1998 SYLLABUS FURTHER DEFINED
Earl Mitchelle

At the grading of the advanced placement calculus examinations in Clemson in June of this year, officials discussed information which further defined the revised syllabus and the 1998 examination.

The 1998 examination will be 15 minutes longer with the extra time being devoted entirely to multiple choice questions. The two weeks of advanced placement testing will begin one week later in 1998 which will give students additional time to prepare. Beginning with the 1998 examination, students who take the BC examination will receive both an AB and a BC score. A teacher's guide for the new 1998 syllabus will be available in the fall of this year. The guide will contain 10 sample syllabi which teachers can use to plan their courses for the 1997-98 academic year.

The new course description for the AB examination does not include any of the current A topics, i.e., Functions and Graphs, and students will not be tested directly on these topics. L'Hospital's Rule and integration by parts have also been deleted from the AB course.

Any topic which is contained in both the AB and BC syllabi will be tested at the same level of difficulty on both examinations. There will be more emphasis on numerical and graphical data, separable differential equations, broader applications of the definite integral, e.g., AB3/BC3, part (c) on the 1996 examination, and functions defined by integrals including composite functions.

The BC examination will cover only the indeterminant forms for L'Hospital's Rule. New topics for the BC examination include the logistics equation, Euler's Method, geometrical interpretah~n of solutions of differential equations, and slope fields. Simpson's Rule and epsilon-delta proofs have been deleted from the BC course.

A preliminary copy of the "Acorn Book" for the 1998.examination has recently been released by The College Board. The final edition will be released in the spring of 1997. A summary of the changes for the 1998 examination prepared by Steven W. Olson of Hingham High School and Northeastern University in Massachusetts follows this article. Olson recently retired from the Test Development Committee.

Changes in the Course Description
1997-1998

AB

No A topics
No L'Hopital's Rule
No Antidifferentiation by Parts
B Topics tested at the same level in AB and BC
More Emphasis on numerical and graphical data
Separable Differential Equations really are in the course
Broader range of applications of the Definite Integral
Functions defined by an integral indude composite functions

BC

L'Hopital's Rule only for
Logistic Equation
Euler's method
Geometric interpretation of solutions to Differential Equations
Slopefields
No Epsilon Deltas
No Simpson's Rule

 

SUMMARY OF CHANGES
Prepared By Steven W. Olson

The following table is an attempt to distill the changes in emphasis dawn to a few recurring themes.

Five graphing calculator active multiple choice questions

WHAT DO GRAPHING CALCULATORS (GCs)
MEAN FOR AP EXAM QUESTIONS?

1. Same old problems, exotic or approximate inputs/answers. e.g.,

i. If f(x) = x^(3/5), then f'(32.0001) = ?

ii. Find the area of the region bounded by the axes, y = x, and y= cos x.

2. Same old problems, exotic functions (for which traditional analytic methods fail).

Find the x-coordinate of a local maximum on the graph of y = 10^x - x¹⁰

3. New problems that students couldn't (or wouldn't) be asked before (because students couldn't or wouldn't solve them!)

  a. Given a table of values ...

• predict from function values (and knowledge that the function is a cubic polynomial, for instance) the location of a point of inflection.

• distinguish among function, 1st derivative, 2nd derivative

• distinguish among function, derivative, antiderivative (note that this is really the same question as above)

  b. Given a graph or graphs ...

• distinguish among function, 1st derivative, 2nd derivative

• distinguish among function, derivative, anti-derivative

• of the 2nd derivative, predict location of:

extrema on 1st derivative graph; point of inflection on 1st derivative; point of inflection on function

• of the 1st derivative, predict location of:

extrema on graph of function; points of inflection of function;

(these questions have already appeared, of course, but will likely remain important as calculator neutral questions.)

  c. determine the error in a definite integral approximation or difference quotient approximation

4. Calculator inactive questions, for which the student who reflexively picks up a calculator is at a disadvantage.e.g.,

i. Evaluate the integral from 1.0001 to 1.0002 of 
  1/(sin(arcsin(1/2x)))

Note: Questions can be contrived to foil numeric integrators, and students may need to know when to make their calculator surrender, and then try to use their brain!

ii. limit of (1 - cos x⁵)/x¹⁰ as x goes to 0

5. Calculator "neutral" questions, eg generic function descriptions:

i. d/dx (e^(g(x)) =

ii. Given f(-x) = f(x) and the integral from 0 to 2 of f(x) is 4, find the integral from -2 to 2 of f(x).

6. Questions that require students to apply new skills they'd acquire because they use GCs regularly. For example, a function whose first derivative is extremely ugly and unyielding to traditional analytic methods could appear. Students must learn to graph the derivative on their calculator and use information from the graph of the derivative to analyze the behavior of the original function. They might also be required to use a root finder on some monstrous derivative to determine critical points. These are similar to questions in category 3 above.

7. More emphasis on setting up definite integrals to solve application problems, instead of evaluating them. Or questions will specifically state that an analytic solution is required and that the calculator can only be used to confirm the answer.

8. ... What do you think?

The All-Purpose Calculus Problem
Dan Kennedy
Chair of the Mathematics Department
Baylor School, Chattanooga, TN
and chair of the AP Calculus Committee.

Here's a caculus problem to end all calculus problems. (And you thought your professor assigned you hard ones!) See how many familiar themes you can find embedded in this problem.

A particle starts at rest and moves with velocity along a 10-foot ladder, which leans against a trough with a triangular cross-section two feet wide and one foot high. Sand is flowing out of the trough at a constant rate of two cubic feet per hour, forming a conical pile in the middle of a sandbox which has been formed by cutting a square of side x from each conier of an 8" by 15" piece of cardboard and folding up the sides. An observer watches the particle from a lighthoouse one mile offshore, peering through a window shaped like a rectangle; surmounted by a semicircle.

This problem originally appeared in the
spring of 1994 in Math Horizons.

(a) How fast is the tip of the shadow moving?
(b) Find the volume of the solid generated when the trough is rotated about the y-axis.
(c) Justify your answer.
(d) Using the information found in parts (a). (b) ,and (c) sketch the curve on a pair of coordinate axes.

BLOCK SCHEDULING: THE BEAT GOES ON
Earl Mitchelle

College Board hosted a question and answer session on block scheduling in conjunction with the grading of the advanced placement calculus examinations at Clemson University in June.

Some of the points made during the session are as follow:

• One teacher reported that contact time in a block schedule was reduced by 45 hours or 25% of the academic year.

• A 90 minute class provides plenty of time to cover a topic and to give tests.

• The amount of material covered each year is reduced, and the cumulative effect over the years is that students know less and are less prepared for the next level.

• Teachers need to be more creative and use more teaching techniques when they have students for 90 minutes.

• A number of schools schedule class time for advanced placement courses over the entire year. In order to do this some schools schedule AP classes for 80-90 minute periods every other day for the entire year rather than 80-90 minutes every day for a semester.

• AP scores in some schools dropped the first year the block schedule was used.

• When a student misses several days of class in a semester block schedule because of illness, field trips, athletic events, etc. the impact can be quite negative.

• When teaching in a block schedule system, teachers need to schedule out the entire year to be sure that all course goals are achieved. One teacher observed that the work done in two periods in a traditional schedule can not just be compressed into one period in a block schedule.

• Some teachers feel that an average student should not be subjected to more than one hour of anything each day.

• Teachers of remedial students felt that an 80-90 minute period was very helpful.

There are still a lot of unanswered questions about block scheduling and its impact on advanced placement courses. Experiences over the next couple of years should help to answer some of these questions.