Fall 2017 MAT 319: Foundations of AnalysisFall 2017 MAT 320: Introduction to Analysis
ScheduleTuTh 10:00-11:20 Heavy Engineering 201 (through 10/5: joint lectures in Math P-131)TuTh 10:00-11:20 Math P-131
InstructorSamuel GrushevskyRobert Hough
Office hoursMW 4-5pm, F 10-11am in Math 4-118
RecitationMW 11:00-11:53 Library E4330MW 11:00-11:53 Math P-131
TAFangyu ZouAleksandar Milivojevic
Office hoursTBDM 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
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 10/5 meet in Math P-131.
First recitation on Wed 8/30, second recitation Wed 9/6.
During joint lectures through 10/4, 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 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.
Tue 8/291. Joint class: Introduction, motivation: what are real numbers? (Grushevsky)Read pages 1-19
Thu 8/312. 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 9/73. 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 9/145. Joint class: Limit of a sequence. (Grushevsky)HW due 9/20: 5.2, 5.6, 7.3, 7.4, 8.1ac
Tue 9/196. Joint class: Limit laws for sequences. (Hough)Read pages 39-55
Thu 9/217. 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 9/268. Joint class: Monotone and Cauchy sequences. (Hough)Read pages 56-65
Thu 9/289. Joint class: Subsequences. (Hough)No HW: prepare for the midterm
Tue 10/3Joint Midterm I in Math P-131.Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions
Thu 10/510.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

The following syllabus below is only for MAT 320, in Math P131
Tue 10/1011. lim sup, lim inf, topological considerationsRead pages 78-94
Thu 10/1212. Topological considerationsHW due 10/18: 12.1, 12.8, 12.13, 12.14, 13.1, 13.3, 13.7, 13.9, 13.10, 13.12, 13.14
Tue 10/1713. Series, alternating series and integral test Read pages 95-122
Thu 10/1914. Decimal expansions HW due 10/25: 14.1, 14.4, 14.7, 14.8, 14.12, 15.2, 15.6, 15.7, 16.1, 16.7, 16.9
Tue 10/2415. Continuous functions, properties Read pages 123-163
Thu 10/2616. Uniform continuity and limits of functions HW due 11/1: 17.2, 17.4, 17.12, 17.13, 18.4, 18.8, 19.2, 19.4, 20.18
Tue 10/3117. Metric spaces, continuityRead pages 164-186
Thu 11/218. ConnectednessHW due 11/8:21.4, 21.5, 21.10, 21.11, 22.1, 22.2, 22.4, 22.5, 22.9
Tue 11/7Midterm 2 Practice midterm, Practice midterm solutions, Midterm 2, Midterm 2 solutions
Thu 11/919.Power series, uniform convergenceRead pages 187-222
Tue 11/1420.Differentiation and integration, Weierstrass approximation theoremHW due 11/20: 23.1, 23.5, 23.7, 24.2, 24.13, 25.6, 25.10, 26.5, 27.5
Thu 11/1621.Properties of the derivativeRead pages 223-248
Tue 11/2122.Mean Value Theorem, L'Hospital's TheoremHW due 11/27: 28.2, 28.16, 29.3, 29.5, 29.17, 29.18, 30.1, 30.3, 30.5
Thu 11/23No class - Happy Thanksgiving!
Tue 11/2823.Taylor's theoremRead pages 249-290
Thu 11/3024.Riemann integral, properties of the integralHW due 12/4: 31.1, 31.2, 31.6, 31.8, 32.5, 32.8, 33.7, 33.8, 33.15
Tue 12/525.Fundamental thm of calculus, exponents and logarithmsRead pages 291-297, 339-366
Thu 12/726.Continuous nowhere, differentiable functions
Final Exam: Friday December 15, 11.15AM-1.45PM
Practice final, Practice final solutions
Review session by Aleksandar: Wednesday, December 13, 10AM-12PM in Math P-131.

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