Fall 2017 MAT 319: Foundations of Analysis | Fall 2017 MAT 320: Introduction to Analysis | |

Schedule | TuTh 10:00-11:20 Heavy Engineering 201 (through 10/5: joint lectures in Math P-131) | TuTh 10:00-11:20 Math P-131 |

Instructor | Samuel Grushevsky | Robert Hough |

Office hours | MW 4-5pm, F 10-11am 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 | TBD | 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 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 8/29 | 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 Math P-131. | 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. | lim sup, lim inf, topological considerations | Read pages 78-94 |

Thu 10/12 | 12. | Topological considerations | HW 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 10/17 | 13. | Series, alternating series and integral test | Read pages 95-122 |

Thu 10/19 | 14. | 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 10/24 | 15. | Continuous functions, properties | Read pages 123-163 |

Thu 10/26 | 16. | 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 10/31 | 17. | Metric spaces, continuity | Read pages 164-186 |

Thu 11/2 | 18. | Connectedness | HW due 11/8:21.4, 21.5, 21.10, 21.11, 22.1, 22.2, 22.4, 22.5, 22.9 |

Tue 11/7 | Midterm 2 | Practice midterm, Practice midterm solutions, Midterm 2, Midterm 2 solutions | |

Thu 11/9 | 19. | Power series, uniform convergence | Read pages 187-222 |

Tue 11/14 | 20. | Differentiation and integration, Weierstrass approximation theorem | HW due 11/20: 23.1, 23.5, 23.7, 24.2, 24.13, 25.6, 25.10, 26.5, 27.5 |

Thu 11/16 | 21. | Properties of the derivative | Read pages 223-248 |

Tue 11/21 | 22. | Mean Value Theorem, L'Hospital's Theorem | HW due 11/27: 28.2, 28.16, 29.3, 29.5, 29.17, 29.18, 30.1, 30.3, 30.5 |

Thu 11/23 | No class - Happy Thanksgiving! | ||

Tue 11/28 | 23. | Taylor's theorem | Read pages 249-290 |

Thu 11/30 | 24. | Riemann integral, properties of the integral | HW due 12/4: 31.1, 31.2, 31.6, 31.8, 32.5, 32.8, 33.7, 33.8, 33.15 |

Tue 12/5 | 25. | Fundamental thm of calculus, exponents and logarithms | Read pages 291-297, 339-366 |

Thu 12/7 | 26. | Continuous nowhere, differentiable functions | |

Practice final, Practice final solutions

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