

* indicates the problem will be collected 
Monday Feb 8 
SOS: Chapter 3 THEOREMS ON CONTINUITY p52
SOS: p6 (Bounds) Axioms for the Real numbers The intermediate value theorem OPTIONAL Spivak: Ch 1 pp 120122 
0. SOS: 1.21(a,b); 1.64(a,b); 1.65(a,b) *1.Give a statement the Extreme Value Theorem with examples illustrating how it can be false if each of the hypotheses is not satisfied, i.e. a. The interval is bounded but not closed, f is continuous on the interval. b. The interval is closed but not bounded, f is continuous on the interval. c. The interval is closed and bounded, f is not continuous on the interval. *2. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form with for = 1 to n and a set of real numbers to . Prove that there are numbers p and q , where for all . *3. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form with for = 1 to n and finite set of real numbers to . Using the Intermediate Value Theorem and the Extreme Value Theorem, prove that f (D) = {y : y = f (x) for some x in D} is also the union of a finite number of intervals of the form with for to and a finite set of real numbers (possibly empty) to . 
Monday Feb 1 
SOS: Chapter 1 Up through POINT SETS, INTERVALS (p5) Chapter 4 MEAN VALUE THEOREMS.(p7778) [Proofs can be found in the solved problems 4.19, 4.20(Note error in F.)] OPTIONAL Spivak: Chapter 1; pp 190192 
*1.Give a statement and proof of the Mean Value
Theorem and of Rolle's Theorem. [You may use other results
for your proof. If you use Rolle's theorem to prove the
MVT prove Rolle's Theorem without the MVT!] *2. Prove: Rolle's Theorem is equivalent to the Mean Value Theorem. *3. For any real number x let x be the absolute value of x, i.e., x = x when x>0,x =x when x< 0, and x = 0 when x =0. Prove: For any real numbers a and b, ` a+b <= a + b`. *4. Prove the following: (i) `ab <= a + b`. (ii) `a  b < = ab`. (iii) `( a  b ) <= ab`. *5. Suppose m(x,y) = yx for x and y real numbers.Prove that for any real numbers, a,b, and c: (i) m(a,b) = 0 if and only if a=b. (ii) m(a,b) = m (b,a). (iii) `m(a,b) <= m(a,c) + m(b,c)` (iv) If m(a,b) = m(a,c) + m(b,c) then either a= c, b=c, a<c< b, or b<c<a. 
Monday, Jan 25 
Review Methods of Proof [Math 240?] See Daniel Solow's links about thinking and structure in Mathematics. 
1. Make a list of 5 theorems that you feel are critical
to understanding and using the material you learned in the
first year of calculus. Be sure to state clearly the theorem with all its hypotheses and conclusions. 2. For each of the 5 theorems you gave, provide at least one example of how the theorem can fail if only one of the hypotheses is not satisfied. 
Chapter
1:Analysis WebNotes by John Lindsay Orr
Introduction
