Some Useful Integrals of Exponential Functions
|
|
- Grace Flowers
- 7 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 right-ngld 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 x-intgrl nd th y-intgrl, 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 informationAC Circuits Three-Phase Circuits
AC Circuits Thr-Phs Circuits Contnts Wht is Thr-Phs Circuit? Blnc Thr-Phs oltgs Blnc Thr-Phs Connction Powr in Blncd Systm Unblncd Thr-Phs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml
More informationInstruction: Solving Exponential Equations without Logarithms. This lecture uses a four-step process to solve exponential equations:
49 Instuction: Solving Eponntil Equtions without Logithms This lctu uss fou-stp 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 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 Shortst-Pth 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 AL-Zlzlh nd Bsl M. AL-Eidh Kuwit Univrsity, Kuwit Abstrct. In this
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 12-13)
con 37: Answr Ky for Problm St (Chaptr 2-3) 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationQuality and Pricing for Outsourcing Service: Optimal Contract Design
Qulity nd Pricing for Outsourcing Srvic: Optiml Contrct Dsign Smr K. Mukhopdhyy Univrsity of Wisconsin-Milwuk Co-uthor: Xiowi Zhu, Wst Chstr Univrsity of PA Third nnul confrnc, POMS Collg of Srvic Oprtions
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 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 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 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 informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 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 informationLast time Interprocedural analysis Dimensions of precision (flow- and context-sensitivity) Flow-Sensitive Pointer Analysis
Flow-Insnsitiv Pointr Anlysis Lst tim Intrprocurl nlysis Dimnsions of prcision (flow- n contxt-snsitivity) Flow-Snsitiv Pointr Anlysis Toy Flow-Insnsitiv Pointr Anlysis CIS 570 Lctur 12 Flow-Insnsitiv
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 informationVersion 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (A-lvl) 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 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 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 not-for-profit 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 rel-vlued
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 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 informationWarm-up for Differential Calculus
Summer Assignment Wrm-up 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 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 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 so-clled volumes of
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 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 t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
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 informationAP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos
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 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 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 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 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 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 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
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 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 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 informationChapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlx-valud 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 informationChange Your History How Can Soccer Knowledge Improve Your Business Processes?
Symposium Inuurl Lctur o Hjo Rijrs, VU, 26-6-2015 Chn Your History How Cn Soccr Knowl Improv Your Businss Procsss? Wil vn r Alst TU/ n DSC/ 1970 born Oostrbk 1988-1992 CS TU/ 1992-1994 TS TU/ 1994-1996
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 ral-valud Fourir sris is xplaind and formula ar givn for convrting
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 informationPhysics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves
Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationRemember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D
24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd
More informationWho uses our services? We have a growing customer base. with institutions all around the globe.
not taking xpr Srvic Guid 2013 / 2014 NTE i an affordabl option for audio to txt convrion. Our rvic includ not or dirct trancription rvic from prviouly rcordd audio fil. Our rvic appal pcially to tudnt
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More informationbaby on the way, quit today
for mums-to-be bby on the wy, quit tody WHAT YOU NEED TO KNOW bout smoking nd pregnncy uitting smoking is the best thing you cn do for your bby We know tht it cn be difficult to quit smoking. But we lso
More informationGeometry 7-1 Geometric Mean and the Pythagorean Theorem
Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationCategory 7: Employee Commuting
7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationLogo Design/Development 1-on-1
Logo Dsign/Dvlopmnt 1-on-1 If your company is looking to mak an imprssion and grow in th marktplac, you ll nd a logo. Fortunatly, a good graphic dsignr can crat on for you. Whil th pric tags for thos famous
More informationCURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
More informationCHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil
More information[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
More informationCapacitance and Dielectrics
Chaptr 5 Capacitanc and Dilctrics 5.1 Introduction...5-3 5. Calculation of Capacitanc...5-4 Exampl 5.1: Paralll-Plat Capacitor...5-4 Intractiv Simulation 5.1: Paralll-Plat Capacitor...5-6 Exampl 5.: Cylindrical
More informationSection 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationAxial flow rate (per unit circumferential length) [m 2 /s] R B, R J =R Bearing Radius ~ Journal Radius [m] S
NOTE 4 TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING Lctur 4 introducs th fundamnts of journal baring analysis. Th long and short lngth baring modls ar introducd. Th prssur fild in a short lngth baring
More informationTheoretical aspects of investment demand for gold
Victor Sazonov (Russia), Dmitry Nikolav (Russia) Thortical aspcts of invstmnt dmand for gold Abstract Th main objctiv of this articl is construction of a thortical modl of invstmnt in gold. Our modl is
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
More informationNerveCenter Protocol and Perl Metrics. November 2014 NCSD-PPM-01
rvcntr Procol nd Prl Mtrics ovbr 2014 CSD-PPM-01 Procol nd Prl Mtrics Strting in rvcntr 6.1 Bld28, th nccd cond lin utility supports gnrting trics for rvcntr Srvr s procol lyr nd Prl intrprtrs. Cling upon
More informationFEE-HELP INFORMATION SHEET FOR DOMESTIC FULL FEE STUDENTS
FEE-HELP INFORMATION SHEET FOR DOMESTIC FULL FEE STUDENTS This is n infomtion sht poducd by th Monsh Lw Studnts Socity Juis Docto Potfolio to ssist full f pying studnts (domstic) in undstnding th issus
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
More information