Alright, I’m bitter. In calc class the other day, we were learning about related rates. Boring. Then, I noticed something: the two numbers a and b whose sum equal x produce the greatest product when a = b = x/2. I doodled a square vs. a rectangle, and convinced myself that this would always be true. I also noticed that in a realated rate problem with a rectangle missing one edge/box missing one side, the rectangle/box was half a square/cube, with the missing edge/side being as large as possilbe. In doing so, I missed the actual lecture about taking the derivative of the equation after solving to get a single variable. After class, I showed my work to the teacher, who was impressed, and said that my method did appear to be valid.
Come the test, we had a box with surface area 1200, and needed to maximize volume. A quick sketch of a half-cube box and application of my rule gave me side lengths 20, 20, and 10, or 4000 units.
I got the question 4/5ths wrong. Apparently, although I had the right answer and work, I didn’t have the right work. I pointed out to the teacher that the directions stated to do the problems as we learned in class, and, regardless of what was taught that day, I had * learned * this method. She conceded the semantic point, but didn’t give me my 4 points back.
So, because I actually engaged in the ‘making connections’ style of learning rather than rote facts, I was penalized. Boo.
Welcome to the cruel world of first- and second-year calculus.
I suspect that giving the application of your ‘rule’ didn’t include any kind of proof or, for that matter, anything other than empirical evidence, which indicates that you missed the general point of the lesson. Instead, you came up with a rule for two specific cases, but didn’t apply the general rule. What are you going to do if the next fence is a parabola?
This mentality is implicit in teaching this kind of stuff, largely because only about 10% of the class cares / will ever use it again, and partly because profs get tired of telling class after class of 200 students why they are being taught something that they don’t care about.
The average first-year calc student is a total moron.
Hey, KarmaComa, as a calculus teacher I can sympathize somewhat, but lighten up a little. If they were total morons, they wouldn’t have made it to first-year calc. I try to teach under the assumption that it’s at least possible to find the stuff interesting for its own sake, even if they don’t ever use it again.
robertliguori’s approach was, in one way, commendable: it’s a well-established method of mathematical discovery to doodle, notice something, and wonder if it would always be true. And then yes, you do have to come up with a proof (i.e. reason why it has to be always true). And you have to be clear under what conditions your “always true” rule applies, and that those conditions are satisfied in any particular case you try to apply your rule to.
If the teacher said
maybe he did have such a reason, or at least was led to believe that he did. (Hence the thread title.)
Writing math tests can be tricky. You want to make the problems somewhat similar to what’s already been done in class, but not too similar lest people rely on unwarranted assumptions or blindly go through the steps without knowing what they’re doing just because that’s what worked last time. And you have to be careful writing the directions, so you don’t have to agonize later over how much credit to give someone who misunderstood what you were asking for or did things the “wrong” way.
Why not approach the prof after class and say “Hey, I was fiddling around with some stuff last night and I came up with this method that worked for some sample problems. Could you look at it and tell me if it actually works or it was just a fluke?”
I think the teacher did the right thing. The purpose of the test was to test your knowledge of Related Rates, not to demonstrate creative methods of solving a problem. Not that there’s anything wrong with inventing creative methods to solve a problem, but that’s not what the test was for. You may have gotten the correct answer (and thus the gracious 1/5 of a point) but you did not prove that you were paying attention in class.
I taught calculus a couple of times as a grad student many years ago. I’m afraid I agree with your teacher, robertliguori. A certain method was taught. The test problem had the purpose of finding out whether your knew theat method. If I’m reading the OP right, you didn’t learn that method. So you deserved not to get points on that problem.
Your reason for not learning that method doesn’t really matter. You had several ways to learn the material that was presented while you were working out your formulas. E.g., you could have read the text on your own, gotten a classmate to help you, or asked the teacher.
According to a literal reading of the directions, I solved the problem accurately. Since I’m not in law school, I didn’t expect that argument to work.
I shouldn’t have extrapolated from the remark that it ‘seems to work’ that it would be an accepted method. But darn it, it’s so elegant!
IANAGeometer, but I’m pretty sure that if you want to maximize volume/surface area, then you want to approximate a sphere/dome as closely as possible. Squares approximate better than rectangles, etc. I would therefore posit that the most efficent hyperbola fence would be shaped like [-1, 1], y = x^2.
My calculus skills are way rusty, but some quick back-of envelope calculations have convinced me that the straight-edge version of your parabola (a V-shaped trough) is most efficient when the top edge of the V is four times its height (ie y=1/2x) .
Would your method have predicted this? Or would you have gone for “y=x” as the best bound for that one?
IMO it would have been better for the teacher to specify the required method in the question, or alternatively make a question that is not “applied”. Confusion is otherwise inevitable, especially when the teacher commended robertliguori’s idea.
Nonsense. This is a calculus class for Buddha’s sake.
Related Rates problems are all nearly the same damn thing. It’s bloody obvious what skill the students were expected to demonstrate on the question. Tests are not just about finding answers. They are about demonstrating skills. This is why good math teachers will give partial credit for an answer with a stupid error where the correct technique was demonstrated.