because the calculator evaluates numerical derivatives using the difference
quotient (or the symmetric difference quotient). For example, if f (x) = |x|, then
f'(0) does not exist. Yet the calculator may find the value of the derivative as
Remember: always be sure f is differentiable at a before asking the calculator
to evaluate f'(a).
• IMPROPER INTEGRALS. Most calculators can compute only definite integrals.
Avoid using yours to obtain an improper integral, such as
• FINAL ANSWERS TO SECTION II QUESTIONS. Although we usually express a
final answer in this book in simplest form (often evaluating it on the calculator),
this is hardly ever necessary on Section II questions of the AP Examination.
According to the directions printed on the exam, “unless otherwise specified” (1)
you need not simplify algebraic or numerical answers; (2) answers involving
decimals should be correct to three places after the decimal point. However, be
aware that if you try to simplify, you must do so correctly or you will lose credit.
• USE YOUR CALCULATOR WISELY. Bear in mind that you will not be allowed to
use your calculator at all on Part A of Section I. In Part B of Section I and Part A
of Section II only a few questions will require one. As repeated often in this
section, the questions that require a calculator will not be identified. You will
have to be sensitive not only to when it is necessary to use the calculator but also
to when it is efficient to do so.
The calculator is a marvelous tool, capable of illustrating complicated
concepts with detailed pictures and of performing tasks that would otherwise be
excessively time-consuming—or even impossible. But the completion of
calculations and the displaying of graphs on the calculator can be slow.
Sometimes it is faster to find an answer using arithmetic, algebra, and analysis
without recourse to the calculator. Before you start pushing buttons, take a few
seconds to decide on the best way to attack a problem.
GRADING THE EXAMINATIONS
Each completed AP examination paper receives a grade according to the
following five-point scale: