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School: Duke
Final Exam - Math 139 Dec 16th Instructor Web page Mauro Maggioni www.math.duke.edu/ mauro/teaching.html You have 3 hours. You may not use books, internet, notes, calculators. The exam should be stapled, written legibly, with your name written at the top
School: Duke
Math 135, Fall 2011: HW 6 This homework is due at the beginning of class on Wednesday October 19th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 135, Fall 2011: HW 2 This homework is due at the beginning of class on Wednesday September 7th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Course: Ord & Prtl Diff Equations
1 Physics and Measurement CHAPTER OUTLINE 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Standards of Length, Mass, and Time Matter and Model-Building Density and Atomic Mass Dimensional Analysis Conversion of Units Estimates and Order-ofMagnitude Calculations Significant F
School: Duke
Course: Real Analysis
HOMEWORK 3 SOLUTIONS Problem 1 A basic property of Poisson random variables is that if cfw_Xk are i.i.d. Poisson(), then Sn is Poisson(n). Thus, e-n (n)k = P(Sn na) = P(Sn - n n(a - ). k! >0 P Hence, if a < we have P(Sn - n n(a - ) P(|Sn - n| n( - a) 0,
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #2, due September 7, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 55, #4. Royden, page 58, #6,7,8. Royden, page 64, #10,13. Note. Number 13 is long but rewarding. The previous
School: Duke
Course: Order And Partial Differential Equations
MATH 353: ODE AND PDE NOTES "0s 0 oo0 ! "0 " s (learning starts with the meaning of names) Stephanos Venakides January 15, 2014 Contents 1 VECTORS 4 1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Linear comb
School: Duke
Course: Order And Partial Differential Equations
Worked problems from end of Wednesday Wednesday, May 05, 2010 3:11 PM Final Review Page 1 Final Review Page 2
School: Duke
Course: Order And Partial Differential Equations
Final Review, Thursday Thursday, May 06, 2010 11:38 AM Final Review Page 1 Final Review Page 2 Final Review Page 3 Related problems: Final Review Page 4
School: Duke
Course: Order And Partial Differential Equations
Final Review, Tuesday Tuesday, May 04, 2010 12:30 PM Final Review Page 1 Final Review Page 2 Final Review Page 3 Final Review Page 4 Final Review Page 5
School: Duke
Course: Order And Partial Differential Equations
Final Review, Wednesday Wednesday, May 05, 2010 11:16 AM Final Review Page 1 Final Review Page 2 Final Review Page 3
School: Duke
Course: Linear Algebra And Differential Equations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
School: Duke
Course: Probability
The Nomos of the Earth The Nomos of the Earth in the International Law of the Jus Publicum Europaeum Carl Schmitt Translated and Annotated by G. L. Ulmen Telos Press Publishing 2006 Copyright 2003 by Telos Press, Ltd. All Rights Reserved. No portion of th
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle DemandforGoodsandServices Wewilldistinguishbetweenthreedifferentkindsofdemand relationships(orcurves): 1. Incomedemandcurves therelationshipbetween(exogenouslygiven)incomeandthe quantityofagoodthati
School: Duke
Course: Probability
CombiningBudgetswithTastes Choice arisesfromindividualsconfrontingtheireconomiccircumstances (i.e.budgets)withtheirtastes thuschoosingthebestpossible bundlefromwhatisaffordable. Wethereforebeginwiththe budget,assumingherethatpants aretwiceasexpensiveasshi
School: Duke
Course: Probability
EndowmentsandIncome Sofarwehave(mostly)assumedexogenousresources Youstartwithafixed amountof$ Thenallocate$ betweengoods But$arisefrompreviousdecisions Sellsome endowment Collect$for consumption EndowmentsandIncome Examplesof Endowments Ourtimeandskill w
School: Duke
Course: Probability
TwoFundamentalRationality AssumptionsaboutTastes 1) CompleteTastes: Rationality Axioms Individualscanalwaysmakecomparison betweenbundles.GivenbundlesA andB, eitheryoupreferA toB orB toA,oryou areindifferentbetweenthetwo. 2) TransitiveTastes: Given(1),ifbu
School: Duke
Course: Probability
School: Duke
6 A v 9P27P2j!u v X4qP2u5C Squ64d r v U 9Hdd7B5v$!#j9j 5$wgB14(!vAsqX954h6!mA1j Bq5 S5G! 3PH32U75qmqPjxP5#5FhCu2#CBPx95 UvC h pqs(G6e61 A6e6j15U(v(17& v X(Gs(G(a% o e '& ) ' % ) %
School: Duke
Course: Real Analysis
Lecture 8: Weak Law of Large Numbers 8-1 Lecture 8 : Weak Law of Large Numbers References: Durrett [Sections 1.4, 1.5] The Weak Law of Large Numbers is a statement about sums of independent random variables. Before we state the WLLN, it is necessary to de
School: Duke
Course: Real Analysis
Lecture 7 : Product Spaces 7.3 Product spaces and Fubinis Theorem i Denition 7.3.1 If (i , Fi ) are measurable spaces, i I (index set), form For simplicity, i = 1 . i i . i (write for this) is the space of all maps: I 1 . For i i , = (i : i I, i i ). is e
School: Duke
Course: Real Analysis
Lecture 6 : Distributions Theorem 6.0.1 (Hlders Inequality) If p, q [1, ] with 1/p + 1/q = 1 then o E(|XY |) |X |p |Y |q (6.1) Here |X |r = (E(|X |r )1/r for x [1, ); and |X | = inf cfw_M : P(|X | > M ) = 0. Proof: See the proof of (5.2) in the Appendix o
School: Duke
Course: Real Analysis
Lecture 5: Inequalities 5-1 Lecture 5 : Inequalities 5.7 Inequalities Let X, Y etc. be real r.v.s dened on (, F , P). Theorem 5.7.1 (Jensens Inequality) Let be convex, E(|X |) < , E(|(X )|) < . Then (E(X ) E(X ) (5.11) Proof Sketch: As is convex, is the s
School: Duke
Final Exam - Math 139 Dec 16th Instructor Web page Mauro Maggioni www.math.duke.edu/ mauro/teaching.html You have 3 hours. You may not use books, internet, notes, calculators. The exam should be stapled, written legibly, with your name written at the top
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2014 Summer Term 2, Yuan Zhang. You have 75 minutes. No books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the policies an
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2013 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Math 135, Fall 2011: HW 6 This homework is due at the beginning of class on Wednesday October 19th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 135, Fall 2011: HW 2 This homework is due at the beginning of class on Wednesday September 7th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 112L.03 WEEK 03 HOMEWORK ADDENDUM Spring 2014 Kathryn plays a game with Shara. They alternate turns ipping a coin (Kathryn starts), and whoever gets heads rst wins. 1. Assuming the coin is fair, what is the probability that Shara wins on her rst ip?
