MATH 401/501  Advanced Calculus

Fall 2009 

Professor: Dr. Janet Vassilev

Office: Humanities 467

Office Hours: TWTh 11-12

Telephone: (505) 277-2214

Email: jvassil@math.unm.edu

webpage: http://www.math.unm.edu/~jvassil

Text :  Analysis With and Introduction to Proof, 4th Edition by Steven Lay 

Course Meetings:  The course lectures will be held 132 Dane Smith Hall on Tuesday and Thursday from 12:30-1:45 p.m.   There is also a Discussion Session on Wednesday from 12-12:50 p.m.

Topics: Definition and topology of real numbers, sequences, limits, functions, continuity, differentiation and integration with an emphasis on rigorous proofs.

Homework (200 points):  Homework will be assigned weekly on Thursdays and will be collected the following Thursday.  Late homework will be penalized 20% off per each day that it is late.  Homework will not be graded unless it is written in order and labeled appropriately.   The definitions and theorems in the text and given in class are your tools for the homework proofs.  If the theorem has a name, use it.  Otherwise, I would prefer that you fully describe the theorem in words that you plan to use, than state by Theorem 3.  Each week I will select about 4 or 5 of the assigned problems to grade. The weekly assignments will each be worth 20 points.  I will drop your lowest two homework scores and the remaining homework will be averaged to get a score out of 200. 

Exams (400 points):  I will give two midterms (100 points) and a final (200 points). There are no make up exams. If a test is missed, notify me as soon as possible on the day of the exam. For the midterms only, if you have a legitimate and documented excuse, your grade will be recalculated without that test.  The Midterms are tentatively scheduled for Tuesday, September 29 and Tuesday, November 10.  The Final is on Thursday, December 17, from 10 am-12 noon. 

Grades:  General guidelines for letter grades (subject to change; but they won't get any more strict): 90-100% - A; 80-89% - B; 70-79% - C; 60-69% - D; below 60% - F.  In assigning Final Grades for the course, I will compare your grade on all course work (including the Final)  and your grade on the Final Exam.  You will receive the better of the two grades.

Tentative Schedule (for Dr. Vassilevís Advanced Calculus):

Date

Section

Topic

Homework

8/25

3 and 4

Proof Techniques

 

8/27

5 and 6

Sets and Relations

 3.1, 3.6(a,b,d,g,i), 3.7(a,b,c), 3.9, 4.4, 4.8, 4.11, 4.17, 4.19, 5.7, 5.14, 5.18, 5.25(a,b), 6.10(a,d), 6.12(a,b,c), 6.14, 6.21, 6.25, 6.27(b,d,e)

9/1

7

Functions

 Homework 1 Solutions

9/3

8 and 10

Cardinality and Natural numbers

 7.4(b,c), 7.6(a,b), 7.7(b,g), 7.8, 7.13, 7.14, 7.17, 7.26, 7.27, 7.32, 8.3(a,c,d), 8.13, 8.18, 8.20, 8.22

9/8

10 and 11

Induction and Ordered Fields

 Homework 2 Solutions

9/10

12

Completeness

 10.3, 10.4, 10.6, 10.7, 10.14, 10.20, 10.21b, 11.4, 11.6, 11.8, 11.9

9/15

13

Topology of Reals

 Homework 3 Solutions Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7

9/17

14

Compact Sets

 12.3(b,h,l), 12.4(b,h,l), 12.6, 12.8, 12.12(a,b), 13,3(a,b), 13.4(a,b), 13.5(c,d), 13.7(a,b,f), 13.12, 13.20,(a,c), 13.21(b,d)

9/22

15

Metric Spaces

 Homework 4 Solutions

9/24

16

Convergence

 

9/29

 

Midterm 1

 

10/1

17

Limit Theorems

14.3(a,c), 14.4, 14.5a, 14.8, 14.9, 14.11, 14.12

10/6

18

Monotone and Cauchy Sequences

 Homework 5 Solutions

10/8

19

Subsequences

16.6(b,c), 16.8b, 16.9a, 16.13, 16.15, 17.5(d,f), 17.6(a,b), 17.7, 17.8, 17.18, 18.3b, 18.4, 18.14

10/13

20

Limits of Functions

 Homework 6 Solutions

10/15

21 and 22

Continuous Functions

 

10/20

23

Uniform Continuity

 

10/22

25

The Derivative

 19.3(a,d), 19.4(a,b), 19.8, 19.9, 19.17, 19.19, 20.13, 20.14, 20.20, 21.8, 21.13, 21.16, 22.3(a,e,g,h), 22.4, 22.7, 22.9

10/27

26

The Mean Value Theorem

 Homework 7 Solutions

10/29

27

LíHopitalís Rule

 23.3(a,e), 23.5, 23.6, 23.10, 23.15, 25.6, 25.8, 25.11, 25.12, 26.6, 26.9, 26.11, 26.13, 26.23

11/3

28

Taylorís Theorem

 Homework 8 Solutions

11/5

29

The Riemann Integral

 

11/10

 

Midterm 2

 

11/12

30

Properties of the Riemann Integral

 

11/17

31

Fundamental Theorem of Calculus

 

11/19

32

Convergence of Series

 26.6, 26.8, 26.18, 26.23, 27.6, 28.9, 28.12, 29.8, 29.9, 29.11a, 29.13, 29.15, 20.5, 30.10, 30.11, 30.21

11/24

33

Convergence Tests

 Homework 9 Solutions

11/26

34

Power Series

 

12/1

35

Pointwise and Uniform Convergence

 

12/3

36

Applications of Uniform Convergence

 32.3, 32.5 (e,k), 33.6, 33.9, 33.11, 33.13, . 34.9, 34.10, 35.3, 35.7, 35.10, 35.14, 35.15

12/8

37

Uniform Convergence of Power Series

 Homework 10 Solutions

12/10

 

Review

 

12/17

 

Final Exam