Online study resources available anywhere, at any time
High-quality Study Documents, expert Tutors and Flashcards
Everything you need to learn more effectively and succeed
We are not endorsed by this school |
- Course Hero has verified this tag with the official school catalog
We are sorry, there are no listings for the current search parameters.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
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: 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
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
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: 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: 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: 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: 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: 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: Elem Differential Equat
Math 356.01 Homework 29, 30, 31, and 32 Homework 29, due Wednesday, November 13 Find the Fourier series in each of the rst three problems by calculating the integrals for the coecients. If the function is odd or even, you can use that fact to simplify the
School: Duke
Course: Elem Differential Equat
Math 356.01 Homework 19, 20, and 21 Homework 19, due Friday, Oct. 18 9.2: 33, 34, 41 and 9.3: 10, 11, 12, 14 When you look at the gures in Sec. 9.3, look at the blue arrows and try to ignore the black arrows! Your sketches should convey information but do
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: Elem Differential Equat
Math 356.01, last three homework assignments Homework 33, due Monday, December 2 Section 13.2, page 644, do problems 14 and 16. Notice the boundary condition! Also do the ones below. Make your answers as simple as possible. 1. ut = 4uxx , 0 x 1, t > 0, ux
School: Duke
Course: Elem Differential Equat
The Method of Undetermined Coefficients for Systems of Differential Equations We can use methods like those in Sec. 4.5 to nd particular solutions of systems of the form y = Ay + f (t), the same kind of equation as in Sec. 9.9, where y(t) is the column ve
School: Duke
Course: Elem Differential Equat
Systems of Second Order Differential Equations On pages 356-57, Section 8.4, Example 4.8, our book set up the dierential equations describing the motion of two masses connected to three springs. (See the gure on page 356 and equations on pp. 356-57.) Each
School: Duke
Course: Elem Differential Equat
A proof of Theorem 7.15 on page 81 Theorem 7.15 (slightly rephrased): Suppose f (t, x) is a function dened on a rectangle R in the tx-plane. Suppose f and f /x are continuous on R, and also suppose that, for some number M and all (t, x) in R, f (t, x) M .
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Quiz #6 Solutions Let f (t) = sin(t2 ). If the interval is [2, 3] and we want 2 subdivisions, then x0 = 2, x1 = 2.5, x2 = 3. 1 1 1 1 (a) LHS= f (x0 ) + f (x1 ) = [f (2) + f (2.5)] = sin(22 ) + sin(2.52 ) .39499 2 2 2 2 1 1 1 1 (d)
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #1 Solutions 1. (a) 3 3 (b) 2 1 (c) 1 + x2 2. Let = tan1 x. Then tan = x. Dierentiating both sides with respect to x gives d tan = dx x, or 1 d = 1. cos2 dx Hence, 2 1 d 1 2 2 1 = cos = cos (tan x) = = 2 dx 1 + x2 1+x d dx
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #4 Solutions 1. (a) The possible values for X are: 0, 1, or 2. (b) With probability 2/6, the player rolls 1 or 6. If that happens then with probability 1/4, the player gets 0 heads (TT), with probability 1/4 the player gets 2
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #2 Solutions 1. (a) y(x) = ln(2 + x) ln 2 (b) y(x) = tan(tan x) 2. 1 (a) S(t) = 150 e 50 t so that S(20) = 100.55 kg. (b) After a very long time the concentration is 0 kg/L. This is not a surprise because we would expect the
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #3 Solutions 1. 2 Let f (x) = eax . (a) LHS= 1 f (0) + 1 f (1/2) = 1 (1 + ea/4 ) 2 2 2 1 1 (b) RHS= 2 f (1/2) + 1 f (1) = 2 (ea/4 + ea ) 2 (c) LHS= 1 100 (d) RHS= 99 f (k/100) = k=0 1 100 100 f (k/100) = k=1 1 100 1 100 99 ek
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
Ea RF hCG | Cj1 v 1 w1 v 1 v 1 v 1 x 1 v 1 v 1 g H TP R gtSCu cHR pQpu$ ycfw_ yz yy y x y 75 yw y v y C1 W E X R a GrCg3 y q E aP V p E a gP T RP H V R g $ r T c 9 q E H g E V p E a s c F nP F E k @Ghr7tGhruCSQSCgtdxf@bQqG7tGhmorpGSbG`gg gP T RP H c WP l
School: Duke
Course: Introductory Calculus
