The scene where Sheldon and Leonard observe Penny through her open apartment door would be repeated as an homage in the series' 100th episode. The characters would even be dressed similarly, right down to Penny wearing the exact same blue top that she wears here.
While Penny has a pleasant outgoing personality, Katie from the unaired pilot opening was meaner and mocked them. Penny does have a temper and some of this personality came over from the first pilot when she's really mad.
Sheldon worries that his offspring via the sperm bank might not know whether to use a differential or an integral to determine the area under a curve. The correct procedure is to use an integral. A differential would measure the slope of the curve.