Documents Found!
As seen in
Less Work, Better Grades
Join
Course Hero
Access
best resources
Ace
your classes
Ace your courses with Course Hero!
|
|
|
Study Smarter, Score Higher
Here are the top 5 related documents
...1
MATH 1007D
TEST 3 (blue). Solution
This test paper is worth 40 marks. Test duration: 50 min. PART I: SHORT QUESTIONS. Questions 1-5 are short questions. For these questions, only the answer will be marked.
Question 1. [3 Marks] Find the most ...
...MATH 4111 (2009): ADDITIONAL NOTES
FRANCESCO SICA
This is a collection of supplementary material (extra theory, solution to tricky exercises etc.). 1. Applications of Zorns Lemma In the following, almost all means all except nitely many. Recall that...
...Homework 1 MATH4805/COMP4805/MATH5605 Theory of Automata
Fall 2008 Due on October 23rd (1) Let A = {0, 1}. Determine a regular expression for each of the following languages: (i) {w A |w has at least two 0's} (ii) {w A |w has at exactly two 0's} (...
Document Content (unformatted)
Course Hero has millions of student submitted documents similar to the one
below including study guides, homework solutions, papers, exam answer keys and textbook solutions.
213, MATH Fall 2008 Solutions for the Pop Quiz This quiz is not going to be graded. I am assigning it to gauge your background. You do not to have sign your name. 3. Compute /4 cos x dx. 0 /4 cos x dx = sin x 0 /4 0 = sin 1 1 sin 0 = 0 = . 4 2 2
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more.
Course Hero has millions of course related materials that will enable you to learn better,
faster and get an A in all your courses.
Below is a small sample set of documents:
Below is a small sample set of documents:
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 9 6.1.2. Suppose, for the sake of contradiction, that f (x) = x1/3 is dierentiable at 0, with the derivative equal to L. This is equivalent to saying that f (x) f (0) =L x0 x0 lim Plugging in the expression for f , we see tha...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 2 1.3.8. For k N let Ak = {1, 2, . . . , k}. Then |Ak | = k, yet Ak = N. k=1 1.3.12. Let Fm be the family of subsets of N with precisely m elements. Let Nm = N . . . N (m times) be the m-fold Cartesian product of N with its...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 5 3.3.13. (d) Let xn = (1 1/n)n , yn = (1 + 1/n)n , and zn = xn yn = (1 1/n2 )n . We know that lim yn = e. Show that lim zn = 1. Indeed, zn < 1, and, by Bernoullis Inequality (p. 29 of the textbook), zn 1 + n (1/n2 ) = 1 1...
University of Illinois, Urbana Champaign >> HW >> 10 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 10 6.2.2. (b) Let g(x) = x/(x2 + 1). By the quotient rule, g (x) = (1 x2 )/(x2 + 1)2 . Thus, g (x) 0 i x [1, 1], and g (x) 0 i x (, 1] [1, ). Therefore, g increases on [1, 1], and decreases on the intervals (, 1] and [...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 10 6.2.2. (b) Let g(x) = x/(x2 + 1). By the quotient rule, g (x) = (1 x2 )/(x2 + 1)2 . Thus, g (x) 0 i x [1, 1], and g (x) 0 i x (, 1] [1, ). Therefore, g increases on [1, 1], and decreases on the intervals (, 1] and [...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 7 5.1.8. By continuity, f (x) = lim f (xn ) = 0 (as f (xn ) = 0 for each n, and x = lim xn ), hence x S. 5.1.9. (a) By the continuity of f , for every > 0 there exists > 0 s.t. |f (x) f (c)| < for any x B (c , c + ). A...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 3 2.1.9. (a) Suppose x1 = 1 + t1 2 and x2 = s2 + t2 2, with t1 , t2 Q. Then s x1 +x2 = (s1 +s2 )+(t1 +t2 ) 2. As sums of rational numbers are rational, x1 +x2 K. Moreover, x1 x2 = (s1 s2 + 2t1 t2 ) + (s1 t2 + s2 t1 ) 2. Pr...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 4 3.1.5. (b) Pick > 0, and show that there exists N N s.t. |2n/(n + 1) 2| < for n > N. A simple calculation shows that |2n/(n + 1) 2| = 2/(n + 1). We know that there exists N N s.t. 1/N < /2. Then |2n/(n + 1) 2| < for n...