School: Duke
Duke University Due: Monday, October 21st, 2013 Math 31L.09: Laboratory Calculus and Functions I Instructor: Hangjun Xu Problem Set 3 Name: I have adhered to the Duke Community Standard in completing this exam. Signature: Problem Points 1 12 2 6 3 8 4 10
School: Duke
Duke University Due: Monday, November 4th, 2013 Math 31L.09: Laboratory Calculus and Functions I Instructor: Hangjun Xu Problem Set 4 Name: I have adhered to the Duke Community Standard in completing this exam. Signature: Problem Points 1 10 2 8 3 8 4 12
School: Duke
Course: Order And Partial Differential Equations
Additional Homework Problems Math 353, Fall 2013 These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a dierential equation? 2. What does it m
School: Duke
Course: Probability
Udvalgte lsninger til Probability (Jim Pitman) http:/www2.imm.dtu.dk/courses/02405/ 17. december 2006 1 02405 Probability 2004-2-2 BFN/bfn IMM - DTU Solution for exercise 1.1.1 in Pitman Question a) 2 3 Question b) 67%. Question c) 0.667 Question a.2) 4 7
School: Duke
Course: Laboratory Calculus I
Tiffany Labon Michael Mclennon Osagie Obanor Bryce Pittard Varying Density Lab Report Part 1: 1. The mosquitoes in the park are most concentrated closest to the river and the least concentrated furthest from the river. This is clear from the function
School: Duke
Math 32L Professor Bookman Will Park Jeff Chen Ralph Nathan Air Pollution: Particulate Matter Lab Report Part 4 1.) M Volume Density 4 p g M 1012 cm3 1.5 3 3 2 cm M 2.5( ) p 3 1013 g 3 2.) Total Mass = Mass of particle * number of particles
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
Course: Multivariable Calculus
CALCULUS II Sequences and Series Paul Dawkins Calculus II Table of Contents Preface . ii Sequences and Series . 3 Introduction . 3 Sequences . 5 More on Sequences .15 Series The Basics .21
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Calculus 1
Math 212, Section 15 Fall 2013 Lecture: Instructor: Oce: E-mail: Oce Hours: Web Page: Text: MWF 8:459:35, Physics 259 Nick Addington Physics 246 adding@math.duke.edu Tuesdays 12:001:00, and by appointment. http:/math.duke.edu/adding/courses/212/ Calculus
School: Duke
Course: Single Variable Calculus
Fall 2013 Math 122L Syllabus (Friday Lab) Textbook: Calculus: Concepts & Contexts (4th ed), by James Stewart Day 1-1 Date 27-Aug Topic Review of AP AB Differentiation topics Lab: L'Hopital's Rule and Relative Rates of Growth Riemann Sums Lab: Riemann Sums
School: Duke
Course: Single Variable Calculus
Fall 2013 Math 122L Syllabus (Thursday Lab) Textbook: Calculus: Concepts & Contexts (4th ed), by James Stewart Day 1-1 Date 26-Aug Topic Review of AP AB Differentiation topics Lab: L'Hopital's Rule and Relative Rates of Growth Riemann Sums Lab: Riemann Su
School: Duke
Course: Calculus
Math 32 Section 6 Fall 2008 Instructor: Tim Stallmann Oce: 037 Physics, West Campus email: tmstallm@math.duke.edu Oce Hours: by appointment About this Course Math 32 is a second-semester calculus course which covers with rigor Riemann sums and the denitio
School: Duke
Course: Multivariable Calculus
Homework Homework problems are assigned for every lecture, and students should ideally complete each assignment on the day of the lecture. The assigned problems for each lesson will be listed on the syllabus. (Note, we might find ourselves behind or ahead
School: Duke
Course: Multivariable Calculus
Syllabus for Math 102, Spring 08-09, Clark Bray Mathematics for Economists, Simon and Blume; Notes on Integrals for Math 102, Bray (Note: New homework problems will be added throughout semester; be sure you are looking at a current version!) Linear Algebr
School: Duke
Final Exam - Math 139 Dec 16th Instructor Web page Mauro Maggioni www.math.duke.edu/ mauro/teaching.html You have 3 hours. You may not use books, internet, notes, calculators. The exam should be stapled, written legibly, with your name written at the top
School: Duke
Math 135, Fall 2011: HW 6 This homework is due at the beginning of class on Wednesday October 19th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 135, Fall 2011: HW 2 This homework is due at the beginning of class on Wednesday September 7th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Course: Ord & Prtl Diff Equations
1 Physics and Measurement CHAPTER OUTLINE 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Standards of Length, Mass, and Time Matter and Model-Building Density and Atomic Mass Dimensional Analysis Conversion of Units Estimates and Order-ofMagnitude Calculations Significant F
School: Duke
Course: Real Analysis
HOMEWORK 3 SOLUTIONS Problem 1 A basic property of Poisson random variables is that if cfw_Xk are i.i.d. Poisson(), then Sn is Poisson(n). Thus, e-n (n)k = P(Sn na) = P(Sn - n n(a - ). k! >0 P Hence, if a < we have P(Sn - n n(a - ) P(|Sn - n| n( - a) 0,
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #2, due September 7, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 55, #4. Royden, page 58, #6,7,8. Royden, page 64, #10,13. Note. Number 13 is long but rewarding. The previous
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #1, due August 31, 2009 Write clear proofs that have good grammar and consist of complete sentences. 1. Prove that a function from a Metric Space < X, > to a metric space < Y, > is continuous at x X if and only if, given and >
School: Duke
Course: CALCULUS 2
Name:_ Day 12 Graded Homework Due Day 13 The curves with equations x n + y n = 1 , n = 4, 6, 8,., are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the arclength L2 k of the fat circle with n = 2
School: Duke
Course: Probability
p. 113, n. 60. There is a 50-50 chance the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability the queen
School: Duke
Course: Introductory Calculus II
SSM: Linear Algebra Section 9.1 Chapter 9 9.1 1. x(t) = 7e5t , by Fact 9.1.1. 3. P (t) = 7e0.03t , by Fact 9.1.1. 5. y(t) = -0.8e0.8t, by Fact 9.1.1. 7. x-2 dx = dt -x-1 = t + C 1 - x = t + C, and -1 = 0 + C, so that 1 -x = t - 1 x(t) = 1 1-t ;
School: Duke
Course: Real Analysis
HOMEWORK 2 SOLUTIONS Problem 1 Fix i1 , i2 , ., ik I. We shall show that Yi1 , .Yik are independent by showing that k Pcfw_Yi1 ti , ., Yik tk = l=1 P(Yil tl ), t1 , t2 , .tk R. Step1: For any bounded continuous functions f1 , f2 , ., fk , we have k k E l
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #3, due September 14, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 70, #19,21,23,24. Royden, page 73, #29,30.
