ime of abou 14.7 minues. Tha s a saving of more han 5 minues from he wo pure sraegies Wha made his eceedingly simple argumen work was he fac ha we could swich ovens a any ime a no cos in erms of eiher los ime or emperaure loss. So here s no reason no o epec o be always in he mode ha has he highes rae of emperaure increase for he curren emperaure. A global argumen. A few sudens seem more naurally o go righ away for a global argumen. For his we need o find a mehod ha acually looks a and compares he enire emperaure-ime rajecory for differen sraegies. 4.1 alligaor egg page 2

3 Here s an eample of such an argumen. Suppose ha we sar in he boiling waer, keep he egg here for minues, and hen swich he egg o he microwave and keep i here unil i s done. Wha does he resuling emperaure-ime rajecory look like? Tha is, draw he graph of he acual emperaure of he egg agains ime. Okay. The graph will depend on he value of he swich poin. To have a paricular eample, ake = 4. Tha means we follow he waer graph ill ime = 4. Bu hen wha? Well hen we swich o he microwave where he emperaure increases a a consan rae of 4º per minue. Thus, from =4 on, he graph is a sraigh line of slope 4. This las piece of he graph is a line parallel o he microwave graph, aking off from he =4 poin on he waer graph. Where does his line end? The egg is cooked when i aains 9º and hus he end poin of his line is found where i has heigh 9. This seems o be close o ime = 16. Thus he oal cooking ime is 16 minues. Tha was a paricular eample, bu i illusraes he general form. For any swich poin, he las par of he graph is always a line of slope 4 aking off from he swich poin. And he graph ends when he line ges o heigh 9. Now le s consider he opimizaion problem. We wan he shores cooking ime, and ha means we wan ha final piece of sraigh line o aain heigh 9 as far o he lef as possible. I should be clear ha his will happen when he line is as high as possible so wha we wan is he highes possible line of slope 4 which inersecs he waer graph. One way o find ha is o ake he microwave line and keep raising i unil i no longer inersecs he waer graph. I should be clear ha a his final poin i will be angen o he waer graph. Thus he opimal swich poin is he poin where he graph has slope 4. This gives us he same poin 8.2 ha we found wih he local analysis. Our opimal rajecory follows he boilingwaer curve o he angen poin and hen moves up along he angen. Wha is he oal cooking ime for his swich poin? i s where he line his T = 9 and ha s seen o be beween = 14 and = 15, somewha closer o 15. Tha s also in line wih our esimae of 14.7 from he local argumen. I s imporan o noice ha he same essenial consrucion appears in boh he local and he global argumens. Bu he sory is quie differen in he wo cases alligaor egg page 3

4 (b) We are given ha he difference D in emperaure beween he boiling waer (1º) and he egg B() is an equaion of eponenial decay of he form D() = 1 B() = Ar where r < 1. To evaluae he consans, use he wo endpoins of he graph, = and =2: D() = 1 B() = 1 1 = 9 D(2) = 1 B(2) = 1 9 = 1 This gives us he equaions: Ar = 9 Ar 2 = 1 Solving: A = 9 9r 2 = 1 r 2 = 1/ D () B () / 2 r = (1/ 9) =.896 Hence: D() = Ar = 9(.896) and he boiling waer emperaure equaion is: B() = 1 D() = 1 9(.896). An eponenial equaion always has a consan muliplier. Here he muliplier is.896. The emperaure difference D is cu by 1.4% every minue. Sudens end o prefer o work wih decimals, as above, bu my own preference is ofen for a more eac form and ha s he case here (parly because he numbers work ou). I would wrie: = 1/ 2 r = (1/ 9) ( 1/ 2 / 2 (1/ 9) ) (1/ 9) r = 1 B( ) = 1 Ar = In fac, I can resis a simplificaion: 1 B( ) = /1 / 2 This displays, no he 1-minue muliplier, bu he 1-minue muliplier. Over any 1 minue period, he emperaure difference beween he waer and he egg is muliplied by 1/ alligaor egg page 4