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 2 Problems assigned up to and including Friday, 09/05. The solutions are due on Friday, 09/12. Assigned Wednesday, 09/03. Section 1.3: 8, 12. Problem A: Denote by S the set of all subsets A N for which both A and N\\A are innite (not uncoun...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 9 Problems assigned up to and including Friday, 10/31. The solutions are due on Friday, 11/07. Assigned Monday, 10/27. Section 6.1: 2, 4, 7, 10, 11(a,c). Problem A (bonus): Recall (see Section 5.1) that the Thomae function f : R R is dened...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 3 Problems assigned up to and including Friday, 09/12. The solutions are due on Friday, 09/19. Assigned Monday, 09/08. Section 2.1: 9. Section 2.2: 15,16(a). Assigned Wednesday, 09/10. Section 2.3: 6, 9. Problem A: Prove that 2 + 3 is irr...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 1 Problems assigned up to and including Friday, 08/29. The solutions are due on Friday, 09/05. Assigned Monday, 08/25. Section 1.1: 12, 15, 20. Section 1.2: 5, 7, 15. Assigned Wednesday, 08/27. Bonus problem A: Suppose we have n lines in a ...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 7 Problems assigned up to and including Friday, 10/17. The solutions are due on Friday, 10/24. Assigned Monday, 10/13. Section 5.1: 8, 9, 10, 11, 12. Problem A (bonus): Suppose the function f : R R is such that |f (x) f (y)| (x y)2 for ...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 4 Problems assigned up to and including Friday, 09/19. The solutions are due on Friday, 09/26. Assigned Monday, 09/15. Section 3.1: 5(b), 10, 14, 16. Assigned Wednesday, 09/17. Section 3.2: 7, 9, 12, 16, 18(d), 19. Assigned Friday, 09/19. S...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 6 Problems assigned up to and including Friday, 10/03. The solutions are due on Friday, 10/17. Assigned Wednesday, 10/01. Section 4.1: 6, 11(a, d), 14. Section 4.2: 3, 5, 12. Back to the syllabus. Back to the main page of the course. The so...
University of Illinois, Urbana Champaign >> MATH >> 444 (Fall, 2008)
MATH 444 FINAL: SOLUTIONS FOR PRACTICE PROBLEMS 1. Suppose a < c < b, and a function f is continuous on (a, b), and dierentiable on (a, b)\\{c}. Suppose, furthermore, that limxc f exists, and equals L. Prove that f is dierentiable at c, and f (c) = ...
University of Illinois, Urbana Champaign >> MID >> 444 (Fall, 2009)
MATH 444 MIDTERM 1: SOLUTIONS FOR PRACTICE PROBLEMS The test will be given on Wednesday, October 8. It will be based on Homeworks 1-5 (Chapters 1-3). In preparing for the test, practice solving the problems from this list. In addition, take a look at...
University of Illinois, Urbana Champaign >> MID >> 444 (Fall, 2009)
MATH 444: SOLUTIONS FOR MIDTERM 1 1 (10 points): Compute the following limits. (a) lim 4n2 + 4n 2n . 4n2 + 4n + 2n 4n2 + 4n 2n = 4n2 + 4n 2n 4n2 + 4n + 2n 4 4n = . = 4n2 + 4n + 2n 4 + 4/n + 2 lim hence lim n But 4 + 4/n = lim(4 + 4/n) = 4 ...
University of Illinois, Urbana Champaign >> MATH >> 444 (Fall, 2008)
MATH 444: SOLUTIONS FOR FINAL 1 (10 points): Prove that, for any positive integer n, n k=1 (k n k=1 (k Denote the statement 1)/k! = 1 1/n! by P (n). We show by induction that P (n) is true for any positive integer n. The basic step consists of ver...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
HOMEWORK 8 Problems assigned up to and including Friday, 10/24. The solutions are due on Friday, 10/31. Assigned Monday, 10/20. Section 5.4: 2, 4, 7, 8, 9, 13 (bonus). Assigned Friday, 10/24. Section 5.6: 12, 13, 15. Back to the syllabus. Back to th...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 3 4.1.58. This is a bonus problem. Very little partial credit will be given. We say that n lines in a plane are in the general position if no two lines are parallel, and no three lines have a common point. The proof proceeds b...
