# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

38 Pages

### 2mid245

Course: MATH 245, Fall 2010
School: Delaware
Rating:

Word Count: 103

#### Document Preview

2, Name: Midterm Math 245 - Fall 2010 Duration: 50 minutes To get full credit you should explain your answers. Each of the rst three questions is worth 5 points. The bonus questions is worth 3 points. 1. Find all the integer solutions (x, y ) of the equation 19x + 71y = 5. # Score 1 2 3 4 Total 2. How many natural numbers with x x 500 have no divisor in {2, 3, 5} ? 3. State and give a short proof of the...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Delaware >> Delaware >> MATH 245

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
2, Name: Midterm Math 245 - Fall 2010 Duration: 50 minutes To get full credit you should explain your answers. Each of the rst three questions is worth 5 points. The bonus questions is worth 3 points. 1. Find all the integer solutions (x, y ) of the equation 19x + 71y = 5. # Score 1 2 3 4 Total 2. How many natural numbers with x x 500 have no divisor in {2, 3, 5} ? 3. State and give a short proof of the binomial ( formula.) Use the binomial formula to 2 20 25 determine the coecient of x in the expansion of x2 x3 . 4. Bonus Question! Prove that every set of 8 distinct integers contains a pair whose sum or dierence is divisible by 12.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Delaware - MATH - 245
MATH 245, Fall 2010University of DelawareThis le contains some suggested problems from the textbook that may help your understanding and your writting skills.Please do NOT hand in solutions for these problems. Chapter 1: 1.3, 1.7, 1.9, 1.15, 1.30, 1.3
Delaware - MATH - 245
NOTES ON INEQUALITIESFELIX LAZEBNIKOrder and inequalities are fundamental notions of modern mathematics. Calculus and Analysis depend heavily on them, and properties of inequalities provide themain tool for developing these subjects. Often students are
Delaware - MATH - 245
Homework 1, Math 245, Fall 2010Due Wednesday, September 15 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. Let A, B, C be sets. Prove that A (B
Delaware - MATH - 245
Homework 1, Math 245, Fall 2010Due Wednesday, September 15 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. Let A, B, C be sets. Prove that A (B
Delaware - MATH - 245
Homework 2, Math 245, Fall 2010Due Wednesday, September 22 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. In a tournament with n 2 teams, ever
Delaware - MATH - 245
Homework 2, Math 245, Fall 2010Due Wednesday, September 22 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. In a tournament with n 2 teams, ever
Delaware - MATH - 245
Homework 3, Math 245, Fall 2010Due Friday, October 1 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. The Fibonacci numbers (Fn )n1 are dened as
Delaware - MATH - 245
Homework 3, Math 245, Fall 2010Due Friday, October 1 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. The Fibonacci numbers (Fn )n1 are dened as
Delaware - MATH - 245
Homework 4, Math 245, Fall 2010Due Wednesday, October 20 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. A club of 8 men and 10 women has to se
Delaware - MATH - 245
Homework 4, Math 245, Fall 2010Due Wednesday, October 20 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. A club of 8 men and 10 women has to se
Delaware - MATH - 245
Homework 5, Math 245, Fall 2010Due Monday, November 1 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. Show that among any 20 distinct numbers f
Delaware - MATH - 245
Homework 5, Math 245, Fall 2010Due Monday, November 1 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. Show that among any 20 distinct numbers f
Delaware - MATH - 245
Homework 6, Math 245, Fall 2010Due Wednesday, December 1 in class.The solution of each exercise should be at most one page long. If you can, try to writeyour solutions in LaTex. Each question is worth 2 points.1. Find all integers x that satisfy the f
Delaware - MATH - 245
