For me, the "why" came from the word (or story) problems -- the exact problems that most people really hated. There is one problem from calculus that really st

For me, the "why" came from the word (or story) problems -- the exact problems that most people really hated. There is one problem from calculus that really stuck with me, and made me really understand limits: "You are on an island, which is 10 miles from the shoreline. You can row a boat 4 miles an hour, and you can run 7 miles an hour. You have a friend in need that lives on the shore, 6 miles down from the part of the (straight) shoreline that is closest to the island. Where on the shore do you need to land the boat to minimize your travel time?" So obviously, aiming the boat directly to your destination minimizes your distance, but you are only going to be traveling at 4 mph. And if you aim straight to the shore you minimize your time in the slow boat, but your distance is longer. The visual solution is to plot the total time based on various landing points on the shore, then find which one is minimal. The exact solution is to make a formula that represents both legs of the trip with the landing point being your variable, take a derivative of the formula and solve for 0 (if I remember correctly).