University of Illinois, Urbana Champaign >> HW >> 10 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 10 7.2.24. We are dealing with the linear non-homogeneous recurrence relation an = 2an1 + 2n . (a) We have to show that an = n2n is a solution to this recurrence relation. In other words, we have to show that n2n = 2 (n 1)2n...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 10 7.2.24. We are dealing with the linear non-homogeneous recurrence relation an = 2an1 + 2n . (a) We have to show that an = n2n is a solution to this recurrence relation. In other words, we have to show that n2n = 2 (n 1)2n...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 5 5.4.10 We can write 1 x+ x 100 100 = k=0 100 k 1 x k x 100k 100 = k=0 100 1002k x . k The number 100 2k is always even, and lies between 100 and 100. Therefore, the coecient of xs of (x + 1/x)100 equals 0 if s is odd,...
University of Illinois, Urbana Champaign >> HW >> 11 (Fall, 2009)
SOLUTIONS FOR QUIZ 11 Compute the number of integers solutions to the equation x1 +x2 +x3 = 12, where 0 x1 5, 0 x2 6, and 0 x3 4. Show your work! Hint. The number of nonnegative integer solutions to the equation t1 + . . . + tk = n equals n+k1...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 11 Compute the number of integers solutions to the equation x1 +x2 +x3 = 12, where 0 x1 5, 0 x2 6, and 0 x3 4. Show your work! Hint. The number of nonnegative integer solutions to the equation t1 + . . . + tk = n equals n+k1...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 4 5.2.10. We have to prove the existence of distinct i and j s.t. both (xi + xj )/2 and (yi + yj )/2 are integer, which happens if and only if both xi + xj and yi + yj are even. However, a + b is even (with a, b Z) i a and b ...
University of Illinois, Urbana Champaign >> HW >> 4 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 4 5.2.10. We have to prove the existence of distinct i and j s.t. both (xi + xj )/2 and (yi + yj )/2 are integer, which happens if and only if both xi + xj and yi + yj are even. However, a + b is even (with a, b Z) i a and b ...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 7 6.3.2. p(E F ) = p(F |E)p(F ) = 5/82/3 = 5/12, hence p(F |E) = p(E F )/p(F ) = (5/12)/(3/4) = 5/9. 6.3.8. Denote by E the event that a person is sick, and F that the person tests positive. Then p(E) = 104 , p(F |E) = 0.999, a...
University of Illinois, Urbana Champaign >> HW >> 7 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 7 6.3.2. p(E F ) = p(F |E)p(F ) = 5/82/3 = 5/12, hence p(F |E) = p(E F )/p(F ) = (5/12)/(3/4) = 5/9. 6.3.8. Denote by E the event that a person is sick, and F that the person tests positive. Then p(E) = 104 , p(F |E) = 0.999, a...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 6 6.1.12. A deck contains 52 cards, four of them aces. The total number of poker hands (that is, the number of collections of 5 cards) equals 52 . This is the cardinality of 5 our sample space S. The number of hands containing ...
University of Illinois, Urbana Champaign >> HW >> 6 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 6 6.1.12. A deck contains 52 cards, four of them aces. The total number of poker hands (that is, the number of collections of 5 cards) equals 52 . This is the cardinality of 5 our sample space S. The number of hands containing ...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 8 6.4.28. This is a bonus problem very little partial credit is given. Recall the following fact: for a1 , . . . , an R, n a1 + . . . + an 2 = k=1 a2 + 2 k 1k<n ak a . 2 Let X = n Xk . Then X 2 = n Xk + 2 1k<n Xk X . B...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 9 A sequence (an ) satises the recurrence relation an = an1 + 6an2 (n 2), and the initial conditions a0 = 3, a1 = 1. Find an explicit formula for an . Answer: an = 3n + 2(2)n. First, write down the characteristic equation: r 2 r...
University of Illinois, Urbana Champaign >> HW >> 10 (Fall, 2009)
SOLUTIONS FOR QUIZ 10 Suppose the events E1 , E2 , and E3 are such that p(E1 ) = p(E2 ) = p(E3 ) = 1/3, p(E1 E2 ) = 1/6, p(E1 E3 ) = p(E2 E3 ) = 1/12, and E1 E2 E3 = . Compute p(E1 E2 E3 ). Answer: p(E1 E2 E3 ) = 2/3. Apply the Inclusion-Ex...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 10 Suppose the events E1 , E2 , and E3 are such that p(E1 ) = p(E2 ) = p(E3 ) = 1/3, p(E1 E2 ) = 1/6, p(E1 E3 ) = p(E2 E3 ) = 1/12, and E1 E2 E3 = . Compute p(E1 E2 E3 ). Answer: p(E1 E2 E3 ) = 2/3. Apply the Inclusion-Ex...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 3 How many license plates can be made using either three letters followed by three digits or four letters followed by two digits? Note: this is Exercise 5.1.28 from the homework. Denote by n1 (n2 ) the number of license plates mad...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 6 6 cards are chosen at random from a 52-card deck. What is the probability this selection contains all four cards of some kind (say, four kings, a 10, and a 4, or four aces and two 6s)? Recall that a card deck has 52 cards, of 13...
University of Illinois, Urbana Champaign >> HW >> 6 (Fall, 2009)
SOLUTIONS FOR QUIZ 6 6 cards are chosen at random from a 52-card deck. What is the probability this selection contains all four cards of some kind (say, four kings, a 10, and a 4, or four aces and two 6s)? Recall that a card deck has 52 cards, of 13...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 4 In a certain class, there are 10 math majors, 12 physics majors, and 18 computer science majors. How many ways are there to select a group of 15 students, consisting of 4 math majors, 5 physics majors, and 6 computer science maj...
University of Illinois, Urbana Champaign >> HW >> 4 (Fall, 2009)
SOLUTIONS FOR QUIZ 4 In a certain class, there are 10 math majors, 12 physics majors, and 18 computer science majors. How many ways are there to select a group of 15 students, consisting of 4 math majors, 5 physics majors, and 6 computer science maj...
University of Illinois, Urbana Champaign >> HW >> 12 (Fall, 2009)
HOMEWORK 12 Problems assigned up to and including Wednesday, 12/03. The solutions are due on Wednesday, 12/10. Assigned Friday, 11/21. Section 8.1: 3(b,c,f), 24, 30, 32(a,e,g), 34(c), 45(e), 48(e). These problems have been moved from Homework 11. As...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 12 Problems assigned up to and including Wednesday, 12/03. The solutions are due on Wednesday, 12/10. Assigned Friday, 11/21. Section 8.1: 3(b,c,f), 24, 30, 32(a,e,g), 34(c), 45(e), 48(e). These problems have been moved from Homework 11. As...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 4 Problems assigned up to and including Friday, 09/19. The solutions are due on Friday, 09/26 . Assigned Wednesday, 09/15. Section 5.2. 10, 14, 32. Problem A (bonus): Suppose we are given n integers. Prove that the sum of several (several m...
University of Illinois, Urbana Champaign >> HW >> 4 (Fall, 2009)
HOMEWORK 4 Problems assigned up to and including Friday, 09/19. The solutions are due on Friday, 09/26 . Assigned Wednesday, 09/15. Section 5.2. 10, 14, 32. Problem A (bonus): Suppose we are given n integers. Prove that the sum of several (several m...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 2 Problems assigned up to and including Friday, 09/05. The solutions are due on Friday, 09/12. Assigned Wednesday, 09/03. Section 2.3. 14(d,e), 26(c), 36, 37. Section 4.1. 6, 21, 36, 40, 48. Bonus problem: 2.3.14(b). For the students using ...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 7 Problems assigned up to and including Friday, 10/17. The solutions are due on Friday, 10/24. Assigned Monday, 10/13. Section 6.3: 2, 8, 16. Assigned Friday, 10/17. Section 6.4: 7, 12, 16, 20. For the students using the Fifth edition: Sec...
University of Illinois, Urbana Champaign >> HW >> 7 (Fall, 2009)
HOMEWORK 7 Problems assigned up to and including Friday, 10/17. The solutions are due on Friday, 10/24. Assigned Monday, 10/13. Section 6.3: 2, 8, 16. Assigned Friday, 10/17. Section 6.4: 7, 12, 16, 20. For the students using the Fifth edition: Sec...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 2 Consider the function f : Z Z Z, dened by f (m, n) = m n. (a) Is f onto? (b) Is f 1 1? Justify your answers! (a) YES. We need to show that, for every k Z, there exists (m, n) Z Z s.t. f (m, n) = k. For k 0, f (k, 0) = k....
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 5 Problems assigned up to and including Friday, 09/26. The solutions are due on Friday, 10/03. Assigned Monday, 09/22. Section 5.4: 10, 14, 16, 22, 30. Assigned Friday, 09/26. Section 5.5: 10(a,c,d), 20, 30, 38, 46. For the students using t...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
HOMEWORK 9 Problems assigned up to and including Friday, 10/31. The solutions are due on Friday, 11/07. Assigned Monday, 10/27. Section 7.1: 4, 8(d), 22, 44. Assigned Friday, 10/31. Section 7.2: 4(a), 8, 14, 24, 28. Update (Monday, 11/03): Problems ...
University of Illinois, Urbana Champaign >> MID >> 213 (Fall, 2009)
MATH 213 MIDTERM 2: SOLUTIONS FOR PRACTICE PROBLEMS The test will be given on Wednesday, November 19. It will be based on Homeworks 6-10, covering the material from Sections 6.1-7.5 (probability, recurrence relations, and inclusion-exclusion). In pre...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 1 2.1.20. Yes. Recall that the elements of P (A) are subsets of A. Then A = XP (A) X. If P (A) = P (B), then A = XP (A) X = Y P (B) Y = B. Alternative solution. c P (A) has exactly one element if and only if c = {a}, for some ...
University of Illinois, Urbana Champaign >> HW >> 11 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 11 7.1.5. Suppose P = (a = x0 < x1 < . . . < xn = b) is a partition, with Ik = [xk1 , xk ], and P is the same partition equipped with tags tk Ik . (a) Suppose u Ik = [xk1 , xk ], and tk [c1 , c2 ]. Then xk1 tk c2 , hence ...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 11 7.1.5. Suppose P = (a = x0 < x1 < . . . < xn = b) is a partition, with Ik = [xk1 , xk ], and P is the same partition equipped with tags tk Ik . (a) Suppose u Ik = [xk1 , xk ], and tk [c1 , c2 ]. Then xk1 tk c2 , hence ...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 1 1.1.12. First show that f (E) f (F ) = f (E F ). Suppose b B belongs to f (E) f (F ). Then b belongs to either f (E) or f (F ). In the former case, there exists a E s.t. b = f (a), while in the latter case, there exists...
University of Illinois, Urbana Champaign >> HW >> 444 (Fall, 2009)
SOLUTIONS FOR HOMEWORK 8 5.4.2. (a) Note that, for x, y R\\{0}, f (x) f (y) = 1 y 2 x2 (y x)(y + x) 1 1 1 2 = = = (y x) 2 + 2 . x2 y x2 y 2 x2 y 2 x y xy If x, y A, then x, y 1, hence 1/(x2 y) + 1/(xy 2) 2, and |f (x) f (y)| 2|x y|. In pa...
University of Illinois, Urbana Champaign >> HW >> 213 (Fall, 2009)
SOLUTIONS FOR QUIZ 8 Suppose the random variable X can be equal to either 2, 4, or 6, with p(X = 4) = 1/2, and p(X = 2) = p(X = 6) = 1/4. Compute the expected value and the variance of X. Answer: E(X) = 4, V(X) = 2. First compute the expected value ...
