MA 426 Test Preparation: Metric Spaces, Derivatives, and Optimization, Study notes of Mathematics

Download MA 426 Test Preparation: Metric Spaces, Derivatives, and Optimization and more Study notes Mathematics in PDF only on Docsity! Material to Review for the Second MA 426 Test Steve Schecter November 7, 2002 In the following, when I say you should know a definition or the statement of something, that doesn’t mean you should know it word-for-word; it just means that this is something that you should be able to use. When I say you should know a proof, again I don’t mean word-for-word, but that you should be able to prove this thing if asked. The problems listed are ones you should be able to do. Whenever the book mentions metric spaces in definitions, theorems, etc., you can substitute R n. • 5.1. Pointwise and uniform convergence. Definitions 5.1.1 and 5.1.2; the lemma proved in lecture for showing that a sequence fk converges uniformly; proof of Proposition 5.1.4; problems 5.1.1, 5.1.2, 5.1.3. • 6.1. Definition of the derivative. Definition 6.1.1. Problems 6.1.2 and 6.1.4; p. 384 problem 2; problem 6.4.1 as we did it; proof that if f(x) = Ax + b then Df(x) = A. • 6.2. Matrix representation. Definition 6.2.1. Proof of Theorem 6.2.2. Problems 6.2.1, 6.2.2. • 6.3. Continuity of differentiable functions. Proofs of the following: (1) if L : Rn → Rm is linear, then there is a constant M such that ‖ L(x) ‖≤ M ‖ x ‖; (2) if f : A → Rm is differentiable at x0, then f is continuous at x0. • 6.4. Conditions for differentiability. The example that begins the section (figure 6.4.1). Statement of Theorem 6.4.1. Definition 6.4.2. Example 6.4.3. Problem 6.4.2 as we did it. • 6.5. Differentiation rules. Proof of the “multiplication by a constant rule” and the “sum rule.” Proofs of properties of O and o (done in lecture and homework). Proof of the chain rule. Relation of the chain rule to the directional derivative formula (Example 6.5.4 as done in lecture). Problems 6.5.2, 6.5.3, 6.5.5. 1