MA 425 content specifications

Posted on Posted in teaching

The content specifications are:

A B C
sequential characterization

(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

(5:3.2.5, 4)

 use 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

(2:5,4:2, 4:4)
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
  • prove two definitions of continuity are equivalent
  • use the \epsilon-\delta 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

(6:11, 7:5)
derivatives use Taylor's Theorem
  • use Caratheodory's Criterion
  • use Mean Value 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,

2: 1.4.8,

6: 6, 3.3.7

9: 7b

10: 7.3.9(d), 8
topology 7:2

10:10