If this isn't the place, I apologize:
At the end of my calculus class, we asked the students (among other things) what some applications of calculus methods are. Disappointingly, many focused on the word problems they'd seen all semester. Answers included finding out how fast a ladder was slipping and estimating your time of arrival when driving.
Question, then, for those of you with interdisciplinary/applied experience: what are some challenging questions that you solved using analytic skills you learned in your first year of undergrad?