No, even if you didn't buy applecare, you should be able to go to
http://support.apple.com -- go to the bottom of the page, and enter your serial number in the box for checking support options. It should come back with text stating when it was purchased.
For instance, my iBook, which never had purchased Applecare.
Note this number will be accurate if you purchased from Apple, but if you purchased from a retail store such as Best Buy or Circuit City, and particularly if you never registered it, it may be less accurate.