# a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

Save this PDF as:

Size: px
Start display at page:

Download "a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points." ## Transcription

1 Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You do not know how ny oles of the gs re present Red over steps (- (4 elow nd sketh the pth of the yle on P plot on the grph elow Lel ll pproprite points ( In the first of four steps, to, n idel gs is opressed fro to while no het is llowed to flow into or out of the syste he opression of the gs rises the teperture fro n initil teperture nd to finl teperture During this proess the quntity γ = onstnt, where γ = 5/3 Wht is the pressure P nd volue of the stte of the gs fter the opression is finished? Wht is the hnge in internl energy of the gs during this hnge of stte? Wht is the work done y the gs during this opression? ( he gs is now llowed to expnd isotherlly fro to, fro volue to volue d Express the work done y the gs in this proess W nd the ount of het Q tht ust e dded fro the het soure t in ters of P,,,, nd

2 Is this het positive or negtive? Explin whether it is dded to the syste or reoved e Wht is the pressure P of the gs fter the expnsion is finished? (3 When the gs hs rehed point is expnds fro to d while no het is llowed to flow into or out of the syste he expnsion of the gs lowers the teperture nd pressure fro n initil teperture to finl teperture During this proess the quntity γ = onstnt f Wht is the pressure P nd volue d of the stte d of the gs fter the d expnsion is finished? g Wht is the hnge in internl energy of the gs during this hnge of stte? h Wht is the work done y the gs during this expnsion? (4 he gs is now opressed isotherlly fro d to t onstnt fro volue d k to i Find the work done y the syste on the surroundings W d nd the ount of het Qd tht flows etween the syste nd the surroundings Are these quntities positive or negtive? Explin whether het is dded to the syste or reoved fro the het soure t otl Cyle: j Wht is the totl work W yle done y the gs during this yle? k Wht is the totl het Q yle ( fro drwn fro the higher teperture het soure during this yle? l Wht is the effiieny of this yle ε x = W yle / Q yle ( fro? Prole 4 Het pup A reversile het engine n e run in the other diretion, in whih se it does negtive work W yle on the world while puping het Q yle (into into reservoir t n upper teperture,, fro lower teperture, he het gin of this yle, defined to e

3 g Q yle (into /W yle = (/ ε x where ε = ( x / is the xiu therodyni effiieny of het engine he refrigertor perforne is defined to e K Q yle ( fro / W yle = /( Consider tht you hve lrge swiing pool nd pln to het your house with het pup tht pups het fro the pool into your house A lrge plte in the wter will rein t 0 o C due to the fortion of ie You pik to e 50 o C, whih will e the teperture of the (lrge rditors used to het your house Assue tht your het pup hs the xiu effiieny llowed y therodynis Wht is the het gin nd the refrigertor perforne for this yle? Be reful to use units of Kelvin for teperture If your house forerly urned 00 gllons of oil in winter (t \$00/gllon, how uh will the eletriity ost (t \$00 per kilowtt-hour to reple this het using 8 the het pup? A gllon of oil hs ss 34 kg nd ontins 4 0 J gl - he ie ue tht ppers in your pool over the winter will e how ny eters on 6 eh side? (It tkes J to elt one kg of ie; it tkes up this uh het when 3 freezing he density of ie is kg -3 his would e gret for ooling your house in the suer even if the pool wred up enough to swi in it, you ould still ool your house y running the het pup in reverse s n ir onditioner! More prtilly, you ight e le to use ground wter (nd the dirt round it s the het sink

4 Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You do not know how ny oles of the gs re present Red over steps (- (4 elow nd sketh the pth of the yle on P plot on the grph elow Lel ll pproprite points ( In the first of four steps, to, n idel gs is opressed fro to while no het is llowed to flow into or out of the syste he opression of the gs rises the teperture fro n initil teperture nd to finl teperture During this proess the quntity γ = onstnt, where γ = 5/3 Wht is the pressure P nd volue of the gs fter the opression is finished? Answer: Aording to the idel gs lw, = n R nd = n R so So the pressure = P = P he opression stisfies γ = γ so using the ove result for pressure P, we get 6

