Fall 2014 MAT 319: Foundations of AnalysisFall 2014 MAT 320: Introduction to Analysis
ScheduleTuTh 10:00-11:20 Library E4310 (CHANGE) (through 10/2: joint lectures in Math P-131)TuTh 10:00-11:20 Math P-131
InstructorSamuel GrushevskyDavid Ebin
Office hoursTu 11:30-12:30 in Math P-143, Tu 2:00-3:30 and Th 11:30-12:00 in Math 3-109Tu, Th 11:30-1:00 in Math P-143
RecitationMW 11:00-11:53 Harriman 112MW 11:00-11:53 Lgt Engr Lab 152
TAApratim ChakrabortyChengjian Yao
Office hoursMW 2:00-3:00, Th 4:00-5:00 in MLCW 4:00-6:00 in MLC, F 2:00-3:00 in Math 2-105
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, required for math majors. 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
TextbookKenneth 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.
GradingHomework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%.

Syllabus/schedule (subject to change)
All joint lectures through 10/2 meet in Math P-131.
First recitation on Wed 8/27, second recitation Wed 9/3.
During joint lectures through 10/2, students with last names starting A-O attend recitation in Harriman 112, students with last names P-Z attend recitation in Lgt Engr Lab 152

Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.
Tue 8/261. Joint class: Introduction, motivation: what are real numbers? (Ebin)Read pages 1-19
Thu 8/282. Joint class: Properties of numbers; induction; concept of a field. (Ebin)HW due 9/3: 1.3, 1.4, 1.10, 1.12, 2.2, 2.5, 3.1, 3.4, 3.6
Tue 9/2 No class: day after Labor Day
Thu 9/43. Joint class: Completeness axiom for real numbers; Archimedean property. (Ebin)Read pages 20-27;
HW due 9/10: parts eghimr of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.14
Tue 9/9 4. Joint class: Infinity, unboundedness. Intro to sequences. (Ebin)Read pages 28-38
Thu 9/115. Joint class: Limit of a sequence. (Ebin)HW due 9/17: 5.2, 5.6, 7.3, 7.4, 8.1ac
Tue 9/166. Joint class: Limit laws for sequences. (Grushevsky)Read pages 39-55
Thu 9/187. Joint class: Divergence to infinity, more formal proofs. (Grushevsky)HW due 9/24: 8.3, 8.6, 8.8, 8.10, 9.1, 9.3, 9.5, 9.12, 9.14
Tue 9/238. Joint class: Monotone and Cauchy sequences. (Grushevsky)Read pages 56-65
Thu 9/259. Joint class: Subsequences. (Grushevsky)No HW: prepare for the midterm
Tue 9/30Joint Midterm I in Math P-131.Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions
Thu 10/210.Joint class: Subsequences. (Grushevsky)HW due 10/8: 10.1, 10.2, 10.5, 10.8, 10.9, 11.2, 11.4, 11.5, 11.8, 11.9

The following syllabus below is only for MAT 319, in Library E4310
Tue 10/711. Subsequences, lim sup, lim infRead pages 66-83
Thu 10/912. Series introductionHW due 10/15: 12.2, 12.3, 12.6, 12.8, 12.11, 12.12
Tue 10/1413. Series convergence testsRead pages 95-109
Thu 10/1614. Functions and continuity HW due 10/22: 14.2, 14.5, 14.7, 14.10, 14.14, 15.1, 15.6, 15.7
Tue 10/2115. Continuous functions Read pages 123-139
Thu 10/2316. Properties of continuous functions HW due 10/29: 17.2, 17.4, 17.5, 17.6, 17.8, 17.12, 17.16
Tue 10/2817. Continuous functions and limitsRead pages 153-164
Thu 10/3018. LimitsNo HW: prepare for the midterm
Tue 11/4Midterm 2 Practice midterm
Thu 11/619.DerivativeHW due 11/12: 28.4, 28.5, 28.7, 28.8, 28.11, 28.14, 28.15, 28.16
Tue 11/1120.Derivative and Mean Value TheoremRead pages 223-240
Thu 11/1321.Mean Value TheoremHW due 11/19: 29.3, 29.5, 29.6, 29.14, 29.17
Tue 11/1822.Riemann IntegralRead pages 269-290
Thu 11/2023.Properties of the integralHW due 12/3: 32.1, 32.2, 33.4, 33.7, 33.14, 34.3, 34.5, 34.11
Tue 11/2524.Fundamental theorem of calculusRead pages 291-297
Thu 11/27No class - Happy Thanksgiving!
Tue 12/225.Power seriesRead pages 187-192
Thu 12/426.Taylor seriesRead pages 249-268
Final Exam: Friday December 12, 11.15AM-1.45PM
Practice final for 319
Review session by Apratim: Wednesday December 10, 11AM-12PM, in Math P1131

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