AP Physics - Heat Engines

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "AP Physics - Heat Engines"

Transcription

1 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 engines, &t. Just out nything tht urns fuel to generte het is het engine. You ould e onsidered to e low-teh het engine (ut relly nie one, I m sure we ll gree). et is dded to the engine t high temperture. rt of the het is used to generte work nd the rest of the het is sent to some low temperture environment. Fny terminology used for het engines involves things like: high temperture het reservoir (the soure of the input het), low temperture het reservoir (where you send the het you don t use) sometimes this is lled the low temperture het sink. For gret mny het engines, the het sink will e the tmosphere think of gsoline r engine. Sttionry eletri power plnts might use wter from river. A stem ship would use the se, nd so on. Most engines hve yli opertion steps tht re repeted over nd over in order to do the work. A r engine hs pistons tht repetedly move up nd down produing power from the motion of the piston. Effiieny: he het tht is not onverted to work is lled the wste het. All het engines produe wste het. he wste het is sent to the het sink. he first lw tells us tht the engine n never produe more work thn the het tht went into it. After ll, energy hs to e onserved, orret? he rtio of output work to input work/energy is lled effiieny. e W Q out his is written s in W e for the A hysis est. Q his simply sys tht the effiieny is the solute vlue of the rtio of output work to input het. he first lw sys tht the effiieny nnot e greter thn one. But we ve lredy sid tht some of the input het does not eome work. Doesn t this men tht the effiieny hs to e less thn one? Well, so it is. his ws disovered in 1824 y Sdi Crnot (tully Niols-Leonrd-Sdi Crnot he hd lot of nmes for Frenh physiist). If n engine ouldn t e one hundred perent effiient, wht ws the est you ould do? Crnot set out to find the nswer to tht question. In his pursuit, he invented thermodynmis. 723

2 Crnot imgined the est possile engine, this would e n idel engine it wouldn t hve frition etween its moving prts nd it wouldn t lose het through the wlls of its ylinders, et. An idel engine would hve no hnge in its internl energy. For the engine: U Q W U 0 so Q W he work done y this idel engine is simply equl to the hnge in het, Q. W Q Q C Q is the high temperture nd Q C is the low temperture. We plug this into the effiieny eqution (for like one yle): W done in one yle e Q dded in one yle e W Q lug in vlue for work we figured out: In W Q QC e But Q in is simply Q. Q Q In in Q Q e Q Q is proportionl to so: e C C his is the effiieny of n idel energy. It is the est effiieny tht it is possile to otin. e C he tempertures must e in Kelvins! No rel engine n hieve this effiieny, ut it is possile to ome lose. he eqution is importnt euse it shows us tht the effiieny depends on the two temperture extremes used y the engine. he higher the temperture the engine opertes t nd the lower the temperture of the het reservoir, the higher the effiieny will e. 724

3 olkien & ehnology: A uli Rdio Commentry y Bill mmk J.R.R. olkien's Lord of the Rings enhnts euse it let's us espe into nother world lled "Middle Erth." Yet, odd s this fntsy world is, it rries n importnt messge for our world. olkien pled t the enter of his sg the question of how tehnology fits into our lives. he story ppers to e out the Quest to destroy ring with inredile powers, ut hidden not elow the surfe is ler messge out tehnology. hroughout the Lord of the Rings olkien often hrterizes evil s tehnology. For exmple, one of the mjor villins, Wizrd lled Srumn, lives in ple olkien lls "Isengrd." olkien, who ws n Oxford rofessor of Anglo-Sxon, knew Isengrd ment "iron yrd", wht we might ll n industril prk. Inside tht iron yrd the evil Wizrd Srumn spends his dys uilding mills, hopping down forest, nd lowing things up. e retes system of tunnels nd dms, nd vents for poisonous gses nd fires. olkien writes tht "wheels nd engines nd explosions lwys delight" Surmn nd his followers. he ide of mhines ppers gin when he desries the evil Srumn s hving " mind of metl nd wheels." In ontrst to this evil were the oits. A simple, smll people who hve n grrin eonomy. olkien one wrote to friend "I m in ft oit (in ll ut size). I like grdens, trees nd unmehnized frmlnds; I smoke pipe, nd [I] like good plin food (unrefrigerted)... olkien lived life s opposed s possile to tehnology. During his lifetime he rejeted trins, television nd refrigerted food. e did own r, ut sold it t the eginning of World Wr II. By tht time olkien pereived the dmge rs nd their new rods were doing to the lndspe. e me to think of the internl omustion engine s the gretest evil ever put upon this Erth. is experienes with wr olored his view of tehnologil hnge. e served in the trenhes during World Wr I nd experiened tehnology s fighter plnes, tnks, omings, nd flmethrowers. "By 1918," he one sid, "ll ut one of my lose friends were ded." Smll wonder he disliked the immense power ehind tehnology. In mny wys the gret theme of the Lord of the Rings is tht no one should hve dominion over the world. he Lord of the Rings is n nti-quest, with its gol to destroy universl power forever. erein lies olkien's messge to us, wht mke his Lord of the Rings still ring true tody. e refused to let the mteril world drw the oundries of life, nd though his smll oits he sserted the individul's right nd responsiility to shpe the deisions nd strutures tht determine their life. Diesel engines re more effiient thn gsoline engines euse they operte t higher temperture. It is expensive to uild mhines tht n operte t high tempertures you need ostly metl lloys tht n hndle the temperture, the engine must e stronger euse it is doing more work, so its omponents hve to e eefed up, nd so on. Engine designers hve to lne trdeoffs like tht etween effiieny nd mnufturing expense. A stem engine opertes on wrm 28.0 C dy. he sturted stem opertes t temperture of C. Wht is the idel effiieny for this engine? 725