University of Illinois, Urbana Champaign >> MID >> 213 (Fall, 2009)
MATH 213: SOLUTIONS FOR MIDTERM 1 1 (10 points): Prove that A (B A) = A B (A and B are sets). We shall prove two inclusions: A (B A) A B and A (B A) A B. If x A (B A), then x belongs to either A, or to B A, hence to B, Either way, x ...
University of Illinois, Urbana Champaign >> MATH >> 213 (Fall, 2008)
MATH 213 FINAL: PRACTICE PROBLEMS The comprehensive test will be given on Tuesday, December 16, 1:30 - 4:30, in our usual classroom (AH 147). In preparing for the test, practice solving the problems from this list. In addition, take a look at the hom...
University of Illinois, Urbana Champaign >> MATH >> 213 (Fall, 2008)
MATH 213: SOLUTIONS FOR FINAL n k=1 (k 1 (10 points): Prove that, for any positive integer n, 1)/k! = 1 1/n!. Denote the statement n (k 1)/k! = 1 1/n! by P (n). We show by induction k=1 that P (n) is true for any positive integer n. The basic ...
University of Illinois, Urbana Champaign >> MATH >> 213 (Fall, 2008)
MATH 213 FINAL: PRACTICE PROBLEMS The comprehensive test will be given on Tuesday, December 16, 1:30 - 4:30, in our usual classroom (AH 147). In preparing for the test, practice solving the problems from this list. In addition, take a look at the hom...
University of Illinois, Urbana Champaign >> MATH >> 231 (Fall, 2008)
Selected answers to Merit Worksheet #21 1. The series in (a) and (c) converge; those in (b) and (d) diverge. 2. The series may converge or diverge, based upon what x is. If |x| < 1, the series converges, while if |x| 1, the series diverges. 3. The s...
University of Illinois, Urbana Champaign >> ACCY >> 411 (Fall, 2009)
ACCY 411 Class Notes Concepts of Risk and Uncertainty Risk: The chance of something happening that will have an impact on objectives. It is measured in terms of likelihood and consequences. Pure Risk: No possibility of gain, only the possibility of l...
University of Illinois, Urbana Champaign >> ACCY >> 411 (Fall, 2009)
IX - Futures F ORWARD AND F UTURES C ONTRACTS F orward and Future contracts are legal agreements for the delivery of goods, services, or assets at a specified price, under specified conditions, where the specified date of delivery and payment is so...
University of Illinois, Urbana Champaign >> CS >> 173 (Fall, 2008)
CS 173 Section 2 Handout, week of 09/10/07 Fall 2007 1. Which of these is a valid argument? (a) If Boris becomes a pastry chef, then if he gives in to his desire for chocolate mousse, then his waistline will suffer. If his waistline suffers, then ...
University of Illinois, Urbana Champaign >> CS >> 473 (Fall, 2008)
CS CS 473g: Algorithms, Fall 2007 Homework 3 (due Tuesday, October 23, 2007 at 11:59.99 p.m.) Version 1.02 Required Problems 1. The good, the bad, and the middle. [20 Points] Suppose youre looking at a ow network G with source s and sink t, and you w...
University of Illinois, Urbana Champaign >> CS >> 473 (Fall, 2008)
CS CS 473g: Algorithms, Fall 2007 Homework 2 (due Tuesday, October 9, 2007 at 11:59.99 p.m.) Version 1.02 Name: Net ID: Name: Net ID: # 1. 2. 3. 4. 5. Total Alias: Alias: Score Grader Neatly print your name(s), NetID(s), and the alias(es) you use...
University of Illinois, Urbana Champaign >> CS >> 231 (Spring, 2008)
Instruction encoding Weve already seen some important aspects of processor design. A datapath contains an ALU, registers and memory. Programmers and compilers use instruction sets to issue commands. Now lets complete our processor with a control ...
University of Illinois, Urbana Champaign >> CS >> 173 (Fall, 2008)
CS 173 Homework 0 (due 8/28/07 in class) Fall 2007 CS 173: Discrete Mathematical Structures, Fall 2007 Homework 0 Due in class on Tuesday, August 28, 2007 This homework has two purposes. First, we want to verify that you have already mastered the...
What are you waiting for?