Arithmetic and Geometric SequencesFelix LazebnikThis collection of problems1 is for those who wish to learn about arithmeticand geometric sequences, or to those who wish to improve their understandingof these topics, and practice with related problems
Delaware - MATH - 245
Math 245 An Introduction to ProofFall 2010Department of Mathematical SciencesUniversity of DelawareInstructor: Sebastian CioabaLecture Time and Place: MonWedFri, 10.10am-11am, Colburn Lab 104.Oce Hours: MonWedFri 11-noon, Ewing 506 or by appointmen
Delaware - CIS - 604
CISC 404/604Homework 1 Solutions1a. (B (C D) is valid and C is unsatisable. To prove B D is valid:Suppose bwoc B D is not valid. a truth-value assignment, say 1 , such that 1 (B ) = T (1)&amp; 1 (D) = F [by defns of validity and ]. (2)Since (B (C D) is
Delaware - CIS - 604
CISC 404/604Homework 1Due on Thursday, March 4, 2010No Late Submissions1. (25 points) Let B , C , and D be formulae and 1 , 2 be sets of formulaeProve or disprove the following statements. To disprove a statement, provide appropriatecounter-examples
Delaware - CIS - 604
CISC 404/604Homework 2 Solutions1a. To show that res (A B ) (B C ) (C A) B ):First, negate what were trying to prove (i.e., to show res consider ):(A B ) (B C ) (C A) B )Second, convert from statement-form into a set of clauses:cfw_[(A B ) (B C ) (C
Delaware - CIS - 604
CISC 404/604Homework 2Due on Thursday, March 18, 20101. Showres (AB ) (B C )(C A)B ).b. cfw_(A B ) (C D) res (A (C D) (B (C D)a.c. cfw_(C B )B ), (C A), (B C )A) res (B C ).2. You may use any of the Theorems/Lemmas/Propositions from the text book
Delaware - CIS - 604
CISC 404/604Homework 3 Solutions1a. (x)P (x) (x)Q(x) (x)(P (x) Q(x)To satisfy: QM = DMExample: D = cfw_0, 1, . . . (henceforth referred to as N),P M = cfw_0, 2, 4, . . ., andQM = D M .M = P M = QMTo falsify: DExample: D = N,P M = cfw_0, 2, 4, .
Delaware - CIS - 604
CISC 404/604Homework 3Due on Thursday, April 22, 20101. (35 points) For each of the following wfs, give a structure that satisesthe wf and another structure that falsies it.a. (x)P (x) (x)Q(x) (x)(P (x) Q(x).b. (x)(P (x) P (a).c. (x)(y )[P (x, y )
Delaware - CIS - 604
CISC 404/604Homework 4 Solutions1a. To show z xP (x, z, f (x, z ) z xyP (x, z, y ) is valid using resolution.Negate: (z xP (x, z, f (x, z ) z xyP (x, z, y )Convert to prenex: z1 z2 x2 y x1 (P (x1 , z1 , f (x1 , z1 ) P (x2 , z2 , y )Skolemize: z2 y x1
Delaware - CIS - 604
CISC 404/604Homework 4Due date Thursday May 4, 2010NO LATE SUBMISSIONS1. (20 points) Using resolution, prove the following are valid:a. z xP (x, z, f (x, z ) z xyP (x, z, y )b. x(P (x) Q(x) (xP (x) xQ(x).2. (10 points) Using resolution, show that (
Delaware - CIS - 604
CISC 404/604Unication and ResolutionLet L be a set of literals that we wish to unify.Let sub [ ].While | Lsub |= 1Scan left to right till a disagreement is foundIf neither term (at place of disagreement) is a variable then return(not uniable)elseL
Delaware - MATH - 242
M ath 242 Final Exam, Spring 2011Sections 012, 014, 050NAME:KEYThere are 12 questions on 12 pages.The points per page are 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6.The 13th page has some formulas you can refer to if needed.[5 points] Question 1. Evaluate
Delaware - MATH - 242
Math 242 Homework 1Sections 012 and 014: Due Wednesday Feb 16thSection 050: Due Thursday Feb 17th1. Evaluate xlimln(ln x).x(ln x)2.xx2. Evaluate limsin x x.x0x33. Evaluate lim24. Evaluate lim xx .+x05. Find the area bounded by the cur
Delaware - MATH - 242
Math 242 Homework 2Sections 012 and 014: Due Wednesday Feb 23rdSection 050: Due Tuesday Feb 22nd1. Find the area bounded by the curves y = sinxand y = x.22. Let A be the region bounded by the curves x = 1 + y 2 and y = x 3.Find the volume obtained
Delaware - MATH - 242
Math 242 Homework 3Sections 012 and 014: Due Wednesday March 2ndSection 050: Due Tuesday March 1stEvaluate these integrals by hand. (If you have time, you can use Maple todouble-check your answers afterward.)51. Evaluatexex dx.02. Evaluatet sin
Delaware - MATH - 242
Math 242 Homework 4Sections 012 and 014: Due MONDAY March 7thSection 050: Due Tuesday March 8thQuestions 1 to 5 should be done by hand.For questions 6 to 8, you can use Maple as much or as little as you like. /61. Evaluatecos3 x dx.02. Evaluates