5 γ = P γ = γ his eoes using γ = 5/3 / 3 = /3 he volue is then 3/ = hus the rtio of the volues is 3/ = So the pressure P is 5/ P = P Wht is the hnge in internl energy of the gs during this hnge of stte? Answer: he hnge in internl energy is 3 3 ( U U = n R = Wht is the work done y the gs during this opression? Answer: Sine no het is exhnged Q = 0 U U = W + Q = W = 3 ( So 3 ( W = < 0 7

6 he surroundings do work opressing the gs ( he gs is now llowed to expnd isotherlly fro to, fro volue to volue d Express the work done y the gs in this proess W nd the ount of het Q tht ust e dded fro the het soure t in ters of P,,,, nd Is this het positive or negtive? Explin whether it is dded to the syste or reoved Answer: his is n isotherl expnsion so the teperture does not hnge the internl energy is onstnt, =0 hus 3 U U = n R = 0 he gs does work on the surroundings euse it is expnding he pressure is not onstnt during this expnsion Sine the gs is expnding y n isotherl proess, the Idel Gs Lw reltes the pressure nd volue vrition ording to nr P = herefore the work done y the gs on the surroundings is the integrl d W = n R = n R ln( / Using the result for the volue fro prt the work is 3/ =, = n R d = n R ln( 3/ W / Rell tht the volues re relted ording to R = / > 0 nd nr = P / so the work done is positive nd given y 8

7 W = n R ln( / = ln( 3/ R > 0 Fro he First Lw of herodynis, 0 = U U = W + Q, hus the het tht flows into the syste fro the het soure t teperture is equl to the work done y the expnding gs Q = W = ln( 3/ R > 0, Note tht this het flow ust flow fro the higher teperture het soure into the syste euse s the gs expnds it should lose internl energy nd would derese its teperture unless het flows into the syste keeping the internl energy nd hene the teperture onstnt e Wht is the pressure P of the gs fter the expnsion is finished? Answer: = n R = hus P = = P R (3 When the gs hs rehed point is expnds fro to d while no het is llowed to flow into or out of the syste he expnsion of the gs lowers the teperture nd pressure fro n initil teperture to finl teperture During this proess the quntity γ = onstnt f Wht is the pressure P d nd the volue d of the stte d of the gs fter the expnsion is finished? Answer: his lultion is identil to prt, with stte d repling stte, nd stte repling stte So the volue is then 9

8 3/ = d hus the rtio of the volues is So the pressure P is hene d 3/ = 5/ P = P d 5/ P d = P g Wht is the hnge in internl energy of the gs during this hnge of stte? Answer: he derese in the internl energy is due to the teperture derese of the idel gs during expnsion 3 ( U d U = P h Wht is the work done y the gs during this expnsion? Answer: Sine no het is exhnged Q d = 0 U d U = W d + Q = W d = 3 ( d P So 3 ( W d = P > 0 he gs does work on the surroundings sine the gs is expnding (4 he gs is now opressed isotherlly fro d to t onstnt fro volue d k to i Find the work done y the syste on the surroundings W d nd the ount of het Q tht flows etween the syste nd the surroundings Are these quntities d 0

9 positive or negtive? Explin whether het is dded to the syste or reoved fro the het soure t Answer: When the gs undergoes opression it will inrese its internl energy ut het flows out of the syste intining onstnt internl energy, U = 0 nd hene the opression is isotherl he lultion of the work nd het is siilr to step ( exept the teperture is held t he work done y the syste on the surroundings is negtive nd is given y the integrl W d d = n R = n R ln( / = P ln( / = P ln(r d d d 3/ Fro prt f the volue d = so the work done is d 3/ 3/ W d = n R = n R ln( / = P ln( / = P ln( R d d Aording to the First Lw this is equl to the het tht flows into the syste whih is lso negtive whih ens tht it tully flows out of the syste into the surroundings t teperture, 3/ Q d = W d = P ln( R otl Cyle: j Wht is the totl work W yle done y the gs during this yle? Answer: he work done y the het engine on the surroundings during the yle is positive nd given y W yle = P ln( 3/ R P 3/ 3/ ln( R = P ln( R k Wht is the totl het Q yle ( fro drwn fro the higher teperture het soure during this yle? Answer: he het tht flowed fro the higher teperture het soure ourred during step ( isotherl expnsion,

10 totl Q tken fro het soure t = P ln( 3/ R l Wht is the effiieny of this yle ε x = W yle / Q yle ( fro? Answer: he effiieny is given y rtio of the work done divided y the eht flowing into the syste fro the higher teperture het soure 3/ 3/ ε x = W yle / Q yle ( fro = P ln( R / P ln( R ε = x / = = le : Sury of Het Engine Proess U f U i W f,i Q f,i 3 ( 0 diti 3 ( opression isotherl expnsion d diti expnsion d isotherl opression otl positive 0 3 ( negtive 0 0 negtive 3/ ln( R positive 3 ( positive 3/ ln( R negtive 3/ R 3/ ln( R ln( positive 3/ R ln( positive fro 0 3/ ln( R negtive, into 3/ R 3/ ln( R ln( positive, fro into

### Study Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines Msschusetts Institute of Technology Deprtment of Physics 8.0T Fll 004 Study Guide Finl Exm The finl exm will consist of two sections. Section : multiple choice concept questions. There my be few concept

### Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

### CHAPTER 20: Second Law of Thermodynamics CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het

### AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

### 1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

### Thermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam April 08 Therml energy Soures of het Trnsport of het How to use het The First Lw of Thermoynmis U W Or, if we wnt to otin externl work: U W 009 vrije Universiteit msterm Close yle stem power plnt The trik

### U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( ) 0-9-0 he First Lw of hermoynmis Effiieny When severl lterntive proesses involving het n work re ville to hnge system from n initil stte hrterize y given vlues of the mrosopi prmeters (pressure p i, temperture

### Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

### Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

### Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

### ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 17 FORCES OF ELECTRIC ORIGIN ENERGY APPROACH(1) ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 17 FORCES OF ELECTRIC ORIGIN ENERGY APPROACH(1) Aknowledgent-These hndouts nd leture notes given in lss re bsed on teril fro Prof. Peter Suer s ECE 330

### Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

### Solutions to Assignment 1 MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

### Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

### 6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

### CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

### Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

### GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

### Chapter 19: The Second Law of Thermodynamics hpter 9: he Seon Lw of hermoynmis Diretions of thermoynmi proesses Irreversile n reversile proesses hermoynmi proesses tht our in nture re ll irreversile proesses whih proee spontneously in one iretion

### Magnetically Coupled Coil Mgnetilly Coupled Ciruits Overview Mutul Indutne Energy in Coupled Coils Liner Trnsformers Idel Trnsformers Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Mgnetilly Coupled Coil i v L

### LESSON 11: TRIANGLE FORMULAE . THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.

### Übungen zur Theoretischen Physik Fa WS 17/18 Krlsruher Institut für ehnologie Institut für heorie der Kondensierten Mterie Übungen zur heoretishen Physik F WS 17/18 Prof Dr A Shnirmn Bltt 4 PD Dr B Nrozhny Lösungsvorshlg 1 Nihtwehselwirkende Spins:

### University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

### p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

### Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into

### Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

### PHY 5246: Theoretical Dynamics, Fall Assignment # 5, Solutions. θ = l mr 2 = l PHY 546: Theoreticl Dynics, Fll 15 Assignent # 5, Solutions 1 Grded Probles Proble 1 (1.) Using the eqution of the orbit or force lw d ( 1 dθ r)+ 1 r = r F(r), (1) l with r(θ) = ke αθ one finds fro which

### CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

### 18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106 8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

### (a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

### SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

### System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

### NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

### Ideal Gas behaviour: summary Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive)

### Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

### Chapter 14. Gas-Vapor Mixtures and Air-Conditioning. Study Guide in PowerPoint Chpter 14 Gs-Vpor Mixtures nd Air-Conditioning Study Guide in PowerPoint to ccopny Therodynics: An Engineering Approch, 5th edition by Yunus A. Çengel nd Michel A. Boles We will be concerned with the ixture

### Chemical Equilibrium Chpter 16 Questions 5, 7, 31, 33, 35, 43, 71 Chemil Equilibrium Exmples of Equilibrium Wter n exist simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property ssoited

### Instructions to students: Use your Text Book and attempt these questions. Instrutions to students: Use your Text Book nd ttempt these questions. Due Dte: 16-09-2018 Unit 2 Chpter 8 Test Slrs nd vetors Totl mrks 50 Nme: Clss: Dte: Setion A Selet the est nswer for eh question.