4 An importnt thing to rememer is tht the tempertures in the eqution must e in Kelvins. hot = C = 373 K ool = 28.0 C = 301 K e h h C 373 K 301 K e 373 K he effiieny n e left s deiml frtion or it n e expressed s perent. o inrese the effiieny of rel engine, one n: Inrese the temperture of the high temperture het reservoir. Derese the het sink temperture. Do oth. he other thing tht n e done is to try nd mke it ehve s lose to n idel engine s possile. his n e done y eliminting frition, deresing the mss of moving prts, &t. No rel engine n hve this effiieny. Eliminting frition nd ll the other good engineering things tht n e done will help the engine pproh the idel effiieny. Unfortuntely it will never reh this level of performne. Wht shme. Digrms: ressure/volume grphs re of tremendous vlue in nlyzing the performne of het engines. Let s look t prt of thermodynmi yle. We will look t the work for eh step in the proess. 1 2 ere is the first step - the proess represents n isori ompression of gs. From to, the pressure remins onstnt its vlue is 1. he volume derese from to 2. his represents work done on the system. It tkes work to ompress the gs. he work is the re under the urve, whih is onveniently shded in so you 2 n see it. he work is: 1. W Now let s look t the next step in the yle. You n see the digrm to the right

5 roess is n isohori ompression. Isohori euse the volume does not hnge ut the pressure inreses. he work for this step is zero. his is euse is zero. W d he next step is proess d. his is n isori expnsion. he gs expnds from to 2, doing work s it expnds. he mount of work is equl to the re under the urve. W he finl step to omplete the yle is proess d. his is n isohori expnsion. he volume stys onstnt, so no work. he pressure dereses. 2 d W he whole yle looks like the grph to the right. he net work done y the mhine in one yle is the re enlosed y the urve. It is the sum of the work, positive or negtive, for eh step in the yle. 2 d he work is just the differene in the two res under the urve tht we hd. W he work done in yle depends on the pth. Let s look t the previous yle nd ompre it with one tht hs different pth

6 2 d Cyle 1 Cyle 2 Cyle 1 is the one we just looked t. he gs is ompressed from 2 to, ompressed from 1 to 2. hen it is expnded from to 2 nd expnded from 2 to 1. retty muh the sme thing hppens in yle 2, exept tht the two expnsions tke ple in only one step insted of two. his differene is signifint however s the net work for yle 1 is greter (twie the mgnitude) thn the net work for yle 2. A het engine s yle is shown in the digrm to the right. 1 = 345 k, 2 = 245 k, 3 = 125 k, nd 4 = 225 k. = 35.0 L nd 2 = 85.0 L. Wht is the net work done during one yle of the engine? We n solve this y finding the re under the urve for the step. hen we find the re under the urve for the d step. he net work is the differene in the two res. Find volume in m 3 : m L 35 x10 m m 1 L m L m 1 L Are for : W A Are Ari Are m m J 1 Ari d 2 728

7 1 Ari m m J W J J J Are for d: Wd Ad Are Ari 3 3 Are m m J 1 Ari Ari m m J W J J J W J J J 6.00 kj Net A sustne undergoes yli proess shown in the grph. A work output ours long pth, work input is required long pth, nd no work is involved in onstnt volume proess. et trnsfer ours during eh proess in the yle. () wht is the work output during proess? () how muh work input is required during proess? () Wht is the net work done during the yle? W A A A () re ri x10 10 m x10 10 m W 1 tm 40 L 4 tm 40 L 1tm 1 L 2 1tm 1 L W 122 x10 J 1.22 x10 J Expnsion proess, so negtive work (tm) (L) () his is positive work. W = re of retngle x10 10 m 2 3 W 1 tm 40 L 40.5 x10 J 4.05 x10 J 1tm 1 L () Wnet W W W 729

