Some Useful Integrals of Exponential Functions


 Grace Flowers
 2 years ago
 Views:
Transcription
1 prvious indx nxt Som Usful Intgrls of Exponntil Functions Michl Fowlr W v shown tht diffrntiting th xponntil function just multiplis it by th constnt in th xponnt, tht is to sy, d x x Intgrting th xponntil function, of cours, hs th opposit ffct: it divids by th constnt in th xponnt: x 1 s you cn sily chck by diffrntiting both sids of th qution An importnt dfinit intgrl (on with limits) is x, 0 x 1 Notic th minus sign in th xponnt: w nd n intgrnd tht dcrss s x gos towrds infinity, othrwis th intgrl will itslf b infinit To visuliz this rsult, w plot blow x nd 3x Not tht th grn lin forms th hypotnus of rightngld tringl of r 1, nd it is vry plusibl from th grph tht th totl r undr th x curv is th sm, tht is, 1, s it must b Th 3x curv hs r 1/3 undr it, ( 3)
2 Now for somthing bit mor chllnging: how do w vlut th intgrl x I? ( hs to b positiv, of cours) Th intgrl will dfinitly not b infinit: it flls off qully fst in both positiv nd ngtiv dirctions, nd in th positiv dirction for x grtr thn 1, it s smllr thn x, which w know convrgs To s bttr wht this function looks lik, w plot it blow for 1 (rd) nd 4 (blu)
3 3 Notic first how much fstr thn th ordinry xponntil x this function flls wy Thn not tht th blu curv, 4, hs bout hlf th totl r of th 1 curv In fct, th r gos s 1/ Th grn lins hlp s tht th r undr th rd curv (positiv plus ngtiv) is somwht lss thn, in fct it s 177 pproximtly But it s not so sy to vlut! Thr is trick: squr it Tht is to sy, writ ( ) x y I dy Now, this product of two intgrls long lins, th xintgrl nd th yintgrl, is xctly th sm s n intgrl ovr pln, th (x, y) pln, strtching to infinity in ll dirctions W cn rwrit it x y x y r dy dy dy whr r x + y, r is just th distnc from th origin (0, 0) in th (x, y) pln Th pln is dividd up into tiny squrs of r dy, nd doing th intgrl mounts to finding th vlu of r for ch tiny squr, multiplying by th r of tht squr, nd dding th contributions from vry squr in th pln In fct, though, this pproch is no sir thn th originl problm th trick is to notic tht r th intgrnd hs circulr symmtry: for ny circl cntrd t th origin (0, 0), it hs th sm vlu nywhr on th circl To xploit this, w shouldn t b dividing th (x, y) pln up
4 4 into littl squrs t ll, w should b dividing it into rgions hving ll points th sm distnc from th origin Ths r clld nnulr rgions: th r btwn two circls, both cntrd t th origin, th innr on of rdius r, th slightly biggr outr on hving rdius r + dr W tk dr to b vry smll, so this is thin circulr strip, of lngth r (th circumfrnc of th circl) nd brdth dr, nd thrfor its totl r is rdr (nglcting trms lik dr, which bcom ngligibl for dr smll nough) So, th contribution from on of ths nnulr rgions is ovr th whol pln is: r rdr, nd th complt intgrl ( I ) 0 r rdr This intgrl is sy to vlut: mk th chng of vribl to u r, du rdr giving so tking squr roots u I du ( ) 0 x I
5 5 Som Intgrls Usful in th Kintic Thory of Gss W cn sily gnrt mor rsults by diffrntiting I() bov with rspct to th constnt! Diffrntiting onc: so w hv x x d 1 d x d d x 1 x nd diffrntiting this rsult with rspct to givs 4 x x 3 4 Th rtio of ths two intgrls coms up in th kintic thory of gss in finding th vrg kintic nrgy of molcul with Mxwll s vlocity distribution 4 x x x x Finding this rtio without doing th intgrls: It is intrsting to not tht this rtio could hv bn found with much lss work, in fct without vluting th intgrls fully, s follows: Mk th chng of vribl y x, so dy nd 3/ x y 3/ x y dy C whr C is constnt indpndnt of, bcus hs compltly dispprd in th intgrl ovr y (Of cours, w know C /, but tht took lot of work) Now, th intgrl with x 4 in plc of x is givn by diffrntiting th x intgrl with rspct to, nd multiplying by 1, s discussd bov, so, diffrntiting th right hnd sid of th bov qution, th x 4 intgrl is just ( 3/) C 5/, nd th C cncls out in th rtio of th intgrls
6 6 Howvr, w do nd to do th intgrls t on point in th kintic thory: th ovrll normliztion of th vlocity distribution function is givn by rquiring tht 1 f( v) dv 4 A v mv /kt nd this in fct dtrmins A, using th rsults w found bov, giving 3/ m mv /kt f() v 4 v kt On lst trick W didn t nd this in th kintic thory lctur, but is sms pity to rviw xponntil intgrls without mntioning it It s sy to do th intgrl It cn b writtn x bx I b (, ) x b b /4 b /4 x b b /4 ( / ) ( / ) I, b / whr to do th lst stp just chng vribls from x to y x  b/ This cn vn b usd to vlut for xmpl x cosbx by writing th cosin s sum of xponntils prvious indx nxt
Higher. Exponentials and Logarithms 160
hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions
More informationChapter 3 Chemical Equations and Stoichiometry
Chptr Chmicl Equtions nd Stoichiomtry Homwork (This is VERY importnt chptr) Chptr 27, 29, 1, 9, 5, 7, 9, 55, 57, 65, 71, 75, 77, 81, 87, 91, 95, 99, 101, 111, 117, 121 1 2 Introduction Up until now w hv
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More informationAC Circuits ThreePhase Circuits
AC Circuits ThrPhs Circuits Contnts Wht is ThrPhs Circuit? Blnc ThrPhs oltgs Blnc ThrPhs Connction Powr in Blncd Systm Unblncd ThrPhs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml
More informationInstruction: Solving Exponential Equations without Logarithms. This lecture uses a fourstep process to solve exponential equations:
49 Instuction: Solving Eponntil Equtions without Logithms This lctu uss foustp pocss to solv ponntil qutions: Isolt th bs. Wit both sids of th qution s ponntil pssions with lik bss. St th ponnts qul to
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationFundamentals of Tensor Analysis
MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006 Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss;
More informationSHAPES AND SHAPE WORDS!
