# SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

Save this PDF as:

Size: px
Start display at page:

Download "SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx"

## Transcription

1 SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x of the iterval [ 3, 0]. Observe that for all k,,..., 3}, if x + : x k x x k } x k k + k 5 +, ad similarly sup x + : x k x x k } 5 + k. Thus, lettig f x x +, L f, P 3 k k 3 3, x 3 0} Similarly, U f, P 3 k 5 + k Thus, 6 3 L f, P L f U f U f, P By the Order Limit Theorem ad the algebraic limit theorem with, we have 6 L f U f 6, so L f U f 0 3 x + dx 6.

2 REAL ANALYSIS I FALL 006 b 3 x dx Cosider the partitio P x 0, x +, x +,..., x k + k,..., x 3 + 3, x 3 } of the iterval [, ]. Observe that 3 x x, so the fuctio f x 3 x is decreasig for x 0 ad icreasig for x 0. The, for all k,,..., 3}, sup } 3 x 3 + : x k x x k k if k 3 + k if + k 3, ad if } 3 x 3 + : x k x x k if k k 3 + k if + k 3 Thus, L f, P k k 3 + k k 3 k + 3 k+ 3 k Similarly, U f, P k k 3 + k k + 4 k 3 k k+ + 4 k 3 k Thus, k k 3 k k L f, P L f U f U f, P 6 5. By the Order Limit Theorem ad the algebraic limit theorem with, we have

3 6 L f U f 6, so L f U f c x dx 3 Cosider the partitio SAMPLE QUESTIONS FOR FINAL EXAM 3 3 x dx 6. P x 0 3, x 3 +, x 3 +,......, x , x 4 } of the iterval [ 3, ]. Lettig ψ x x, observe that xk if k 3 because x is decreasig here if ψ x : x [x k, x k ]} x k if k > 3 because x is icreasig here 3 + k if k k. if k > 3 Thus, L ψ, P Note that 3 k 3 k 3 + k + 6 k + 4 k k3+ k 4 k3 4 k3+ 6 k 3 + k k k 4 4 7, k 3 k k k3+ k. so L ψ, P Similarly, 3 + k sup ψ x : x [x k, x k ]} if k k if k > 3,

4 4 REAL ANALYSIS I FALL 006 so ad U ψ, P 4 k3+ 3 k 3 k 3 + k + 6 k k 4 k k 3 k k 4 k3+ 4 k k 6 + k k3+ k, which implies 7 +, U ψ, P Sice L ψ, P L ψ U ψ U ψ, P, we have L ψ U ψ for all N, so that we ca use the order limit theorem as to coclude that the fuctio ψ is itegrable ad that L ψ U ψ 3 x dx 0. 5 Usig the defiitio of Riema itegral, fid ψ x 0 if x 3 7 if x 3 ψ x dx if it exists, if For k N, cosider the partitio P k,, +, } of the iterval [, ]. The 3 k 3 k [ if ψ x : x 3 k, 3 + ]} 0, ad k [ sup ψ x : x 3 k, 3 + ]} 7; k the ifima ad suprema of ψ over the other itervals are zero. Thus, L f, P k U f, P k k k. Sice L f, P k L f U f U f, P k, we have 0 L f U f 4 k

5 SAMPLE QUESTIONS FOR FINAL EXAM 5 for all k N. Usig the order limit theorem as k, we have that L f U f ψ x dx 0. x 6 Prove usig the defiitio that lim x+3. x 3 x Proof: For ay ε > 0, let δ mi ε, } > 0. If 0 < x 3 < δ, the < x < 3 ad x 3 < ε, which implies x x 3 < ε, so that x x 3 x x < ε, or x x + 3 x < ε. 7 Usig the defiitio, prove that each fuctio below is cotiuous o its domai. a g x x+ Proof: For ay ε > 0, if c, let δ mi ε c + 4, c + } > 0. If 4 x c < δ, the 4 c + < x c < c +, so 4 c c + < x < c + c +, or c + c + < x + < c + + c +, so that x + > c + > 0, or c + < x + We also have that x c < δ implies c x < ε 4 c +, which implies c x < ε x + c +, so that c x x + c + < ε, or x + c + < ε. Thus, g is cotiuous at ay c such that c. b h x 4x + 3 Proof: Let c be ay elemet of [ 3,, the domai of h. 4 Case : If c 3, the for ay ε > 0, let δ ε > 0. If x c 4 4 x < δ ad x 3, the 4 4 x < ε, so that 4x + 3 h x h 4 3 < ε. Thus, h is cotiuous at 3 4.

