Thanks guys for the help, but I'm still not sure I grasp it, so bear with me as I think aloud and perhaps ramble a bit.
WillJ,
We might be playing around the court and not seeing each other. Consider the money supply fixed in the short term (which it is), and the effect of interest in buying said money. That increases its value but also sends a signal to increase overall supply (wheras the opposite should occur on the other end). That would help stop the band stretching, so to speak, and at some point, the band would contract back.
Sorry, you completely lost me there.

By "That increases its value," are you referring to the higher interest rate causing the higher exchange rate? If so, that's the very part I'm having almost all the trouble with. By "sends a signal to increase overall supply," do you mean increase the money supply, or the number of dollars on the forex market, or what? If the former, I don't understand why that is, and if the latter, it seems to me that should read quantity supplied, not supply.
Isn't it just because you can buy dollars in a matter of seconds, but it takes a lot longer for companies to shift from buying stuff from country X (that has raised interest rates) to country Y (that has lower interest rates)? And it costs a lot of money to do that. So companies hold off from making the switch, in the hopes that interest rates / exchange rates will eventually come back down; if they don't, or if the central bank raises rates so much that the cost of staying with X is greater than the cost of switching to Y, then companies would switch to Y. Or am I missing the point entirely...
I don't think you're on the wrong track; we can probably extend that logic even further, since now that I think about it, I might have been wrong about the demand lowering for non-interest bearing stuff. Assuming (as an approximation) a completely open world economy, the interest rate within a country doesn't affect the demand for goods/services in that country, since people can always borrow money elsewhere to buy them. For the purpose of answering my question, I think we can safely assume that a change in the interest rate just affects people's attitudes toward borrowing and lending, not buying stuff.
But still, lending
and borrowing, not just lending (lending being the only thing my textbook wants us to consider).
A couple things make this confusing. For one, to repeat what I said/asked earlier:
Within the US, an increase in the interest rate doesnt mean a change in the supply of or demand for money, rather the quantity supplied and the quantity demanded (supply and demand for money affect the interest rate, not vice versa). Whys it different internationally?
Perhaps the reason lies somewhere in the fact that there are two conceptually distinct markets for dollars: one in which people sell (loan) dollars and others buy (borrow) them by giving the sellers more dollars in the future (the market whose "price" is the interest rate). The other market is one in which people sell dollars and others buy them by giving the sellers other currencies (the market whose "price" is the exchange rate).
Not only is there the division between those two markets, theres a supposed division between the two countries. To what extent are we supposed to conceptualize them as two different economies versus a single open economy?
If we step back and consider something simpler, the market for cows:
If the US cow supply is halved (due to a huge farming disaster), ceteris paribus, the price of cows in the US goes up and the price of cows in other countries stays the same. Americans buy more cows from abroad. Due to the increase in demand for foreign cows, prices eventually equalize. In this case, Americans needed to exchange dollars for foreign currency, so the dollar depreciated.
Back to the markets for money:
If the US money supply is halved (by the Federal Reserves actions), ceteris paribus, the interest rate in the US goes up and the interest rate in other countries stays the same. Now what happens? Im having a hard time drawing parallels with the cows.
Since it comes down to borrowing and lending, I think my earlier numerical example actually does capture the heart of the issue. An increase in the interest rate leads to an appreciation of the currency simply because of the different sizes of the movements of currencies across the countries when the incentives for borrowing and lending changes. The numerical example kind of makes sense to me, but it also seems kind of silly, and Im not sure how to fit it into economic theory.
Note that if we change the numerical example to one in which the US interest rate is
dramatatically higher than the German rate, the oppostite happens (and I think, empirically, currencies do depreciate when the interest rate goes up too much):
Exchange rate 1:1
interest rate in US: 100%, Germany: 5%
If American, borrow 1000 Euros
buy 1000 dollars with them
have 2000 dollars
buy 50 Euros and pay back
If German, borrow 1000 Euros
buy 1000 dollars with them
have 2000 dollars
buy 2000 Euros with them
pay 50 Euros
total bought on foreign exchange: 2000 dollars, 2050 Euros (again, ratio of Americans spending on this to Germans spending might not be 1:1)