Your goal when meeting your new supervisor is to clearly demonstrate you're not a problem. You're smart, but not so smart as to possibly be construed as any kind of threat. If you're given the opportunity, make a full verbal presentation and then leave behind a formal memo outlining what you presented. If you're told there's no time, make your elevator pitch and set up another appointment; don't leave behind the full memo since it could be used to avoid a full meeting. If you're confronted with a problem that you anticipated, frame it as an obstacle and launch into your pitch. If you're surprised, explain that the problem is news to you and offer to get back to the supervisor with information as soon as possible.
This script can be modified to: