Dealing with quantitative data and problem solving life is a story problem! Attacking Quantitative Problems

Size: px
Start display at page:

Download "Dealing with quantitative data and problem solving life is a story problem! Attacking Quantitative Problems"

Transcription

1 Daling with quantitati data and problm soling lif is a story problm! A larg portion of scinc inols quantitati data that has both alu and units. Units can sa your butt! Nd handl on mtric prfixs Dimnsional Analysis Dis calculation using conrsion factors so unndd units cancl and ncssary units ar lft bhind. How many cm ar in km? Appndix A-5 Practic, Practic, Practic!!! Attacking Quantitati Problms. Rad th qustion. Idntify th rquird quantity (and units). Dis a way to us gin information to gt rquird quantity (and units) st up ncssary calculations Chck to b sur you gt th corrct units!. Insrt alus into quations and sol 5. Look at th answr is it rasonabl (ordr of magnitud, unit)? 6. Rad th qustion again Exampl: If th diamtr of a singl carbon atom is 5 pm, how many atoms would b in a lin 00.0 m long (about th diamtr of a human hair)? How much would this lin wigh if a singl carbon atom wighs.99 x 0 - g?

2 What can a rsult tll us? What is th "truth"? W don't (and can't) dtrmin th "tru alu" of any componnt in a mixtur. All masurmnts ha som associatd rror (uncrtainty). t Rprsntation of rsults must illustrat this rror; othrwis alu is uslss! W can work to undrstand (and impro) th prcision of a masurmnt. Prcision = Comparing multipl mthods incrass confidnc in th accuracy of our rsults. Accuracy = Rprsntation of Quantitati Data Significant Figurs: Numbr of digits ncssary to prsnt a rsult with th appropriat accuracy. Last digit has uncrtainty, at last! Rsults ar prsntd to th appropriat numbr of significant figurs, and oftn accompanid by th rror in th final digit. Valu Error.g ppm Whn rounding, look at all digits byond th last plac ndd. Always round alus that ar xactly halfway to th narst n digit

3 Whn looking at a numbr: Guidlins for Sig. Figs.. all nonzro digits ar significant,. all zros that t appar btwn nonzro digits it of a numbr ar significant,. all zros at th nd of a numbr on th right hand sid of th dcimal point ar significant,. all othr zros ar not significant. Exampls: How many significant figurs in ach alu? Sig. Figs. and Calculations In doing calculations, th numbr sig figs is limitd by th last crtain pic of data. Addition/Subtraction Multiplication/Diision Exampl: How many sig. figs. should b in th answrs to th following: = 58.6 x. = NOTE: Wait to truncat rsults until th nd of th calculation! WHY??? Indicat xtra digits by subscripting: Exact numbrs don t impact sig. figs. 6

4 Typs of Error (Uncrtainty) Uncrtaintis fall into two main classs: Systmatic Error: Random Error Two gnral approachs to quantifying rror: Absolut Uncrtainty Rlati Uncrtainty, (% Rlati Uncrtainty) 7 Statistics and Quantitati Data Statistics allow an stimation of uncrtainty in our data. AS LONG AS: Foundation of statistics: Normal Distribution Two paramtrs dfin th distribution: : : quncy of Obsrations Fr Statistical analyss rlat xprimntal data to this idal distribution. BUT, thr ar som challngs! Obsrd Valu 8

5 Statistics in Practic: Rmmbr th ky assumption! Dscribing th prcision of our data. Arag, or Arithmtic Man: n is th numbr of sampls xi i x n Sampl Standard Diation: s i x i x n 9 Dscribing th uncrtainty in our data. How good is it? Whn w dtrmin an arag (with som associatd rror), how sur ar w that th "tru alu" is clos to this arag? What factors influnc this confidnc? Th most common statistical tool for dtrmining that th "tru" alu is clos to our calculatd man is th confidnc intral. Th confidnc intral prsnts a rang about th man, within which thr is a fixd probability of finding th tru alu, m. ts x n 0 5

6 Mor on Confidnc Limits Valus for t ar tabulatd basd on sral confidnc lls and arious numbrs of dgrs of frdom. Confidnc Ll (%) Dgrs of Frdom NOTE: n though th numbr of masurmnts (n) is usd in th CI calculation, t is dtrmind basd on th dgrs of frdom (n-). Lt s look at som data: What is th arag? What is th standard diation (s)? What is th 95% confidnc limit? Error Propagation (aka propagation of uncrtainty) How do rrors in indiidual alus affct an analysis? Effcts dpnd on how th data is usd. Ruls com from calculus! l Error Propagation in Addition and Subtraction: Orall rror is basd on th absolut uncrtaintis of indiidual alus. Exampl: ??? 6

7 7 Mor Error Propagation Error Propagation in Multiplication and Diision Concpt is th sam as addition and subtraction, xcpt rlati uncrtaintis ar usd rlati uncrtaintis ar usd % % % %

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy

More information

Mor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12 Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

General Notes About 2007 AP Physics Scoring Guidelines

General Notes About 2007 AP Physics Scoring Guidelines AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation

More information

Numbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.

Numbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design. Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th

More information

Calculus concepts derivatives

Calculus concepts derivatives All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving

More information

Applied Statistics II - Categorical Data Analysis Data analysis using Genstat - Exercise 2 Logistic regression

Applied Statistics II - Categorical Data Analysis Data analysis using Genstat - Exercise 2 Logistic regression Applid Statistics II - Catgorical Data Analysis Data analysis using Gnstat - Exrcis 2 Logistic rgrssion Analysis 2. Logistic rgrssion for a 2 x k tabl. Th tabl blow shows th numbr of aphids aliv and dad

More information

1 Isoparametric Concept

1 Isoparametric Concept UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric

More information

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration

More information

Where k is either given or determined from the data and c is an arbitrary constant.

Where k is either given or determined from the data and c is an arbitrary constant. Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is

More information

4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.

4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16. . 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts

More information

Data Assimilation 1. Alan O Neill National Centre for Earth Observation UK

Data Assimilation 1. Alan O Neill National Centre for Earth Observation UK Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal

More information

Observer Bias and Reliability By Xunchi Pu

Observer Bias and Reliability By Xunchi Pu Obsrvr Bias and Rliability By Xunchi Pu Introduction Clarly all masurmnts or obsrvations nd to b mad as accuratly as possibl and invstigators nd to pay carful attntion to chcking th rliability of thir

More information

Gradebook & Midterm & Office Hours

Gradebook & Midterm & Office Hours Your commnts So what do w do whn on of th r's is 0 in th quation GmM(1/r-1/r)? Do w nd to driv all of ths potntial nrgy formulas? I don't undrstand springs This was th first lctur I actually larnd somthing

More information

Systems of Equations

Systems of Equations CHAPTER 4 Sstms of Equations 4. Solving Sstms of Linar Equations in Two Variabls 4. Solving Sstms of Linar Equations in Thr Variabls 4. Sstms of Linar Equations and Problm Solving Intgratd Rviw Sstms of

More information

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero. SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain

More information

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting

More information

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim. MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function

More information

2013 Specialist Mathematics GA 3: Written examination 2

2013 Specialist Mathematics GA 3: Written examination 2 0 0 Spcialist Mathmatics GA : Writtn xamination GENERAL COMMENTS Th 0 Spcialist Mathmatics xamination comprisd multipl-choic qustions (worth marks) and fiv xtndd qustions (worth 8 marks). Th papr smd accssibl

More information

Text: WMM, Chapter 5. Sections , ,

Text: WMM, Chapter 5. Sections , , Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013 18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:

More information

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs

More information

Massachusetts Institute of Technology Department of Mechanical Engineering

Massachusetts Institute of Technology Department of Mechanical Engineering Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our

More information

Linear Non-Gaussian Structural Equation Models

Linear Non-Gaussian Structural Equation Models IMPS 8, Durham, NH Linar Non-Gaussian Structural Equation Modls Shohi Shimizu, Patrik Hoyr and Aapo Hyvarinn Osaka Univrsity, Japan Univrsity of Hlsinki, Finland Abstract Linar Structural Equation Modling

More information

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,

More information

Recursive Estimation of Dynamic Time-Varying Demand Models

Recursive Estimation of Dynamic Time-Varying Demand Models Intrnational Confrnc on Computr Systms and chnologis - CompSysch 06 Rcursiv Estimation of Dynamic im-varying Dmand Modls Alxandr Efrmov Abstract: h papr prsnts an implmntation of a st of rcursiv algorithms

More information

Math 34A. Final Review

Math 34A. Final Review Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right

More information

MATH 1080 Test 2-SOLUTIONS Spring

MATH 1080 Test 2-SOLUTIONS Spring MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =

More information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold

More information

Answer Homework 5 PHA5127 Fall 1999 Jeff Stark

Answer Homework 5 PHA5127 Fall 1999 Jeff Stark Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw

More information

u r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C

u r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin

More information

CE 530 Molecular Simulation

CE 530 Molecular Simulation CE 53 Molcular Simulation Lctur 8 Fr-nrgy calculations David A. Kofk Dpartmnt of Chmical Enginring SUNY Buffalo kofk@ng.buffalo.du 2 Fr-Enrgy Calculations Uss of fr nrgy Phas quilibria Raction quilibria

More information

Problem Set #2 Due: Friday April 20, 2018 at 5 PM.

Problem Set #2 Due: Friday April 20, 2018 at 5 PM. 1 EE102B Spring 2018 Signal Procssing and Linar Systms II Goldsmith Problm St #2 Du: Friday April 20, 2018 at 5 PM. 1. Non-idal sampling and rcovry of idal sampls by discrt-tim filtring 30 pts) Considr

More information

SCHUR S THEOREM REU SUMMER 2005

SCHUR S THEOREM REU SUMMER 2005 SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation

More information

MATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations

MATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,

More information

1 Minimum Cut Problem

1 Minimum Cut Problem CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms

More information

ph People Grade Level: basic Duration: minutes Setting: classroom or field site

ph People Grade Level: basic Duration: minutes Setting: classroom or field site ph Popl Adaptd from: Whr Ar th Frogs? in Projct WET: Curriculum & Activity Guid. Bozman: Th Watrcours and th Council for Environmntal Education, 1995. ph Grad Lvl: basic Duration: 10 15 minuts Stting:

More information

DIFFERENTIAL EQUATION

DIFFERENTIAL EQUATION MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous

More information

Quasi-Classical States of the Simple Harmonic Oscillator

Quasi-Classical States of the Simple Harmonic Oscillator Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats

More information

Content Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.

Content Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles. Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of

More information

5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd

5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd 1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as

More information

September 23, Honors Chem Atomic structure.notebook. Atomic Structure

September 23, Honors Chem Atomic structure.notebook. Atomic Structure Atomic Structur Topics covrd Atomic structur Subatomic particls Atomic numbr Mass numbr Charg Cations Anions Isotops Avrag atomic mass Practic qustions atomic structur Sp 27 8:16 PM 1 Powr Standards/ Larning

More information

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018 Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs

More information

Abstract Interpretation: concrete and abstract semantics

Abstract Interpretation: concrete and abstract semantics Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion

More information

Sara Godoy del Olmo Calculation of contaminated soil volumes : Geostatistics applied to a hydrocarbons spill Lac Megantic Case

Sara Godoy del Olmo Calculation of contaminated soil volumes : Geostatistics applied to a hydrocarbons spill Lac Megantic Case wwwnvisol-canadaca Sara Godoy dl Olmo Calculation of contaminatd soil volums : Gostatistics applid to a hydrocarbons spill Lac Mgantic Cas Gostatistics: study of a PH contamination CONTEXT OF THE STUDY

More information

Southern Taiwan University

Southern Taiwan University Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:

More information

2.3 Matrix Formulation

2.3 Matrix Formulation 23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,

More information

ECE602 Exam 1 April 5, You must show ALL of your work for full credit.

ECE602 Exam 1 April 5, You must show ALL of your work for full credit. ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b

More information

Strongly Connected Components

Strongly Connected Components Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts

More information

The following information relates to Questions 1 to 4:

The following information relates to Questions 1 to 4: Th following information rlats to Qustions 1 to 4: QUESTIN 1 Th lctrolyt usd in this ful cll is D watr carbonat ions hydrogn ions hydroxid ions QUESTIN 2 Th product formd in th ful cll is D hydrogn gas

More information

Solution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:

Solution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers: APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding

More information

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)

More information

Alpha and beta decay equation practice

Alpha and beta decay equation practice Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

ECE 344 Microwave Fundamentals

ECE 344 Microwave Fundamentals ECE 44 Microwav Fundamntals Lctur 08: Powr Dividrs and Couplrs Part Prpard By Dr. hrif Hkal 4/0/08 Microwav Dvics 4/0/08 Microwav Dvics 4/0/08 Powr Dividrs and Couplrs Powr dividrs, combinrs and dirctional

More information

CO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS

CO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS CO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS Stig Holst ABB Automation Products Swdn Bapuji S Palki ABB Utilitis India This papr rports

More information

MA 262, Spring 2018, Final exam Version 01 (Green)

MA 262, Spring 2018, Final exam Version 01 (Green) MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in

More information

MEASURING HEAT FLUX FROM A COMPONENT ON A PCB

MEASURING HEAT FLUX FROM A COMPONENT ON A PCB MEASURING HEAT FLUX FROM A COMPONENT ON A PCB INTRODUCTION Elctronic circuit boards consist of componnts which gnrats substantial amounts of hat during thir opration. A clar knowldg of th lvl of hat dissipation

More information

Appendix. Kalman Filter

Appendix. Kalman Filter Appndix A Kalman Filtr OPTIMAL stimation thory has a vry broad rang of applications which vary from stimation of rivr ows to satllit orbit stimation and nuclar ractor paramtr idntication. In this appndix

More information

10. Limits involving infinity

10. Limits involving infinity . Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of

More information

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0 unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr

More information

As the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.

As the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B. 7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and

More information

CS 6353 Compiler Construction, Homework #1. 1. Write regular expressions for the following informally described languages:

CS 6353 Compiler Construction, Homework #1. 1. Write regular expressions for the following informally described languages: CS 6353 Compilr Construction, Homwork #1 1. Writ rgular xprssions for th following informally dscribd languags: a. All strings of 0 s and 1 s with th substring 01*1. Answr: (0 1)*01*1(0 1)* b. All strings

More information

dx equation it is called a second order differential equation.

dx equation it is called a second order differential equation. TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld

More information

Continuous probability distributions

Continuous probability distributions Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd

More information

DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P

DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,

More information

SUMMER 17 EXAMINATION

SUMMER 17 EXAMINATION (ISO/IEC - 7-5 Crtifid) SUMMER 7 EXAMINATION Modl wr jct Cod: Important Instructions to aminrs: ) Th answrs should b amind by ky words and not as word-to-word as givn in th modl answr schm. ) Th modl answr

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

Laboratory work # 8 (14) EXPERIMENTAL ESTIMATION OF CRITICAL STRESSES IN STRINGER UNDER COMPRESSION

Laboratory work # 8 (14) EXPERIMENTAL ESTIMATION OF CRITICAL STRESSES IN STRINGER UNDER COMPRESSION Laboratory wor # 8 (14) XPRIMNTAL STIMATION OF CRITICAL STRSSS IN STRINGR UNDR COMPRSSION At action of comprssing ffort on a bar (column, rod, and stringr) two inds of loss of stability ar possibl: 1)

More information

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002 3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or

More information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,

More information

Calculation of Morse Potential Parameters of bcc Crystals and Application to Anharmonic Interatomic Effective Potential, Local Force Constant

Calculation of Morse Potential Parameters of bcc Crystals and Application to Anharmonic Interatomic Effective Potential, Local Force Constant VNU Journal of Scinc: Mathmatics Physics, Vol. 31, No. 3 (15) 3-3 Calculation of Mors Potntial Paramtrs of bcc Crystals and Application to Anharmonic Intratomic Effctiv Potntial, Local Forc Constant Nguyn

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

PHA 5127 Answers Homework 2 Fall 2001

PHA 5127 Answers Homework 2 Fall 2001 PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring

More information

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals

More information

Sliding Mode Flow Rate Observer Design

Sliding Mode Flow Rate Observer Design Sliding Mod Flow Rat Obsrvr Dsign Song Liu and Bin Yao School of Mchanical Enginring, Purdu Univrsity, Wst Lafaytt, IN797, USA liu(byao)@purdudu Abstract Dynamic flow rat information is ndd in a lot of

More information

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real. Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual

More information

Problem Statement. Definitions, Equations and Helpful Hints BEAUTIFUL HOMEWORK 6 ENGR 323 PROBLEM 3-79 WOOLSEY

Problem Statement. Definitions, Equations and Helpful Hints BEAUTIFUL HOMEWORK 6 ENGR 323 PROBLEM 3-79 WOOLSEY Problm Statmnt Suppos small arriv at a crtain airport according to Poisson procss with rat α pr hour, so that th numbr of arrivals during a tim priod of t hours is a Poisson rv with paramtr t (a) What

More information

CS 361 Meeting 12 10/3/18

CS 361 Meeting 12 10/3/18 CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm

More information

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved. 6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b

More information

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th

More information

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7. Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation

More information

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables.

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables. Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv

More information

MATHEMATICS (B) 2 log (D) ( 1) = where z =

MATHEMATICS (B) 2 log (D) ( 1) = where z = MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +

More information

AS 5850 Finite Element Analysis

AS 5850 Finite Element Analysis AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form

More information

Thermodynamical insight on the role of additives in shifting the equilibrium between white and grey tin

Thermodynamical insight on the role of additives in shifting the equilibrium between white and grey tin hrmodynamical insight on th rol of additivs in shifting th quilibrium btwn whit and gry tin Nikolay Dmntv Dpartmnt of Chmistry, mpl Univrsity, Philadlphia, PA 19122 Abstract In this study mthods of statistical

More information

Unit 6: Solving Exponential Equations and More

Unit 6: Solving Exponential Equations and More Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that

More information

Electrochemistry L E O

Electrochemistry L E O Rmmbr from CHM151 A rdox raction in on in which lctrons ar transfrrd lctrochmistry L O Rduction os lctrons xidation G R ain lctrons duction W can dtrmin which lmnt is oxidizd or rducd by assigning oxidation

More information

Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review

Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG

More information

Searching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.

Searching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list. 3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if

More information

From Elimination to Belief Propagation

From Elimination to Belief Propagation School of omputr Scinc Th lif Propagation (Sum-Product lgorithm Probabilistic Graphical Modls (10-708 Lctur 5, Sp 31, 2007 Rcptor Kinas Rcptor Kinas Kinas X 5 ric Xing Gn G T X 6 X 7 Gn H X 8 Rading: J-hap

More information