1. ## Advice for Tougher Questions

So yesterday, we had our second test in Calc 1. I thought that I was totally prepared, and I was based on the problems that we had been assigned and that I had studied. I studied not only problems in our text book, but also ones that I found online. I was in really good shape.

But then, I got to the test, and there were questions on it that we had never been exposed to. For example, we had this question on the Mean Value Theorem:

"Use the Mean Value Theorem on the function $\displaystyle f(x) = \sqrt{1+x}$ for $\displaystyle 0\leq {x} \leq {1}$, to show that $\displaystyle \sqrt{2} < \frac{3}{2}$"

It had never been suggested to us that this theorem could be applied in this way. Not in any of our homework, lectures, or even in any of the problems in the textbook that we weren't assigned. I could have studied for another month and still not have been ready for this question because I never had a hint that this theorem could be applied in this way.

Maybe if I was smarter I could have figured this out on the fly, but I'm not and I didn't. I'm not complaining (even though I question whether this was a fair question to throw on a test out of the blue, but set that aside). What I am doing is looking for advice about resources that I can look at for future tests to try to anticipate future questions that I can't otherwise anticipate. Forget this question - it's water under the bridge. I just raise it as an example.

How do I anticipate these kind of abstract / higher level application types of questions that probably won't appear on our homework. Anybody have any advice for me?

Thanks a lot. Weird question, I know.

2. I would suggest studying problem-solving in general. There's the book How to Solve It, by George Polya, that I'd highly recommend. You could also check out How to Solve It: Modern Heuristics, by Michalewicz and Fogel. Both are very fun and informative.

You might also check out the application questions in your calculus book. These tend to be end-of-the-chapter questions that bring a lot of different topics together.

All of this will help get your thinking to be more creative and imaginative. Because really, the most important quality a mathematician can possess (or anyone desiring to be good at math) is a good imagination. So you need to train your imagination. That means reading good books. I don't mean the latest junk, I mean classics like Homer, Virgil, Dante, Milton, Austen, Dickens, Tolstoy, and Tolkein. Take a look here for a good start. Training your imagination also means cutting down drastically on video games and TV (both regular TV and movies).

You can train yourself to be better. I'm just saying I think it'll take a while, and a lot of hard work.

3. Originally Posted by joatmon
"Use the Mean Value Theorem on the function $\displaystyle f(x) = \sqrt{1+x}$ for $\displaystyle 0\leq {x} \leq {1}$, to show that $\displaystyle \sqrt{2} < \frac{3}{2}$"

.
What the mean value theorem tells you is that there exists a $\displaystyle$$c$ in $\displaystyle (0,1)$ such that:

$\displaystyle f'(c)=\dfrac{f(1)-f(0)}{1-0}=\sqrt{2}-1$

But $\displaystyle f'(x)=\frac{1}{2\sqrt{1+x}}$ which is a decreasing function so $\displaystyle f'(c) < 1/2$ on $\displaystyle (0,1)$. Hence

$\displaystyle \dfrac{1}{2}>\sqrt{2}-1$

and the result follows.

CB