Problems/Topics You Had Trouble With

I figured we could discuss what went wrong in the old tests, quizzes, homeworks, etc and go over things we felt unsure about.

To begin, I felt iffy about showing that Differentiable functions shows continuity on exam two. I wrote down a few things but it would be nice to come to a nice solid proof that we all feel comfortable with. Please add your own problems and concerns, sometimes it simply takes a different way to explain something for someone to understand perfectly!

@TheBearMinimum, if I am not mistaken you are referring to the proof of Thm. 5.2.3., which was problem 3 on Exam 2 Part 1 (Wed. portion). He gave us a proof in class similar to the following:

Assume that $f$ maps the open interval $I$ to $\mathbb{R}$, that $c \in I$, and that $f$ is differentiable at $c$. Prove that $f$ is continuous at $c$.

proof:

To show $f$ is continuous at $c$, we use Limit Characterization of Continuity and W.T.S. $\lim_{x\rightarrow c} f(x) = f(c)$ or equivalently that $\lim_{x\rightarrow c} (f(x) - f(c)) = 0$ since we can assert by the Algebraic Limit Thm. that:

$$\lim_{x\rightarrow c} (f(x) - f(c)) = \lim_{x \rightarrow c} f(x) - \lim_{x \rightarrow c} f(c) = \lim_{x \rightarrow c} f(x) - f(c)$$

Since $f$ is differentiable, we can use the definition of a derivative and the Algebraic Limit Thm. to assert the following is true:

$\lim_{x\rightarrow c} (f(x) - f(c)) = \lim_{x\rightarrow c} \Bigg(\frac{f(x)-f(c)}{x-c} *(x-c)\Bigg) = \lim_{x\rightarrow c} \Bigg(\frac{f(x)-f(c)}{x-c}\Bigg)*\lim_{x\rightarrow c} (x-c) = f'(c) *0 = 0$

Thus $f$ is continuous at c.

I have had some trouble this semester with determining when we can use the rules we know and love from calculus, and when we had to use the formal definition. Case in point is the limit question from the second half of the 2nd exam. I apologize for being vague, but it was the question where Dr. Mcclure said that we could just use the limit laws instead of the epsilon delta proof.
I have also been struggling with the same for derivatives: when do I use the formal definition, and when can I use the rules I learned throughout calculus? I don't know that there is a hard and fast rule, but if anyone could help with some basic guidelines for these types of problems it would be much appreciated.
Thanks in advance for the help everybody, discourse has been a life saver this semester .

My general sense has been that for example-based questions, we can use any tools we know. I have certainly used every trick I thought helpful to generate examples of functions, and used standard derivative rules to compute derivatives, etc. As far as I know, this is fine.

On the other hand, if we're proving a particular property of, say, derivatives, we shouldn't use rules which we haven't explored much in this class (particularly ones which would be challenging to prove formally). The limit-based proofs (rather than epsilonics) were fair game, as I understand it, because we formally proved that they are equivalent to the definitions, and so we could be more comfortable with applying them on their own. I suspect the reason we proved them in the first place was because they would make proofs (like the one on the test) much easier to deal with later. But, for example, l`Hopital's rule is something we haven't really gone into, so I wouldn't suspect we would want to employ it in a proof.