MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.

Size: px
Start display at page:

Download "MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c."

Transcription

1 MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st

2 Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ v Noti that v Now b th funamntal thorm of alulus μ v v his is a sparabl iffrntial quation with initial onition v Sparating th variabls an intgrating [rmmbr that osh sinh ] w fin that v sinhμ/ Intgrating, w gt osh μ C, μ whr C must b /μ sin an osh hus w hav μ osh h shap of th graph of this funtion is in -shap μ μ [Draw it!] Qustion h onstant C has units of /tim Solving th quation [ithr as linar first orr or as a sparabl quation] w gt M M Ct Clarl will approah M mor rapil if C is larg; that is, C masurs how rapil th stunt is abl to larn hus th quation in prsss th ia that th stunt s

3 prforman improvs mor slowl as sh approahs hr maimum possibl prforman As th ars go b an th stunt boms mor familiar with th mthos of mastring mathmatis, hr rat of larning nw things might b pt to improv; but surl thr is an uppr boun to how muh sh an improv h tanh funtion is a simpl wa of rprsnting this sin it alwas inrass but is boun abov [Rmin th stunts of th shap of tanh if nssar] hn K rprsnts hr maimum possibl sp of larning [sin tanh is asmptoti to ], an masurs th amount of tim rquir for hr to ralis hr maimum potntial [Not that K has units of /tim, whil of ours has units of tim, so K is imnsionlss] h quation an now b writtn as K tanh t / M It s onvnint now to fin Q M, so th quation is Q K tanh t / Q his is a first-orr quation with intgrating fator osh K t /, so w hav Q osh K t / A, whr A is a onstant Sin w ar assuming that, w hav Q - M, so w hav A - M, thus finall K [ sh t / ] M Stunts ar nourag to graph ampls of suh qustions using [for ampl] th softwar availabl fr at Qustion 4 h onstant K masurs th rapiit with whih th rumour will spra It pns on how intrsting th rumour is, how muh th stunts lik to gossip, how gullibl th ar, t h right han si of th quation is sign to b small both nar R an nar R, whn in th rumour an b pt to spra slowl ithr baus not nough or too man popl hav har it W hav R KR KR

4 his is a Brnoulli quation as isuss in th nots W solv it b fining Z /R, whih transforms th quation into a linar on: Z KZ K, with solution R C p Kt h problm sas that this highl intrsting rumour was start b on stunt, so R hus C 99/ Hn R 99 p Kt Of ours as t tns to infinit, R tns to Qustion 5 Suppos w writ th quation govrning th ranium as k whr rprsnts th numbr of ranium atoms, t, as usual h half-lif of ranium an b us to omput th a rat onstant, as in th ltur nots, an similarl for horium From th ltur nots w hav k k k [ p k k t] an so if / is w know vrthing in this quation pt t Solving for t, ou shoul fin that th answr is approimatl 95 thousan ars In th son part of th problm, th a rat onstant is assum to b a givn funtion of tim a Now th quation for boms t / b h units of a ar larl /ars an thos of b must b ars [rsumabl b is vr larg; othrwis w woul hav noti this fft hat is, w gt bak th usual quations for raioativit in th limit as b tns to infinit] Sparating th variabls an oing th intgration w gt [using an a bit of simpl algbra]

5 p[ at ] t / b aking th limit as t tns to infinit, w fin that tns to p[-ab]; that is, ranium woul nvr ompltl isappar if this thor wr right [Sin b is supposl vr larg, howvr, th surviving amount is vr small] OIONAL NOE ON HE EQAION IN QESION : Consir a small lmnt of th abl, with th lft n loat at a point,, an th right at, Lt tan θ /, an lt not th tnsion in th abl h horiontal fors ating ar just os θ an os θ os θ at th two ns of th small lmnt, so if ths fors ar to balan w must hav os θ onstant h vrtial fors ar sin θ an sin θ sin θ an also μ s, whr s is th lngth of th lmnt For ths to balan w must hav sin θ μ s, or an θ μ s Intgrating this quation from [whr / ] w obtain th stat quation

( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition

( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of

More information

VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS

VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f

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

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv

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

Differential Equations

Differential Equations Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations

More information

MSLC Math 151 WI09 Exam 2 Review Solutions

MSLC Math 151 WI09 Exam 2 Review Solutions Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y

More information

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts

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

a 1and x is any real number.

a 1and x is any real number. Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars

More information

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b) 4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y

More information

nd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).

nd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1). Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial

More information

ENGR 323 BHW 15 Van Bonn 1/7

ENGR 323 BHW 15 Van Bonn 1/7 ENGR 33 BHW 5 Van Bonn /7 4.4 In Eriss and 3 as wll as man othr situations on has th PDF o X and wishs th PDF o Yh. Assum that h is an invrtibl untion so that h an b solvd or to ild. Thn it an b shown

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

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

Numerical methods, Mixed exercise 10

Numerical methods, Mixed exercise 10 Numrial mthos, Mi ris a f ( ) 6 f ( ) 6 6 6 a = 6, b = f ( ) So. 6 b n a n 6 7.67... 6.99....67... 6.68....99... 6.667....68... To.p., th valus ar =.68, =.99, =.68, =.67. f (.6).6 6.6... f (.6).6 6.6.7...