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #4, due September 21, 2009 Write clear proofs that have good grammar and consist of complete sentences. 1. Royden, page 89, #3 2. Royden, page 89, #4 3. Royden, page 89, #5 4. Royden, page 89, #6 5. Royden, page 89, #7 6. Royd
School: Duke
Course: Real Analysis
831 Theory of Probability Fall 2009 Homework 6 Due Thursday, November 12 1. IID variables close up. Let cfw_Yk be i.i.d. random variables on R with a common continuous density f . As we put more and more points Yk down they tend to concentrate so let us
School: Duke
Course: Real Analysis
831 Theory of Probability Fall 2009 Homework 5 Due Tuesday, October 27 1. Let cfw_Xn , X be real random variables. d (a) Suppose Xn X in probability. Show that then also Xn X . (b) Suppose Xn converges in distribution to a constant c. Show that then Xn c
School: Duke
Course: Real Analysis
831 Theory of Probability Fall 2009 Homework 4 Due Tuesday, October 13 1. Monte Carlo integration. Assume given a continuous function u on [0, 1] such that 0 u(x) 1. We use u to create an array of cfw_0, 1-valued random variables as follows: let cfw_Xn,k
School: Duke
Course: Real Analysis
831 Theory of Probability Fall 2009 Homework 3 Due Thursday, October 1 1. Let > 0. Show that, for a > 0, lim en (n)k k! k : 0kna = 0, if a < 1, if a > . n Hint: apply the WLLN to Poisson random variables. 2. For each n N let cfw_Xn,k : 1 k n be IID random
School: Duke
Course: Real Analysis
831 Theory of Probability Fall 2009 Homework 2 Due Tuesday, September 22 1. Let cfw_Xn,i : n N, i I be real-valued random variables on (, F, P ). Assume that for each n N, the variables cfw_Xn,i : i I are independent. Assume that for each i I there is a r
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #10, due November 4, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 291, #1. Royden, page 298, #3, 4. Royden, page 310, #21,24,25,29. 8. (a) Let f (x) = x1 sin x for x > 0 and f
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #9, due October 28, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 258, #5a,b,c Royden, page 267, # 19, 21, 22 Royden, page 275, #27, 28 Royden, page 279, #33, 35
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #8, due October 21, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 122, #7 Royden, page 126, # 9, 12, 14, 15, 16, 17 Royden, page 135, #22
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #7, due October 12, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 101, #1,4. Royden, page 104, # 7, 10. Royden, page 110, #12, 14, 15
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #6, due October 7, 2009 Write clear proofs that have good grammar and consist of complete sentences. Royden, page 40, #22 Royden, page 46, #37 Royden, page 50, #48 Royden, page 64, #14 Hint: follow the Beale notes given out in
School: Duke
Course: Real Analysis
Mathematics 241, Problem Set #5, due September 28, 2009 (dont turn in) Write clear proofs that have good grammar and consist of complete sentences. Royden, page 96, #20,21,24 Royden, page 94, #16, 17
School: Duke
Course: Order And Partial Differential Equations
Exam 2 Math 108 - 06 Name: December 9, 2011 Grade: / 100 Each problem is worth 20 points. You must show all work and explain all reasoning to receive credit. Clarity will be considered in grading. Problem 1. Consider the following equation t y(t )e2 d,
School: Duke
Course: Order And Partial Differential Equations
Math 108 (01) Exam 2 Name: March 28, 2012 Grade: / 100 You must show all work and explain all reasoning to receive credit. Clarity will be considered in grading. Write your nal answer in a box. Problem 1 (10pts). Solve the initial value problem y + y =
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2014 Summer Term 2, Yuan Zhang. You have 75 minutes. No books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the policies an
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2013 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2013 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2014 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2013 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 103, 2012 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2014 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2013 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
School: Duke
Course: Real Analysis
HOMEWORK 4 SOLUTIONS Problem 1 First observe that (1) n1 ESn = n1 1kn EXn,k = 1kn n 1 u k n which in particular is the Riemann sum of u for the partition 1 k : 1 k n . Since u is n integrable, we know that that the Riemann sum converges to the integral of
School: Duke
Oce Problem 1 - Math 139 Due Oct 13th Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Oce consultation: Oct. 7th, 8th, 1pm-3pm, usual oce hours on Oct. 11th. Possibly other ti
School: Duke
Homework 13 - Math 139 Due Wed Dec. 8th Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Reading: from Reeds textbook: Section 6.3, 6.4 Problems: 6.2: #1,5,8,9 6.3: #3,6
School: Duke
Homework 12 - Math 139 Due Thu Dec. 2nd Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Reading: from Reeds textbook: Section 5.7, 6.1 (review), 6.2. I would suggest also read
School: Duke
Homework 11 - Math 139 Due Nov. 17th Instructor Oce Oce hours Web page Mauro Maggioni 293 Physics Bldg. Monday 1:30pm-3:30pm. www.math.duke.edu/ mauro/teaching.html Reading: from Reeds textbook: Section 5.3,5.6 Problems: 5.1: #1,4,11 5.2: #9,15 Additional
School: Duke
Course: Order And Partial Differential Equations
MATH 353: ODE AND PDE NOTES "0s 0 oo0 ! "0 " s (learning starts with the meaning of names) Stephanos Venakides January 15, 2014 Contents 1 VECTORS 4 1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Linear comb
School: Duke
Course: Order And Partial Differential Equations
Worked problems from end of Wednesday Wednesday, May 05, 2010 3:11 PM Final Review Page 1 Final Review Page 2
School: Duke
Course: Order And Partial Differential Equations
Final Review, Thursday Thursday, May 06, 2010 11:38 AM Final Review Page 1 Final Review Page 2 Final Review Page 3 Related problems: Final Review Page 4
School: Duke
Course: Order And Partial Differential Equations
Final Review, Tuesday Tuesday, May 04, 2010 12:30 PM Final Review Page 1 Final Review Page 2 Final Review Page 3 Final Review Page 4 Final Review Page 5
School: Duke
Course: Order And Partial Differential Equations
Final Review, Wednesday Wednesday, May 05, 2010 11:16 AM Final Review Page 1 Final Review Page 2 Final Review Page 3
School: Duke
Course: Linear Algebra And Differential Equations
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School: Duke
Course: Probability
Yale Law Journal April 2006 Essay *1346 THE CORE OF THE CASE AGAINST JUDICIAL REVIEW Jeremy Waldron [FNa1] Copyright (c) 2006 Yale Law Journal Company, Inc.; Jeremy Waldron *1347 ESSAY CONTENTS I. II. III. IV. V. VI. VII. INTRODUCTION. DEFINITION OF JUDIC
School: Duke
Course: Calculus
Calculus I Review 1. Fundamental stu: (a) Write the denition of derivative of f (x) at x = a. (b) Write the denition of the denite integral b a f (t)dt. (c) Write the rst fundamental theorem of calculus. 2. (a) Draw the graphs of y = ex and y = ln x on th
School: Duke
Course: Differential Geometry
Midterm, Math 206 Differential Geometry: Curves and Surfaces in R3 Instructor: Hubert L. Bray March 3, 2011 Your Name: Honor Pledge Signature: 1 Instructions: This is a 75 minute, closed book exam. You may bring one 8 2 11 piece of paper with anything you
School: Duke
Course: Differential Geometry
Midterm, Math 421 Differential Geometry: Curves and Surfaces in R3 Instructor: Hubert L. Bray February 26, 2013 Your Name: Honor Pledge Signature: 1 Instructions: This is a 75 minute, closed book exam. You may bring one 8 2 11 piece of paper with anything
School: Duke
Course: Differential Geometry
Final Exam, Math 421 Differential Geometry: Curves and Surfaces in R3 Instructor: Hubert L. Bray Monday, April 29, 2013 Your Name: Honor Pledge Signature: 1 Instructions: This is a 3 hour, closed book exam. You may bring one 8 2 11 piece of paper with any
School: Duke
Course: Differential Geometry
Final Exam, Math 206 Differential Geometry: Curves and Surfaces in R3 Instructor: Hubert L. Bray Saturday, May 7, 2011 Your Name: Honor Pledge Signature: 1 Instructions: This is a 3 hour, closed book exam. You may bring one 8 2 11 piece of paper with anyt
School: Duke
Course: Introductory Calculus
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School: Duke
Course: Introductory Calculus
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School: Duke
Course: Linear Algebra
EXAM 3 Math 107, 2011-2012 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Linear Algebra
EXAM 2 Math 107, 2011-2012 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Linear Algebra
EXAM 1 Math 107, 2011-2012 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Linear Algebra
EXAM 3 Math 107, 2011-2012 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Linear Algebra
EXAM 2 Math 107, 2011-2012 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Linear Algebra
EXAM 1 Math 107, 2011-2012 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Linear Algebra
EXAM 3 Math 107, 2010-2011 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Linear Algebra
EXAM 2 Math 107, 2010-2011 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Fall 2006-2007, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number "I have adhered to t
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Fall 2006-2007, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number "I have adhered to t
School: Duke
Course: Calculus 1
EXAM 2 Math 103, Summer 2006, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. (/20 points) 2. (/20
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Summer 2006, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. (/20 points) 2. (/20
School: Duke
Course: Calculus 1
EXAM 3 Math 103, Spring 2006, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. (/30 points) 2. (/20
School: Duke
Course: Calculus 1
EXAM 2 Math 103, Spring 2006, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. (/20 points) 2. (/15
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Spring 2006, Clark Bray. You have 50 minutes. No notes, no books. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. (/20 points) 2. (/20 points) "I have
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, Fall 2009-2010, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. 2. "I have adhere
School: Duke
Course: Mathematics For Economists
EXAM 1 Math 102, 2010-2011 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, 2009-2010 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, 2009-2010 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. 2. 3. 4. 5. 6.
