Spring 2012 MAT 319: Foundations of AnalysisSpring 2012 MAT 320: Introduction to Analysis
ScheduleTuTh 10:00-11:20 Earth&Space 131TuTh 10:00-11:20 SB Union 237 (through 10/4: joint lectures in Earth&Space 131)
InstructorSamuel GrushevskyTony Phillips
Office hoursTu 11:30-1 Math 3-109, and Th 2:30-4 MLCW 2-4 Math 3-113
During the joint lectures please attend the office hours of the professor lecturing
RecitationMW 11:00am-11:53am Harriman 112MW 11:00am-11:53am SB Union 237
TAJoseph AdamsJonathan Hales
Office hoursMW 12-1 in MLC, W 1-2 in 2-104
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
TextbookKenneth Ross Elementary Analysis: The Theory of Calculus
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.
GradingHomework: 20%, Midterm I: 25%, Midterm II: 20%, Final: 35%.

Syllabus/schedule (subject to change)
All joint lectures through 10/4 meet in Earth&Space 131.
First recitation on Wed 8/29.
During joint lectures through 10/4, students with last names starting A-M attend recitation in Harriman 112, students with last names N-Z attend recitation in SB Union 237
Tue 8/281. Joint class: Introduction, motivation: what are real numbers? (Grushevsky)Read pages 1-15
Thu 8/302. Joint class: Properties of numbers; induction; concept of a field. (Grushevsky)HW due 9/5: 1.3, 1.4, 1.5, 1.12, 2.1, 2.3, 3.1, 3.5, 3.6
Tue 9/4 No class: Labor Day
Thu 9/63. Joint class: Completeness axiom for real numbers; Archimedean principle. (Phillips)Read pages 19-25;
HW due 9/12: parts ehlmno of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.15
Tue 9/11 4. Joint class: Infinity, unboundedness. Definition and examples of sequences. (Phillips)Read pages 27-41
Thu 9/135. Joint class: Limit of a sequence. (Phillips)HW due 9/19: 5.1, 5.2, 5.4, 7.2, 7.4, 7.5ab
Tue 9/186. Joint class: Limit laws for sequences. (Grushevsky)Read pages 43-53
Thu 9/207. Joint class: Divergence to infinity, more formal proofs. (Grushevsky)HW due 9/26: 8.2ace, 8.4, 8.9, 8.10, 9.1, 9.4, 9.6, 9.9, 9.12, 9.13, 9.15
Tue 9/258. Joint class: Monotonic sequences, lim sup and lim inf. (Phillips)Read pages 54-63
Thu 9/279. Joint class: Cauchy sequences, and decimal expansion. (Phillips)No HW: prepare for the midterm
Tue 10/2Joint Midterm I in Earth&Space 131.
Thu 10/410.Joint class: Subsequences; monotonic subsequences. (Grushevsky)HW due 10/10: 10.1, 10.2, 10.4, 10.6, 10.10, 11.2, 11.3, 11.5, 11.7, 11.10

The following syllabus below is only for MAT 319, in Earth & Space 131
Tue 10/911.Lim sup and lim infRead pages 75-77
Thu 10/1112.Review of sequences; seriesHW due 10/17: 8.8,9.16,10.8,11.6,12.1,12.3,12.8,12.10
Tue 10/1613.SeriesRead pages 90-98,103-104
Thu 10/1814.Alternating series; continutyHW due 10/24: 14.2,14.5,14.7,14.10,14.12,14.14,15.1,15.6
Tue 10/2315.ContinuityRead pages 115-131
Thu 10/2516.Continuous functionsHW due 11/5 (date change):17.1,17.4,17.9,17.12,17.13,17.14,18.4,18.6,18.9
Tue 10/30No Class: hurricane Sandy
Thu 11/1No Class: hurricane Sandy
Tue 11/617.Properties of continuous functions
Thu 11/818. Continuity and limitHW due 11/19 11/14: 18.4,18.6,18.9,18.12,20.2,20.4,20.6,20.8
Tue 11/13Midterm II (another date change!) covers up to and including section 17 18
Thu 11/1519. Limits, derivativeRead pages 144-154,205-211
Tue 11/2020. Mean value theoremRead pages 213-220 HW due 11/28: 28.2,28.4,28.6,28.8,28.14,28.15,29.1,29.7
Thu 11/22 No class: Thanksgiving
Tue 11/2721. L'Hospital's ruleRead pages 222-229
Thu 11/2922. Riemann integral HW due 12/5: 29.11,29.14,29.16,29.18
Tue 12/423. Properties of the integralRead pages 243-266
Thu 12/624. Fundamental theorem of calculus
Wed 12/12 10am-noon Review Session in Math P-131: JoePractice Final
Thu 12/13 4pm-6pm Review Session in Math P-131: Sam
Final Exam: Friday, December 14, 11:15AM-1:45PM in ESS 131 (the usual classroom)

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