Economics 204 Lecture 11Monday, August 14, 2006 Revised 8/14/06, Revisions marked by * Sections 4.1-4.3, Unied Treatment Denition 1 Let f : I R, where I R is an open interval. f is dierentiable at x I if f (x + h) f (x) =a lim h0 h for some a R.
Econ 204Summer/Fall 2006 Corrections to de la Fuente
1. On page 23, de la Fuente presents two denitions of correspondence. In the second denition, de la Fuente requires that for all x X, (x) = . The rst denition simply says that is a function from
Economics 204 Summer/Fall 2006 Lecture 1Monday July 31, 2006 Revised August 1, 2006; revisions marked by * Bob Anderson Zack Grossman Website: http:/emlab.berkeley.edu/users/anderson/Econ204 /204index.html Lectures will often run past 3:00, dont sche
Economics 204 Lecture 2, August 1, 2006 Revised 8/1/06, Revisions Marked by * Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2N, the set of all subsets of N, is not countable. Proof: Suppose 2N is countable. Then there is a bijection f : N 2N .
Economics 204 Lecture 3Wednesday, August 2, 2006 Revised 8/3/06, solely to mark denitions Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in Rn Denition 1 A metric space is a pair (X, d), where X is a set and d : X X
Economics 204 Lecture 4August 3, 2006 Revised 8/4/06, Revisions indicated by * Section 2.4, Open and Closed Sets Denition 1 Let (X, d) be a metric space. A set A X is open if xA>0B (x) A A set C X is closed if X \ C is open. Example: (a, b) is ope
Economics 204 Lecture 5Friday, August 4, 2006 Revised 8/7/06, revisions marked by * Section 2.6 (Continued) Properties of Real Functions Theorem 1 (6.23, Extreme Value Theorem) Let f be a continuous real-valued function on [a, b]. Then f assumes its
Economics 204 Lecture 6Monday, August 7, 2006 Revised 8/8/06, Revisions indicated by * Section 2.8, Compactness Denition 1 A collection of sets U = {U : } in a metric space (X, d) is an open cover of A if U is open for all and U A ( may be nite
Economics 204 Lecture 7Tuesday, August 8, 2006 Revised 8/8/06, Revisions indicated by * *Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets Denition 1 Two sets A, B in a metric space are separated if AB =
Economics 204 Lecture 8Wednesday, August 9, 2006 Revised 8/9/06, Revisions indicated by * Chapter 3, Linear Algebra Section 3.1, Bases Denition 1 Let V be a vector space over a eld F . A linear combination of x1, . . . , xn is a vector of the form y=
Economics 204 Lecture 9Thursday, August 10, 2006 Revised 8/10/06, Revisions indicated by * Section 3.3, Isomorphisms Denition 1 Two vector spaces X, Y over a eld F are isomorphic if there is an invertible (recall this means one-to-one and onto) T L(
Economics 204 Lecture 10Friday, August 11, 2006 Revised 8/11/06Revisions indicated by * Section 3.5, Change of Basis, Similarity Let X be a nite-dimensional vector space with basis V . If T L(X, X) it is customary to use the same basis in the domain
Economics 204 Lecture 12Tuesday, August 14, 2007 Revised 8/14/07, revisions indicated by * Section 4.4 (Cont.): Taylors Theorem in Rn Denition 1 X Rn, X open, f : X Rm . f is continuously dierentiable on X if f is dierentiable on X and dfx is a c
Economics 204 Lecture 13Wednesday, August 15, 2007 Revised 8/15/07, revisions indicated by * Section 5.5 (Cont.) Transversality Theorem The Transversality Theorem is a particularly convenient formulation of Sards Theorem for our purposes: Theorem 1 (
University of California, Berkeley Economics 204 First Midterm Test Tuesday August 27, 2002; 6-9pm Each question is worth 20 of the total Please use separate bluebooks for Parts I and II Part I
1. Consider the function
ux; y; z = xy + yz + xz
a Find
Economics 204 First Midterm Test Friday October 19 6-9pm
1. Consider the linear transformation T satisfying T 1; 1 = 0; 2 and T ,1; 1 = 1; 0. a Compute the matrix representation of T with respect to the standard basis f1; 0; 0; 1g. b Compute the kern
Economics 204 Lecture 12Tuesday, August 15, 2006 Revised 8/16/06, Revisions indicated by * Section 4.4, Taylors Theorem Theorem 1 (1.9, Taylors Theorem in R1 ) Let f : I R be n-times dierentiable, where I R is an open interval. If x, x + h I, then
Economics 204 Lecture 13Wednesday, August 16, 2006 Revised 8/16/06, Revisions indicated by * Sections 5.2 (Cont.), Transversality and Genericity Denition 1 Suppose A Rn. A has Lebesgue measure zero if, for every > 0, there is a countable collection
Economics 204 Lecture 14Thursday, August 17, 2006 Revised 8/17/06, Revisions indicated by * Dierential Equations Existence and Uniqueness of Solutions Denition 1 A dierential equation is an equation of the form y (t) = F (y(t), t) where F : Rn R Rn
Economics 204 Lecture 15Friday, August 18, 2006 Revised 8/18/06, Revisions indicated by * Getting Real Solutions from Complex Solutions If M is a real matrix and is a complex eigenvalue, corresponding eigenvector(s) must be complex. If y is a real
Econ 204Fall 2006 List of Theorems for the Final Exam
You are responsible for the statements of all theorems stated in the text or in class. Understanding the proofs of theorems may help you to answer some questions, so you are advised to try to unde
Problem Set 1Revised 8/2/06, problem 8
Economics 204 - August 2006 Due Friday, 4 August in Lecture
1. Set Theory (a) Determine whether this formula is always right or sometimes wrong. Prove it if it is right. Otherwise give both an example and a coun
Problem Set 1 Solutions
Economics 204 - August 2006
1. Set Theory (a) Determine whether this formula is always right or sometimes wrong. Prove it if it is right. Otherwise give both an example and a counterexample and state (but dont prove) an addit
Economics 204 - August 2006 Due Tuesday, 8 August in Lecture
1. Theorem 4 from the lim inf/lim sup handout: Let {xn } be a sequence of real numbers. Then lim xn = x R {, }
x
Problem Set 2
if and only if lim inf xn = lim sup xn = x.
x x
Prove thi
Problem Set 2 Solutions
Economics 204 - August 2006
1. Theorem 4 from the lim inf/lim sup handout: Let {xn } be a sequence of real numbers. Then lim xn = x R {, }
x
if and only if lim inf xn = lim sup xn = x.
x x
Prove this theorem for the case
Economics 204 - August 2006 Due Friday, 11 August in Lecture
1. Connectedness (a) Suppose we dene two sets as being separated if neither contains a limit point of the other and that a set in a metric space is connected if it cannot be written as the
Economics 204 - August 2006
1. Connectedness (a) Suppose we dene two sets as being separated if neither contains a limit point of the other and that a set in a metric space is connected if it cannot be written as the union of two disjoint non-empty s
Economics 204 - August 2006 Due Tuesday, 15 August in Lecture Revised 8/13/06, see * in Problem 4
1. Let be the set of all continuous functions whose domain is the unit interval [0, 1] and range is R. Let be the subset consisting of all real polyno
Problem Set 4 - Solutions
Economics 204 - August 2006
1. Let be the set of all continuous functions whose domain is the unit interval [0, 1] and range is R. Let be the subset consisting of all real polynomials (whose domain is restricted to the uni
Economics 204 Lecture 14Thursday, August 16, 2007 Revised 8/16/07, revisions indicated by * Dierential Equations Existence and Uniqueness of Solutions Denition 1 A dierential equation is an equation of the form y (t) = F (y(t), t) where F : U Rn whe