MAT 3751 Analysis II Winter

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Introduction …nobody (well, almost nobody) understands calculus fully (especially the theoretical parts) the first time around; multiple exposures are required.

Introduction …taking the first course in advanced calculus or real analysis in which students grapple with limits, and often at the same time as learning to prove, has never been exactly easy for anybody.

Introduction …Like many others, I found real analysis the hardest of the math major requirements; it took me half the semester to catch on. So don’t worry: just keep at it, be patient, and have fun.

Introduction This is a continuation of MAT 3749 We will mainly look into integral calculus

Prerequisites Introduction to Analysis

Text Bloch, The Real Numbers and Real Analysis

Objectives By the end of the course, a student should know the basic properties of the topology of the real line understand the concept of uniform continuity understand the concept of Riemann Integrability through Riemann Sums comprehend the characterization of Riemann integrability through the lower and upper sums comprehend the properties of Riemann Integrable functions

Objectives understand and able to prove the two versions of Fundamental Theorem of Calculus understand and able to derive the integration formulas such as Integration by Substitution, and Integration by Parts understand the precise definition of the limit of a sequence and state properties associated with convergence and divergence of sequences understand and able to prove and apply the Squeeze Theorem for Sequences

Objectives understand and able to prove and apply the Monotone Convergence Theorem understand and able to prove and apply Bolzano– Weierstrass Theorem understand and able to prove and apply Cauchy Completeness Theorem understand and able to prove and apply the Nested Interval Theorem understand and able to apply the Sequential Characterization of Limits and Continuity

Expectations Able to provide written explanations of the ideas behind key concepts. Able to clearly present and explain solutions to problems in both written and verbal form. Read and write proofs appropriate at this level. Able to work as a team to solve problems. Able to work independently to solve problems. Able to apply knowledge to new situations.

Homework Homework problem sets will be assigned. All work must be typed. The ONLY references you can use are the textbook and the lecture note. You cannot use any other resources such as other books, software, and the internet.

Homework *.doc  Type with Equation Editor *.docx  Saved to.docx to type with MS Equation Editor

Homework Group HW Individual HW

Group Homework You are required to work together in a group of 2 or 3.

Individual Homework No discussion with any other person, except may be the instructor. Discussing or copying homework is considered as an act of academic dishonesty

Homework Your homework must be neat and easy to read. Otherwise, no points will be given. The instructor may make you redo your homework sets (again and again) until the presentations are acceptable. Homework must be written with proper logical format.

Homework Staple your Homework. Points will be taken off if you fail to do so. Homework is due at the beginning of the class. Absolutely no late homework.

Quizzes Daily Short Quizzes (5 -10 min.) Cover the materials discussed in the last class session as well as your reading assignment. This is to encourage you to study alone the way, instead of spending 15 straight hours the night before exam. It is a substantial portion of your grade!

WARNING! You will not get a class grade higher than D+ if you do not have at least a 60% grade in quizzes. You will not get a class grade higher than A- if you do not have at least a 93% grade in quizzes.

Class Participation: 1. I will ask questions during the class period. 2. There are group classwork/labs in some class sessions. 3. You are expected to have printed handouts. At the end of the quarter, your grades on class participation will be determined by the above activities and other observations by the instructor.

Exams 1 mid-term exams and a final Mid term is a in-class exam Final consists of 2 parts Part I Take Home Exam Part II In-class Exam

Points Distribution Exam 12/580 points Final3/19, 10:30-12:30120 points QuizzesDaily100 points HomeworkDaily80 points Class ParticipationDaily20 points

Final Class Grade* 90%A Range 80%B Range 70%C Range 60%D Range Below 60%E *You will not get a class grade higher than D+ if you do not have at least a 60% grade in quizzes.

Handouts You need to print your handouts prior to the class time. Handouts will be finalized by 10 pm the night before.