Despite the fact that I'm working on a master's degree in math, it's still been decades since I've had to use calculus in more than a simple way. The bottom line is, my calculus is rusty. Oh, I can still help students in 1st semester Calculus AB, but after that I usually have to study a little bit before I can help.
Before school this morning I saw one of my former students working on some calculus problems. We talked for a few minutes; I shared what I'm learning in Discrete Optimization, which he found quite interesting, and he showed me what he was working on.
One of his problems was "find the shortest distance from this function to this point". I told him that I'd find the slope of the tangent line to the function (basic calculus), calculate the negative reciprocal to get a perpendicular, and then use the point and that slope--in other words, I'd take an algebraic approach with a little bit of calculus. I asked how he'd do it. His approach was to use the distance formula from a generic point on the function, and find the minima.
My approach was very BFI (brute force and ignorance), his was more "elegant" (to use a word math people like to use). Recent experience counts!
And now, I have to study for this afternoon's test!