Xt eY cfw_C` C1 q1 1 q1 q1 q1 q1 q1 q1 z a gc e fC vvae d$ 75 C1 p Xv q e t `C3 X tc i X t zc g ec a iv e z $ g v 9 X a z X i X t vv v Y c Y X ~ @`cfw_7`cfw_CfdfCw%yw@ud`7`cfw_`fu`sz zc g ec a v pc e t ~ ~ v v v Yc t X p i g v z z Xv g y x t e u vv v u
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: Stochastic Calculus
Math 219, Stochastic Calculus: Introductory Lecture As in ordinary calculus much of our time will be spent on stochastic integration and stochastic dierential equations. We begin with some nonrandom examples written in slightly dierent notation. Exponenti
School: Duke
Course: Stochastic Calculus
Chapter 5 Convergence to SDE 5.1 Weak convergence We begin with a treatment of weak convergence on a general space S with a metric , i.e., a function with (i) (x, x) = 0, (ii) (x, y ) = (y, x), and (iii) (x, y ) + (y, z ) (x, z ). Open balls are dened by
School: Duke
Course: Stochastic Calculus
Chapter 4 SDE as Markov processes 4.1 Discrete state space To prepare for our discussion of diusion processes, we recall the analogous results for a countable state space S . A Markov chain on S is described by giving the rate q (i, j ) at which jumps for
School: Duke
Course: Stochastic Calculus
CHAPTER 3. SEMIMARTINGALE INTEGRATION 62 3.5 OUSDE Explict solutions of SDE Example 3.3. Ornstein Uhlenbeck Process. Let Bt be a one dimensional Brownian motion and consider (3.10) dXt = Xt dt + dBt OUsde which describes one component of the velocity of a
School: Duke
Course: Stochastic Calculus
Chapter 3 Semimartingale Integration 3.1 Basic denitions The solutions to our SDEs have the form t Xt X0 = t (Xs ) dBs + 0 b(Xs ) ds 0 where for the moment we suppose Xt , b(Xt ), and (Xt ) are real numbers. and b and are continuous. To have a useful int
School: Duke
Course: Stochastic Calculus
Chapter 1 Brownian motion Brownian motion is a process of tremendous practical and theoretical signicance. It originated (a) as a model of the phenomenon observed by Robert Brown in 1828 that pollen grains suspended in water perform a continual swarming m
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: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Quiz #6 Solutions Let f (t) = sin(t2 ). If the interval is [2, 3] and we want 2 subdivisions, then x0 = 2, x1 = 2.5, x2 = 3. 1 1 1 1 (a) LHS= f (x0 ) + f (x1 ) = [f (2) + f (2.5)] = sin(22 ) + sin(2.52 ) .39499 2 2 2 2 1 1 1 1 (d)
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #1 Solutions 1. (a) 3 3 (b) 2 1 (c) 1 + x2 2. Let = tan1 x. Then tan = x. Dierentiating both sides with respect to x gives d tan = dx x, or 1 d = 1. cos2 dx Hence, 2 1 d 1 2 2 1 = cos = cos (tan x) = = 2 dx 1 + x2 1+x d dx
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #4 Solutions 1. (a) The possible values for X are: 0, 1, or 2. (b) With probability 2/6, the player rolls 1 or 6. If that happens then with probability 1/4, the player gets 0 heads (TT), with probability 1/4 the player gets 2
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #2 Solutions 1. (a) y(x) = ln(2 + x) ln 2 (b) y(x) = tan(tan x) 2. 1 (a) S(t) = 150 e 50 t so that S(20) = 100.55 kg. (b) After a very long time the concentration is 0 kg/L. This is not a surprise because we would expect the
School: Duke
Course: Laboratory Calculus And Functions II
Math 026L.04 Spring 2002 Test #3 Solutions 1. 2 Let f (x) = eax . (a) LHS= 1 f (0) + 1 f (1/2) = 1 (1 + ea/4 ) 2 2 2 1 1 (b) RHS= 2 f (1/2) + 1 f (1) = 2 (ea/4 + ea ) 2 (c) LHS= 1 100 (d) RHS= 99 f (k/100) = k=0 1 100 100 f (k/100) = k=1 1 100 1 100 99 ek
School: Duke
Course: Linear Algebra
EXAM 3 Math 107, 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: Linear Algebra
EXAM 1 Math 107, 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: Linear Algebra
EXAM 1 Math 107, 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: Linear Algebra
EXAM 1 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 3 Math 107, 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: Linear Algebra
EXAM 2 Math 107, 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: Calculus 1
EXAM 3 Math 103, Fall 2004, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/40 points) 2. (/30 points) "I have adhered to the Duke Commu
School: Duke
Course: Calculus 1
EXAM 2 Math 103, Fall 2004, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/20 points) 2. (/40 points) "I have adhered to the Duke Commu
School: Duke
Course: Calculus 1
EXAM I Math 103, Fall 2004, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/40 points) "I have adhered to the Duke Community Standard in
School: Duke
Course: Calculus 1
EXAM 4 Math 103, Fall 2004, Clark Bray. You have 50 minutes. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/40 points) 2. (/30 points) "I have adhered to the Duke Commu
