MA 425 content specifications
The content specifications are:
(2:1.4.2, 3:4, 5:2, 5:5, 5:3.4.6)
|formulate and prove||prove at least 2||prove at least 1|
|Cauchy criteria||formulate and prove||prove at least 1
|Archimedean Principles||prove all are equivalent
(2:6 and 2:8 together)
|use each of AP1, AP2, AP3|
|forms of completeness||use Nested Intervals||prove 2 are equivalent
|compactness||use open-covers version||use Heine-Borel version||use sequential compactness|
|limits and continuity||use topological version of continuity to show a function is continuous||
||use the sequential criterion for continuity to show a function is continuous|
|major/named theorems||prove 2 ``at infinity" versions of named theorems
(7:8, 8:3, 9: 1-6)
|prove an ``at infinity" version of a named theorem
(7:8, 8: 3, 9: 1-6)
|prove at least one
|derivatives||use Taylor's Theorem||
||use the limit definition of the derivative|
|integration||compute 3 integrals with proof||compute 2 integrals with proof||compute 1 integral with proof|
|convergence of functions||prove a sequence converges pointwise but not uniformly|
|series||prove series versions of 2 named theorems for sequences||prove series version of a named theorem for sequences|
|explanation points||3 points||2 points||1 point|
|pathologies & exotica points||3 points||2 points||1 point|
|topology points||3 points||2 points||1 point|
- Numbers in italics represent problems that definitely satisfy that specification. 3:4 means problem 4 from homework 3. 2:1.4.8 means Abbott's exercise 8 from chapter 1.4, which was assigned on homework 2.
- formulate and prove means you must come up with the statement of the theorem, in addition to proving it.
- use means to use the result in question in the course of a proof
- a named theorem is one with a person's name attached (such as the Bolzano-Weierstrass Theorem) or a phrase (such as the Mean Value Theorem)
- a sequential characterization or a sequential criterion is a theorem of the form [statement not involving sequences] if and only if [statement involving sequences]. Examples include:
- The sequential characterization of supremum.
- The sequential criterion for continuity.
- The sequential characterization of closed sets.
- forms of completeness are: the Monotone Sequence Property, the Least Upper Bound Property, the Greatest Lower Bound Property, and the Bolzano-Weierstrass Property
Points can be earned by completing the following problems. Topology points may also be earned by using arguments that reference open, closed, and/or compact sets. Explanation points may also be earned by giving a very clear unsolicited explanation as part of one of your proofs.
|explanation||2:2, 3:2, 8:4, 10:9|
|pathologies & exotica||1:3,
6: 6, 3.3.7
10: 7.3.9(d), 8