1 Pintbl Activity Pg 1 SAPES AND SAPE WORDS! (bst fo 1 o plys) Fo ch child (o pi of childn), you will nd: wo copis of pgs nd Cyons Scissos Glu stick 10 indx cds Colo nd Mk Shp Cds! Giv ch child o pi of
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More informationReading. Minimum Spanning Trees. Outline. A File Sharing Problem. A Kevin Bacon Problem. Spanning Trees. Section 9.6
Rin Stion 9.6 Minimum Spnnin Trs Outlin Minimum Spnnin Trs Prim s Alorithm Kruskl s Alorithm Extr:Distriut ShortstPth Alorithms A Fil Shrin Prolm Sy unh o usrs wnt to istriut il monst thmslvs. Btwn h
More informationImportant result on the first passage time and its integral functional for a certain diffusion process
Lcturs Mtmátics Volumn 22 (21), págins 5 9 Importnt rsult on th first pssg tim nd its intgrl functionl for crtin diffusion procss Yousf ALZlzlh nd Bsl M. ALEidh Kuwit Univrsity, Kuwit Abstrct. In this
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationhttp://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationLong run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
More informationCayley s Formula. Graphs  II The number of labeled trees on n nodes is n n2. Planar Graphs. Is K 5 planar? Outline. K 5 can be embedded on the torus
Grt Thortil Is In Computr Sin Vitor Amhik CS 15251 Crngi Mllon Univrsity Cyly s Formul Grphs  II Th numr of ll trs on n nos is n n2 Put nothr wy, it ounts th numr of spnning trs of omplt grph K n. 4
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationStatistical Machine Translation
Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More informationMATH 181Exponents and Radicals ( 8 )
Mth 8 S. Numkr MATH 8Epots d Rdicls ( 8 ) Itgrl Epots & Frctiol Epots Epotil Fuctios Epotil Fuctios d Grphs I. Epotil Fuctios Th fuctio f ( ), whr is rl umr, 0, d, is clld th potil fuctio, s. Rquirig
More informationFundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY
Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl
More informationDecember Homework Week 1
Dcmbr Hmwrk Wk 1 Mth Cmmn Cr Stndrds: K.CC.A.1  Cunt t 100 by ns nd by tns. K.CC.A.2  Cunt frwrd bginning frm givn numbr within th knwn squnc (instd f hving t bgin t 1). K.CC.B.4.A  Whn cunting bjcts,
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationCIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)
6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationQuality and Pricing for Outsourcing Service: Optimal Contract Design
Qulity nd Pricing for Outsourcing Srvic: Optiml Contrct Dsign Smr K. Mukhopdhyy Univrsity of WisconsinMilwuk Couthor: Xiowi Zhu, Wst Chstr Univrsity of PA Third nnul confrnc, POMS Collg of Srvic Oprtions
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More information7 Timetable test 1 The Combing Chart
7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationSPECIAL VOWEL SOUNDS
SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationChapter 5 Capacitance and Dielectrics
5 5 5 Chaptr 5 Capacitanc and Dilctrics 5.1 Introduction... 53 5. Calculation of Capacitanc... 54 Exampl 5.1: ParalllPlat Capacitor... 54 Exampl 5.: Cylindrical Capacitor... 56 Exampl 5.3: Sphrical
More informationGot diabetes? Thinking about having a baby?
Gt dibts? Thinking but hving bby? U.S. Dprtmnt f Hlth nd Humn Srvics Cntrs fr Diss Cntrl nd Prvntin Dibts nd Prgnncy Dibts is cnditin in which yur bdy cnnt us sugrs nd strchs (crbhydrts) frm fd t mk nrgy.
More informationRational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and
Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationProjections  3D Viewing. Overview Lecture 4. Projection  3D viewing. Projections. Projections Parallel Perspective
Ovrviw Lctur 4 Projctions  3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl  Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog
More informationLast time Interprocedural analysis Dimensions of precision (flow and contextsensitivity) FlowSensitive Pointer Analysis
FlowInsnsitiv Pointr Anlysis Lst tim Intrprocurl nlysis Dimnsions of prcision (flow n contxtsnsitivity) FlowSnsitiv Pointr Anlysis Toy FlowInsnsitiv Pointr Anlysis CIS 570 Lctur 12 FlowInsnsitiv
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationHANDOUT E.19  EXAMPLES ON FEEDBACK CONTROL SYSTEMS
MEEN 64 Parauram Lctur 9, Augut 5, HANDOUT E9  EXAMPLES ON FEEDBAC CONTOL SSTEMS Exampl Conidr th ytm hown blow Th opn loop tranfr function i givn by Th clod loop tranfr function i Exampl Conidr th ytm
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationVersion 1.0. General Certificate of Education (Alevel) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (Alvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationNetwork Analyzer Error Models and Calibration Methods
Ntwork Anlyzr Error Modls nd Clirtion Mthods y Doug Rytting Pg This ppr is n ovrviw of rror modls nd clirtion mthods for vctor ntwork nlyzrs. Prsnttion Outlin Ntwork Anlyzr Block Digrm nd Error Modl ystm
More informationBatteries in general: Batteries. Anode/cathode in rechargeable batteries. Rechargeable batteries
Bttris i grl: Bttris How bsd bttris work A rducig (gtiv) lctrod A oxidizig (positiv) lctrod A  th ioic coductor Rchrgbl bttris Rctios ust b rvrsibl Not too y irrvrsibl sid rctios Aod/cthod i rchrgbl
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationc d b a f(a,b,c,d,e) d e d e e b b c
85 Chptr 6 Exri 6. From Exri 5., w know tht th inglrror ttor for 2outof5 o (; ; ; ; ) i implmnt y th xprion: E(; ; ; ; ) = + + + + + + + + + + + + + Uing only gt from Tl 4. of th txtook w n gnrt ll
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationCPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.
Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More information3. Building a Binary Search Tree. 5. Splay Trees: A SelfAdjusting Data Structure
Chptr 10 BINARY TREES 1. Gnrl Binry Trs 2. Binry Srch Trs 3. Builing Binry Srch Tr 4. Hight Blnc: AVL Trs 5. Sply Trs: A SlfAjusting Dt Structur Outlin Trnsp. 1, Chptr 10, Binry Trs 243 1999 PrnticHll,
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationBasis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationSchedule C. Notice in terms of Rule 5(10) of the Capital Gains Rules, 1993
(Rul 5(10)) Shul C Noti in trms o Rul 5(10) o th Cpitl Gins Ruls, 1993 Sttmnt to sumitt y trnsror o shrs whr thr is trnsr o ontrolling intrst Prt 1  Dtils o Trnsror Nm Arss ROC No (ompnis only) Inom Tx
More informationLesson 12.1 Trigonometric Ratios
Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml
More informationModern Portfolio Theory (MPT) Statistics
Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationSection 55 Inverse of a Square Matrix
 Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationPage 1 of 16 Fall 2011: 5.112 PRINCIPLES OF CHEMICAL SCIENCE Problem Set #4 solutions This problem set was graded out of 99 points
Pag of 6 Fall 0: 5. PRINCIPLES OF CHEMICAL SCIENCE Problm St #4 solutions This problm st was gradd out of 99 points Qustion (out of 7 points) (i) For ach pair, dtrmin which compound has bonds with gratr
More informationUses for Binary Trees  Binary Search Trees
CS122 Algorithms n Dt Struturs MW 11:00 m 12:15 pm, MSEC 101 Instrutor: Xio Qin Ltur 10: Binry Srh Trs n Binry Exprssion Trs Uss or Binry Trs Binry Srh Trs n Us or storing n rtriving inormtion n Insrt,
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationVan der Waals Forces Between Atoms
Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu
More informationRenewable Energy Sources. Solar Cells SJSUE10 S John Athanasiou
Rnwabl Enrgy Sourcs. Solar Clls SJSUE10 S2008 John Athanasiou 1 Rnwabl Enrgy Sourcs Rnwabl: Thy can last indfinitly 1. Wind Turbin: Convrting th wind nrgy into lctricity Wind, Propllr, Elctric Gnrator,
More informationChange Your History How Can Soccer Knowledge Improve Your Business Processes?
Symposium Inuurl Lctur o Hjo Rijrs, VU, 2662015 Chn Your History How Cn Soccr Knowl Improv Your Businss Procsss? Wil vn r Alst TU/ n DSC/ 1970 born Oostrbk 19881992 CS TU/ 19921994 TS TU/ 19941996
More informationChapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlxvalud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationAP Calculus MultipleChoice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus MultiplChoic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More information