TYPES OF QUESTIONS FOR EXAM 1 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- State the In Class version of the Completeness Axiom, Archimedean Property, and Denseness of Q
• 1 -- Proof by Induction
• 1 -- Prove a number is irrational using the Rational Zero's Theorem
• 1 -- Proof(s) (up to two) using the Ordered Field Axioms and Corresponding Theorems (3.1 and 3.2) Note: The Axioms and Theorems will be given to you.
• 1 -- Finding max/min and sup/inf for a list (up to 3) of sets
• 1 -- Compute limits (up to 3) using any method you know
• 1 -- Prove a limit converges to a given value using the definition (epsilon-N property)
• 2 -- Other
• 2 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 1.

• 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 2.) No notes, books, calculator, scratch paper, or classmates may be used as resources for this exam.
• 3.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.
• 4.) You will be graded on proper use of limit, sequence, and set notation.
• 5.) You are free to use any theorem in the book, any theorem presented in class, or any result of an assigned homework problem without proof. The only exception is if it trivializes the problem. For example, if I ask you to prove the Denseness of Q, you CANNOT invoke the Denseness of Q to prove it.
• 6.) If you use a named theorem, you MUST cite the name when invoking the theorem. Common abbreviations are fine (i.e. PMI for Principle of Mathematical Induction).

TYPES OF QUESTIONS FOR EXAM 2 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- Prove Lemma 10.9 using the IN CLASS proof
• 1 -- Prove the limit of a sequence is (or is not) infinity (or -infinity) using the definition
• 1 -- Compute limit of sequence using Theorems/Lemmas 9.2-9.10
• 1 -- Use definition of limit of sequence to prove one of the following Theorems/Lemmas: 9.2, 9.3, 9.4, 9.5, or 9.9
• 1 -- Determine properties of sequences (up to 3) like non-increasing/non-decreasing, lim inf/sup, set of subsequential limits
• 1 -- Prove a sequence defined by a recursion relation converges
• 1 -- Subsequence(s) Problem
• 2 -- Others
• 1 or 2 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 2.

• 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 2.) No notes, books, calculator, scratch paper, or classmates may be used as resources for this exam.
• 3.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.
• 4.) You will be graded on proper use of limit, sequence, and set notation.
• 5.) You are free to use any theorem in the book, any theorem presented in class, or any result of an assigned homework problem without proof. The only exception is if it trivializes the problem. For example, if I ask you to prove the Denseness of Q, you CANNOT invoke the Denseness of Q to prove it.
• 6.) If you use a named theorem, you MUST cite the name when invoking the theorem. Common abbreviations are fine (i.e. PMI for Principle of Mathematical Induction).

TYPES OF QUESTIONS FOR FINAL (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- Prove Section 13 Example 6 (pg 92) using the IN CLASS proof
• 1 -- Proof by Induction
• 1 -- Prove a limit converges to a given value or not using the definition (epsilon-N property)
• 1 -- Compute limit of sequence using Theorems/Lemmas 9.2-9.10
• 1 -- Prove a sequence defined by a recursion relation converges
• 1 -- Determine absolute convergence, conditional convergence, or diverge of (up to 3) specific series using tests
• 1 -- Proving convergence or divergence for an abstract series
• 1 -- Show a given function d(x,y) is or is not a metric
• 1 -- Detemine properties of sets (up to 3) like open, closed, bounded, closure, interior, boundary, compact, etc...
• 1 -- Prove a metric space is complete or not
• 3 -- Others
• 2 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR FINAL.

• 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 2.) No notes, books, calculator, scratch paper, or classmates may be used as resources for this exam.
• 3.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.
• 4.) You will be graded on proper use of limit, sequence, and set notation.
• 5.) You are free to use any theorem in the book, any theorem presented in class, or any result of an assigned homework problem without proof. The only exception is if it trivializes the problem. For example, if I ask you to prove the Denseness of Q, you CANNOT invoke the Denseness of Q to prove it.
• 6.) If you use a named theorem, you MUST cite the name when invoking the theorem. Common abbreviations are fine (i.e. PMI for Principle of Mathematical Induction).