I've studied the Stack Exchange API to some extent (although I'm a newbie to such things), and it appears I can query the database using many criteria. But does Stack Exchange store any "trends"? For example, among the set of tags [A, B, C, and D], which one had the most questions and/or the greatest increase since last week? I'm guessing I have to keep my own local database and draw my own conclusions by running a set of queries on a regular basis. Am I right?
For trends like this, the Stack Exchange Data Explorer is a better option (if you know SQL; if you don't, check out the awesome tutorial). About the only downside is that it is refreshed once per week, on Sunday morning; the API gives you real-time results.
The API would be able to provide the number of questions asked in the last week for certain tags, e.g. this call gives you all support questions on Stack Apps since July 20th (your question is the only one):