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, Fall 2009-2010, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. 2. 3. 4. 5. 6. "I
School: Duke
Course: Mathematics For Economists
EXAM 1 Math 102, 2009-2010 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. Good luck! Name ID number 1. 2. "I have adhe
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, 2010-2011 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, 2010-2011 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the pol
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Course: Mathematics For Economists
EXAM 1 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, 2010-2011 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Mathematics For Economists
EXAM 1 Math 102, 2010-2011 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, 2010-2011 Spring, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplified. All of the p
School: Duke
Course: Probability
The Nomos of the Earth The Nomos of the Earth in the International Law of the Jus Publicum Europaeum Carl Schmitt Translated and Annotated by G. L. Ulmen Telos Press Publishing 2006 Copyright 2003 by Telos Press, Ltd. All Rights Reserved. No portion of th
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle DemandforGoodsandServices Wewilldistinguishbetweenthreedifferentkindsofdemand relationships(orcurves): 1. Incomedemandcurves therelationshipbetween(exogenouslygiven)incomeandthe quantityofagoodthati
School: Duke
Course: Probability
CombiningBudgetswithTastes Choice arisesfromindividualsconfrontingtheireconomiccircumstances (i.e.budgets)withtheirtastes thuschoosingthebestpossible bundlefromwhatisaffordable. Wethereforebeginwiththe budget,assumingherethatpants aretwiceasexpensiveasshi
School: Duke
Course: Probability
EndowmentsandIncome Sofarwehave(mostly)assumedexogenousresources Youstartwithafixed amountof$ Thenallocate$ betweengoods But$arisefrompreviousdecisions Sellsome endowment Collect$for consumption EndowmentsandIncome Examplesof Endowments Ourtimeandskill w
School: Duke
Course: Probability
TwoFundamentalRationality AssumptionsaboutTastes 1) CompleteTastes: Rationality Axioms Individualscanalwaysmakecomparison betweenbundles.GivenbundlesA andB, eitheryoupreferA toB orB toA,oryou areindifferentbetweenthetwo. 2) TransitiveTastes: Given(1),ifbu
School: Duke
Course: Probability
School: Duke
Course: Probability
ConsumerChoiceSetsandBudget Constraints Peoplemakethebestchoicestheycan giventheircircumstances Constraints CurrentIncome Assets(fromSavings) Skills Time FutureIncome(for Borrowing) ConsumerChoiceSetsandBudget Constraints Consumersaresmall,havenopowerove
School: Duke
Course: Probability
WhatIsMicroeconomics? Microeconomics isthescience thatinvestigatesthe socialconsequences oftheinteractionof rational beingsthatpursuetheirperceived selfinterest. EconomicsasaScience RationalityandSelfInterest SocialConsequences WhatIsMicroeconomics? Ec
School: Duke
Course: Probability
+r o a 0) q ,h F o o $r d r-i +) ,1 - hn i ) o d tri q) Ecfw_ h Sr t ) . trl ol Fcfw_ cfw_-) a G) d ?1 r-f t) H o +.t o o q) f) d Tcfw_ ) o d cfw_rJ .Fcfw_ (J F * e r-( br) .h F o o L rlt4 Fcfw_ >( o c) .-L d H r-l JlJ l-) A o I t) +) .F( -r > >r *A O a L
School: Duke
Course: Probability
Recent Titles in Global Perspectives in History and Politics Macro-Nationalisms: A History of the Pan-Movements Louis L. Snyder The Myth of Inevitable Progress Franco Ferrarotti Power and Policy in Transition: Essays Presented on the Tenth Anniversary of
School: Duke
Course: Probability
CR4-3 14 12/23/04 4:47 PM Page 1 The Theory of the Partisan A Commentary/Remark on the Concept of the Political CARL SCHMITT Translated by A. C. Goodson, Michigan State University, East Lansing Dedicated to Ernst Forsthoff on his 60th Birthday, 13 Septemb
School: Duke
Course: Probability
The Unknown Donoso Cortsj Carl Schmitt When one attempts to place Donoso Cortes in the history of 19th cen- tury political thought, one must begin with an apology and many reserva- tions. Here one is dealing with a man whose name is hardly recognized outs
School: Duke
Course: Probability
Donoso Corts in Berlin (1849)1 Carl Schmitt The l'ifreuzzeirz/zng2 published the following tribute to Donoso Cortes in a dispatch from Paris on May 4, 1853: Not only Spain, but all of Christianity has suffered a great loss as a result of the death of the
School: Duke
Course: Probability
A PanEuropean Interpretation ofDonoso Cortsl Carl Schmitt I The men of the German National Assembly in Frankqu in 1848 wanted to create an empire whose very existence would be tantamount to a European revolution. This sentence was written in 1849. Its aut
School: Duke
Course: Probability
Page iii Political Romanticism Carl Schmitt translated by Guy Oakes Page iv This translation 1986 by the Massachusetts Institute of Technology This book was originally published as Politische Romantik, 1919, 1925 by Duncker & Humblot, Berlin. All rights r
School: Duke
Course: Probability
CARL SCHMITT HAMLET OR . My father's brother - but no more like my father Than 1 to Hercules. I . 152-153 HECUBA Hamlet. Act ,5C.11, 1HE IRRUPTION OF TIME INTO PLAY EDITED. TRANSLATBD AND WITH A POSTPACB by SIMONA DRAGHICI, PhD. , , ", ." 1" < , PlUTARCH
School: Duke
Course: Probability
Studies in Contemporary German Social Thought Thomas McCarthy, General Editor Political Theology Theodor W. Adorno, Against Epistemology: A Metacritique Theodor W . Adorno, Prism Karl-Otto Apel, Understanding and Explanation: A Trawcendental-Pragmatic Per
School: Duke
Course: Probability
OWEN M. FISS Groupsand the EqualProtection Clause This is an essay about the structure and limitations of the antidiscrimination principle, the principle that controls the interpretation of the Equal Protection Clause. To understand the importance of that
School: Duke
Course: Probability
DAVID DYZENHAUS HOBBES AND THE LEGITIMACY OF LAW1 (Accepted 13 February 2001) ABSTRACT. Legal positivism dominates in the debate between it and natural law, but close attention to the work of Thomas Hobbes the founder of the positivist tradition reveals a
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle EconomicsCostsandSunkCosts Anexpensebyafirmisnotaneconomiccostunlessthereis somethingthefirmcandotoavoidtheexpense. Anythingelseisasunkcost anexpensethatcannotbe recoveredtomatterwhatthefirmdoes.