More information

Junction Tree Algorithm 1. David Barber

Junction Tree Algorithm 1. David Barber Juntion Tr Algorithm 1 David Barbr Univrsity Collg London 1 Ths slids aompany th book Baysian Rasoning and Mahin Larning. Th book and dmos an b downloadd from www.s.ul.a.uk/staff/d.barbr/brml. Fdbak and

More information

Modeling with first order equations (Sect. 2.3).

Modeling with first order equations (Sect. 2.3). Moling with first orr quations (Sct. 2.3. Main xampl: Salt in a watr tank. Th xprimntal vic. Th main quations. Analysis of th mathmatical mol. Prictions for particular situations. Salt in a watr tank.

More information

Hydrogen Atom and One Electron Ions

Hydrogen Atom and One Electron Ions Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial

More information

Additional Math (4047) Paper 2 (100 marks) y x. 2 d. d d

Additional Math (4047) Paper 2 (100 marks) y x. 2 d. d d Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional

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

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

Evans, Lipson, Wallace, Greenwood

Evans, Lipson, Wallace, Greenwood Camrig Snior Mathmatial Mthos AC/VCE Units 1& Chaptr Quaratis: Skillsht C 1 Solv ah o th ollowing or x: a (x )(x + 1) = 0 x(5x 1) = 0 x(1 x) = 0 x = 9x Solv ah o th ollowing or x: a x + x 10 = 0 x 8x +

More information

Calculus II (MAC )

Calculus II (MAC ) Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.

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

Multiple Short Term Infusion Homework # 5 PHA 5127

Multiple Short Term Infusion Homework # 5 PHA 5127 Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300

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

Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours

Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn

More information

3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here.

3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here. PHA 5127 Dsigning A Dosing Rgimn Answrs provi by Jry Stark Mr. JM is to b start on aminophyllin or th tratmnt o asthma. H is a non-smokr an wighs 60 kg. Dsign an oral osing rgimn or this patint such that

More information

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform

More information

Differential Equations: Homework 3

Differential Equations: Homework 3 Diffrntial Equations: Homwork 3 Alvin Lin January 08 - May 08 Sction.3 Ercis Dtrmin whthr th givn quation is sparabl, linar, nithr, or both. Not sparabl and linar. d + sin() y = 0 Ercis 3 Dtrmin whthr

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

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

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6

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

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

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 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 th

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

AP Calculus BC Problem Drill 16: Indeterminate Forms, L Hopital s Rule, & Improper Intergals

AP Calculus BC Problem Drill 16: Indeterminate Forms, L Hopital s Rule, & Improper Intergals AP Calulus BC Problm Drill 6: Indtrminat Forms, L Hopital s Rul, & Impropr Intrgals Qustion No. of Instrutions: () Rad th problm and answr hois arfully () Work th problms on papr as ndd () Pik th answr

More information

CS 491 G Combinatorial Optimization

CS 491 G Combinatorial Optimization CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl

More information

Sundials and Linear Algebra

Sundials and Linear Algebra Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.

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

MAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP

MAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP EXERCISE - MAXIMA-MINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6.

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

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

Assignment 4 Biophys 4322/5322

Assignment 4 Biophys 4322/5322 Assignmnt 4 Biophys 4322/5322 Tylr Shndruk Fbruary 28, 202 Problm Phillips 7.3. Part a R-onsidr dimoglobin utilizing th anonial nsmbl maning rdriv Eq. 3 from Phillips Chaptr 7. For a anonial nsmbl p E

More information

are given in the table below. t (hours)

are given in the table below. t (hours) CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th

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

N1.1 Homework Answers

N1.1 Homework Answers Camrig Essntials Mathmatis Cor 8 N. Homwork Answrs N. Homwork Answrs a i 6 ii i 0 ii 3 2 Any pairs of numrs whih satisfy th alulation. For xampl a 4 = 3 3 6 3 = 3 4 6 2 2 8 2 3 3 2 8 5 5 20 30 4 a 5 a

More information

Lecture 14 (Oct. 30, 2017)

Lecture 14 (Oct. 30, 2017) Ltur 14 8.31 Quantum Thory I, Fall 017 69 Ltur 14 (Ot. 30, 017) 14.1 Magnti Monopols Last tim, w onsidrd a magnti fild with a magnti monopol onfiguration, and bgan to approah dsribing th quantum mhanis

More information

3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3 2x. 3x 2.   Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc

More information

Section 11.6: Directional Derivatives and the Gradient Vector

Section 11.6: Directional Derivatives and the Gradient Vector Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th

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

Solutions to Homework 5

Solutions to Homework 5 Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()

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

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Combinatorial Networks Week 1, March 11-12

Combinatorial Networks Week 1, March 11-12 1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl

More information

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018 CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs

More information

1973 AP Calculus AB: Section I

1973 AP Calculus AB: Section I 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=

More information

Integral Calculus What is integral calculus?

