Math 2400/2410: Honors
Calculus With Theory, Fall 2011 and Spring 2012 |
Return to
Pete's home page |

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

Week 1 Lecture Notes (12 pages) click here

Mathematical Induction (26 pages): click here

Polynomial and Rational Functions (6 pages): click here

Continuity and Limits (18 pages): click here

Differentiation (33 pages): click here

Completeness (19 pages): click here

Differential Miscellany (21 pages): click here

Integration (32 pages): click here

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

Practice problems for the first midterm exam: click here.

Solutions to the first set of practice problems: click here.

Practice problems for the second midterm exam: click here.

Solutions to the second set of practice problems: click here

Midterm Exam 1, with solutions: click here.

Midterm Exam 2, with solutions: click here.

Midterm Exam 3, with solutions: click here

Practice problems for the first midterm exam: click here

Practice problems for the second midterm exam: click here

Midterm Exam 1, with solutions: click here

Midterm Exam 2, with solutions: click here

HOMEWORK

required: Chapter 1: 1, 2, 5, 6, 10, 11, 13, 14, 15, 20, 25

extra: Chapter 1: 16, 17, 22

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

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

c) Extend the results of parts a) and b) to the case where a is any integer.

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

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

All of the following problems are required.

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

Chapter 5: Exercise 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

required: Chapter 5: 7, 8, 10, 15, 16, 18, 20, 21; Chapter 6: 3, 13, 14

extra: Chapter 6: 7

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

required: Chapter 10: 7, 9, 11, 16, 20, 23, 24; Chapter 11: 1, 2, 6

extra: Chapter 10: 31, 33, 34

required: Chapter 11: 7

extra: Let J be a nonempty interval which is

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.)

Please read Chapters 7 and 13 of Spivak's text.

Required: Chapter 7: 2, 5, 7, 10, 14, 16

Extra: Chapter 7: 12, 13, 17, 21

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.

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.

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) = x

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

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.

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

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:

Extra Credit 3.1: Prove by

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

Chapter 19: 1 (do any four parts), 2 (do any five parts), 3 (do any five parts).

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.

Chapter 23: 1 (any ten parts), 4, 5, 7.

Rquired: Chapter 23: 1 (please do six more parts disjoint from the ten you did last time), 3, 6, 7, 8, 13, 24, 25.

Also do the following

Problem 7.1: For which positive integers p,q,r does the series sum

Extra Credit: The title of Chapter 12 in my lecture notes is

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 Major in Mathematics?

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