School: Duke
Course: Probability
HOBBES AND THE PRINCIPLE OF PUBLICITY 447 HOBBES AND THE PRINCIPLE OF PUBLICITY JEREMY WALDRON Abstract: A common view is that Hobbesian authoritarianism is quite indifferent to whether people know or understand the truth about politics and political arra
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle AggregationandWelfare Weusemarketdemandandsupplycurvestopredict marketoutcomes. Butweoftentreatthesemarketcurvesasiftheyemergedfromasingle individual arepresentativeagentthatstandsinforagroup. For
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle Equilibrium Thesituationthatariseswhenpeoplearealldoingthebesttheycan giventheircircumstancesiscalledanequilibrium. Anequilibriumiscompetitive ifallagents consumers,workers, firms,etc. aresmallrela
School: Duke
Course: Probability
ClicktoeditMastertitlestyle ClicktoeditMastertitlestyle Generalvs.PartialEquilibrium PartialEquilibriumModels considerasinglemarketinisolation. Butoftenchangesinonemarketspilloverintoothermarkets,givingriseto incomeorwealtheffectsandalteringpriceselsewher
School: Duke
Course: Probability
Political Economy of Institutions and Decisions Continuing his groundbreaking analysis of economic structures, Douglass North here develops an analytical framework for explaining the ways in which institutions and institutional change affect the performan
School: Duke
Course: Probability
Seminar on the Politics of Institutional Change Political Science 618S.01 Gross Hall 111 -Wed. 4:40-7:10 PM Professor Karen L. Remmer Duke University E-mail: remmer@duke.edu Office: Gross Hall 205 Phone: 919-660-4309 Office hours: Thurs.2-3:30pm "The prob
School: Duke
Course: Probability
Alternative visions of change in Douglass North's new institutionalism Fiori, Stefano Journal of Economic Issues; Dec 2002; 36, 4; ProQuest Central pg. 1025 Reproduced with permission of the copyright owner. Further reproduction prohibited without permiss
School: Duke
Course: Probability
Articles Informal Institutions and Comparative Politics: A Research Agenda Gretchen Helmke and Steven Levitsky Mainstream comparative research on political institutions focuses primarily on formal rules. Yet in many contexts, informal institutions, rangin
School: Duke
Course: Probability
Journal of Theoretical Politics 15(2): 123144 09516928[200304]15:2; 123144; 031645 Copyright & 2003 Sage Publications London, Thousand Oaks, CA and New Delhi INSTITUTIONALISM AS A METHODOLOGY Daniel Diermeier and Keith Krehbiel ABSTRACT We provide a denit
School: Duke
Course: Probability
Carey / PARCHMENT, EQUILIBRIA, AND INSTITUTIONS COMPARATIVE POLITICAL STUDIES / August-September 2000 Institutions are rules that constrain political behavior. Although there is consensus that being written down is neither necessary nor sufficient for an
School: Duke
Course: Probability
Do Constitutions Constrain? Adam Przeworski Department of Politics New York University October 30, 2003 THE QUESTION BEFORE US IS DO CONSTITUTIONS CONSTRAIN?. MY ANSWER COMES IN TWO PARTS: (1) IT IS OBVIOUS THAT AT TIMES THEY DO NOT. (2) IT IS VERY HARD T
School: Duke
Course: Probability
GETTING INSTITUTIONS RIGHT Dani Rodrik Harvard University April 2004 There is now widespread agreement among economists studying economic growth that institutional quality holds the key to prevailing patterns of prosperity around the world. Rich countries
School: Duke
6 A v 9P27P2j!u v X4qP2u5C Squ64d r v U 9Hdd7B5v$!#j9j 5$wgB14(!vAsqX954h6!mA1j Bq5 S5G! 3PH32U75qmqPjxP5#5FhCu2#CBPx95 UvC h pqs(G6e61 A6e6j15U(v(17& v X(Gs(G(a% o e '& ) ' % ) %
School: Duke
Course: Real Analysis
Lecture 8: Weak Law of Large Numbers 8-1 Lecture 8 : Weak Law of Large Numbers References: Durrett [Sections 1.4, 1.5] The Weak Law of Large Numbers is a statement about sums of independent random variables. Before we state the WLLN, it is necessary to de
School: Duke
Course: Real Analysis
Lecture 7 : Product Spaces 7.3 Product spaces and Fubinis Theorem i Denition 7.3.1 If (i , Fi ) are measurable spaces, i I (index set), form For simplicity, i = 1 . i i . i (write for this) is the space of all maps: I 1 . For i i , = (i : i I, i i ). is e
School: Duke
Course: Real Analysis
Lecture 6 : Distributions Theorem 6.0.1 (Hlders Inequality) If p, q [1, ] with 1/p + 1/q = 1 then o E(|XY |) |X |p |Y |q (6.1) Here |X |r = (E(|X |r )1/r for x [1, ); and |X | = inf cfw_M : P(|X | > M ) = 0. Proof: See the proof of (5.2) in the Appendix o
School: Duke
Course: Real Analysis
Lecture 5: Inequalities 5-1 Lecture 5 : Inequalities 5.7 Inequalities Let X, Y etc. be real r.v.s dened on (, F , P). Theorem 5.7.1 (Jensens Inequality) Let be convex, E(|X |) < , E(|(X )|) < . Then (E(X ) E(X ) (5.11) Proof Sketch: As is convex, is the s
School: Duke
Course: Real Analysis
Lecture 4: Expected Value 4-1 Lecture 4 : Expected Value References: Durrett [Section 1.3] 4.5 Expected Value Denote by (, F , P) a probability space. Denition 4.5.1 Let X : R be a F\B -measurable random variable. The expected value of X is dened by E(X )
School: Duke
Course: Real Analysis
Lecture 3 : Random variables and their distributions 3.1 Random variables Let (, F ) and (S, S ) be two measurable spaces. A map X : S is measurable or a random variable (denoted r.v.) if X 1 (A) cfw_ : X ( ) A F for all A S One can write cfw_X A or (X A)
School: Duke
Course: Real Analysis
Lecture 2 : Ideas from measure theory 2.1 Probability spaces This lecture introduces some ideas from measure theory which are the foundation of the modern theory of probability. The notion of a probability space is dened, and Dynkins form of the monotone
School: Duke
Course: Real Analysis
Lecture 1 : Introduction We will start with a simple combinatorial problem. Consider cfw_1, 11000 . How many elements x cfw_1, 11000 satisfy 1000 xi 50? i=1 More generally, for any n N and > 0 how many elements x cfw_1, 1n satisfy n xi n? i=1 The answer i
School: Duke
Final Exam - Math 139 Dec 16th Instructor Web page Mauro Maggioni www.math.duke.edu/ mauro/teaching.html You have 3 hours. You may not use books, internet, notes, calculators. The exam should be stapled, written legibly, with your name written at the top
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2014 Summer Term 2, Yuan Zhang. You have 75 minutes. No books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the policies an