Integral Calculus What is integral calculus? Intgral Calulus What is intgral alulus? In diffrntial alulus w diffrntiat a funtion to obtain anothr funtion alld drivativ. Intgral alulus is onrnd with th opposit pross. Rvrsing th pross of diffrntiation

More information

The second condition says that a node α of the tree has exactly n children if the arity of its label is n.

The second condition says that a node α of the tree has exactly n children if the arity of its label is n. CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is

More information

Background: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.

Background: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals. Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby

More information

Case Study Vancomycin Answers Provided by Jeffrey Stark, Graduate Student

Case Study Vancomycin Answers Provided by Jeffrey Stark, Graduate Student Cas Stuy Vancomycin Answrs Provi by Jffry Stark, Grauat Stunt h antibiotic Vancomycin is liminat almost ntirly by glomrular filtration. For a patint with normal rnal function, th half-lif is about 6 hours.

More information

Examples and applications on SSSP and MST

Examples and applications on SSSP and MST Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs

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

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht

More information

Calculus Revision A2 Level

Calculus Revision A2 Level alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ

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

Problem 22: Journey to the Center of the Earth

Problem 22: Journey to the Center of the Earth Problm : Journy to th Cntr of th Earth Imagin that on drilld a hol with smooth sids straight through th ntr of th arth If th air is rmod from this tub (and it dosn t fill up with watr, liquid rok, or iron

More information

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)

More information

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS

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

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=

More information

Handout 28. Ballistic Quantum Transport in Semiconductor Nanostructures

Handout 28. Ballistic Quantum Transport in Semiconductor Nanostructures Hanout 8 Ballisti Quantum Transport in Smionutor Nanostruturs In this ltur you will larn: ltron transport without sattring (ballisti transport) Th quantum o onutan an th quantum o rsistan Quanti onutan

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

Digital Signal Processing, Fall 2006

Digital Signal Processing, Fall 2006 Digital Signal Prossing, Fall 006 Ltur 7: Filtr Dsign Zhng-ua an Dpartmnt of Eltroni Systms Aalborg Univrsity, Dnmar t@om.aau. Cours at a glan MM Disrt-tim signals an systms Systm MM Fourir-omain rprsntation

More information

Functions of Two Random Variables

Functions of Two Random Variables Functions of Two Random Variabls Maximum ( ) Dfin max, Find th probabilit distributions of Solution: For an pair of random variabls and, [ ] [ ] F ( w) P w P w and w F hn and ar indpndnt, F ( w) F ( w)

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

ME311 Machine Design

ME311 Machine Design ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform

More information

Finite Element Analysis

Finite Element Analysis Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis

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

Math-3. Lesson 5-6 Euler s Number e Logarithmic and Exponential Modeling (Newton s Law of Cooling)

Math-3. Lesson 5-6 Euler s Number e Logarithmic and Exponential Modeling (Newton s Law of Cooling) Math-3 Lsson 5-6 Eulr s Numbr Logarithmic and Eponntial Modling (Nwton s Law of Cooling) f ( ) What is th numbr? is th horizontal asymptot of th function: 1 1 ~ 2.718... On my 3rd submarin (USS Springfild,

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

Basic Polyhedral theory

Basic Polyhedral theory Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist

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

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl

More information

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7

More information

Self-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016

Self-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016 Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac

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

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

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local

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

3 a b c km m m 8 a 3.4 m b 2.4 m

3 a b c km m m 8 a 3.4 m b 2.4 m Chaptr Exris A a 9. m. m. m 9. km. mm. m Purpl lag hapr y 8p 8m. km. m Th triangl on th right 8. m 9 a. m. m. m Exris B a m. m mm. km. mm m a. 9 8...8 m. m 8. 9 m Ativity p. 9 Pupil s own answrs Ara =

More information

4 x 4, and. where x is Town Square

4 x 4, and. where x is Town Square Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and

More information

Carriers Concentration in Semiconductors - VI. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Carriers Concentration in Semiconductors - VI. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India Carrirs Conntration in Smionutors - VI 1 Prof.P. Ravinran, Dpartmnt of Pysis, Cntral Univrsity of Tamil au, Inia ttp://folk.uio.no/ravi/smi01 P.Ravinran, PHY0 Smionutor Pysis, 17 January 014 : Carrirs

More information

Constants and Conversions:

Constants and Conversions: EXAM INFORMATION Radial Distribution Function: P 2 ( r) RDF( r) Br R( r ) 2, B is th normalization constant. Ordr of Orbital Enrgis: Homonuclar Diatomic Molculs * * * * g1s u1s g 2s u 2s u 2 p g 2 p g

More information