> On the graphing calculator questions, AB #18 is "improved" so that you > actually need to put it in to the calculator now. That is, the old > and new questions both ask for the average value on [-1,1] of > e^(-x^2), but the old test gave answer choices of .37, .75, 1, 1.49, > and 1.81, while the new test gives .70, .75, .80, .85, .90. The old > question was easy to do by just noting that the function was bounded > between 1/e^2 and 1. And the old question caught the likely error, > which is to find the integral with your calculator but not divide by > two to find the average value. > > The new question MUST be done by mechanically plugging in to a > calculator. The only error that might lead you to one of the other > answer choices is using too coarse a partition when you numerically > integrate. That is, they've replaced a question which allows thinking > and conceptual understanding OR mechanical plugging in to a calculator > with a question that tests ONLY ability to plug in to the calculator. > I think this is a very BAD change.
Excellent observation. Thanks for making it. I whole-heartedly agree.
> Another new question, #19, in my opinion asks for lack of > understanding. They give a table of values of f(1.7), f(1.8), f(1.9), > f(2.0). They tell you f is differentiable on [0,3] and ask for the > best approximation for f'(1.7). The correct answer SHOULD be that we > have no idea, because we don't know whether f is oscillating rapidly > between the given points. We need more information about f! But of > course, the AP people just want us to plug in delta-f over delta-x > without thinking about these issues.
Another excellent point. I think we see here the negative influence of Harvard Calculus which has questions of this type. I agree with you that this question asks for lack of understanding.
> And not only that, but the wrong answers they give are foolish! They > do give the answer choice that would come from just taking delta-f > without dividing by delta-x. But they do NOT have the wrong answer > that would come from using f(2.0) - f(1.7) / .3 instead of > f(1.8) - f(1.7) / .1 ... that is, they don't test the one idea that > they should be, which is that the best approximation to the derivative > comes from using the point that's CLOSEST. They do have the answer > that comes from using 1.9, though.
It's hard to say that f(2.0) - f(1.7) / .3 is necessarily wrong (as you point out above).
> And of course, the epsilon-delta problem is gone, since that's no > longer in the AP curriculum at all.
That's a shame. These questions are excellent for promoting logical thinking and clear understanding of quantifiers like: and, or, for every, there exists, etc.
> There are some nice new questions on the B part: Find the limit as > x -> infinity of > integral from 1 to x of sqrt(4 - e^-t) dt, all over x. > > I think this problem reflects the new ideas very well, because > understanding what this function looks like will get you far. It's an > easy question to answer just with intution and no calculation. And my > colleague Ted Alper points out that you could solve it very nicely > with L'Hopital's rule. What worries me is that people will do it by > approximating with a calculator (what's the value when x = 100? > 1000?) and then deciding that it looks convergent to them. I'd rather > see this problem on the non-calculator part of the test.
It's interesting that from the reform context this type of question is regarded as "new." It is standard fare in traditional calculus books. E.g., in sect. 9.1 of CALCULUS by Percell and Varberg you find several quesitons of this type along with more challenging ones. I think it's a good idea to leave it in the calculator section to filter out students who let their calculators do their thinking for them.
> A couple of the new problems seem very traditional to me: find the > max area of a rectangle that fits under the curve of cos x. Find the > area of the region in the first quadrant enclosed by xe^x, x=0, and > x=k, in terms of k. Why are problems like these new for this year?
I don't know, but I think it's good to have questions like this.
Thanks for the interesting observations.
David Klein Math Dept. Calif. State University, Northridge