On AP Calculus Examinations
9/26/2002, 10/7/2002, 2/14/2007]
This is Mr. Hansen’s annotated version of the list found on page
14 of the AP Course Description.
1. Graph a function within an arbitrary viewing window. Implicit
within this is the ability to compute a function value at a point without
showing any work, since you could always use TRACE to find the value of a function
at a point. Other ways to find the value of a function at a point must
therefore be allowed: using the Y1(X) notation in the 8-line mode of
your calculator, constructing a TI-83 table, using STAT EDIT to define a
computed column, etc. Use whichever method is best for the task at hand.
However, use standard math notation, never calculator notation, for writing
2. Find zeros of functions (MATH 0 or 2nd CALC zero). Implicit within
this is the ability to find intersection points of any two functions f
and g, since f and g intersect precisely at the zeros of
the function (f – g).
3. Find numeric derivative at a point (nDeriv,
or MATH 8, or 2nd CALC 6 ENTER). However, remember that you cannot use this
notation in your written work; use dy/dx
or “prime” notation instead.
4. Find numeric value of a definite integral (fnInt,
or MATH 9, or 2nd CALC 7). Again, remember that you must use standard integral
notation, not calculator notation.
else must we remember?
- Final answers must be
accurate to at least 3 places after the decimal point.
- Be careful not to write an
equal sign (=) if the more appropriate symbol is the “approximately equal”
especially in problems involving linear approximation or Taylor series
- Do not round any intermediate
results. Maintain the full accuracy of your calculator (14 significant
digits, though normally only 10 are displayed) until you round once at the
very end. You can certainly use the »
sign or the ellipsis ( . . . ) to avoid wasting
time writing down all those digits. The STO key on your calculator is also
useful as a time-saver.
- You cannot use a maximum or
minimum finder such as the one provided on the TI-83. In other words, you must
use traditional methods of the calculus to find maxima and minima: set
derivative to 0 or DNE, check second derivative (or behavior of first
derivative in a neighborhood), check endpoints, and make sure there are no
other critical points.
- You cannot use a graph to prove
continuity, the existence of a cusp, the existence of local maxima or
minima, or almost anything else that you might be tempted to see on a
graph. Reason for this seemingly arbitrary rule: There are many examples
of functions that can fool calculators. Graphs are for illustration and
exploration only. For example, if you are supposed to locate a point of
inflection, you must use traditional methods of the calculus; you cannot
“eyeball” the point of inflection, nor can you use nDeriv
of nDeriv to plot the second derivative as a
function and then run the root finder on that. Of course, before you start
writing up your solution, you could use any technique you wish to help you
understand the problem, even awkward techniques such as taking the nDeriv of nDeriv. The bottom
line is that your writeup cannot be based solely
on the calculator; you have to interpret and present what the calculator
says, and you have to show all the algebraic steps in standard notation. Don’t
try to use a calculator-produced graph as a proof of anything.
Side note: Interestingly enough, you can reason from a sketch
accompanied with a sentence of explanation—provided the sketch is based on an
analysis of the signs of the function and its derivatives. For example, a sketch
showing that f '' is continuous, positive for all x >
3, and negative for all x < 3 is ironclad proof that f has a
point of inflection at x = 3, provided that you show clear algebraic
support for those inequalities and
explain what it is about the sign change that leads you to draw your
conclusion. For example, your explanation would be adequate if you wrote,
“There is a point of inflection at x
= 3 because f '' changes
sign there.” However, as noted above, a calculator-produced plot of f
'' showing an apparent x-axis crossing at 3 proves nothing.
Similarly, “sign charts” unaccompanied by words of explanation are no longer
accepted by the AP graders. See this
article from the AP site if you desire the full details.
- You cannot use a computer
algebra system (such as the TI-89) to perform polynomial division for you
or to remove all the intermediate steps in the simplification of an
algebraic expression. Except for features 1-4 listed above, you must show
all algebraic steps in your solutions. You can use the TI-89 to check your
work, of course, but you still have to write out the work manually.
- You cannot use any computer
or calculator that has a QWERTY keyboard, nor can you use a stylus- or
pen-based computer (Palm Pilot, etc.). The College Board publishes a list
of approved electronic devices (including, fortunately, both the TI-83 and
the TI-89), but most other devices are prohibited. Even “el cheapo”
non-graphing scientific calculators are prohibited! Obviously, cell phones
- Although you can bring to the
examination anything you like in your calculator’s memory, including
programs and text comments stored as programs, you cannot take away any
secrets when you leave. In other words, you must not store any problems or
portions of problems into your calculator’s memory, since the entire
examination is treated as privileged information. Not even your teacher is
allowed to know the problems. (Free-response problems are publicly
released a few days later, but multiple-choice problems are released only
once every several years.)