<p>My teacher marked me as having a question wrong on a multiple choice test even though I put down the correct letter. She said I didn't solve it the correct way but I'm not sure I made a mistake. Can anyone give me their thoughts on how to solve the following problem? </p>
<p>f(x)= x^(-1/2) + (2-x)^(-1/2) calculate the area under the curve, above the x-axis, and on the interval [0,2]</p>
<p>NOTE: f(x)=1/sqrt(x) + 1/ sqrt(2-x)</p>
<p>The teacher says I need to use limits to solve because of discontinuities, but there are not discontinuities after integrating and I still arrived at the right answer. Thanks.</p>
<p>How did you solve it? Because I can think of a few ways that you could get the right answer with an invalid method, but I don’t know if you actually used any of them.</p>
<p>First off, let me say that I think your way makes perfect sense. Having said that, I also get what your teacher wanted, and it is true that your way isn’t entirely rigorous. </p>
<p>The thing is, it is not in general true that the definite integral of f(x) across [a, b] is equal to F(b) - F(a); it is true only if f(x) is continuous throughout the interval (and in a few other cases, but they aren’t relevant here). Consider for instance f(x) = 1/x. The integral across [0, 2] in that case cannot possibly be F(2) - F(0), because F(0) isn’t actually defined.</p>
<p>So what you’re truly doing is slightly more complicated. You’re taking the limit of the definite integral of your f(x) across the interval [x, y], ** as x goes to 0 and y goes to 2 **. It may seem unnecessarily complicated, but unlike the simpler method it’s fully correct whether or not the integral actually exists.</p>
<p>ok i get that. I forgot f(x) must be continuous. I was thinking that F(x) must be continuous. I know how to use limits to do it I just thought I would be able to use basic methods for this integral. Thanks.</p>
<p>This is a problem with any multiple choice exam. The correct answer may be determined using math that is not rigorously correct (above), by compounding errors (wrong X wrong = right), by plugging in answers (a classic SAT strategy), or even with wild guessing (“I like pachyderms so I will choose answer E”).</p>
<p>A better MC question would have resulted in the wrong answer using the “typical wrong method”.</p>
<p>But the teacher made the decision to give a multiple choice test. You should not be penalized for getting the right multiple-choice answer using a method that the teacher didn’t want you to use or finds faulty. Open-ended questions are a fair means of asking a student to supply a correct line of reasoning to reach the correct answer.</p>
<p>If you have some philosophical objection to the idea of multiple choice tests where you must show work, don’t frame it that way. Pretend that his teacher gave a free response test, and helped them along by giving them a hint as to what their solution should be. Because really, arguing that a teacher shouldn’t be ALLOWED to have the test format they do is kinda pointless.</p>
<p>It isn’t a philosophical objection to an abstract situation, it’s an objection to an actual teaching method. One use of multiple choice on a high school calculus test is in preparation for the AP exam, which of course has no such “show work” edict. MC questions are also used simply to make grading easier. Both of these are very reasonable. I suppose that if the teacher spelled out that the work must be shown to get credit for the MC questions, then that would resolve the fairness issue and I would agree with the “free response + hint” equivalence, but zman5’s surprise here suggests that these instructions were not made clear. Of <em>course</em> the teacher is allowed to do this; I just don’t think it is a particularly good teaching method.</p>
<p>I was annoyed because she just said it was multiple choice test. I don’t have a problem with her grading that way, but she should at least tell us that is how she will grade.</p>