8 4 4 3 W 1.22 x10 J x10 J x10 J net he Crnot Cyle: Sdi Crnot envisioned perfet mhine tht would hve the gretest possile effiieny tht it ould possily hve. We ve lredy seen how the eqution for this effiieny ws developed. But wht kind of mhine ould do tht? Well, the mhine tht Crnot me up with is simple piston/ylinder devie. he operting sequene of the thing is lled the Crnot yle. ere is the devie (the drwing ove). he sides nd top of the ylinder re insulted so het nnot flow in or out of the system. he ottom is mde of n idel ondutor so tht het n flow in or out of the system through the ottom of the ylinder. hree stnds re ville for ylinder plement; hot stnd, old stnd, nd perfet insultor stnd. Let s see how the yle works. Step1: Isotherml expnsion. he ylinder is pled on high temperture het sink tht is t. he het is onduted through the ottom of the ylinder nd the gs sors het. We ll this het Q in. As result of the dded het, the gs expnds, pushing the piston upwrd. his step does some work. We ll this step isotherml expnsion euse the temperture stys onstnt (isotherml mens onstnt temperture) nd the volume inreses. he work done is equl to. his step is represented s the urve AB on the digrm elow. Step 2: Aditi expnsion: he ylinder is immeditely moved from the hot stnd to the insulted stnd. One it is pled on the insulted stnd, het n no longer flow into the system (or out of it). he gs ontinues to expnd, ut sine het is no longer entering or leving, this is n diti expnsion. he pressure in the ylinder drops to its lowest vlue. his is represented y the urve BC on the digrm. Work ontinues to e done y the system during this step. Step 3: Isotherml ompression: he ylinder is immeditely ple on the low temperture het sink. et, whih we ll Q out, flows from the ylinder into the het sink. he system loses 730

9 het, the volume dereses, nd the gs is ompressed isothermlly. his is represented y the urve CD on the digrm. Step 4: Aditi ompression: he ylinder is ple on the insulted stnd. et no longer n enter or leve, so the system undergoes diti ompression k to its originl stte long the urve AD. he het engine is now redy to undergo nother yle. ere is the digrm for the Crnot yle. A to B -- isotherml expnsion. inreses nd dereses B to C -- diti expnsion. C to D -- isotherml ompression D to A -- diti ompression Work is done long the urves AB nd BC. he effiieny of the Crnot yle, how well it opertes, depends on the sorption of het nd the loss of het in the respetive steps of the yle. One of the key ftors tht ontrols the flow of het is the temperture differene. his would e the temperture of the hot stnd nd the old stnd. When pled on the hot stnd, het flows into the ylinder nd it rehes (if left long enough) the sme temperture s the hot stnd. When pled on the old stnd, the temperture differene is equl to the hot temperture minus the old temperture (of the old stnd). If is inresed, het will flow fster nd the mhine will operte more effiiently. So the higher the hot temperture reservoir (the hot stnd), the greter the mount of het sored y the system. Also, if we derese the old temperture reservoir, this too will inrese the mount of het tht flows. Inrese the het flow nd you inrese the effiieny of the system. 731

10 Der Dotor Siene, You know those little irds filled with red stuff tht o up nd down drinking from glss of wter? I n't mke mine stop, even when I tke wy his wter. e won't stop. lese help! -- Joey erry from Springfield, MO. Dr. Siene responds: hose little irds re tully perpetul motion mhines. One you get them strted, they'll never stop -- ever -- not until the sun winds down nd our glxy goes nov. Even then, somewhere in the imploded lk holes tht ws one our solr system, little irds will e o-o-oing -- even in the noiseless vuum of spe. he "red stuff" inside these irds is tully neutrino solution kept in ple y mysterious new fore in the universe lled hyperhrge, whih is ross etween nuler ond nd superglue. Fortuntely, most of these irds re wht siene lls "roken" nd will not o t ll. his is true, t lest for most of us, who ought these little irds t irport gift shops nd we n only thnk the powers-tht-e. Otherwise, they'd just give us the reeps. Boing, oing, oing through eternity, through entropy, oing, oing, oing, oing... Solitude Ell Wheeler Wilox Lugh, nd the world lughs with you; Weep, nd you weep lone. For the sd old erth must orrow its mirth, But hs troule enough of its own. Sing, nd the hills will nswer; Sigh, it is lost on the ir. he ehoes ound to joyful sound, But shrink from voiing re. Fest, nd your hlls re rowded; Fst, nd the world goes y. Sueed nd give, nd it helps you live, But no mn n help you die. here is room in the hlls of plesure For long nd lordly trin, But one y one we must ll file on hrough the nrrow isles of pin. Rejoie, nd men will seek you; Grieve, nd they turn nd go. hey wnt full mesure of ll your plesure, But they do not need your woe. Be gld, nd your friends re mny; Be sd, nd you lose them ll. here re none to deline your netred wine, But lone you must drink life s gll. 732

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

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?

