Instructor: Associate Professor Pete L. Clark, Ph.D., pete (at) math (dot) uga (dot) edu

Course webpage: http://www.math.uga.edu/~pete/MATH2400F2011.html (i.e., right here)

Office Hours: Boyd 502, Monday 2:30 - 3:30 pm, Wednesday 5:15 - 6 pm, Thursday 12:15 - 1:15 pm, Friday 2:30 - 3:30 pm, and by appointment

Course Text (required): Calculus, by Michael Spivak (4th edition).

For information on grades, exams and other procedural matters, please consult the course syllabus.

SUPPLEMENTAL LECTURE NOTES: These notes are provided free of charge, free of obligation and free of warranty.

All of the Above, Plus Sequences and Series (283 pages) : click here
COURSE ANNOUNCEMENTS :

Welcome to the course!

The first midterm exam will be given Thursday, September 29.
The second midterm exam will be given Thursday, November 3 Monday, November 7.
REVIEW MATERIALS:

FALL SEMESTER

SPRING SEMESTER
HOMEWORK

Assignment 1: Due in class, Thursday August 25, 2011

required: Chapter 1: 1, 2, 5, 6, 10, 11, 13, 14, 15, 20, 25
extra: Chapter 1: 16, 17, 22

Assignment 2: Due 4 pm, Friday September 2, 2011

required: Chapter 2: 1,2, 3 parts a through d, 5, 6 parts (i) and (ii), 8, 9, 12, 13, 14, 16, 19, 26
extra: Chapter 2: 3 part e, 4, 6 parts (iii) and (iv), 17, 18, 20, 27, 28

Extra Extra Problem 1 (Division Theorem):
a) Let a and b be positive integers. Show that there exists natural numbers q and r such that a = qb + r and 0 <= r < b.
b) Show that the q,r of part a) are unique: i.e., if there is any other pair (q',r') of natural numbers satisfying the same property, then q = q' and r = r'.
c) Extend the results of parts a) and b) to the case where a is any integer.

Assignment 3: Due in class, Friday September 9, 2011

required: Chapter 3: 1-4, 6, 8; Chapter 4: 1-4, 6, 11

Assignment 4: Due in class, Friday September 16, 2011

required: Chapter 3: 9, 12, 13, 26; Chapter 4: 11, 12, 14, 17 parts (i) and (ii)
Hint for Problem 3.13: For any function f, f(x) + f(-x) is even

Assignment 5: Due in class, Friday September 23, 2011
All of the following problems are required. Please read the instructions carefully.
Chapter 9: For these problems, you may (for now!) treat limits intuitively, as we did in Week 4 of class. Exercises: 1,2,3,7,9
Chapter 6: For these problems, you need not prove your answers (for now!) using the epsilon-delta defintion of continuity. Exercises 1,4,5
Chapter 5: Exercise 1
Required Problem V.1:
a) Write down the formal, epsilon-delta definition of continuity of a function f at a point x = c. Draw a picture to accompany your definition.
b) Write down the formal definition of a continuous function.
c) Show that for any real numbers m,b, the function y(x) = mx+b is continuous.
d) Give an example -- with proof! -- of a function which is not continuous at some point x = c of its domain.

Assignment 6: Due in class, Thursday October 6, 2011
required: Chapter 5: 7, 8, 10, 15, 16, 18, 20, 21; Chapter 6: 3, 13, 14
extra: Chapter 6: 7

Assignment 7: Due in class, Friday October 14, 2011
required: Chapter 9: 5, 13, 16, 19, 23; Chapter 10: 1, 2 [feel free to ignore the somewhat obnoxious comment about not taking too much longer than 20 minutes: take as long as you want and need!], 3, 4, 5
extra: Chapter 9: 28

Assignment 8: Due in class, Friday October 21, 2011
required: Chapter 10: 7, 9, 11, 16, 20, 23, 24; Chapter 11: 1, 2, 6
extra: Chapter 10: 31, 33, 34

Assignment 9: Due by 5pm, Monday October 31, 2011
required: Chapter 11: 7 through 14; Chapter 12: 1, 4, 5, 7, 15
extra: Let J be a nonempty interval which is not of the form [a,b]. Let J be any nonempty interval. Show that there is a continuous function f: I --> R with image f(I) = J.

Assignment 10: Due in class, Wednesday November 9 Thursday, November 10, 2011
required: Chapter 11: 23, 25, 27, 29, 30, 33, 38, 62; Chapter 12: 14, 20
extra: Chapter 11: 26, 31
Extra Extra Problem 2: Prove the Lambda-Vee Lemma. (See page 24 of the lecture notes on differentiation.)

Assignment 11: Due in class Friday, November 18, 2011
Required: Chapter 7: 2, 5, 7, 10, 14, 16
Extra: Chapter 7: 12, 13, 17, 21

Assignment 12: Due in class Monday, November 28, 2011
Please read Chapter 8 of Spivak's text. (Extra: take a look at my lecture notes on integration.) Also, please solve and turn in Exercise 1.2 on page 4 from my lecture notes on completeness.