5 Finally, since we have an equaion for he B-graph, we can use his o check our graphical answer for he swich poin. This is he poin a which he slope of he B-graph is he same as he microwave slope: 4 /minue. Calculus will give us he ools of he derivaive o do his (par (c)) bu even wihou calculus we can ge a good idea from he slopes of shor secans (which are average raes of change). In par (a) we esimaed he slope-4 poin o be close o = 8.2. A he righ, we abulae he values of B() beween =8 and =9. On an inerval of lengh.1, we wan o look for an increase of.4. We ge eacly ha (o 3 decimal places) on he inerval [8.2, 8.3]. Tha ells us ha he swich poin will be somewhere in his inerval. [Acually a small argumen is needed for his. Can you find i?] We see ha on he previous inerval [8.1, 8.2] he change is a bi more han.4, and on he following inerval [8.3, 8.4] he change is a bi less han.4. Tha fis wih he concave-down form of he graph. (c) Our problem is o find he swich poin which minimizes he oal cooking ime. Le s give his a name: we ll call i Toal cooking ime: τ = τ(). Now we need o find an epression for τ(). Well i s he sum of he imes he egg spends in each phase. The boiling waer is easy enough minues. The microwave requires a bi more hough. The firs phase brough he emperaure of he egg up o B(). To finish cooking, we need an era 9 B() degrees, and a 4 /minue his will ake (9 B())/4 minues. The oal ime is hen 9 B( ) τ ( ) = + 4 The calculus approach is o say ha a an inerior minimum he derivaive of τ mus vanish: B ( ) τ ( ) = 1 = 4 This solves as B ( ) = (.896) = 4 (.896) =.45 To solve his for we ake he naural log of boh sides: ln(.896) = ln(.45) = ln.45 ln.896 = We should swich o he microwave afer 8.24 minues. This confirms our graphical analysis ime emperaure B() The derivaive condiion will always find an inerior minimum, bu his mus be checked agains he end-poins, = and =2, bu we already know ha hese can be opimal as hey boh give cooking imes of 2 minues (see Problem 1). The derivaive of B From our calculaions above: B ( ) = 1 9(.896) B ( ) = 9(.896) ln(.896) = 9.88(.896) 4.1 alligaor egg page 5

6 Problems 1. Use he fac ha B(1) = 7 o argue immediaely ha he end-poin soluions, = and =2, can be opimal. 2. Recen advances in cooking have shown ha alligaor eggs can safely be heaed in a microwave a a slighly higher seing, giving a emperaure increase of 5 /min, hus requiring only a 16-minue cooking ime. Supposing ha he boiling waer mehod is sill available, follow he various approaches of he eample o find he opimal poin o swich from he boiling waer o he microwave. Illusrae your seps on a copy of he graph, as is done in he Eample, and esimae he oal cooking ime. Full eplanaions should be given. 3. I has been deermined ha he delicae flavour of an alligaor egg will be bruised if he rae of change of emperaure of he egg ever eceeds 8 /minue. Use a graphical approach o find he cooking program which minimizes he oal ime, bu does no allow he rae of emperaure increase o eceed 8 /minue. As usual you can assume ha swiches beween mehods happen insananeously wih no emperaure loss. 4. I have a porable heaing coil which I can plug in (say in a moel room) and pu ino a large mug of waer and hea i up o make ea. If I pop a super ea cozy over he mug while I m doing his, hen here is no hea loss o he room, and he waer emperaure increases a a consan rae, going from 2 o 1 in 5 minues. (a) Draw he graph of he waer emperaure agains ime. (b) Now suppose ha I ve go he waer o 1, bu insead of making he ea, I ge waylaid by a desire o perform an eperimen and I simply leave he mug sanding in he room wihou he cozy. Draw wha you hink is a reasonable graph of he emperaure of he waer over a 4-minue period. Here are a couple of poins you can mark on your graph during he firs wo minues, he emperaure drops from 1 o 7, and during he second wo minues i drops o. (c) Now, wih he waer emperaure a, I pu he heaing coil back in he mug and aemp o hea i back up o 1 bu wihou he cozy. Do I succeed? No! I find ha afer a long ime he emperaure of he waer sabilizes a a level somewha shor of boiling. Show ha an analysis of your wo graphs will allow you o predic he emperaure a which my mug of waer will sabilize. 5. I have a hole in my ire which makes i lose pressure a he consan percenage rae of 4% every half minue. Saring a 4 kpa, he graph a he righ shows he pressure rajecory over a 2 minue period. Suppose ha, in spie of he hole, I ry o keep he ire inflaed by pumping air in a a slow consan rae. Wha I know is ha wihou he hole, he pump will inflae he ire from o 4 kpa in 2 minues. Bu if I use he same pump on he ire wih he hole, saring a kpa, i never reaches 4 kpa bu approaches a consan pressure. Draw a rough graph of he pressure agains ime for his case, eplaining how you have deermined he limiing pressure. Then use he ools of calculus o calculae his limiing pressure eacly. pressure P (kpa) ime (min) 4.1 alligaor egg page 6

