**Instructor:**
Roman Vershynin

Office: 4844 East Hall

E-mail: romanv "at" umich "dot" edu

**Class meets:** MWF 2:10 - 3:00 pm in 4088 EH.

**Office hours:** M: 12:45 - 2:00 pm, W: 10:45 am - 12:00 noon, in 4844 EH.
Please do come to my office hours, I will be there to help you. It is an effective way to
improve your command of the material. You can also grab me after a class if you
have a quick question. I won't be able to hold office hours at any other times than those posted.
Instead you can come to the office hours of Professor Mark Rudelson who is teaching a different section
of Math 451 this Fall. Professor Rudelson's office hours are T 2:00 - 3:00 pm, W: 5:30 - 7:30 pm, in 3834 EH.
We have an agreement with Prof. Rudelson so you don't need to ask him in advance, just come.

**Prerequisites:**
A thorough understanding of Calculus and one of 217, 312, 412.

**Course Description:**
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of Calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are **rigor** and **proof**; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a **strong commitment** to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451.

**Textbook:** Kenneth A. Ross, *Elementary analysis: the theory of calculus*.
Springer, Corr. 10, 1998. ISBN: 9780387904597.
The course covers most of the material in the book except the starred (optional) sections,
in the following order (tentative): 1 - 20, 28, 29, 32 - 34, 23 - 26, 31, concluding with a full week of review.

**Homework, Exams and Grading:**
There will be one in-class midterm exam, one take-home exam and one final exam.
You are welcome and encouraged to discuss homework with other students, but you must write your solutions individually.
To form study groups and for online collaboration, feel free to use the chat room and forums on the
Ctools class page.
The course grade will be determined as follows:

- Homework (25%). Will be assigned every class.
**Due every Friday**before class. One homework with the lowest score will be dropped. Scroll down to see the homework. - Midterm Exam 1 (20%). October 26, in class. Covers the material in Sections 1 - 12 that we covered in class. Books and calculators are allowed. Solutions.
- Midterm Exam 2 (20%), take-home. Due November 21 at the beginning of class. Solutions.
- Final Exam (35%). Friday, December 16, 1:30 - 3:30 pm, in class. Covers the whole course.
Books and calculators are allowed.
Solutions.
**Warning: Statement of Problem 3 is false.**

**Missing/late work:**
**Missed exams** - there will be no make-up for the exams for any reason.
A missed midterm exam counts as zero points, with the following exception.
If you miss a midterm exam due to a documented medical or family emergency,
the exam's weight will be added to the weight of the final exam.
**Late homework** (and late take-home exam) can not be accepted. In extenuating circumstances,
you may e-mail me your scanned or typed homework (as a single pdf file) on the day
when the homework is collected by 8:00 p.m.

**Lecture Schedule and Homework:**
It is a useful practice to read ahead the sections to be covered.
Solutions are posted for some (but not all) homework problems.
Please come to the office hours to discuss other problems.

If you type your homework in LaTeX, you may (but are not required to) use the following simple template: homework-template.tex.

- Wednesday, September 7
- Section 1 (begin).
- No homework today.
- Friday, September 9
- Section 1 (end). As always, study the examples in that section.
- Homework 1, due Friday, September 16: 1.1, 1.2, 1.3, 1.5, 1.9, 1.11. Extra 10%: 1.12. Please remember to always write your arguments using complete English sentences (similar to how I do this in class).
- Monday, September 12
- Skip Section 2. Section 3 up to Theorem 3.2. Study the proofs of the parts of Theorems 3.1 and 3.2 that we have not covered in class.
- Homework 1, due Friday, September 16: 3.1, 3.3, 3.4. Solutions to 3.4 and 1.12.
- Wednesday, September 14
- Section 3, starting from Theorem 3.2. Study the proof of Theorem 3.5(ii).
- Homework 2, due Friday, September 23: 3.5(a), 3.6, 3.7(c). Extra 10%: 3.8.
- Friday, September 16
- Section 4 up to Example 3.
- Homework 2, due Friday, September 23: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6.
- Monday, September 19
- Section 4 up to Theorem 4.6.
- Homework 2, due Friday, September 23: 4.8, 4.10, 4.14. Selected solutions to Homework 2.
- Wednesday, September 21
- Section 4 (Theorem 4.7), Section 5.
- Homework 3, due Friday, September 30: 4.12, 4.16, 5.2.
- Friday, September 23
- Sections 7, 8. These are the crucial sections in this course. Please invest your time to understand the definition of the limit now. Specifically (as usual) study all examples in Sections 7,8. Also do all odd-numbered exercises for Section 7 and exercise 8.1. There are solutions and hints for these exercises in the end of the book.
- Homework 3, due Friday, September 30: 7.4, 8.2, 8.4.
- Monday, September 26
- Section 8 (Example 4), the Squeeze Theorem (Exercise 8.5a), Section 9 (Theorem 9.1). Study Exercises 8.7, 8.5b.
- Homework 3, due Friday, September 30: 8.6, 8.8, 8.10. Selected solutions to Homework 3.
- Wednesday, September 28
- Section 9 (limit theorems). I will be away from town; Professor Mark Rudelson will teach the class.
- Homework 4, due Friday, October 7: 9.4, 9.8.
- Friday, September 30
- Section 9 (9.7 begin, examples that follow).
- Homework 4, due Friday, October 7: Click here to see additional required problems.
- Monday, October 3
- Section 9 (9.7 through the end of section), Ratio Test (Exercise 9.12), Exponential vs polynomial limit (Exercise 9.14).
- Study infinite limits (9.8 throught the end), in particular the proofs of Theorems 9.9 and 9.10.
- Homework 4, due Friday, October 7: 9.10, 9.16 (a),(b), 9.18. Selected solutions to Homework 4.
- Wednesday, October 5
- 10.1 - 10.5. The number e as a limit. Class notes.
- Study the proof of Theorem 10.4 (try ot prove it yourself by modifyng the proof of Weierstrass theorem).
- Homework 5, due Friday, October 14: 9.6, 10.2.
- Friday, October 7
- The Nested Intervals Theorem. Bolzano-Weierstrass theorem (11.5). Class notes. A note on continued fractions.
- Homework 5, due Friday, October 14: 10.8, 10.10; give a complete proof for the theorem on continued fractions discussed in class (see the note above).
- Monday, October 10
- Cauchy sequences (10.8 - 10.11). Banach fixed point theorem. Class notes.
- Homework 5, due Friday, October 14: 10.4, Exercise on p.34 of class notes. Selected solutions to Homework 5.
- Wednesday, October 12
- Lim inf and lim sup (parts of Sections 10, 11). Class notes.
- Complete the argument of Theorem on p.37 of class notes (for infinite s,t and for lim inf) and the argument of Theorem 10.7 on p.38 (for lim inf). Study the last theorem on p.38 of the class notes.
- Homework 6, due Friday, October 21: 11.2(c,d,e), 12.2, 12.4, 12.14(a).
- Friday, October 14
- 14.1 - 14.6. Class notes.
- Study Exercise 14.5 and the Comparison Test for infinite series (14.6.ii).
- Homework 6, due Friday, October 21: 14.6(a), 14.8. Click here to see additional required problems.
- Selected solutions to Homework 6.
- Wednesday, October 19
- Root Test and Ratio Test (14.7 through the end of Section 14). Class notes.
- Study all examples in Section 14.
- Homework 7, due Friday, October 28: 14.2, 14.4, 14.12.
- Friday, October 21
- Integral Test, Limit Comparison Test, Alternating Series (Section 15). Class notes.
- Study Example (b) on p.48 of the class notes.
- Homework 7, due Friday, October 28: 15.2, 15.4, 15.6.
- Selected solutions to Homework 7.
- Monday, October 23
- Two definitions of continuity (Section 17 up to Example 3).
- Study Examples 1 - 3 in Section 17.
- No additional homework problems today.
- Wednesday, October 26
- Midterm Exam 1. Solutions.
- Friday, October 28
- Operations with continuous functions (from 17.3 to the end of Section 17). Extreme Value Theorem (18.1).
- Study Examples 4, 5 in Section 17. Solve Exercises 17.3, 17.9 (the textbook has solutions to these exercises).
- Homework 8, due Friday, November 4: 17.2, 17.10(a,b,c), 17.12.
- Monday, October 31
- Intermediate Value Theorem (18.2), Inverse Function Theorem (18.5).
- We proved the Intermediate Value Theorem for y=0 in class. Deduce it for general y (the statement is Theorem 18.2). Study Example 2.
- Homework 8, due Friday, November 4: 18.4, 18.6, 18.10.
- Selected solutions to Homework 8.
- Wednesday, November 2
- The class was cancelled due to a power outage.
- Friday, November 4
- Uniform continuity (Section 19 up to Example 6).
- Study all examples in Section 19 except 19.3 and Example 10.
- Homework 9, due Friday, November 11: 19.2, 19.4, 19.10.
- Monday, November 7
- Limit of functions (Section 20 up to Corollary 20.7; only two-sided limits were discussed). Class notes.
- Study Examples 2, 3, 5, 7.
- Homework 9, due Friday, November 11: 20.18; additional required problems.
- Selected solutions to Homework 9.
- Wednesday, November 9
- One-sided limits and limits at infinity (end of Section 20). Asymptotic behavior of functions. Class notes.
- Do the exercises suggested in class (see class notes).
- Homework 10, due Friday, November 18: 20.2, 20.4, 20.6, 20.8, 20.12, 20.16.
- Friday, November 11
- Differentiation (28.1-28.2). Derivatives of polynomial, exponential, logarithmic funcitons and sin(x). Class notes.
- Homework 10, due Friday, November 18: 28.2(a), 28.8, 28.14.
- Selected solutions to Homework 10.
- Monday, November 14
- Differentiation rules (28.3). Derivative in Landau's notation. Chain rule (28.4). Class notes.
- Study the proof of parts (i), (ii), (iii) of Theorem 28.3 (try to prove those by yourself)
- Wednesday, November 16
- Midterm Exam 2 is posted. It is due Monday, November 21, at the beginning of class. Solutions.
- Criterion of local extrema (29.1), Rolle's Theorem (29.2), Mean Value Theorem (29.3). Consequences of Mean Value Theorem: derivative in Landau's notation; uniqueness of antiderivative (Corollaries 29.4-29.5). Class notes.
- Friday, November 18
- Derivative of the inverse function (29.9). L'Hopital's Rule (Section 30).
- Please study Corollary 29.7. Review the examples on L'Hopital's Rule in Section 30. Class notes.
- Monday, November 21
- Taylor's theorem (31) for n=1. Remainder of linearization in Cauchy and Lagrange forms. Class notes.
- Homework 11, due Friday, December 2: 28.4 (this is an example of a differentiable but not continuously differentiable function), 28.16 (this justifies change of variable in limits), 29.2 (do not try to use trigonometric identities), 29.12, 29.14, 30.2.
- Wednesday, November 23
- Taylor's theorem (31) for general n. Lagrange and Peano forms of the remainder. Taylor series. Taylor series of e^x, sin(x), cos(x). Computing limits using Taylor's theorem. Class notes.
- Homework 11, due Friday, December 2: 31.2. Additional required problem: Suppose f(x) is an even function which satisfies the conditions of Taylor's theorem; show that Taylor's series of f about 0 contains only even powers of x.
- Selected solutions to Homework 11.
**Happy Thanksgiving!**- Monday, November 28
- Definition of the Riemann integral (32 up to 32.4). Class notes.
- Homework 12, due Friday, December 9: 32.6, 32.8. (You may use Theorem 32.5 which will be proved on Wednesday).
- Wednesday, November 30
- Criterion of integrability (32.5). Class notes.
- Friday, December 2
- Partitions with small mesh (Theorem 32.7). Riemann sums (Theorem 32.9). Class notes.
- Homework 12, due Friday, December 9: 33.4, 33.10, 33.8 (you may use Theorem 33.3).
- Monday, December 5
- Integrability of continuous (33.2) and monotonic (33.1) functions. Integrals of sums and products (33.3). Comparison of intergals (33.4). Class notes.
- Homework 12, due Friday, December 9: 33.2, 33.6.
- Selected solutions to Homework 12.
- Wednesday, December 7
- Intermediate value theorem for integrals (33.4). Integral of the absolute value of a function (33.5). Additivity property with respect to the interval (33.6). Integrability of piecewise continuous functions (33.8). Lebesgue criterion of integrability (without proof). Class notes.
- Friday, December 9
- Fundamental theorem of calculus (34.1, 34.3). Antiderivative and indefinite integral. Class notes.
- Practice problems for the last sections for which no homework was assigned: 33.1, 33.3, 34.2 (express the limits as derivatives), 34.5, 34.11.
- Monday, December 12
**The class will meet in 3088 EH today.**- Review. No office hours today. Instead, office hours will be on Wednesday 10-12.
- Friday, December 16
- Office hours 10-12.
**Final Exam.**4088 EH, 1:30 - 3:30 p.m.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2011-12/451/451.html