| Fall 2017 MAT 319: Foundations of Analysis | Fall 2017 MAT 320: Introduction to Analysis | |
| Schedule | TuTh 10:00-11:20 Heavy Engineering 201 | TuTh 10:00-11:20 Math P-131 ( through 10/5: joint lectures in Heavy Engineering 201) |
| Instructor | Samuel Grushevsky | Robert Hough |
| Office hours | Tu 2:30-3:30, W 1:30-2:30, Th 11:30-12:30 in Math 3-109 | MW 4:00-5:00, F 10:00-11:00 in Math 4-118 |
| Recitation | MW 11:00-11:53 Library E4330 | MW 11:00-11:53 Math P-131 |
| TA | Fangyu Zou | Aleksandar Milivojevic |
| Office hours | WF 10:00-11:00 in Math 2-109, F 11:00-12:00 in MLC | M 12:00-1:00, W 10:00-11:00 in Math 3-104, M 10:00-11:00 in MLC |
| Description | A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series. | A careful study of the theory underlying calculus. The real number system. Basic properties of functions of one real variable. Differentiation, integration, and the inverse theorem. Infinite sequences of functions and uniform convergence. Infinite series. |
| Overview | The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements. | An introductory course in analysis, it provides a closer and more rigorous look at material which most students encountered on an informal level during their first two semesters of Calculus. Students learn how to write proofs. Students (especially those thinking of going to graduate school) should take this as early as possible. |
| Prerequisites |
C or higher in MAT 200 or permission of instructor; C or higher in one of the
following: MAT 203, 205, 211, 307, AMS 261, or A- or higher in MAT 127, 132, 142,
or AMS 161. Math majors are required to take either MAT 319 or MAT 320 | |
| Textbook | Kenneth Ross Elementary Analysis: The Theory of Calculus, 2nd edition | |
| Homework | Weekly problem sets will be assigned, and collected in Wednesday recitation. The emphasis of the course is on writing proofs, so please
try to write legibly and explain your reasoning clearly and fully. You are encouraged to discuss the homework problems with others, but your write-up must be your own work.
Late homework will never be accepted, but under documented extenuating circumstances the grade may be dropped. Your lowest homework grade will be dropped at the end of the class. | |
| Grading | Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. | |
Syllabus/schedule (subject to change)
All joint lectures through 10/5 meet in Heavy Engineering 201.
First recitation on Wed 8/30, second recitation Wed 9/6.
During joint lectures through 10/5, students with last names starting A-O attend recitation in Library E4330, students with last names P-Z attend recitation in Math P-131
Recommendations on choosing MAT 319 versus MAT 320 will be made based upon your performance on the first midterm and homework to that date.
| Tue 8/29 | 1. | Joint class: Introduction, motivation: what are real numbers? (Grushevsky) | Read pages 1-19 | |
| Thu 8/31 | 2. | Joint class: Properties of numbers; induction; concept of a field. (Hough) | HW due 9/6: 1.3, 1.4, 1.10, 1.12, 2.2, 2.5, 3.1, 3.4, 3.6 | |
| Tue 9/5 | No class: day after Labor Day | |||
| Thu 9/7 | 3. | Joint class: Completeness axiom for real numbers; Archimedean property. (Grushevsky) | Read pages 20-27; HW due 9/13: parts eghimr of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.14 | |
| Tue 9/12 | 4. | Joint class: Infinity, unboundedness. Intro to sequences. (Grushevsky) | Read pages 28-38 | |
| Thu 9/14 | 5. | Joint class: Limit of a sequence. (Grushevsky) | HW due 9/20: 5.2, 5.6, 7.3, 7.4, 8.1ac | |
| Tue 9/19 | 6. | Joint class: Limit laws for sequences. (Hough) | Read pages 39-55 | |
| Thu 9/21 | 7. | Joint class: Divergence to infinity, more formal proofs. (Hough) | HW due 9/27: 8.3, 8.6, 8.8, 8.10, 9.1, 9.3, 9.5, 9.12, 9.14 | |
| Tue 9/26 | 8. | Joint class: Monotone and Cauchy sequences. (Hough) | Read pages 56-65 | |
| Thu 9/28 | 9. | Joint class: Subsequences. (Hough) | No HW: prepare for the midterm | |
| Tue 10/3 | Joint Midterm I in Heavy Engineering 201. | Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions | ||
| Thu 10/5 | 10. | Joint class: Subsequences. (Grushevsky) | HW due 10/11: 10.1, 10.2, 10.5, 10.8, 10.9, 11.2, 11.4, 11.5, 11.8, 11.9 | |
| Tue 10/10 | 11. | Subsequences, lim sup, lim inf | Read pages 66-83 |
| Thu 10/12 | 12. | Series introduction | HW due 10/18: 12.2, 12.3, 12.6, 12.8, 12.11, 12.12 |
| Tue 10/17 | 13. | Series convergence tests | Read pages 95-109 |
| Thu 10/19 | 14. | Functions and continuity | HW due 10/25: 14.2, 14.5, 14.7, 14.10, 14.14, 15.1, 15.6, 15.7 |
| Tue 10/24 | 15. | Continuous functions | Read pages 123-139 |
| Thu 10/26 | 16. | Properties of continuous functions | HW due 11/1: 17.2, 17.4, 17.8, 17.12, 17.16, 18.1, 18.4 |
| Tue 10/31 | 17. | Properties of continuous functions | Read pages 153-164 |
| Thu 11/2 | 18. | Limits | (Shorter) HW due 11/8: 18.8, 18.9, 18.10, 20.1, 20.2 |
| Tue 11/7 | 19. | Limits | HW due 11/15: 20.6, 20.10, 20.14, 20.18, 28.4, 28.5, 28.7, 28.8 |
| Thu 11/9 | Midterm 2 | Practice midterm Another practice midterm | |
| Tue 11/14 | 20. | Derivative | Read pages 223-232 |
| Thu 11/16 | 21. | Derivative and Mean Value Theorem | HW due 11/29: 28.11, 28.14, 28.16, 29.3, 29.5, 29.6, 29.14, 29.17 |
| Tue 11/21 | 22. | Mean Value Theorem | Read pages 233-240,269-279 |
| Thu 11/23 | No class - Happy Thanksgiving! | ||
| Tue 11/28 | 23. | Riemann Integral | HW due 12/6: 32.1, 32.2, 33.4, 33.7, 33.14, 34.3, 34.5, 34.11 |
| Thu 11/30 | 24. | Properties of the Integral | Read pages 280-290 |
| Tue 12/5 | 25. | Fundamental Theorem of calculus | Read pages 291-297 |
| Thu 12/7 | 26. | Intro to metric spaces | Read pages 83-94. There will be a 5pt bonus problem on this on the final |
Practice final for 319.
Review session by Fangyu: Tue December 12, 2.00PM-4.00PM, in Math P-131.
Sam's extra office hours: Wed Dec 13, 2.00PM-4.00PM, Thu Dec 14, 2.30PM-3.30PM, in Math 3-109.
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