MTH361 Final Study Guide

The final examination will be held Wednesday, December 16th at 1:30PM in Duffy 205.

The exam will consist of three sections:

A matching section with some terms and their definitions, also some theorems and their statements
Two proofs you have seen before. Two of the following four proofs will appear on the exam:
- Prove that a subset of R is open if and only if its compliment is closed.
- If a function f is continuous on a non-degenerate interval [a,b] then it is integrable on [a,b] (5.10)
- If a function f is continuous and 1-1 on an interval I, then f is strictly monotonic on I (part of 4.32)
- The Fundamental Theorem of Calculus part 1 (The part with F(b)-F(a)) (5.28 ii)
Two proofs you may or may not have seen before.

Obviously due to time limitations the latter set of proofs will have to be relatively short ones.

The terms will be taken from the following list:

Term	Text Reference
epsilon neighborhood	Topology Introduction (part 1)
open set
limit point
isolated point
closed set
closure of a set
perfect set
compliment of a set
F-sigma set
G-delta set
compact set	Topology Introduction (part 2)
open cover
finite subcover
separated sets
disconnected sets
totally disconnected
infinitely differentiable	The nth derivative of f exists for any n in N
E is dense in R	There is and element of E between any two elements of R
E is nowhere dense	The closure of E contains no nonempty open intervals
differentiable at a point	Definition 4.1
derivative at a point	Definition 4.1
f is real analytic at x=a	f has a Taylor series expansion that converges to f(x) in some neighborhood of a
continuously differentiable	Definition 4.6 ii)
characteristic function of E	takes the value 1 if x is in E, 0 otherwise
monotone	Definition 4.16 iii)
partition	Definition 5.1
upper Riemann sum	Definition 5.3 i)
Lower Riemann sum	Definition 5.3 ii)
Riemann integrable function	Definition 5.9
upper integral	Definition 5.13 i)
lower integral	Definition 5.13 ii)

The theorems (on the matching section) will be taken from the following list:

Term	Text Reference
Baire's theorem	Class notes
Heine-Borel theorem	Class notes
Taylor's theorem	Theorem 4.24
Mean Value theorem	Theorem 4.15 i)
Generalized Mean Value theorem	Theorem 4.15 ii)
Rolle's theorem	Lemma 4.12
Inverse Function theorem	Theorem 4.32
Topological characterization of continuity	Class notes
Fundamental Theorem of Calculus	Theorem 5.28
Chain Rule	Theorem 4.11