One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out. The problem was set up as such:
A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?
Here's what we'll need to do:
1. We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.
2. We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.
3. Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima...
I used to do this and I see a lot of students who do this common mistake when studying. Maybe you are working through old homework problems to prepare for an exam in math or physics and you have the solutions in front of you. You get to a certain point and you get stuck, so you check the solution, see what the next action you have to take is, and then continue working through the problem. Eventually you get an answer that may (or may not) be right and check the solution again. If it is, you feel great and move on. If it isn't you compare the work and see what you did wrong and understand the mistake so you move on. All this is a fine way to start studying, but the major mistake is that most students don't go back to that problem and try to do it again. Even if you were able to understand the solution or the mistake you made, you never actually got through the problem completely without aid. So now if you come to this problem on your test, this will be the first time you actually completely...