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### 6pmquiz2

Course: QUIZ 2, Fall 2009
School: UC Davis
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Coursehero >> California >> UC Davis >> QUIZ 2

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UC Davis - QUIZ - 5
UC Davis - QUIZ - 7
UC Davis - QUIZ - 8
UC Davis - QUIZ - 8
UC Davis - QUIZ - 6
UC Davis - QUIZ - 5
UC Davis - QUIZ - 4
UC Davis - QUIZ - 1
UC Davis - QUIZ - 2
UC Davis - QUIZ - 2
UC Davis - QUIZ - 5
UC Davis - QUIZ - 1
UC Davis - QUIZ - 8
UC Davis - QUIZ - 4
UC Davis - QUIZ - 4
UC Davis - MATH - 130
Solutions to Homework 6By H akan Nordgren Problem 5.13: For each of the following subsets of R2 decide if the set is open in R2 or not and if the set is dense in R2 , or not. Give a reasons for your answer. 1. A = {(x, y) R2 : y &gt; 0}. 2. B = {(x, y
UC Davis - MATH - 20
Math 20E Homework 11.5.10 Let P1 = (3, 4, 7), P2 = (4, -1, 6). P1 P2 = (4 - 3)i + (-1 - 4)j + (6 - 7)k = i - 5j - k. 1.7.1 Let u = 2i + j + 2k, v = 3i - 4k. The angle between u and v is given by: uv -2 -2 cos() = = = . |u|v| (3)(5) 15 So, = cos-1
UC Davis - MATH - 10
Solutions to midterm 2By H akan Nordgren Question 1: Evaluate the integral xe-2x dx.Answer: When the integrand consists of an x or x2 (or x3 , etc) multiplying a function like sin x, cos x or ex , then more likely than not, the best way to do the
UC Davis - MATH - 140
Solutions to homework 3By H akan Nordgren RUDIN PROBLEMS: Question 25 from page 118: Let f be twice dierentiable on [a, b], let f (a) &lt; 0, and let f (b) &gt; 0. Let f (x) &gt; 0 and let 0 f (x) M on [a, b]. Let be the unique point in (a, b) such that
UC Davis - MATH - 140
Solutions to homework 6By H akan Nordgren RUDIN PROBLEMS: Question 1 from page 78: Prove that if (xn ) converges then so does (|xn |). Is the converse true? Answer: This question was not worded as well as it could have been. It is best to assume tha
UC Davis - MATH - 140
Solutions to homework 6By H akan Nordgren RUDIN PROBLEMS: Question 16 from page 168: Let (fn ) be a sequence of equicontinuous functions on a compact set K, which converges pointwise to f . Prove that fn converges uniformly to f . Answer: This argum
UC Davis - MATH - 140
Solutions to homework 5By H akan Nordgren ALSO PROBLEMS: Question a: Let (X, d) be a metric space. Let E X. Show that there are A, B E disjoint, non-empty and relatively closed in E such that E = A B if and only if there is a non-empty G E such
UC Davis - MATH - 140
Solutions to homework 4By H akan Nordgren RUDIN PROBLEMS: Question 1 from page 165: Let (fn ) be a uniformly convergent sequence of bounded functions on a set E. Show that (fn ) is uniformly bounded on E. Answer: Suppose that that the fn converge un
UC Davis - MATH - 140
Solutions to homework 5By H akan Nordgren RUDIN PROBLEMS: In these solutions I will use the following theorem at least twice: THEOREM 1 Let f (x) = lim fn (x) for all x E. Dene Mn := sup {|f (x) fn (x)| : x E} . Then fn f uniformly on E if and o
UC Davis - MATH - 110
Solutions to Homework 7By H akan Nordgren I am going to skip 4.4.3 since its solution is very similar to 4.4.1, for which I have already provided a solution. 4.4.6: Show that a0 +n=1 rn an cos(n) + bn sin(n) ,is a solution of Laplaces equation
UC Davis - MATH - 130
Solutions to Homework 5By H akan Nordgren Problem 1: For each of the following 3 3 matrices, find the eigenvalues and eigenvectors; find the matrix T such that T -1 AT is in canonical form; and find that canonical form. 1. 0 3 1 A = 4 1 -1 . 2 7
UC Davis - MATH - 130
Solutions to Homework 3By H akan Nordgren As always, the sketches are all in a separate pdf. Problem 9.8: For the following systems determine whether the system is a gradient system or a Hamiltonian system, and then draw its phase-portrait. 1. x y 2
UC Davis - MATH - 130
Solutions to Homework 2By H akan Nordgren Problem 2.2: Find the general solution for each of the following linear systems. 1. X= 2. X= 3. X= 4. X= 1 2 0 3 1 2 3 6 1 2 1 0 1 2 3 3X.X.X.X.Solution: We must nd eigenvalues and eigenvectors for
UC Davis - MATH - 130
Solutions to Homework 2By H akan Nordgren As always, all sketches are on a separate pdf. Problem 9.4: For the following system x y z = = = (x + 2y)(z + 1) (x + y)(z + 1) z 3 ,1. Show that (0, 0, 0) is not asymptotically stable when = 0. 2. Show t
UC Davis - MATH - 361
Solutions to homework 4By H akan Nordgren Problem 1: Let X be a vector-space and let 1 and 2 be norms on X. Let (xn ) be a sequence of points in X. We say that 1 and 2 are equivalent if xn - x 1 0 if and only if xn - x 2 0. Show that 1 and 2
UC Davis - MATH - 181
Solutions to Homework 3By H akan Nordgren Problem 3.12.1: Let X be a random variable with probability function pX (k) = 1ent that MX (t) = n(1et ) Solution: By denitionn1 1 nfor k = 0, . . . , n 1. ShowMX (t) = E etX = all k Now we need a lemm