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Topic:
Course description from the college board web site
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Course description from the college board web site
Posted:
May 22, 1997 5:37 PM


I just downloaded the course description PDF file from the college board web site.
I still haven't finished organizing my thoughts on the differences in the topics list, but I'll be sharing those with you soon. Here are my thoughts on the changes in the sample questions.
The freeresponse questions are exactly the same as in the 1997 description, which is also the same as the contents of the 1995 AP test.
The multiple choice questions are almost exactly the same as in the 1997 course description; in particular, they still include the WRONG question, #6 from the AB sample, that they've been using for several years already:
If dy/dx = 9y^4 and if y = 1 when x = 0, what is the value of y when x = 1/3?
Ironically, the failure of this question is exactly the sort of thing that the new AP syllabus is supposed to guard against  brainlessly plugging in to standard formulas and solutions rather than thinking about what's really going on.
Of the new multiple choice questions, AB #1, 8, 9 are all interpreting graphs, and #14 is a nice question that tests for understanding of related rates quite well. Wups! #14 isn't really a new question, it was just moved from the B part to the A part.
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.
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 deltaf over deltax without thinking about these issues.
And not only that, but the wrong answers they give are foolish! They do give the answer choice that would come from just taking deltaf without dividing by deltax. 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.
The other new question looks pretty good: the rate of oil leakage from a tanker is given, and you need to figure out that total amount that's leaked out over a period of ten hours. A good way to check that people understand that they need to integrate.
Now, on to the BC questions! Let's see, some minor changes like replacing "The normal to" with "The line perpendicular to the tangent of".
A L'Hopital's rule question from part A was switched with a Taylor series question from part B. I guess the idea there is to allow students to use the calculator to approximate the limit and figure out a sensible answer (or at least check the answer) for the limit. Also, they point out that some people have calculators that compute Taylor series for them ... I guess they wanted to avoid that.
A really nice fundamental theorem question was moved from the B part to the A part. (was #18, now #7)
There's also a slope field question which tests basic understanding of the concept.
And of course, the epsilondelta problem is gone, since that's no longer in the AP curriculum at all.
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 noncalculator part of the test.
#21 is also a nice, new problem, comparing the 7th degree Taylor polynomial for sin x with the function itself, asking when it's greater and when it's less.
And they've replaced what looks to me like a pretty nice graphing question (which of the following could be the graph of the sum from n = 0 to 25 of (sin x / 2)^n ?) with another fairly reasonable question, #24: g(t) = 100 + 20 sin (pi t / 2) + 10 cos (pi t / 6). For 0 <= t <= 8, g is decreasing most rapidly when t = ? I don't have any particularly strong preference between these two questions. The old one seems more original, the new one seems more hackneyed. On the other hand, the new one seems more likely to come up in realworld practice.
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?
Joshua
[just in case you didn't catch it: the problem with #6 on the AB test is that the differential equation has a pole at x = 1/27 and so we have no idea what the solution is beyond there. You can see this, among other reasons, because y changes sign between the given x = 0 and the "answer" at x = 1/3; but the differential equation itself gives slope zero when y is zero, so how can y change sign? Only by going through positive infinity ...]



