I think the muddiest point in today’s lecture was the relation between continuity, differentiability. The reason why I pick this point was that it is very easy to use the false conclusion when we are working on problems. In this post, I would start with the theorem in the textbook and then use graphs to provide […]
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 […]
I thought that the muddiest point from last Thursday’s class was when we went over proving that . The reason why I thought that this was the muddiest is because the proof required to cases. A case in which k was negative and a case that case was positive . This is often a detail […]
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 […]
The muddiest point for me from the lecture on Tuesday is visualizing a proposition we proved. The proposition states that if a function g from [0,1] to [0,1] is bijective and increasing, then the sum of its integral from 0 to 1 and its inverse’s integral from 0 to 1 is 1. It’s not essentially […]
The problem from the proof outline from 11/28 covered the definition of a nondecreasing function. We learned about the nondecreasing sequences but not nondecreasing functions in this course. In this post, I would like to review the definition of a non-decreasing function and further explore relative definitions of a nonincreasing function, an increasing function, a […]
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 […]
In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as $$ Recall that by definition, A bounded function defined on the interval is Riemann-integrable if . For , the set of all possible […]
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