Assignment 13: Due by office hours on Wednesday, December 7, 2011
Comments on the homework: Let's agree not to use extended real numbers in your solutions to this problem set: e.g., if a subset is unbounded above, you should say that it does not have a least upper bound, not that its least upper bound is infinity. Also you may use real induction if you want, but the problems are not designed for this to be necessary (or even, it seems to me, especially helpful).
Required: Chapter 8: 1, 3, 4a), 5, 6, 7, 10, 14, 17, 20
Extra: Chapter 8: 11, 15, 16
Extra Extra: Looking through the exercises in Chapter 8 I found several that I could not assign because we have covered them in class, or they are in the lecture notes, or they have been covered in solutions to review / exam problems on the course webpage. For one point of extra credit each, find such problems and document your claim by giving a reference to the appropriate course materials.

Spring Semester Homework Assignments:

Assignment 1: Due in class Friday, January 27, 2012
Please read Chatpter 13 of Spivak's text and solve problems 1, 5, 7 from that chapter. Also please solve the following:
Extra 1.1: Use Newton's method to find a rational number x which approximates the square root of 2012 to at least 20 decimal digits of accuracy.
Extra 1.2: Let a > 0, and let f(x) = x2 - a. We showed in class that there exists delta > 0 such that for all x1 in [a-delta,a+delta], the Newton's method sequence converges to the square root of a. Show that this actually holds for all x1 > 0. (Hint: try to find the largest set on which the amelioration function T(x) is contractive.)
Added: This problem turned out to be harder than I expected. So I am making this problem extra credit. Let me ask you instead to show that there exists C < sqrt{a} such that for all x1 greater than or equal to C, the Newton's method sequence converges.
Extra 1.3: In class I said that a convex function defined on an interval I is continuous. In fact this holds only when I is an open interval: i.e., does not contain any endpoints.
a) Find a convex function with domain [0,1] which is not continuous at 0.
b) Let f be a continuous convex function with domain [0,1]. Find all real numbers L such that if we redefine f at 0 to be equal to L, then the resulting function remains convex.
Extra 1.4: For a continuous function f defined on an interval I, show that the epigraph of f is convex iff every secant line lies on or above the graph of f.
Extra 1.5: Prove Jensen's Inequality.
Extra 1.6: Use Jensen's Inequality to prove the Weighted Arithmetic-Geometric Mean Inequality.

Assignment 2: Due in class Friday, February 3, 2012
Required: From Chapter 13 of Spivak's text: 3, 10, 11a),b),c), 19, 20, 23, 26 [Hint for part c): Thomae's Function], 30a),b), 33
Extra Credit: From Chapter 13: 11d), 15

Assignment 3: Due in class Friday, February 10, 2012
Please read Chapter 14 of Spivak's text and do the following problems: 1 (do any four parts), 2, 3 (beware: part b) is tricky!), 4 (hint: you are trying to show the functions are constant: what's the differential calculus approach to this?), 6, 15, 19, 21, 25, 26 (hint: do not use the fundamental theorem!).
Extra Credit 3.1: Prove by Real Induction that if g is continuous and f is integrable, then the composite function g o f is integrable.

Assignment 4: Due in class Wednesday, February 22, 2012
Please read (or at least skim) Chapters 18 and 19 of Spivak's text and do the following problems:
Chapter 18: 1 (do any four parts), 2, 3, 6, 7, 10 (extra credit: this function, the logarithmic integral, is extremely important in mathematics: why??),
Chapter 19: 1 (do any four parts), 2 (do any five parts), 3 (do any five parts).

Assignment 5: Please turn in any ten of the following problems by Friday, March 9, 2012. Then, please turn in eight more of them by Monday, March 19, 2012.
Chapter 22: 1 (any four parts), 2 (any four parts), 3, 4, 5, 6, 7 (parts a and b only), 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 21 (part a only), 22, 23, 24, 25, 27, 28, 30, 31.

Assignment 6: due in class Monday, April 2, 2012.
Chapter 23: 1 (any ten parts), 4, 5, 7.

Assignment 7: due in class Monday, April 16, 2012.
Rquired: Chapter 23: 1 (please do six more parts disjoint from the ten you did last time), 3, 6, 7, 8, 13, 24, 25.
Chapter 20: 1,2,3
Also do the following
Problem 7.1: For which positive integers p,q,r does the series sumn 1/(np (log n)q (log log n)r) converge?
Extra Credit: The title of Chapter 12 in my lecture notes is Taylor Taylor Taylor Taylor. Ostensibly this holds because the chapter has four sections: Taylor's Theorem Without Remainder, Taylor Polynomials, Taylor's Theorem With Remainder, Taylor Series. But more than this I am trying to make an allusion to the character Major Major Major Major from Joseph Heller's great novel Catch 22. I am looking for a passage from the novel concerning this character, containing text that either has some mathematical significance or could be misinterpreted as such. Please help!

Assignment 8>: due in class Monday, April 29, 2012.
Required: Chapter 20: 1, 3, 5, 8, 14, 18
Chaper 24: 1, 3, 5, 8, 14
Extra Credit: Chapter 24: 16, 19, 20

Some interesting -- and perhaps relevant -- links:

Why You Should Be A Math Major

Why Major in Mathematics?

What Can a Math Degree Do For You?, Jaqueline Jensen, Sam Houston State University.