Delaware - MATH - 242
Math 242 Homework 5Sections 012 and 014: Due Wednesday March 16thSection 050: Due Tuesday March 15thAll questions on this homework should be done by hand. (However, you maynd it useful to use Maple to double-check your answers afterwards.)21. Evalua
Delaware - MATH - 242
Math 242 Homework 5Solutions (may be somewhat terse)21. Evaluate2t31dt.t2 1Solution. Try t = sec . Then t2 1 = tan and dt = sec tan d.Also, when t = 2, we have = /4, and when t = 2, we have = /3.The integral becomes /3/4sec tan d =sec3 tan
Delaware - MATH - 242
Math 242 Homework 6Sections 012 and 014: Due Wednesday March 23rdSection 050: Due Tuesday March 22ndQuestions 1 through 5 should be done by hand.For questions 6 through 8, you can use Maple as much or as little as you like.1. Evaluatesin(ln x) dx. (
Delaware - MATH - 242
Math 242 Homework 6Solutions (to the by hand questions)Questions 1 through 5 should be done by hand.For questions 6 through 8, you can use Maple as much or as little as you like.1. Evaluatesin(ln x) dx. (May be a little tricky.)Solution. Let I =sin
Delaware - MATH - 242
Math 242 Homework 7Sections 012 and 014: Due MONDAY April 11thSection 050: Due Tuesday April 12thIn questions 1 through 8, determine whether the series converges or diverges.(Give reasons.)1.1n=1 2n2.lnn=1n2 + 12n2 + 13.1 + 3n2nn=14.nn
Delaware - MATH - 242
Math 242 Homework 7SolutionsIn questions 1 through 8, determine whether the series converges or diverges.(Give reasons.)1.n=112nSolution. This series is2.lnn=11times the harmonic series, so it diverges.2n2 + 12n2 + 1n2 + 11Solution. Not
Delaware - MATH - 242
Math 242 Homework 8Sections 012 and 014: Due Wednesday April 20thSection 050: Due Tuesday April 19thIn Questions 1 to 4, determine whether the series converges or diverges. Givereasons.1.n1nn=1 n 4(1)n sin2.n=1n3.1 + sin n10nn=14.nn3 +
Delaware - MATH - 242
Math 242 Homework 8Solutions (may be somewhat terse)In Questions 1 to 4, determine whether the series converges or diverges. Givereasons.1.n1nn=1 n 4n11 n = n , or limitCONVERGES. Try basic comparison with bn =n44comparison with the same bn
Delaware - MATH - 242
Math 242 Homework 9Sections 012 and 014: Due Wednesday April 27thSection 050: Due Tuesday April 26thIn questions 1 to 5, nd the interval of convergence of the power series.1.xn3n=1 n2.nxnn=13.xnnn=2 4 ln n(1)n4.n=05.(1)nn=0x2n(2n)!(
Delaware - MATH - 242
Math 242 Homework 9Solutions (may be somewhat terse)In questions 1 to 5, nd the interval of convergence of the power series.1.n=1xnn3an+1xn+1n3n3= n = |x| |x|. So |x| &lt; 1. Checkingan(n + 1)3 x(n + 1)3endpoints, we nd that both are includ
Delaware - MATH - 242
Math 242 Homework 10Sections 012 and 014: Due Wednesday May 4thSection 050: Due Tuesday May 3rdex 1dx as a power series.x1. Express the indenite integral0 .21dx as an innite series.1 + x50Then nd the sum of the rst two terms of that series. Co
Delaware - MATH - 242
Math 242 Homework 10Solutionsex 1dx as a power series.x1. Express the indenite integralxx2 x3 x4e =1+ ++++ 1!2!3!4!xx2 x3 x4x+++ e 1= +1!2!3!4!xex 11xx2 x3=+++ x1! 2!3!4!xx2x3x4ex 1dx =++++ x1 1! 2 2! 3 3!
Delaware - MATH - 242
Math 242 Homework 11Sections 012 and 014: Due MONDAY May 9thSection 050: Due Tuesday May 10th1. Consider the parametric curve x = t + ln t, y = t ln t, where 0 &lt; t &lt; .Find dy/dx and d2 y/dx2 . Find where the curve is increasing, decreasing,concave up
Delaware - MATH - 242
Brief solutions for (parts of) Homeworks 3 and 4Homework 4 /6 /6300= uu332. I =1/2=0(1 u2 ) du(1 sin x) cos x dx =cos x dx =1.sin /62sin 01111=.2 2424sec3 x dx =sec x sec2 x dx. Integrate by parts with u = sec x,dv = sec2 x dx
Delaware - MATH - 242
Math 242 Review Part 1In this course, we covered most of the following four chapters. Chapter 6 (Applications of Integration) Chapter 7 (Techniques of Integration) Chapter 11 (Innite Series) Chapter 10 (Parametric and Polar Curves)There were three a
Delaware - MATH - 242
Math 242 Review Part 2In addition to (most of) Chapters 6, 7, 11, and 10, we studied the followingthree miscellaneous topics: Newtons Method (4.8) LHopitals Rule (4.4) Arc Length (8.1)1. Use Newtons method with the specied initial approximation x1 t
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 6 and 81. Evaluate lim x1/x .x12. Evaluate xlim ( x2 + x x).23. Evaluate(3x 2)20 dx.34. Evaluatesin t dt.4
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 13 and 151. Find the area bounded by the curves y = x2 and y =1x22.+12. Find the area bounded by the curves y = |x| and y = x2 2.23. Let R be the region bounded by the curves y = 1 x2 and y = 0. Fi
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 20 and 221. Let A be the region bounded by y = ex , y = 1, and x = 2. Find thevolume obtained by revolving A around the line y = 2.12. Let A be the region bounded by y = 4x x2 and y = 3. Find the volume
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Sept 27 and 291. Evaluatesin4 x dx.12. Evaluatecos5 x dx.2173. Evaluatexex dx.03 /24. Evaluatesin3 x cos11 x dx.04
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 4 and 6101. Evaluate4x26dx.911dx.2 + 2xxHINT: Start by completing the square.2. Evaluate213. Evaluatex x2 + 4 dx.034a. Evaluatexdx.x2 + 14b. Evaluatex2dx.x2 + 14
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 11 and 1321. Use the Trapezoidal Rule and Simpsons rule to approximate0using n = 4 subintervals.11dx1 + x42. Evaluate0(x2xdx.+ 2)223. Find a good approximate root of x3 x 1 = 0 by starting wi
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 18 and 201. By hand, nd an approximation to 71/3 . Choose an appropriate functionf (x), an appropriate rst approximation x1 , and nd x2 .12. Find the length of the curve y = ln(sec x) between x = 0 and x
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Oct 25 and 27This is a longer list of problems than usual. It can also serve as practice forTest 2. /21a. Evaluatecos3 x dx.0 /21b. Evaluatecos2 x dx.012. Evaluatesec3 x dx.223. Evaluate1x2 1d
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 1 and 3D1. Does the series converge or diverge?1nn=1 n + 3D2. Does the series converge absolutely, converge conditionally, or diverge?(1)n+1n=2 n ln n1D3. Does the series converge absolutely, conver
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 8 and 10D1. Does the series converge or diverge?n2 132n=1 n + 2n + 5xnD2. Find the interval of convergence of the power series.n=0 n!1D3. Find the interval of convergence of the power series(2)n
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 15 and 17D1. Find a power series representation of the function f (x) =is the radius of convergence of your power series?1. What1 + x2D2. Find a power series representation of the function f (x) = arct
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 221. Evaluate the integralcos x (1 + sin2 x) dx.2. Evaluate the integralsin3 xdx.cos x113. Evaluate the integral0earctan xdx.1 + x24. Does the series2n n!converge or diverge?n=1 (n + 2)!25
Delaware - MATH - 242
Math 242Suggested lab/discussion problems, Nov 29 and Dec 1converge or diverge? If it converges,nn=1does it converge absolutely or conditionally?(1)n sinD1. Does the seriesnD2. Does the seriesn=123nconverge or diverge?1D3. Does the series
Delaware - MATH - 242
Math 242, U of D, Spring 2011Analytic Geometry and Calculus BInstructor: Idris Mercer, Ewing 529, idmercer@math.udel.eduIn emails, please write Math 242 and your section number.Oce hours: Mondays 5:006:30, Thursdays 9:0011:00, or by appointmentCourse
Delaware - MATH - 242
M ath 242 Section 010: Test #1NAME: _L/CPlease circle your l ab/discussion:020021022This test has 6 questions on 6 pages.The p oints per page are 5, 5, 5, 5, C, 6.Show your &quot;work. A correct answer w ith l ittle or no reasoning may reeeivea low s
Delaware - MATH - 242
M ath 242 Section O il: Test #1NAME: _frPlease circle your lab/discussion:030031032This test has 6 questions on 6 pages.The points per page are 5, 5, 5, 5, G, 6.Show your work. A correct answer with little or no reasoning may receivea low score.