School: Duke
Course: Calculus 1
EXAM 2 Math 103, Summer 2005 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 Good luck! Name ID number 1. (/20 points) 2. (/20 points) "I have adhered to the D
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Summer 2005 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 Good luck! Name ID number 1. (/15 points) 2. (/15 points) 3. (/20 points) "I have
School: Duke
Course: Calculus 1
EXAM 1 Math 103, Summer 2005 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 Good luck! Name ID number 1. (/20 points) 2. (/20 points) "I have adhered to the D
School: Duke
Course: Advanced Calculus 1
Math 431.02: Exam #1 Instructor: Dr. Herzog February 19th, 2014 1. Let cfw_an and cfw_bn be bounded sequences and define sets A, B, and C by A = cfw_an , B = cfw_bn , and C = cfw_an + bn . Prove that sup C sup A + sup B. Give an example to show that str
School: Duke
Course: Advanced Calculus 1
Math 431.01 Spring 2014: Quiz # 2 Instructor: Dr. Herzog January 24th, 2014 1. Suppose that an a and that an b for each n. Prove that a b. Proof. To obtain a contradiction, we suppose that a < b. Let = (b - a)/2 > 0. Since an a, we may pick N N such that
School: Duke
Course: Advanced Calculus 1
Math 431.01 Spring 2014: Quiz # 1 Solutions Instructor: Dr. Herzog January 13th, 2013 1. Prove that if x and y are real numbers, then 2xy x2 + y 2 . Proof. Let x, y R. Observe that 0 (x - y)2 = x2 - 2xy + y 2 . Adding 2xy to both sides of the previous ine
School: Duke
Course: Advanced Calculus 1
Math 431.01 Spring 2014: Quiz # 3 Solutions Instructor: Dr. Herzog January 27th, 2014 1. Suppose that cfw_an is a Cauchy sequence. Prove that cfw_a2 is also a n Cauchy sequence. Is the converse true? Proof. Since an is Cauchy, an a for some a R. By the
School: Duke
Course: Advanced Calculus 1
Math 431.01 Spring 2014: Quiz # 4 Instructor: Dr. Herzog February 5th, 2014 1. Let cfw_dn be a sequence of limit points of the sequence cfw_an and suppose that dn d. Prove that d is also a limit point of cfw_an . Proof. Let > 0 and N N be given. We must
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: 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
Math 230.01, Fall 2012: HW 5 Solutions Due Thursday, October 4th, 2012. Problem 1 (p.158 #2). Let X and Y be the numbers obtained in two draws at random from a box containing four tickets labeled 1, 2, 3, 4. Display the joint distribution table for X and
School: Duke
Course: Probability
Math 230.01, Fall 2012: HW 4 Solutions Problem 1 (p.121 #3). SOLUTION. Each trial consists of rolling 100 dice. We deem the event cfw_25 or more sixes in rolls of 100 dice a success, and we are told its probability is p = 0.022. Since we are repeating the
School: Duke
Course: Probability
Math 230.01, Fall 2012: HW 3 Solutions September 19, 2012 Problem 1. A woman has 2 children, one of whom is a boy born on a Tuesday. What is the probability that both children are boys? (You may assume that boys and girls are equally likely and independen
School: Duke
Course: Probability
Math 230.01, Fall 2012: HW 2 Solutions This homework is due at the beginning of class on Thursday January 26th, 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 wri
School: Duke
Course: Probability
Math 230.01, Fall 2012: HW 1 Solutions Problem 1 (p.9 #2). Suppose a word is picked at random from this sentence. Find: a) the chance the word has at least 4 letters; SOLUTION: All words are equally likely to be chosen. The sentence has 10 words; 7 are 7
School: Duke
Course: General Statistical Probability
Basic Markov Chains The Probability Workbook Computers on the Blink 4/24/13 4:02 PM conditional densities Basic Markov Chains In each of the graphs pictured, assume that each arrow leaving a vertex has a equal chance of being followed. Hence if there are
School: Duke
Course: Stochastic Calculus
Homework of SDE 2 Yuan Zhang Feb/4/2012 Problem I. Show that with probability 1, lim supt Bt /t1/2 = and that lim supn Bn /(n log n)1/2 2 where the second limit is through the integer Proof: For any m > 0, it is easy to see that, for any t, cfw_sups>t Bs
School: Duke
Course: General Statistical Probability
Point of increase - The Probability Workbook http:/sites.duke.edu/probabilityworkbook/point-of-increase/ The Probability Workbook Learning probability by doing ! Home Counting some Examples Generating Functions and Limit Theorems Front Page Home / Bernoul
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: 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