Keep in mind that NSDate doesn't represent a "day" in any form, just a point in time. You'll have to do a little work with NSDateComponents, converted using an NSCalendar (typically the user's default calendar) to figure out the start and end NSDate values, and compare use those to compare your time interval to.
I would start by taking the NSDate that falls within your day, and converting it to an NSDateComponents, but without hours, minutes or seconds. Here's a quick (untested) example to help you get started:
NSDateComponents *comps = [[NSCalendar currentCalendar] components:(NSYearCalendarUnit | NSMonthCalendarUnit | NSDayCalendarUnit) fromDate:[NSDate date]];
NSDateComponents *day = [[[NSDateComponents alloc] init] autorelease];
[day setDay:1];
NSDate *start = [[NSCalendar currentCalendar] dateFromComponents:comps];
NSDate *end = [[NSCalendar currentCalendar] dateByAddingComponents:day toDate:start options:0];
Then you can compare the time interval as suggested by Mr. Jalkut (or another method, NSDate can work with NSTimeIntervals in a few different ways). I would definitely spend some time looking over NSCalendar and NSDateComponents in the docs though, you might find a better method than what I'm suggesting for what you need to do.