6 6 REAL ANALYSIS I FALL 006 Case : If c > 3, the for ay ε > 0, if δ mi ε c + 3, 3 4c + 3} > 0 ad if 6 x c < δ, the 4x x c c > 4 3 4c c 6 4c + 3, 4 so that 4c + 3 < 4x + 3. Further, agai if x c < δ, The 4 x c < ε 3 4c + 3, so that 4x 4x 4c < ε c + 3, or 4x 4x + 3 4c + 3 < ε c x + 3 4c + 3 4x c + 3 < ε, or 4x + 3 4c + 3 < ε. Thus, h is cotiuous at c. 8 Prove or disprove: If φ : [0, ] R is cotiuous, ad if x is a Cauchy sequece such that x [0, ] for all N, the φ x is Cauchy. True: same proof as 0b below. 9 Usig the defiito, prove that the derivative of x is. x Observe that if x > 0, y x lim y x y x lim y x y + x x by the algebraic limit theorem. 0 Prove that the derivative of is, usig the defiitio. x x Proof: Let f x. If x ad y are i the domai of f, that is x, y R \ 0}, the x y x y x x y yx y x yx. We may ow take the limit as y x of the expressio above, usig the algebraic limit theorem ad the cotiuity of g t algebraic cotiuity theorem. Thus, the limit tx exists ad is f x lim y x y x y x lim y x yx x x x. Assumig Rolle s Theorem, prove the Mea Value Theorem. Hit: Apply Rolle s Theorem to the fuctio g x f x fb fa x. b a Determie if the each series below coverges or diverges, ad prove your aswer.

7 SAMPLE QUESTIONS FOR FINAL EXAM 7 a 3 Observe that for, 4, so that 3 4, or 4 3. Thus, for, Sice 0 3 3/ / 4 3 coverges by the p-series test p 3 >, 3/ 3 coverges by the direct compariso test. b Observe that this series is alteratig. Sice 0 < < for all N, sice +3+ lim 0, the Squeeze Theorem implies that lim 0. Next, observe that +3+ for all 4, , so the absolute values of the terms of the series are decreasig. By the alteratig series test, the series coverges. 3 True or False. Prove your respose you may refer to theorems. a If φ : 0, ] R is cotiuous, ad if x is a Cauchy sequece such that x 0, ] for all N, the φ x is Cauchy. False: For example, let φ x which is cotiuous o 0, ] by the algebraic x properties of cotiuous fuctios, ad cosider the sequece x defied by x 0, ]. The x coverges to 0 ad is thus Cauchy, but φ x, ad so h x diverges ad is therefore ot Cauchy. b If φ : [0, ] R is cotiuous, ad if x is a Cauchy sequece such that x 0, for all N, the φ x is Cauchy. True. Sice [0, ] is closed, x coverges to a poit x [0, ]. The, sice φ is cotiuous, φ x coverges to φ x. Thus, φ x is Cauchy, sice it is a coverget sequece. c Every bouded fuctio o [a, b] is cotiuous o [a, b]. False: For example, let a 0, b, ad cosider the fuctio q : [0, ] R defied by if x q x 0 otherwise The q x for all x [0, ], so q is bouded. However, q is ot cotiuous at ; eve though is a limit poit of [0, ], lim q x 0 q 0. x

8 8 REAL ANALYSIS I FALL 006 d Every bouded fuctio o [a, b] is itegrable o [a, b]. False: For example, if if x Q g x 0 otherwise the g is bouded 0 g x x, but each upper sum is b a, ad each lower sum is zero, so the fuctio g is ot Riema itegrable. e If lim x x is a coverget sequece, the the set x N} x} is compact. True: Sice x coverges, the set x N} is bouded, say x B for all N. The every elemet of S x N} x} is bouded i absolute value by max B, x }. Because lim x x, every eighborhood of x cotais all but fiitely may terms of the sequece x. Thus, if y R x}, the if we let ε y x, V ε y V ε x, so that V ε y y} ca cotai at most a fiite umber of terms x,..., x k } of the sequece. Thus, if ε mi ε, y x,..., y x k } > 0, we have that V eε y y}. Thus, y is ot a limit poit of S. We have show that o elemet other tha x is a limit poit of S, so S is closed. Sice S is closed ad bouded, S is compact, by the Heie-Borel Theorem. f Ay bouded set that cotais all of its limit poits is compact. True: Sice such a set would be closed ad bouded, the Heie-Borel Theorem implies that the set is compact. g Ay set that cotais all of its limit poits is compact. False: For example, N is closed but ot bouded, ad so it cotais all of its limit poits there are oe but is ot compact. h The set x R x 3 3x 6 + 7x 7x 00} is closed. True: Call this set A; observe that A f, 00], where f is the fuctio defied by f x x 3 3x 6 +7x 7x. The fuctio f is cotiuous because it is a polyomial, a sum of products of j x x ad costats, ad the algebraic properties of cotiuous fuctios the imply f is cotiuous. The set, 00] is closed, ad its complemet 00, is ope. Observe that B R \ Af 00, is ope because it is the cotiuous iverse image of a ope set. Thus, A R \ B is closed. i If f is cotiuous o,,, the it is also cotiuous o, ] [,. False: Let if x [, ] f x 0 otherwise The fuctio is costat ad thus cotiuous o,,, but f is ot cotiuous at or at. For istace, the defiitio of cotiuous does ot hold at for ε, because for ay δ > 0, f x f > for x δ, eve though x δ < δ. j If the fuctio φ : R R satisfies the itermediate value property, the it is cotiuous. False: For example, let si φ x x if x 0 0 if x 0 The φ is ot cotiuous at 0, because φ δ, δ [, ] for every δ > 0 so the defiitio of cotiuous caot hold at 0 with ε. O the other had, φ satisfies the itermediate value property IVP. Sice φ is cotiuous away from 0,

9 SAMPLE QUESTIONS FOR FINAL EXAM 9 it automatically satisfies the IVP for itervals cotaied i R \ 0}. Next, if [a, b] is ay iterval cotaiig zero, φ [a, b] φ a, b [, ], so there are a ifiite umber of poits x i a, b such that φ x is ay possible itermediate value. k If f is oegative ad itegrable o [0, ] ad if f 0, the f x 0 0 for every x 0,. False: Let f x if x 0 otherwise The this fuctio is oegative, ad it is ot idetically zero o 0,. However, observe that every lower sum will be zero for this fuctio sice every iterval cotais poits x where f x 0. Next, for k >, cosider the partitio P k defied by P k 0,, + },. k k The U f, P k k. k k k Sice L f, P k L f U f U f, P k, ad sice lim U f, P k 0 ad k L f, P k 0, by the algebraic limit theorem, we have 0 L f U f 0, so that f is itegrable ad f 0. 0 l If f ad g are two itegrable fuctios o [, ], the f g f g. True: f g f g f g f g. m If A is bouded, the if A is a limit poit of A. False: For example, the set A, } has o limit poits, but if A. The fuctio 0 if t is ratioal Bubba t t if t is irratioal is cotiuous at 0. True: Give ε > 0, if t 0 < δ ε, the Bubba t Bubba 0 Bubba t 0 0 if t is ratioal t if t is irratioal < ε. o If b f x dx 0, f is cotiuous, ad f x 0 for all x [a, b], the a f x 0 for all x [a, b]. True: Suppose that f x is ot always 0; that is, suppose that f c < 0 for some c [a, b]. The, sice f is cotiuous, lettig ε fc > 0, there exists δ > 0 such

13 Proof: If p k x SAMPLE QUESTIONS FOR FINAL EXAM 3 k 0 f 0! x is the k th partial sum of the Taylor series of si x, the Lagrage Remaider Theorem implies that for each x R, si x p x f + c x +, +! where c is a umber betwee 0 ad x. Observe that sice d si x cos x, d cos x dx dx si x, si x, ad cos x, we have that f + t for all t R. The 0 si x p x f + c +! x+ x + +!. Next, for fixed x R, cosider the series x + x + x + +!. Observe that for ay +! +! x +, which has a limit of zero as. By the ratio test, the series coverges, so by the divergece criterio for series, lim iequality above, + x +! 0, so by the Squeeze Theorem ad the lim si x p x 0, so the Taylor series for si x coverges to si x for all x R. Prove usig basic priciples that a sequece of cotiuous fuctios that coverges uiformly coverges to a cotiuous fuctio. Proof: Suppose f is a sequece of cotiuous fuctios that coverges uiformly to a fuctio f o A R. Let c be ay elemet of A. By the defiitio of uiform covergece, give ay ε > 0, we may choose N N such that for all N, f x f x < ε 3 for all x A. Next, for that specific N, the cotiuity of f N implies that we may choose δ > 0 so that if x A ad x c < δ the f N x f N c < ε. The if x c < δ, we 3 have f x f c f x f N x + f N x f N c + f N c f c f x f N x + f N x f N c + f N c f c < ε 3 + ε 3 + ε 3 ε. Thus f is cotiuous at c. Give a example of a sequece of cotiuous fuctios that coverges but does ot coverge to a cotious fuctio. Justify your respose. For each N, let α : [0, ] R be defied by α x x ; all of these fuctios are cotiuous by the algebraic cotiuity theorem. The let β x lim α x 0 if 0 x < if x Observe that lim β x 0 β, so β is ot cotiuous at. x 3 Let B be a bouded, oempty set of real umbers, ad let b be the least upper boud of B. If b is ot a elemet of B, which of the followig is ecessarily true? Justify your resposes. a B is closed. False: for example, if B 0,, b / B. b B is ot ope. False: see above. c b is a limit poit of B.

14 4 REAL ANALYSIS I FALL 006 True. By the sup lemma, give ay ε > 0, there is a x B such that b ε < x b, but sice b / B we also have b ε < x < b. Thus, every ε-eighborhood of b cotais a poit x B b} B; thus, b is a limit poit of B. d No sequece i B coverges to b. False. By part c, b is a limit poit of B, so there exists a sequece of poits i B b} B that coverges to b. e There is a ope iterval cotaiig b that cotais o poits of B. False. By part c, b is a limit poit of B. Every ope iterval cotaiig b is a ope set, so there exists ε > 0 so that V ε b is a subset of that iterval. But every such V ε b cotais a poit of B b} B, so the ope iterval does as well. 4 Prove that every oempty ope set i R is a uio of ope itervals. Proof: If A is a oempty ope subset of R, the for every x A, there is a ε > 0 such that V ε x A. The cosider B V ε x. x A,V εx A The otatio above implies that the uio is take over all possible x A ad all possible ε > 0 such that V ε x A. The B is a uio of ope itervals, ad y B, the y V ε x A for some ε > 0 ad some x A, so y A. Next, if y A, there exists ε > 0 such that y V ε y A, so y B. Thus, A B is a uio of itervals. 5 Assumig that f ad g are real-valued fuctios o R, egate the followig statemet: For each s R, there exists r R such that if f r > 0, the g r > 0. Negatio: There exists s R such that for all r R such that f r > 0, we have g r 0. φx 6 Suppose that φ : R R is a fuctio such that lim x 0 x is a real umber L ad φ 0 0. Which of the followig are ecessarily true? Justify your resposes. a φ is differetiable at 0. True: defiitio of the derivative: φ φ x φ 0 0 lim L. x 0 x 0 b L 0. False: For example, if φ x x for all x R, the φ 0 L. c lim φ x 0. x 0 True: Differetiable implies cotiuous at zero, so sice 0 is a limit poit of the domai, lim φ x φ 0 0. x 0 7 Provide a defiitio for the statemet lim f x. x c lim f x, the lim x c x c f x 0. The prove that if The statemet lim f x meas that give M > 0, δ > 0 such that 0 < x c < x c δ ad x domai of f implies f x M. If this is true, the, give ε > 0, choose N N such that < ε. The choose δ as above with M N. The 0 < x c < δ N ad x domai of f \ 0} implies that 0 fx < ε. Thus, lim N x c f x 0. 8 Give a example of a fuctio α : R R that is uiformly cotiuous ad a example of a fuctio β : R R that is ot uiformly cotiuous. Solutio: For example, let α t t. The, give ay ε > 0, if t x < δ ε, the α t α x t x < ε. Thus α is uiformly cotiuous.

15 SAMPLE QUESTIONS FOR FINAL EXAM 5 Next, let β x x. Observe that for every δ > 0, there exists such that δ >, or δ >. The + δ < δ, but β + δ β + δ δ + 4 δ >, so the defiitio of uiformly cotiuous does ot hold for ay ε <. 9 Be able to prove the product ad quotiet rules. OK, I m able. 30 Prove that if f is differetiable o a iterval with f x o that iterval, the f ca have at most oe fixed poit. Note: a fixed poit of f is a umber x such that f x x. Proof: Cosider the fuctio h x f x x. The by hypothesis, h is the sum of differetiable fuctios ad is thus differetiable o the iterval, ad h x 0 o that iterval. Suppose that h a 0 ad h b 0 for some a ad b i the iterval ie both a ad b are fixed by f. The, by Rolle s Theorem, if a b, the there exists c betwee x ad y such that h c 0, which is a cotradictio. Thus, there is at most oe fixed poit for f o that iterval.

INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

### a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x

0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

### Analysis Notes (only a draft, and the first one!)

Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### 2-3 The Remainder and Factor Theorems

- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

### http://www.webassign.net/v4cgijeff.downs@wnc/control.pl

Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial

### Math 113 HW #11 Solutions

Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

### Building Blocks Problem Related to Harmonic Series

TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

### MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### Notes on exponential generating functions and structures.

Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES

RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES GEOFFREY GRIMMETT AND SVANTE JANSON Abstract. We study the radom graph G,λ/ coditioed o the evet that all vertex degrees lie i some give subset S of the oegative

### Solutions to Exercises Chapter 4: Recurrence relations and generating functions

Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose

### Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

### Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

### Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

### Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

### Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

### Overview on S-Box Design Principles

Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### A sharp Trudinger-Moser type inequality for unbounded domains in R n

A sharp Trudiger-Moser type iequality for ubouded domais i R Yuxiag Li ad Berhard Ruf Abstract The Trudiger-Moser iequality states that for fuctios u H, 0 (Ω) (Ω R a bouded domai) with Ω u dx oe has Ω

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Factors of sums of powers of binomial coefficients

ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

### Irreducible polynomials with consecutive zero coefficients

Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

### 3. If x and y are real numbers, what is the simplified radical form

lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

### Descriptive Statistics

Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

### 5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

### NOTES ON PROBABILITY Greg Lawler Last Updated: March 21, 2016

NOTES ON PROBBILITY Greg Lawler Last Updated: March 21, 2016 Overview This is a itroductio to the mathematical foudatios of probability theory. It is iteded as a supplemet or follow-up to a graduate course

### Lipschitz maps and nets in Euclidean space

Lipschitz maps ad ets i Euclidea space Curtis T. McMulle 1 April, 1997 1 Itroductio I this paper we discuss the followig three questios. 1. Give a real-valued fuctio f L (R ) with if f(x) > 0, is there

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### 2. Degree Sequences. 2.1 Degree Sequences

2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we

### Research Article Sign Data Derivative Recovery

Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

### Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### A Constant-Factor Approximation Algorithm for the Link Building Problem

A Costat-Factor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus

### PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

### On the L p -conjecture for locally compact groups

Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

### Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

### Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

### Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

### Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal

### The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

### SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES

SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES MATH 57A. Itroductio Our geometric ituitio is derived from three-dimesioal space. Three coordiates suffice. May objects of iterest i aalysis, however, require far

### 19 Another Look at Differentiability in Quadratic Mean

19 Aother Look at Differetiability i Quadratic Mea David Pollard 1 ABSTRACT This ote revisits the delightfully subtle itercoectios betwee three ideas: differetiability, i a L 2 sese, of the square-root

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

### A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:

### Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the Average-Case Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis

### COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE. Richard A. Weida Lycoming College Williamsport, PA 17701 weida@lycoming.

COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE Richard A. Weida Lycomig College Williamsport, PA 17701 weida@lycomig.edu Abstract: Lycomig College is a small, private, liberal

### Universal coding for classes of sources

Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### 4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.

4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### 5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

### 10-705/36-705 Intermediate Statistics

0-705/36-705 Itermediate Statistics Larry Wasserma http://www.stat.cmu.edu/~larry/=stat705/ Fall 0 Week Class I Class II Day III Class IV Syllabus August 9 Review Review, Iequalities Iequalities September

### CHAPTER 11 Financial mathematics

CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

### Question 2: How is a loan amortized?

Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued

### CS85: You Can t Do That (Lower Bounds in Computer Science) Lecture Notes, Spring 2008. Amit Chakrabarti Dartmouth College

CS85: You Ca t Do That () Lecture Notes, Sprig 2008 Amit Chakrabarti Dartmouth College Latest Update: May 9, 2008 Lecture 1 Compariso Trees: Sortig ad Selectio Scribe: William Che 1.1 Sortig Defiitio 1.1.1

### Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

### Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

### MAXIMUM LIKELIHOODESTIMATION OF DISCRETELY SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH. By Yacine Aït-Sahalia 1

Ecoometrica, Vol. 7, No. 1 (Jauary, 22), 223 262 MAXIMUM LIKELIHOODESTIMATION OF DISCRETEL SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH By acie Aït-Sahalia 1 Whe a cotiuous-time diffusio is

### Stochastic Online Scheduling with Precedence Constraints

Stochastic Olie Schedulig with Precedece Costraits Nicole Megow Tark Vredeveld July 15, 2008 Abstract We cosider the preemptive ad o-preemptive problems of schedulig obs with precedece costraits o parallel

### Foundations of Operations Research

Foudatios of Operatios Research Master of Sciece i Computer Egieerig Roberto Cordoe roberto.cordoe@uimi.it Tuesday 13.15-15.15 Thursday 10.15-13.15 http://homes.di.uimi.it/~cordoe/courses/2014-for/2014-for.html

### Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

Caad. J. Math. Vol. 60 (1), 2008 pp. 3 32 Covex Bodies of Miimal Volume, Surface Area ad Mea Width with Respect to Thi Shells Károly Böröczky, Károly J. Böröczky, Carste Schütt, ad Gergely Witsche Abstract.

### Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

### AMS 2000 subject classification. Primary 62G08, 62G20; secondary 62G99

VARIABLE SELECTION IN NONPARAMETRIC ADDITIVE MODELS Jia Huag 1, Joel L. Horowitz 2 ad Fegrog Wei 3 1 Uiversity of Iowa, 2 Northwester Uiversity ad 3 Uiversity of West Georgia Abstract We cosider a oparametric

### 1.3. VERTEX DEGREES & COUNTING

35 Chapter 1: Fudametal Cocepts Sectio 1.3: Vertex Degrees ad Coutig 36 its eighbor o P. Note that P has at least three vertices. If G x v is coected, let y = v. Otherwise, a compoet cut off from P x v

### Ramsey-type theorems with forbidden subgraphs

Ramsey-type theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called H-free if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.

### Integer Factorization Algorithms

Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal

### Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios

### A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece

### TO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC

TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies

### Entropy of bi-capacities

Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

### Exploratory Data Analysis

1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

### A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING

A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING MATTHEW ACTIPES Abstract. This paper begis by defiig a probability space ad establishig probability fuctios i this space over discrete radom variables.

### Lecture 7: Stationary Perturbation Theory

Lecture 7: Statioary Perturbatio Theory I most practical applicatios the time idepedet Schrödiger equatio Hψ = Eψ (1) caot be solved exactly ad oe has to resort to some scheme of fidig approximate solutios,

### ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION

ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION STEVEN BÜRGER AND BERND HOFMANN Abstract. We revisit i L 2 -spaces the autocovolutio equatio x x = y with solutios which

### Nr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996

Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi

### Lehmer s problem for polynomials with odd coefficients

Aals of Mathematics, 166 (2007), 347 366 Lehmer s problem for polyomials with odd coefficiets By Peter Borwei, Edward Dobrowolski, ad Michael J. Mossighoff* Abstract We prove that if f(x) = 1 k=0 a kx

### Journal of Combinatorial Theory, Series A

Joural of Combiatorial Theory, Series A 118 011 319 345 Cotets lists available at ScieceDirect Joural of Combiatorial Theory, Series A www.elsevier.com/locate/jcta Geeratig all subsets of a fiite set with

### Chapter 7: Confidence Interval and Sample Size

Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

### Fast Fourier Transform

18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

### Escola Federal de Engenharia de Itajubá

Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José

### Rényi Divergence and L p -affine surface area for convex bodies

Réyi Divergece ad L p -affie surface area for covex bodies Elisabeth M. Werer Abstract We show that the fudametal objects of the L p -Bru-Mikowski theory, amely the L p -affie surface areas for a covex