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2013 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2013 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2014 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 2013 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 103, 2012 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2014 Summer Term 1, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 2013 Summer Term 2, Clark Bray. You have 75 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the p
School: Duke
Course: Order And Partial Differential Equations
Exam 2 Math 108 - 06 Name: December 9, 2011 Grade: / 100 Each problem is worth 20 points. You must show all work and explain all reasoning to receive credit. Clarity will be considered in grading. Problem 1. Consider the following equation t y(t )e2 d,
School: Duke
Course: Order And Partial Differential Equations
Math 108 (01) Exam 2 Name: March 28, 2012 Grade: / 100 You must show all work and explain all reasoning to receive credit. Clarity will be considered in grading. Write your nal answer in a box. Problem 1 (10pts). Solve the initial value problem y + y =
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
Course: Multivariable Calculus
NAME: MAT 212 Fall 2014 Sections 1 & 2 Test on Chapter 14 Rules: - no books, notes or calculator only your exam and blank sheets that can be obtained from the front; - your exam should be your own work only no exchanging problem scor
School: Duke
Course: Multivariable Calculus
NAME: MAT 212 Fall 2014 Sections 1 & 2 Test on Chapter 13 Rules: - no books, notes or calculator only your exam and blank sheets that can be obtained from the front; - your exam should be your own work only no exchanging problem scor
School: Duke
Course: Multivariable Calculus
1. Lines in 3-dimensional space The coordinates of two points Q1 and Q2 are (2,1,2) (for Q1) and (1,1,1) (for Q2). We also consider two planes, p1 and p2, given by the equations 33+2y222 (forpl) and 2xy+3z=0 (forpg). a) Does Q1 lie in p1? b) Does Q2 lie i
School: Duke
Course: Multivariable Calculus
1. Flux through a surface The surface S is shown below, as Viewed from two different angles, and parametrized as 2 . . cos i sin 2a) sin a cost NIH z a:(u,t)= (1+ it: _ (1+; . If: x \ Ir- x/a/"x - ' zu,t : 1+ \, ( > ( 2 where u ranges from U to E, and i
School: Duke
Course: Multivariable Calculus
1- (10 points each) (a) Find the volume of the solid in the rst octent bounded by the coordinate planes, the cylinder m2+y2 :4, and the plane z+y :3. 12/25? who? WA M:- (g m m w 01wa 9 '"STE " 33-42039 3 cfw_hellgwsali <3 (b) Let S be the surface that is
School: Duke
Course: Multivariable Calculus
EXAM 3 Math 212, 20142015 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the polici
School: Duke
Course: Multivariable Calculus
1. (2 pts each) True or false, you must justify your answer: (a) Ifavbza-cthenb=c. (e) Let r(t) = (mcfw_t),y(t),z(t) and assume 1' t) and r"(t) exist, then%(r(t) x I" : r05) >< r(t). ( ( 2" cfw_(-6 PM We. Wt: Wmi 0" (d) If f (may) =1n(y) then V) = 1/y-
School: Duke
Course: Multivariable Calculus
MATH 212 Midterm Exam 2 Fall 2014 Name: Instructions: Do not open this exam until instructed to do so. No material is allowed to be used in the exam. This includes books, notes, and calculators. You have 75 minutes for the exam. Use your time wisely; i
School: Duke
Course: Multivariable Calculus
EXAM 2 Math 212, 20142015 Fall, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the polici
School: Duke
Course: Multivariable Calculus
EXAM 1 Math 212, 20142015 Fall, Clark Bray. You have 50 minutes. N 0 notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All answers must be simplied. All of the polic
School: Duke
Course: Multivariable Calculus
MATH 212 Midterm Exam 1 Fall 2014 Name: Instructions: Do not open this exam until instructed to do so. No material is allowed to be used in the exam. This includes books, notes, and calculators. You have 75 minutes for the exam. Use your time wisely; i
School: Duke
Course: Mathematics For Economists
EXAM 3 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Course: Mathematics For Economists
EXAM 2 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Course: Mathematics For Economists
EXAM 1 Math 102, Fall 2011-2012, Yuriy Mileyko You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT. CLARITY WILL BE CONSIDERED IN GRADING. All of the policies and guidelines on the cl
School: Duke
Math 135, Fall 2011: HW 6 This homework is due at the beginning of class on Wednesday October 19th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 135, Fall 2011: HW 2 This homework is due at the beginning of class on Wednesday September 7th, 2011. You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem
School: Duke
Math 112L.03 WEEK 03 HOMEWORK ADDENDUM Spring 2014 Kathryn plays a game with Shara. They alternate turns ipping a coin (Kathryn starts), and whoever gets heads rst wins. 1. Assuming the coin is fair, what is the probability that Shara wins on her rst ip?
School: Duke
Duke University Due: Monday, October 21st, 2013 Math 31L.09: Laboratory Calculus and Functions I Instructor: Hangjun Xu Problem Set 3 Name: I have adhered to the Duke Community Standard in completing this exam. Signature: Problem Points 1 12 2 6 3 8 4 10
School: Duke
Duke University Due: Monday, November 4th, 2013 Math 31L.09: Laboratory Calculus and Functions I Instructor: Hangjun Xu Problem Set 4 Name: I have adhered to the Duke Community Standard in completing this exam. Signature: Problem Points 1 10 2 8 3 8 4 12
School: Duke
Course: Order And Partial Differential Equations
Additional Homework Problems Math 353, Fall 2013 These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a dierential equation? 2. What does it m
School: Duke
School: Duke
School: Duke
School: Duke
School: Duke
Math 111L.03 WEEK 08 HOMEWORK ADDENDUM Fall 2013 These problems are due on Monday, October 21 by 4:30pm. Collaboration is allowed, but what you turn in should be your own work. Please organize your work and write clearly. Remember to put your name on your
School: Duke
Math 111L.03 WEEK 07 HOMEWORK ADDENDUM Fall 2013 These problems are due on Wednesday, October 16 by 4:30pm. Collaboration is allowed, but what you turn in should be your own work. Please organize your work and write clearly. Remember to put your name on y
School: Duke
Course: Calculus
Supplemental homework 4 Math 112L.04 Due Friday, March 28, 2014. Show your work clearly. Write your answers on a separate piece of paper. 1. In this problem we will use what we now know about Taylor series to revisit a problem from Test 1. The problem was
School: Duke
Course: Calculus
Supplemental homework 2 Math 112L.04 Due Friday, Jan. 31, 2014. Turn in only problems 1, 2, and 3. Show your work clearly. Write your answers on a separate piece of paper. 1 1 2 k+2 2 k+4 . Find a concise expression for the N th partial sum, SN , for this
School: Duke
Course: Calculus
Supplemental homework 3 Math 112L.04 Due Wednesday, March 5, 2014. Turn in only the problems on this page. Show your work clearly. Write your answers on a separate piece of paper. 1. Consider the following function (a is a positive constant): a(1 + x), 1
School: Duke
Course: Calculus
Supplemental homework 3 Math 112L.04 Due Wednesday, March 5, 2014. Show your work clearly. Write your answers on a separate piece of paper. 1. Consider the following function (a is a positive constant): a(1 + x), 1 p(x) = a , 2 x 0, 1 x 0 0<x1 otherwise (
School: Duke
Course: Calculus
Supplemental homework 1 Math 112L.04 Due Friday, January 17, 2014. Show your work clearly. Write your answers on a separate piece of paper. 1. Suppose we have a weighted coin that comes up heads with probability h, where h is a number between 0 and 1. Let
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #3 SOLUTIONS Problem 1.3.1. Let 1 3 A= 2 . 1 Find A1 or determine that A is not invertible. Solution. The row reduction 1 3 2 1 1 0 0 1 / 1 0 R2 +3R1 R2 PQ WV 1 1 7 R2 R2 `/ 0 PQ WV 1 ` 1 2R2 R1 `R`/ 0 2 7 2 1 0 1 1 3 1/7 3/7 1/7 3/7
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #1 SOLUTIONS Problem 1.1.2. Solve the system 2x + y 2z = 0 2x y 2z = 0 x + 2y 4z = 0 Solution. Note that 2 rref 2 1 1 2 0 1 1 2 0 = 0 2 4 0 0 0 1 0 0 0 1 0 0 0 Hence the only solution is 0 x y = 0 . 0 z Problem 1.1.8. Solve the sy
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #4 SOLUTIONS Problem 1.4.4. Let 1 A= 0 0 0 2 0 0 0 . 1 4 Find A1 . Solution. Use Theorem 1.11 to compute A1 4 = diag (1, 2, 1) 1 4 1 = diag 1, , 1 2 4 = diag 1, 1 ,1 . 16 Problem 1.4.12. Let 1 A= 1 2 2 2 B= 3 4 3 , 1 1 5 . 1 Find B A
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #5 SOLUTIONS Problem 1.5.5. Let 1 1 2 2 A= 4 3 3 2 1 Find det (A) by expanding about column 2. Solution. Use Theorem 1.16 in the book to compute det (A) = (1) 4 3 2 2 (1) 3 1 2 3 + (2) 1 4 3 2 = (1) (4) (1) (3) (2) (1) (2) (1) (3) (3
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #2 SOLUTIONS Problem 1.2.5. Compute A 4B where 1 3 2 A = 3 1 , B = 3 2 1 0 1 2 . 4 Solution. Compute 8 1 3 2 1 1 3 A 4B = 3 1 4 3 2 = 3 1 + 12 0 2 1 0 4 2 1 7 7 18 3+4 = 3 + 12 1 + 8 = 15 7 . 2 15 2 + 0 1 + 16 Problem 1.2.9. Com
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #10 SOLUTIONS Problem 2.5.5. Show that x2 1, x2 + 1, x + 1 are linearly independent on R. Solution. Note that 2 2 w x 1, x + 1, x + 1 x=0 x2 1 = 2x 2 x2 + 1 2x 2 x+1 1 0 2 x=0 1 = 0 2 1 1 0 1 = 4 = 0. 2 0 2 Theorem 2.15 in the book th
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #9 SOLUTIONS Problem 2.4.4. (c) Determine if 0 0 1 1 , 1 0 0 1 , 1 0 0 0 , 0 1 1 0 form a basis for M22 (R). Solution. Considering 1 0 0 1 1 + 2 1 0 0 1 + 3 1 0 0 0 + 4 0 1 1 0 = 0 0 0 0 gives the system 2 + 3 = 0 1 + 4 = 0 4 = 0 1 +
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #6 SOLUTIONS Problem 1.6.4. Use Theorem 1.21 to determine if the matrix 2 1 3 3 A = 1 1 6 0 0 is invertible. Solution. Compute 2 det (A) = 1 6 1 1 0 3 1 3 =6 1 0 3 = 6 (3 + 3) = 0. 3 Hence A is not invertible. Problem 1.6.6. Use the a
School: Duke
Course: Linear Algebra
MATH 107.01 HOMEWORK #8 SOLUTIONS Problem 2.3.7. Determine if the M22 (R) vectors 1 1 0 0 , 1 1 1 1 , 0 1 1 1 are independent. Solution. The linear combination 1 0 0 1 + 2 1 1 1 1 1 + 3 0 1 1 0 = 1 0 0 0 gives the system 1 0 1 0 Since 1 0 rref 1 0 we s
School: Duke
Course: Elem Differential Equat
Math 356.01 Homework 28, due Monday, November 11 Do these problems in the book, and also the one below. 10.5: 2, 6, 22 and 10.6: 31 In problem 22 you should nd that the energy can decrease but cannot increase. Problem S. In our usual model for a spring, t
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 1 I. Problems to be graded on completion. Graphs 1. 2. 4. 5. 8. 2.4 (a) (a) (a) (a) (a) 2. (b) -1. (c) does not exist. (d) -3. -4. (b) -4. (c) -4. (d) 2. 1. (b) . (c) does not exist. (d) 1. -. (b) -. (c) -. (d) 1. 2. (b) 2. (c) 2.
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 6 I. Problems to be graded on completion. 25. We have 180 feet of fence, so 2x + y + (y - 100) = 180, so y = 140 - x. The area of the pen is A = xy = 140x - x2 , which is maximized when 0 = dA = 140 - 2x, so x = 70, so y = 70. But
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 5 I. Problems to be graded on completion. 14. 22 12 + 4(2)(1) = 12(1), so (2, 1) lies on the curve. Now 2xy 2 + 2x2 yy + 4y + 4xy = 12y 4 + 8y + 4 + 8y = 12y y = -2 y-1 = -2 or y = -2x + 5. x-2 16. 0 + cos(1 0) + 3 12 = 4, so (1,
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 4 I. Problems to be graded on completion. 1. a. (1 + x2 )-1 + x[-(1 + x2 )-2 ]2x = (1 + x2 )-1 - 2x2 (1 + x2 )-2 = b. c. d. e. f. 1 - x2 . (1 + x2 )2 1 2. sin(32 ) = sin(30 + 2 ) = sin( + 90 ) sin( ) + cos( ) 90 = 1 = 23 90 = 2 +
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 2 I. Problems to be graded on completion. 1. Substitute u = 4x and v = 2x. As x 0, u 0 and v 0. sin 4x 4x sin 4x 4x = = lim lim sin 2x x0 x0 sin 2x 2x 2x 4x lim x0 2x sin 4x 4x sin 2x lim x0 2x x0 lim = 4 lim x0 2 u0 sin u u sin v
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 3 I. Problems to be graded on completion. 1. b = x. c = v. e = u. g = t. h = a. j = 0. l = q. p = h. r = n. s = f . t = d. u = m. x = k. 2. u1/3 - x1/3 (u1/3 - x1/3 )(u2/3 + u1/3 x1/3 + x2/3 ) = lim ux ux u-x (u - x)(u2/3 + u1/3 x
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 7 I. Problems to be graded on completion. 1. y = x4 - 4x3 + 1, so y = 4x3 - 12x2 = 4x2 (x - 3), so y = 12x2 - 24x = 12x(x - 2). We cannot solve the equation y = 0. When y = 0, x = 0 or x = 3. When y = 0, x = 0 or x = 2. The signs
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 8 1. (a) To go from the second to the third lines, observe that for any numbers m and n, the inequality |m| n is equivalent to the inequality -n m n. If -n m n, then -n m, so n -m, so -m n, and also m n, so |m| n. b b (b) If f (x)
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 13 I. Problems to be graded on completion. 1. a. dy = y sin(2x + 3) dx dy = sin(2x + 3) dx y dy = sin(2x + 3) dx y 1 log y = - cos(2x + 3) + C 2 1 1 - 2 cos(2x+3)+C = eC e- 2 cos(2x+3) . y=e Using the initial condition, 5 = eC e-
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 12 I. Problems to be graded on completion. 1. a. The average value is 1 1 - (-1) 1 x2 dx = -1 1 x3 2 3 1 = -1 1 . 3 1 The function takes this value when x = .577. 3 b. The average value is 1 1 x4 1 x3 dx = 1 - (-1) -1 2 4 The func
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 11 I. Problems to be graded on completion. 1. a. Consider the line 2x + y = 4. If x = 0 then y = 4, so the y-intercept is 4. If y = 0 then 2x = 4, so x = 2, so the x-intercept is 2. Around the x-axis, the radius of a slice is 4 -
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 9 I. Problems to be graded on completion. 2. 2 -1 (x3 - x + 2) dx = 1 x4 x2 - + 2x 4 2 2 -1 1 =8+ . 4 1 4. 1 -3 [0 - (x2 + 2x - 3)] dx = -3 (-x2 - 2x + 3) dx = - x3 - x2 + 3x 3 -3 2 = 10 + . 3 6. If x + 4 = x2 - 2 then 0 = x2 - x
School: Duke
Course: Introductory Calculus
Solutions to Problem Set 10 I. Problems to be graded on completion. 1. Evaluate the following indefinite integrals: a. Let u = x3 + 1, so du = 3x2 dx, so x2 dx = x2 x3 + 1 dx = 1 3 du. 1 u3/2 2 + C = 9 (x3 + 1)3/2 + C 3 2/3 u1/2 1 du = 3 b. Let u = ex , s
School: Duke
Course: Probability
Math 230, Fall 2012: HW 10 Solutions Problems # 2,5 on p.399. Problems # 4,5,7 on p.407. Problems # 5,6,10 on p. 426-427. 1 Problem 1 (p. 399 # 2). SOLUTION: The distribution of G conditional on T = k is Binomial(k, 2 ) for k = 0, 1, 2, 3, 4. So we comput
School: Duke
Course: Probability
Math 230, Fall 2012: HW 9 Solutions Problem 1 (p.345 #4). Let X and Y be independent random variables each uniformly distributed on (0, 1). Find: a) P (|X Y | 0.25); b) P (|X/Y 1| 0.25); c) P (Y X | Y 0.25). SOLUTION. These problems are most easily solved
School: Duke
Course: Probability
Math 230, Fall 2012: HW 8 Solutions Problem 1 (p.309 #5). SOLUTION. Consider nding the cdf of X 2 rst. Let Y = X 2 . Since 1 X 2, 0 Y = X 2 < 4. The cumulative distribution function of Y , FY (y ), is dened as FY (y ) = P (Y < y ) = P (X 2 < y ) = P ( y <
School: Duke
Course: Probability
Math 135, Spring 2012: HW 7 Problem 1 (p. 234 #2). SOLUTION. Let N = the number of raisins per cookie. If N is a Poisson random variable with parameter , then P (N 1) = 1 P (N = 0) = 1 exp() and for this to be at least 0.99, we need ln(0.01) Recall that
School: Duke
Course: Probability
Math 230, Fall 2012: HW 6 Solutions Problem 1 (p.202 #4). SOLUTION. We know that 2 22 V (X1 X2 ) = E X1 X2 (E [X1 X2 ]) By independence, E [X1 X2 ] = 1 2 22 2 2 E X1 X2 = E [X1 ]E [X2 ] and 2 E X1 = V (X1 ) + 2 1 2 = 1 + 2 1 2 2 and similarly E X2 = 2 + 2
School: Duke
Course: Probability
Udvalgte lsninger til Probability (Jim Pitman) http:/www2.imm.dtu.dk/courses/02405/ 17. december 2006 1 02405 Probability 2004-2-2 BFN/bfn IMM - DTU Solution for exercise 1.1.1 in Pitman Question a) 2 3 Question b) 67%. Question c) 0.667 Question a.2) 4 7
School: Duke
Course: Laboratory Calculus I
Tiffany Labon Michael Mclennon Osagie Obanor Bryce Pittard Varying Density Lab Report Part 1: 1. The mosquitoes in the park are most concentrated closest to the river and the least concentrated furthest from the river. This is clear from the function
School: Duke
Math 32L Professor Bookman Will Park Jeff Chen Ralph Nathan Air Pollution: Particulate Matter Lab Report Part 4 1.) M Volume Density 4 p g M 1012 cm3 1.5 3 3 2 cm M 2.5( ) p 3 1013 g 3 2.) Total Mass = Mass of particle * number of particles
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
Course: Multivariable Calculus
CALCULUS II Sequences and Series Paul Dawkins Calculus II Table of Contents Preface . ii Sequences and Series . 3 Introduction . 3 Sequences . 5 More on Sequences .15 Series The Basics .21
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus III or needing a refresher in som
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
Course: Multivariable Calculus
Calculus II Preface Here are a set of practice problems for my Calculus II notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included i
School: Duke
Course: Multivariable Calculus
Calculus II Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Calculus II or needing a refresher in some o
School: Duke
Course: Multivariable Calculus
Calculus II Preface Here are a set of practice problems for my Calculus II notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included i
School: Duke
Course: Multivariable Calculus
Calculus III Preface Here are my online notes for my Calculus III course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus III or needing a refresher in some
School: Duke
LINEAR ALGEBRA IN A NUTSHELL ( A is n by n ) Nonsingular A is invertible The columns are independent The rows are independent The determinant is not zero Ax = 0 has one solution x = 0 Ax = b has one solution x = A-1 b A has n (nonzero) pivots A has full r
School: Duke
Course: Calculus 1
Math 212, Section 15 Fall 2013 Lecture: Instructor: Oce: E-mail: Oce Hours: Web Page: Text: MWF 8:459:35, Physics 259 Nick Addington Physics 246 adding@math.duke.edu Tuesdays 12:001:00, and by appointment. http:/math.duke.edu/adding/courses/212/ Calculus
School: Duke
Course: Single Variable Calculus
Fall 2013 Math 122L Syllabus (Friday Lab) Textbook: Calculus: Concepts & Contexts (4th ed), by James Stewart Day 1-1 Date 27-Aug Topic Review of AP AB Differentiation topics Lab: L'Hopital's Rule and Relative Rates of Growth Riemann Sums Lab: Riemann Sums
School: Duke
Course: Single Variable Calculus
Fall 2013 Math 122L Syllabus (Thursday Lab) Textbook: Calculus: Concepts & Contexts (4th ed), by James Stewart Day 1-1 Date 26-Aug Topic Review of AP AB Differentiation topics Lab: L'Hopital's Rule and Relative Rates of Growth Riemann Sums Lab: Riemann Su
School: Duke
Course: Calculus
Math 32 Section 6 Fall 2008 Instructor: Tim Stallmann Oce: 037 Physics, West Campus email: tmstallm@math.duke.edu Oce Hours: by appointment About this Course Math 32 is a second-semester calculus course which covers with rigor Riemann sums and the denitio
School: Duke
Course: Multivariable Calculus
Homework Homework problems are assigned for every lecture, and students should ideally complete each assignment on the day of the lecture. The assigned problems for each lesson will be listed on the syllabus. (Note, we might find ourselves behind or ahead
School: Duke
Course: Multivariable Calculus
Syllabus for Math 102, Spring 08-09, Clark Bray Mathematics for Economists, Simon and Blume; Notes on Integrals for Math 102, Bray (Note: New homework problems will be added throughout semester; be sure you are looking at a current version!) Linear Algebr