On Tuesday, 12/5, we began class by presenting the HW10 Challenge Problems in their entirety. We then completed proofs both parts of the Fundamental Theorem of Calculus (Theorem 7.5.1). Proof of part i) involved using properties of partitions and lower and upper sums. Proof of part ii) was accomplished by proving two claims: is continuous […]
Class of 11/30 basically talked about different Theorem related to the Properties of Integral. Those Theorem are built around the Fundamental Theorem of Calculus which we will discuss in next class meeting. We talked about Theorem 7.4.1 which states that Assume is bounded, and let . Then, f is integrable on [a, b] if and […]
Class on Tuesday 11/28 began with student presentations of solutions to the Homework 9 challenge problems. We then made the proposition that if is non-decreasing (i.e. , then exists. We then produced an outline of how to prove it, which essentially consisted of 3 steps: a) choose arbitrary b) choose , with a condition that […]
On Thursday 11/16, Dr. Kerckhove lectured in Dr. LeCrone’s absence. The topic of the day was Riemann Integration. Dr. K began with a definition of Riemann Integration and then went through an outline of what we would cover: Computing Riemann sums to approximate Riemann Integral Controlling error Systematic way to improve a given estimate If […]
During the lecture of Thursday 11/9, we started Chapter 6 Sequences and Series of Functions. We first learned the definitions of pointwise convergence and uniform convergence. Then we worked on the template of proving fn → f is uniform on domain A and fn is not convergence uniform on domain A. After learning the definitions, […]
We began class with challenge problem presentations. We introduced Theorem 5.2.5 (Chain Rule) and went over some of its implications. One thing that is important to note about the usage of Chain Rule is that we can control the inputs of a function, specifically so that they are not equal, but we cannot do this […]
On Thursday’s class, we finished Chapter 4 with the Intermediate Value Property. This is a useful property and we could prove it by different ways. On class, Dr. Lecrone showed to us how to use connected sets and continuity to prove this theorem. We also started talking about differentiation, a topic has close relationship with […]
On Thursday 10/19, we began class with a weekly situation, “Are we doing mathematics or philosophy?”. Dr. LeCrone then read a thorough and quite verbose definition of a function from another analysis book. We then switched to the topic of the day: continuity, part of Ch. 4 (Continuity on Compact Sets and Uniform Continuity). We […]
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