### A, Electromagnetic Fields Final Exam December 14, 2001 Solution 304-351, Electrognetic Fiels Finl Ex Deceer 14, 2001 Solution 1. e9.8. In chpter9.proles.extr.two loops, e of thin wire crry equl n opposite currents s shown in the figure elow. The rius of ech loop is

### April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

### - 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting

### Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

### Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

### Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

### This conversion box can help you convert units of length. b 3 cm = e 11 cm = b 20 mm = cm. e 156 mm = b 500 cm = e cm = b mm = d 500 mm = Units of length,, To onvert fro to, ultiply y 10. This onversion ox n help you onvert units of length. To onvert fro to, divide y 10. 100 100 1 000 10 10 1 000 Convert these lengths to illietres: 0 1 2

### AP Calculus AB Unit 4 Assessment Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope

### 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

### Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

### 8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

### AP Physics - Heat Engines A hysis - et Engines et engines operte y onverting het into work. Exmples of het engines ound - ound! Sys the hysis Khun - ll round us. Gsoline engines, jet engines, diesel engines, stem engines, Stirling

### CS241 Week 6 Tutorial Solutions 241 Week 6 Tutoril olutions Lnguges: nning & ontext-free Grmmrs Winter 2018 1 nning Exerises 1. 0x0x0xd HEXINT 0x0 I x0xd 2. 0xend--- HEXINT 0xe I nd ER -- MINU - 3. 1234-120x INT 1234 INT -120 I x 4.

### I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

### INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Spacetime and the Quantum World Questions Fall 2010 Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1

### 1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes. 1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond

### Nondeterministic Finite Automata Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

### 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

### ME 354 Tutorial, Week#11 Non-Reacting Mixtures Psychrometrics Applied to a Cooling Tower ME 5 Tutoril, Week# Non-Recting Mixtures Psychroetrics Applied to Cooling Toer Wter exiting the condenser of poer plnt t 5 C enters cooling toer ith ss flo rte of 5000 kg/s. A stre of cooled ter is returned

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

### 332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006 2:221 Principles of Electricl Engineering I Fll 2006 Nme of the student nd ID numer: Hourly Exm 2 Novemer 6, 2006 This is closed-ook closed-notes exm. Do ll your work on these sheets. If more spce is required,

### Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

### Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233, Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.

### ENSC 461 Tutorial, Week#9 Non-Reacting Mixtures Psychrometrics Applied to a Cooling Tower ENSC 61 Tutoril, Week#9 Non-Recting Mixtures Psychroetrics Applied to Cooling Toer Wter exiting the condenser of poer plnt t 5C enters cooling toer ith ss flo rte of 15000 kg/s. A stre of cooled ter is

### Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011 Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

### Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

### Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

### Chapter 4 The second law of thermodynamics hpter 4 he second lw of thermodynmics Directions of thermodynmic processes et engines Internl-combustion engines Refrigertors he second lw of thermodynmics he rnotcycle Entropy Directions of thermodynmic

### Potential Changes Around a Circuit. You must be able to calculate potential changes around a closed loop. Tody s gend: Potentil Chnges Around Circuit. You must e le to clculte potentil chnges round closed loop. Electromotive force (EMF), Terminl Voltge, nd Internl Resistnce. You must e le to incorporte ll

### CIT 596 Theory of Computation 1. Graphs and Digraphs CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

### Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

### Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

### Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### 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

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:

### 5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

### JURONG JUNIOR COLLEGE JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Sub-topic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by

### T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

### Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx Tutoril 2 Euler Lgrnge In one entene: d Fy = F d Importnt ft: ) The olution of EL eqution i lled eterml. 2) Minmum / Mimum of the "Mot Simple prolem" i lo n eterml. 3) It i eier to olve EL nd hek if we

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

### The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

### AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

### Electromagnetism Notes, NYU Spring 2018 Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system

### A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

### SIMPLE NONLINEAR GRAPHS S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle

### Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

### Prefix-Free Regular-Expression Matching Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

### In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then

### QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

### QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1} S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown