Lots of docs will induce you right after your due date, which is bad practice, but docs prefer a scheduled labor to a spontaneous one often (no getting woken up at 2 am!). You're not actually "overdue" until you're more than 42 weeks. The risks to the baby don't really start increasing until after 42 weeks, and the AVERAGE first time mom goes to 41 weeks and 3 days, so going past your "due date" is normal.
Personally, I would refuse induction if you pass your due date. You can opt for a non-stress test to see how the baby's doing if you want. As long as the baby is healthy, leave it in there until it's ready (as long as it comes on its own before 42 weeks).
Btw, the reason I am anti-induction (unless it's actually medically necessary) is that induction is much more likely to result in a c-section, fetal distress, painful labor, and other complications. Not to say those can't happen in spontaneous labor, of course, but it's proven that they're more common with inductions. So I wouldn't take those risks unless I really needed to, not just because I was technically past my due date (which is just an estimate anyway).