7 Working copy of alligaor egg graph alligaor egg page 7

8 Working copy of ire pressure graph (problem 5). pressure P (kpa) ime (min) 4.1 alligaor egg page 8

### Graphing the Von Bertalanffy Growth Equation

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

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

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

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

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

### 11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

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

MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

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

Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

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

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

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

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

### CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

### Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

### Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

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

1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

### Circuit Types. () i( t) ( )

Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

### Section 7.1 Angles and Their Measure

Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

### 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.

### 2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

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

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

### Fourier Series Solution of the Heat Equation

Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

### AP Calculus AB 2013 Scoring Guidelines

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

### Week #9 - The Integral Section 5.1

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

### 9. Capacitor and Resistor Circuits

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

### Acceleration Lab Teacher s Guide

Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

### Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

### INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

### Relative velocity in one dimension

Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

### Rotational Inertia of a Point Mass

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

### ( ) in the following way. ( ) < 2

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

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

### AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### RC (Resistor-Capacitor) Circuits. AP Physics C

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

### Chabot College Physics Lab RC Circuits Scott Hildreth

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

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

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

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

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

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

Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

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

Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

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

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

### Chapter 2 Kinematics in One Dimension

Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

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

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

### Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

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

. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

### Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing

### 1 HALF-LIFE EQUATIONS

R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

### HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

### AP Calculus AB 2007 Scoring Guidelines

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

### Lectures # 5 and 6: The Prime Number Theorem.

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

Chaper 8 Copyrigh 1997-2004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon

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

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

### 17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

### Period 4 Activity Solutions: Transfer of Thermal Energy

Period 4 Aciviy Soluions: Transfer of Thermal nergy 4.1 How Does Temperaure Differ from Thermal nergy? a) Temperaure Your insrucor will demonsrae molecular moion a differen emperaures. 1) Wha happens o

### Math 201 Lecture 12: Cauchy-Euler Equations

Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

### Permutations and Combinations

Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

### Economics 140A Hypothesis Testing in Regression Models

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

### 5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

### Morningstar Investor Return

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

### A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

### Economics Honors Exam 2008 Solutions Question 5

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

### PHYS245 Lab: RC circuits

PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of

### Chapter 4: Exponential and Logarithmic Functions

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

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

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

Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

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

Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

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

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

### Two Compartment Body Model and V d Terms by Jeff Stark

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

### Newton s Laws of Motion

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

### Section A: Forces and Motion

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

### FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

### THE PRESSURE DERIVATIVE

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

### Motion Along a Straight Line

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

### A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

### A Curriculum Module for AP Calculus BC Curriculum Module

Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

### Graduate Macro Theory II: Notes on Neoclassical Growth Model

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

### Physics 111 Fall 2007 Electric Currents and DC Circuits

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

### 4. International Parity Conditions

4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Differential Equations and Linear Superposition

Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

### TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

### Return Calculation of U.S. Treasury Constant Maturity Indices

Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

### MOTION ALONG A STRAIGHT LINE

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

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### 3 Runge-Kutta Methods

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

### 4.2 Trigonometric Functions; The Unit Circle

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

### Part 1: White Noise and Moving Average Models

Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

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

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

### AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

### 4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

### Discussion Examples Chapter 10: Rotational Kinematics and Energy

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

### Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have

### Chapter 15: Superposition and Interference of Waves

Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when

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

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

### CHAPTER FIVE. Solutions for Section 5.1

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

### Signal Rectification

9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

### RC Circuit and Time Constant

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

### Understanding Sequential Circuit Timing

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

### Using RCtime to Measure Resistance

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

### LAB 6: SIMPLE HARMONIC MOTION

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

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

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

### Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual