Supporting information

Size: px
Start display at page:

Download "Supporting information"

Transcription

1 upportng nformton for Towrds more useful n vtro toxcty dt wth mesured concentrtons by M.B. Herng R.H.M.M. chreurs F. Busser.T. vn der g B. vn der Burg nd J..M. Hermens Contnng 0 pges 2 tbles nd 2 fgures.

2 upplementry nformton on the dscusson of the reltve potences. Tble 3 n the mnuscrpt shows tht the rp nomnl of octylphenol slghtly decreses wth ncresng serum percentge (down to fctor 2 lower). We doubted whether ths ws rel trend or ust concdentl mesurement vrblty. Therefore we modelled the dependency of the nomnl reltve potency on serum content wth respect to bndng to lbumn nd HBG. The relton between rp nomnl nd rp of octylphenol cn be descrbed s n equton : rp ff rpnomn l () ffo In ths equton the frctons re those t the C 50nomnl of the compound n queston nd the nd O n the subscrpts denote estrdol nd octylphenol respectvely. The frcton of lgnd (ff ) cn be derved from equton 2 (see lso ppendx I). ff totl ff totl ff (2) In ths equton ff nd ff re the unoccuped ( ) frcton of lbumn nd HBG. Combnton of equtons nd 2 yelds equton 3. rp no mn l rp O totl totl ff ff O O totl totl ff ff O (3) In ths equton ff nd ff re the unoccuped frctons of lbumn nd HBG respectvely when there s only estrdol present nd ff O nd ff O re those when there s only octylphenol present. Bsed on equton 3 model ws wrtten nd run wth Berkeley Mdonn of whch the scrpt cn be found n ppendx III. rmeter vlues from tble n the rtcle were used s well s vlues for some ddtonl prmeters lsted n tble. Fgure shows how the nomnl reltve potency of octylphenol decreses wth ncresng serum content wth only mnor decrese n the rnge of 5-50% serum s ws used n our experments. To verfy f other compounds wth usully lower ffnty for lbumn mght show more pronounced effect of serum the ffnty constnt of octylphenol ws replced by those of dethylstlbestrol (D) nd pp -DDT (see tble ). Note tht for these two compounds the ntrnsc estrogenc potency of octylphenol ws tken; therefore the bsolute vlue of the nomnl reltve potences clculted for these compounds s unrelble. The trend n nomnl reltve potency however s relstc becuse tht s determned by the compound-specfc proten bndng.

3 Nomnl reltve potency % 0.00% 0.00% 0.0%.0% 0% 00% erum % Fgure. Modelled effect of serum % on nomnl reltve estrogenc potency of octylphenol (thnnest lne) D (mddle lne) nd pp -DDT (thckest lne) bsed on n ntrnsc reltve potency of 0-3 (note tht ths s not the rel reltve potency of D nd pp -DDT). Tble. xtr prmeter vlues for nomnl reltve potency model (equton 3). rmeter Vlue ource rp ff ff ff O tble 3 rtcle clculted * clculted * clculted * ff O 0.98 clculted * (D) M - 2 (pp -DDT) M - 3 * these vlues were clculted wth the bndng model from equton 2 n the rtcle Tble 2 gves the outcome of the model for the nomnl reltve potences t the three serum percentges tested for the three compounds modelled. The clculted vlues for octylphenol correspond well wth the mesured ones (Tble 3 n rtcle) consderng the clculton model s smplfcton wth severl ssumptons. The trend of decresng rp nomnl wth ncresng serum percentge n prtculr s very smlr. Ths shows tht the mesured decrese n rp nomnl ws not concdentl mesurement vrblty but rel trend. 2

4 Tble 2. Clculted nomnl reltve estrogenc potences (rp) of octylphenol D nd pp -DDT t dfferent serum contents. rp nomnl octylphenol rp nomnl D rp nomnl pp -DDT 5% serum 20% serum 50% serum D ctully shows smller decrese n rp nomnl (0%) thn octylphenol (23%) whle pp -DDT does not show ny decrese wth ncresng serum content. pprently compounds wth lower ffnty for lbumn show lower effect of serum. Ths s n contrst to the expectton tht the lter strt of the decrese curve of these compounds would show hgher effect of serum n the rnge of -50% serum f the curve would run prllel to tht of octylphenol. Clerly however the curves of D nd pp -DDT do not run prllel to the curve of octylphenol: they re quenched. s the ffnty of octylphenol for lbumn s hgh for such n unspecfc bndng proten most other compounds wll hve lower ffnty. Therefore we do not expect lrge effects of serum content on nomnl reltve potences of the morty of compounds. Only when there s no serum present t ll n the ssy lke n the yest estrogen screen (Y) 4 very dfferent rp nomnl cn be found for compounds wth hgh lbumn ffnty (fgure : rp nomnl s for octylphenol t 0% serum). Gülden nd co-workers 5 found much lrger effects of vlblty on the nomnl reltve potences (decrese down to 6% nd ncrese up to 3%) but ths ws n effect due to lrger cell number n the suspenson not serum content. Hydrophobcty nd cell membrne content cn therefore probbly ply lrger role n the system-dependency of the rp nomnl thn lbumn ffnty nd serum content do. However ths effect wll not come to expresson n ll n vtro ssys s for exmple n the reporter gene ssy used here the cell content wll not only determne the mount of vlblty loss but sometmes lso the heght of the response. These two effects cn nullfy ech other. The reltve potences n Tble 3 n the rtcle re more or less constnt (no trend) whch s s expected but re tmes hgher thn the nomnl reltve potences. The generl nfluence of the lbumn bndng ffnty of test compound on the dfference between rp nomnl nd rp ws studed n more detl usng smple exposure model. Ths model ws bsed on equtons nd 2 nd the exct scrpt cn be found n ppendx IV. Model clcultons were performed for the condtons of our ssy wth 5% serum nd estrdol s the reference compound. The result of ths smulton s shown n fgure 2. If reltve potences would be the sme s the nomnl reltve potences ll brs of the sme seres would hve the sme heght nd ll lnes would be completely horzontl. Ths s clerly not the cse for compounds wth bndng ffnty hgher thn 0 5 M 3

5 nd nomnl reltve potency hgher thn 0-6 : for these theoretcl compounds the reltve potency ncreses consderbly becomng more thn 0-fold hgher thn the nomnl reltve potences. Compounds wth low nomnl reltve potences do not show ny effect of -vlue here becuse these compounds hve such hgh C 50 -vlue tht the lbumn must be sturted wth lgnd t ths concentrton regrdless of the ffnty of the lgnd for lbumn. Compounds wth low -vlues do not show n effect of bndng on the dfference between nd nomnl reltve potences becuse they do not bnd to sgnfcnt extent. For prortston purposes chemcls re usully rnked bsed on ther nomnl reltve potency. Fgure 2 shows tht the rnkng cn chnge f concentrtons re used to estmte reltve potences. Ths concluson s only bsed on the effect of bndng to lbumn whle bndng to other protens (e.g. HBG) nd other processes my lso led to chnge n rnkng when bsed on reltve potences Free reltve potency log Fgure 2. ffect of the ffnty constnt ( ) of compound for lbumn on the dfference between the nomnl reltve potency nd the reltve potency n n ssy wth 5% serum present n the culture medum. nes connect brs of the sme nomnl reltve potency: 0-6 (blck) 0-4 (drk grey) 0-2 (strped) nd (lght grey). References () Rowlnd M.; Tozer T. N. Clncl phrmcoknetcs; 3 ed.; Wllms & Wlkns: Med hldelph 995. (2) heehn D. M.; Young M. ndocrnology (3) tyl..; Nth. Indn J. xp. Bol (4) rnold. F.; Robnson M..; Notdes. C.; Gullette. J. Jr.; Mcchln J.. nvron. Helth erspect (5) Gülden M.; Mörchel.; ebert H. Toxcol. In Vtro

6 ppendx I. Model dervtons ssumng sngle bndng ste per proten the bndng recton of lgnd () to proten () cn be formulted s (I.) where s the concentrton of lgnd s the concentrton of unoccuped proten nd s the concentrton of lgnd-proten complexes. The w of Mss cton sttes tht t bndng equlbrum the ffnty constnt ( ) cn be expressed s n equton I.2 2. (I.2) The mss blnces of system contnng one proten nd one lgnd re totl totl (I.3) n whch totl nd totl re the totl concentrtons of proten nd lgnd respectvely. The nd bound concentrtons n equton I.2 cn be substtuted wth the followng defntons: ff ff totl ( ff ) (I.4) totl totl n whch ff nd ff re the frctons of lgnd nd proten respectvely. Rerrngement of the resultng equton leds to equton I.5 2 : ff ff totl (I.5) lterntvely f the proten mss blnce s used to substtute n equton I.2 nd the resultng equton s rerrnged equton I.6 cn be obtned whch corresponds wth the ngmur equton: totl (I.6) Ths equton cn be used to substtute n the mss blnce equtons to obtn equton I.7: 5

7 6 totl totl totl totl (I.7) s clculted frst by the modellng softwre usng the lower formul of equton I.7 (for exmple by the GU ROOT functon n Berkeley Mdonn). Wth the obtned the upper formul of equton I.7 cn be used to clculte or ny other desred prmeter. In generl f there re n dfferent lgnds n whch bnd wth bndng ssocton constnt to m dfferent protens m the mss blnces become: totl totl totl totl (I.8) For exmple wth one lgnd nd two protens (lbumn) nd (HBG) the mss blnces re: totl totl totl totl totl totl totl (I.9) nd for two lgnds (estrdol) nd (xeno-estrogen) nd two protens nd they become:

8 7 totl totl totl totl totl totl totl totl totl totl (I.0) References () Rng H..; Rtter J.M.; Dle M.M. hrmcology; Churchll vngstone: New York 998. (2) Rowlnd M.; Tozer T.N. Clncl hrmcoknetcs; Wllms & Wlkns: Med (hldelph) 995.

9 ppendx II. Bndng model scrpt ;Model for bndng of one compound to two serum protens (lbumn nd HBG) MTHOD R4 RNM TIMlogtot RNM TRTTIMlogtot0 RNM TOTIMlogtotf RNM DTMINdmn RNM DTMdmx RNM DTOUTdout ; s log totl concentrton of lgnd logtot0-0.0 logtotf0.0 dmn.e-6 dmx. dout e4 3.7e8 t.3e-4 t0.5e-7 ; s ffnty constnt of lgnd for lbumn defult s for estrdol ; s ffnty constnt of lgnd for HBG defult s for estrdol ; s totl lbumn concentrton defult s for 00% FC ; s totl HBG concentrton defult s for 00% FC fff/tot fff/t ffsf/t fb(tot-f)/tot ; s frcton of lgnd ; s unoccuped frcton of lbumn ; s unoccuped frcton of HBG ; s bound frcton of lgnd tot0.00**logtot f(.0-(*f)/(.0*f))*t f(.0-(*f)/(.0*f))*t ; s conc. of unoccuped lbumn ; s conc. of unoccuped HBG GU ftot/2. ROOT f(.0*t/(.0*f)*t/(.0*f))*f-tot ;s conc. of lgnd IMIT f > 0. IMIT f < tot 8

10 ppendx III. Nomnl reltve potency model scrpt ;Model to study the course of the nomnl reltve potency s functon of proten content nd ;proten sturton MTHOD R4 RNM TIMlogfserum ; s log of frcton of serum present n medum RNM TRTTIMlogfserum0 RNM TOTIMlogfserumf RNM DTMINdmn RNM DTMdmx RNM DTOUTdout RNM DTd logfserum0-6 logfserumf 0 dmn.e-6 dmx dout.e-2 dt e4 3.7e8 O.0e7 O 4.7e5 Rf e-3 ff ff ffo 0.87 ffo 0.98 fserum0**logfserum t 3.7e-4*fserum t 5e-8*fserum ; s ffnty constnt of estrdol for lbumn ; s ffnty constnt of estrdol for HBG ; s ffnty constnt of octylphenol for lbumn ; s ffnty constnt of octylphenol for HBG ; s reltve potency of octylphenol ; s unoccuped frcton of lbumn t nomnl C50 of estrdol ; s unoccuped frcton of HBG t nomnl C50 of estrdol ; s unocc. frcton of lbumn t nomnl C50 of octylphenol ; unoccuped frcton of HBG t nomnl C50 of octylphenol ; s totl conc. of lbumn ; s totl conc. of HBG ffo /( O*t*ffO O*t*ffO) ; s frcton octylphenol ff /( *t*ffo O*t*ffO) ; s frcton estrdol Rt Rf*(ffO/ff) ; s nomnl (totl) reltve potency IMIT Rt < Rf 9

11 ppendx IV. Free reltve potency model scrpt ; Model for effect of on reltve potences ; ths model s mde to clculte "" reltve potences from "totl" (or "nomnl") reltve ;potences for dfferent -vlues of the test lgnd whle the reference compound (estrdol) ;stys constnt. In other words: the effect of the bndng ffnty of screened compound for one ;proten on the dfference between nd totl reltve potency. RNM TIMlog RNM TRTTIMlog0 RNM TOTIMlogf RNM DTMINdmn RNM DTMdmx RNM DTOUTdout RNM DTd ; s log of bndng ffnty log00 logf9 dmn.e-6 dmx. dout d0. t6.5e-6 ; s totl proten concentrton defult t 5% FC Rt.e-6 ; s nomnl (totl) reltve potency s nom.c50 (estrdol)/nom.c50 (test lgnd) ff0.323 ; s frcton of reference (estrdol) t sme proten conc. s test lgnd defult ; s mesured vlue round nomnl C50 t 5% FC t7.76e- ; s nomnl C50 vlue of reference (estrdol) t gven proten ;concentrton fff/tot ; s frcton of test lgnd tott/rt ; s nomnl C 50 of test lgnd fff/t ; s unoccuped frcton of proten 0.00**log f(.0-(*f)/(.0*f))*t RfRt*(ff/ff) ; s conc. of unoccuped proten ; s reltve potency GU ftot/2. ROOT f(.0*t/(.0*f))*f-tot IMIT f > 0. IMIT f < tot ; s conc. of test lgnd 0

Chapter Newton-Raphson Method of Solving a Nonlinear Equation

Chapter Newton-Raphson Method of Solving a Nonlinear Equation Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson

More information

Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC

Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses

More information

Applied Statistics Qualifier Examination

Applied Statistics Qualifier Examination Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng

More information

DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)

DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x) DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng

More information

Remember: Project Proposals are due April 11.

Remember: Project Proposals are due April 11. Bonformtcs ecture Notes Announcements Remember: Project Proposls re due Aprl. Clss 22 Aprl 4, 2002 A. Hdden Mrov Models. Defntons Emple - Consder the emple we tled bout n clss lst tme wth the cons. However,

More information

UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II

UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )

More information

Chemical Reaction Engineering

Chemical Reaction Engineering Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W

More information

4. Eccentric axial loading, cross-section core

4. Eccentric axial loading, cross-section core . Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we

More information

Chemical Reaction Engineering

Chemical Reaction Engineering Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F E F E + Q! 0

More information

Many-Body Calculations of the Isotope Shift

Many-Body Calculations of the Isotope Shift Mny-Body Clcultons of the Isotope Shft W. R. Johnson Mrch 11, 1 1 Introducton Atomc energy levels re commonly evluted ssumng tht the nucler mss s nfnte. In ths report, we consder correctons to tomc levels

More information

6. Chemical Potential and the Grand Partition Function

6. Chemical Potential and the Grand Partition Function 6. Chemcl Potentl nd the Grnd Prtton Functon ome Mth Fcts (see ppendx E for detls) If F() s n nlytc functon of stte vrles nd such tht df d pd then t follows: F F p lso snce F p F we cn conclude: p In other

More information

Online Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members

Online Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members Onlne Appendx to Mndtng Behvorl Conformty n Socl Groups wth Conformst Members Peter Grzl Andrze Bnk (Correspondng uthor) Deprtment of Economcs, The Wllms School, Wshngton nd Lee Unversty, Lexngton, 4450

More information

4. More general extremum principles and thermodynamic potentials

4. More general extremum principles and thermodynamic potentials 4. More generl etremum prncples nd thermodynmc potentls We hve seen tht mn{u(s, X )} nd m{s(u, X)} mply one nother. Under certn condtons, these prncples re very convenent. For emple, ds = 1 T du T dv +

More information

Principle Component Analysis

Principle Component Analysis Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors

More information

Quiz: Experimental Physics Lab-I

Quiz: Experimental Physics Lab-I Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled

More information

Substitution Matrices and Alignment Statistics. Substitution Matrices

Substitution Matrices and Alignment Statistics. Substitution Matrices Susttuton Mtrces nd Algnment Sttstcs BMI/CS 776 www.ostt.wsc.edu/~crven/776.html Mrk Crven crven@ostt.wsc.edu Ferur 2002 Susttuton Mtrces two oulr sets of mtrces for roten seuences PAM mtrces [Dhoff et

More information

Electrochemical Thermodynamics. Interfaces and Energy Conversion

Electrochemical Thermodynamics. Interfaces and Energy Conversion CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):

More information

The Schur-Cohn Algorithm

The Schur-Cohn Algorithm Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for

More information

Haddow s Experiment:

Haddow s Experiment: schemtc drwng of Hddow's expermentl set-up movng pston non-contctng moton sensor bems of sprng steel poston vres to djust frequences blocks of sold steel shker Hddow s Experment: terr frm Theoretcl nd

More information

6 Roots of Equations: Open Methods

6 Roots of Equations: Open Methods HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 6 Roots of Equtons: Open Methods Smple Fed-Pont Iterton Newton-Rphson Secnt Methods MATLAB Functon: fzero Polynomls Cse Study: Ppe Frcton Brcketng

More information

Solubilities and Thermodynamic Properties of SO 2 in Ionic

Solubilities and Thermodynamic Properties of SO 2 in Ionic Solubltes nd Therodync Propertes of SO n Ionc Lquds Men Jn, Yucu Hou, b Weze Wu, *, Shuhng Ren nd Shdong Tn, L Xo, nd Zhgng Le Stte Key Lbortory of Checl Resource Engneerng, Beng Unversty of Checl Technology,

More information

Lecture 4: Piecewise Cubic Interpolation

Lecture 4: Piecewise Cubic Interpolation Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml

More information

GAUSS ELIMINATION. Consider the following system of algebraic linear equations

GAUSS ELIMINATION. Consider the following system of algebraic linear equations Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()

More information

Katholieke Universiteit Leuven Department of Computer Science

Katholieke Universiteit Leuven Department of Computer Science Updte Rules for Weghted Non-negtve FH*G Fctorzton Peter Peers Phlp Dutré Report CW 440, Aprl 006 Ktholeke Unverstet Leuven Deprtment of Computer Scence Celestjnenln 00A B-3001 Heverlee (Belgum) Updte Rules

More information

Chapter Newton-Raphson Method of Solving a Nonlinear Equation

Chapter Newton-Raphson Method of Solving a Nonlinear Equation Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson

More information

The Representation of Multi-component Adsorption in Reservoir Simulation of CO 2 Sequestration in Coal and Enhanced Coalbed Methane Recovery

The Representation of Multi-component Adsorption in Reservoir Simulation of CO 2 Sequestration in Coal and Enhanced Coalbed Methane Recovery The Representton of Mult-component Adsorpton n Reservor Smulton of CO 2 Sequestrton n Col nd Enhnced Colbed Methne Recovery 59 Zhejun Pn nd LD Connell CSIRO Petroleum, Prvte Bg 1, Clyton South, Vctor,

More information

Investigation phase in case of Bragg coupling

Investigation phase in case of Bragg coupling Journl of Th-Qr Unversty No.3 Vol.4 December/008 Investgton phse n cse of Brgg couplng Hder K. Mouhmd Deprtment of Physcs, College of Scence, Th-Qr, Unv. Mouhmd H. Abdullh Deprtment of Physcs, College

More information

Model Fitting and Robust Regression Methods

Model Fitting and Robust Regression Methods Dertment o Comuter Engneerng Unverst o Clorn t Snt Cruz Model Fttng nd Robust Regresson Methods CMPE 64: Imge Anlss nd Comuter Vson H o Fttng lnes nd ellses to mge dt Dertment o Comuter Engneerng Unverst

More information

INTRODUCTION TO COMPLEX NUMBERS

INTRODUCTION TO COMPLEX NUMBERS INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

Demand. Demand and Comparative Statics. Graphically. Marshallian Demand. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert

Demand. Demand and Comparative Statics. Graphically. Marshallian Demand. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert Demnd Demnd nd Comrtve Sttcs ECON 370: Mcroeconomc Theory Summer 004 Rce Unversty Stnley Glbert Usng the tools we hve develoed u to ths ont, we cn now determne demnd for n ndvdul consumer We seek demnd

More information

Chapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations

Chapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations Cter. Runge-Kutt nd Order Metod or Ordnr Derentl Eutons Ater redng ts cter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl eutons nd ow to use t to solve roblems. Wt s te Runge-Kutt

More information

7.2 Volume. A cross section is the shape we get when cutting straight through an object.

7.2 Volume. A cross section is the shape we get when cutting straight through an object. 7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A

More information

13 Design of Revetments, Seawalls and Bulkheads Forces & Earth Pressures

13 Design of Revetments, Seawalls and Bulkheads Forces & Earth Pressures 13 Desgn of Revetments, Sewlls nd Bulkheds Forces & Erth ressures Ref: Shore rotecton Mnul, USACE, 1984 EM 1110--1614, Desgn of Revetments, Sewlls nd Bulkheds, USACE, 1995 Brekwters, Jettes, Bulkheds nd

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus: More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ + + + mn mnmzes χ, nd

More information

Statistics 423 Midterm Examination Winter 2009

Statistics 423 Midterm Examination Winter 2009 Sttstcs 43 Mdterm Exmnton Wnter 009 Nme: e-ml: 1. Plese prnt your nme nd e-ml ddress n the bove spces.. Do not turn ths pge untl nstructed to do so. 3. Ths s closed book exmnton. You my hve your hnd clcultor

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Tests for the Ratio of Two Poisson Rates

Tests for the Ratio of Two Poisson Rates Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

More information

Physics 121 Sample Common Exam 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7. Instructions:

Physics 121 Sample Common Exam 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7. Instructions: Physcs 121 Smple Common Exm 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7 Nme (Prnt): 4 Dgt ID: Secton: Instructons: Answer ll 27 multple choce questons. You my need to do some clculton. Answer ech queston on the

More information

Dennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1

Dennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1 Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg

More information

The steps of the hypothesis test

The steps of the hypothesis test ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of

More information

Definition of Tracking

Definition of Tracking Trckng Defnton of Trckng Trckng: Generte some conclusons bout the moton of the scene, objects, or the cmer, gven sequence of mges. Knowng ths moton, predct where thngs re gong to project n the net mge,

More information

Mechanical resonance theory and applications

Mechanical resonance theory and applications Mechncl resonnce theor nd lctons Introducton In nture, resonnce occurs n vrous stutons In hscs, resonnce s the tendenc of sstem to oscllte wth greter mltude t some frequences thn t others htt://enwkedorg/wk/resonnce

More information

Torsion, Thermal Effects and Indeterminacy

Torsion, Thermal Effects and Indeterminacy ENDS Note Set 7 F007bn orson, herml Effects nd Indetermncy Deformton n orsonlly Loded Members Ax-symmetrc cross sectons subjected to xl moment or torque wll remn plne nd undstorted. At secton, nternl torque

More information

Terminal Velocity and Raindrop Growth

Terminal Velocity and Raindrop Growth Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,

More information

Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers

Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

More information

Chapter 3. Vector Spaces

Chapter 3. Vector Spaces 3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

MATHEMATICAL MODEL AND STATISTICAL ANALYSIS OF THE TENSILE STRENGTH (Rm) OF THE STEEL QUALITY J55 API 5CT BEFORE AND AFTER THE FORMING OF THE PIPES

MATHEMATICAL MODEL AND STATISTICAL ANALYSIS OF THE TENSILE STRENGTH (Rm) OF THE STEEL QUALITY J55 API 5CT BEFORE AND AFTER THE FORMING OF THE PIPES 6 th Reserch/Exert Conference wth Interntonl Prtcton QUALITY 009, Neum, B&H, June 04 07, 009 MATHEMATICAL MODEL AND STATISTICAL ANALYSIS OF THE TENSILE STRENGTH (Rm) OF THE STEEL QUALITY J55 API 5CT BEFORE

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

CENTROID (AĞIRLIK MERKEZİ )

CENTROID (AĞIRLIK MERKEZİ ) CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.

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

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b. Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

Math 426: Probability Final Exam Practice

Math 426: Probability Final Exam Practice Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Name: SID: Discussion Session:

Name: SID: Discussion Session: Nme: SID: Dscusson Sesson: hemcl Engneerng hermodynmcs -- Fll 008 uesdy, Octoer, 008 Merm I - 70 mnutes 00 onts otl losed Book nd Notes (5 ponts). onsder n del gs wth constnt het cpctes. Indcte whether

More information

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1

More information

New data structures to reduce data size and search time

New data structures to reduce data size and search time New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute

More information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

Lecture 5 Single factor design and analysis

Lecture 5 Single factor design and analysis Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke

More information

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B]

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B] Conept of Atvty Equlbrum onstnt s thermodynm property of n equlbrum system. For heml reton t equlbrum; Conept of Atvty Thermodynm Equlbrum Constnts A + bb = C + dd d [C] [D] [A] [B] b Conentrton equlbrum

More information

Data Structures and Algorithms CMPSC 465

Data Structures and Algorithms CMPSC 465 Dt Structures nd Algorithms CMPSC 465 LECTURE 10 Solving recurrences Mster theorem Adm Smith S. Rskhodnikov nd A. Smith; bsed on slides by E. Demine nd C. Leiserson Review questions Guess the solution

More information

Chapter 5 Supplemental Text Material R S T. ij i j ij ijk

Chapter 5 Supplemental Text Material R S T. ij i j ij ijk Chpter 5 Supplementl Text Mterl 5-. Expected Men Squres n the Two-fctor Fctorl Consder the two-fctor fxed effects model y = µ + τ + β + ( τβ) + ε k R S T =,,, =,,, k =,,, n gven s Equton (5-) n the textook.

More information

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

2008 Mathematical Methods (CAS) GA 3: Examination 2

2008 Mathematical Methods (CAS) GA 3: Examination 2 Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

8. INVERSE Z-TRANSFORM

8. INVERSE Z-TRANSFORM 8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil

More information

Department of Mechanical Engineering, University of Bath. Mathematics ME Problem sheet 11 Least Squares Fitting of data

Department of Mechanical Engineering, University of Bath. Mathematics ME Problem sheet 11 Least Squares Fitting of data Deprtment of Mechncl Engneerng, Unversty of Bth Mthemtcs ME10305 Prolem sheet 11 Lest Squres Fttng of dt NOTE: If you re gettng just lttle t concerned y the length of these questons, then do hve look t

More information

Calculus - Activity 1 Rate of change of a function at a point.

Calculus - Activity 1 Rate of change of a function at a point. Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

More information

Chemistry 163B Absolute Entropies and Entropy of Mixing

Chemistry 163B Absolute Entropies and Entropy of Mixing Chemstry 163 Wnter 1 Hndouts for hrd Lw nd Entropy of Mxng (del gs, dstngushle molecules) PPENDIX : H f, G f, U S (no Δ, no su f ) Chemstry 163 solute Entropes nd Entropy of Mxng Hº f Gº f Sº 1 hrd Lw

More information

PLEASE SCROLL DOWN FOR ARTICLE

PLEASE SCROLL DOWN FOR ARTICLE Ths rtcle ws downloded by:ntonl Cheng Kung Unversty] On: 1 September 7 Access Detls: subscrpton number 7765748] Publsher: Tylor & Frncs Inform Ltd Regstered n Englnd nd Wles Regstered Number: 17954 Regstered

More information

Scientific notation is a way of expressing really big numbers or really small numbers.

Scientific notation is a way of expressing really big numbers or really small numbers. Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17 EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

More information

Designing Information Devices and Systems I Discussion 8B

Designing Information Devices and Systems I Discussion 8B Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus

ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem

More information

The Study of Lawson Criterion in Fusion Systems for the

The Study of Lawson Criterion in Fusion Systems for the Interntonl Archve of Appled Scences nd Technology Int. Arch. App. Sc. Technol; Vol 6 [] Mrch : -6 Socety of ducton, Ind [ISO9: 8 ertfed Orgnzton] www.soeg.co/st.html OD: IAASA IAAST OLI ISS - 6 PRIT ISS

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent? 1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the

More information

Rule 1 Rule 2 Rule 3 Rule 4. Balance Weight

Rule 1 Rule 2 Rule 3 Rule 4. Balance Weight 107 Tble 1 Predicted Success (Percentge of Correct Responses) on Different Blnce-Scle Problems for Individuls Using Siegler s (1976) Four Rules Problem Type Level of Performnce Rule 1 Rule 2 Rule 3 Rule

More information

ψ ij has the eigenvalue

ψ ij has the eigenvalue Moller Plesset Perturbton Theory In Moller-Plesset (MP) perturbton theory one tes the unperturbed Hmltonn for n tom or molecule s the sum of the one prtcle Foc opertors H F() where the egenfunctons of

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Two Coefficients of the Dyson Product

Two Coefficients of the Dyson Product Two Coeffcents of the Dyson Product rxv:07.460v mth.co 7 Nov 007 Lun Lv, Guoce Xn, nd Yue Zhou 3,,3 Center for Combntorcs, LPMC TJKLC Nnk Unversty, Tnjn 30007, P.R. Chn lvlun@cfc.nnk.edu.cn gn@nnk.edu.cn

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede Fll Anlss of Epermentl Mesurements B. Esensten/rev. S. Errede Monte Crlo Methods/Technques: These re mong the most powerful tools for dt nlss nd smulton of eperments. The growth of ther mportnce s closel

More information