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

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

More information

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( )

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

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

Chapter 19: The Second Law of Thermodynamics

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

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

NON-DETERMINISTIC FSA

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

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Non Right Angled Triangles

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

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

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

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

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

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

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

More information

QUADRATIC EQUATION. Contents

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,

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Pythagoras theorem and surds

Pythagoras theorem and surds HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors

More information

Chapter 4 The second law of thermodynamics

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

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Naming the sides of a right-angled triangle

Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h

More information

(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.

(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

More information

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

More information

Section 7.1 Area of a Region Between Two Curves

Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt

More information

PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS 211 MIDTERM I 21 April 2004 PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

More information

dy ky, dt where proportionality constant k may be positive or negative

dy ky, dt where proportionality constant k may be positive or negative Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Chapter 6 Continuous Random Variables and Distributions

Chapter 6 Continuous Random Variables and Distributions Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

More information

Nondeterministic Finite Automata

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)

More information

Dense Coding, Teleportation, No Cloning

Dense Coding, Teleportation, No Cloning qitd352 Dense Coding, Teleporttion, No Cloning Roert B. Griffiths Version of 8 Ferury 2012 Referenes: NLQI = R. B. Griffiths, Nture nd lotion of quntum informtion Phys. Rev. A 66 (2002) 012311; http://rxiv.org/rhive/qunt-ph/0203058

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

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

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

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

More information

5.1 Estimating with Finite Sums Calculus

5.1 Estimating with Finite Sums Calculus 5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

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

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3. . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough

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

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

State Minimization for DFAs

State Minimization for DFAs Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

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

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Lecture 7 notes Nodal Analysis

Lecture 7 notes Nodal Analysis Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

For the flux through a surface: Ch.24 Gauss s Law In last chapter, to calculate electric filede at a give location: q For point charges: K i r 2 ˆr

For the flux through a surface: Ch.24 Gauss s Law In last chapter, to calculate electric filede at a give location: q For point charges: K i r 2 ˆr Ch.24 Guss s Lw In lst hpter, to lulte eletri filed t give lotion: q For point hrges: K i e r 2 ˆr i dq For ontinuous hrge distributions: K e r 2 ˆr However, for mny situtions with symmetri hrge distribution,

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb. CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

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

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

Lecture 2 : Propositions DRAFT

Lecture 2 : Propositions DRAFT CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte

More information

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using

More information

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter 8A Length nd perimeter 8E Are of irle 8B Cirumferene of irle 8F Surfe re 8C Are of retngles nd tringles 8G Volume of prisms 8D Are of other qudrilterls 8H Are nd volume onversions SA M PL E Mesurement

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Section 6.1 Definite Integral

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

More information

Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration. Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. = - o t = Δ Δt ccelertion = = - o t chnge of elocity elpsed time ccelertion is ector, lthough

More information

Solutions to Assignment 1

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

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

More information

Quadratic reciprocity

Quadratic reciprocity Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

More information

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

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

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

Effects of Drought on the Performance of Two Hybrid Bluegrasses, Kentucky Bluegrass and Tall Fescue

Effects of Drought on the Performance of Two Hybrid Bluegrasses, Kentucky Bluegrass and Tall Fescue TITLE: OBJECTIVE: AUTHOR: SPONSORS: Effets of Drought on the Performne of Two Hyrid Bluegrsses, Kentuky Bluegrss nd Tll Fesue Evlute the effets of drought on the visul qulity nd photosynthesis in two hyrid

More information

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the

More information

Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}

Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)} Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)

More information

Functions. mjarrar Watch this lecture and download the slides

Functions. mjarrar Watch this lecture and download the slides 9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

More information

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air Deprtment of Mechnicl Engineering ME 3 Mechnicl Engineering hermodynmics Lecture 33 sychrometric roperties of Moist Air Air-Wter Vpor Mixtures Atmospheric ir A binry mixture of dry ir () + ter vpor ()

More information

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

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

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

CS 360 Exam 2 Fall 2014 Name

CS 360 Exam 2 Fall 2014 Name CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output

More information

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

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

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information