Math 409-300 - Test 1 Review - Summer 2014
General Information
Test 1 will be given on Wednesday, 7/2/14, during our usual class
time and in our usual classroom. I will have extra office hours on
Tuesday afternoon, 1-4 pm, and on Wednesday morning, 8:30-9:30 am.
- Bluebooks. Please bring an 8½x11 bluebook.
-
Calculators. You may use scientific calculators to do numerical
calculations — logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.
-
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
- Structure and Coverage The test will have 6 to 8
questions, some with multiple parts. It will cover chapters 1, 2 and
3, except for sections 1.4, 1.6, 2.5, and 3.4. You should expect to
have to work problems similar to homework problems or problems done in
class. Be able to state definitions or theorems from the first list,
and be able to prove the theorems in the secondlist.
- Keys to Test 1 and three quizzes from Summer 2005
Definitions and Statements of Theorems
You are expected to know definitions for, or be able to state, the
following:
-
- Real numbers
- Triangle inequalities
- Completeness axiom
- Bounded set, supremum, infimum
- Approximation Property for Suprema, Infima
- Sequences
- Sequence
- Subsequence
- Limit of a sequence
- Bounded sequences
- Monotone (increasing, decreasing) sequences
- Monotone Convergence Theorem (for sequences)
- Nested sequence of sets
- Nested Interval Theorem
- Bolzano-Weierstrass Theorem
- Cauchy sequence
- Cauchy's Theorem - Every Cauchy sequence is convergent
- Functions of a continuous variable
- Limit of f(x) as x → a; one-sided limits; limits as x →
∞
- Sequential characterization of limits
- Continuous function on a set E
- Sequential characterization of continuity
- Composition of g with f, g∘f. Know that
If f:A → B, g:B→
R, and both f and g are continuous on A and B, respectively,
then g∘f is continuous.
- Bounded function
- Extreme Value Theorem
- Intermediate Value Theorem
Proofs
Be able to prove the following:
- Approximation Property for Suprema, Infima
- Monotone Convergence Theorem for Sequences
- Bolzano-Weierstrass Theorem
- Extreme Value Theorem
Updated 6/29/2014 (fjn)