AP Calculus Advice
Simplifying
Q: Is an unreduced fraction acceptable on the free response
section of the
exam? For example, if the answer is 12/25 and the student
writes 24/50, is
there a deduction?
A: Unless the problem explicitly asks for a particular
format, any
equivalent fraction is fine, 0.480 is fine also. Indeed,
leaving 24/50 is
probably better than trying to simplify and risking loss of
a point for a
stupid mistake. Things like e^2-1, 2/4 and sin(2) are
acceptable.
If your answer is given as a decimal approximation, it
should be correct to
three places after the decimal point.
Do not simplify unless you absolutely must!
However, if the question asks for the answer "to the
nearest dollar", then
the question should not be answered with 12/25.
If you take the derivative of (x^2 + 2x)/(x^3 + 2x^2 + 2x),
you can just
apply the quotient rule and leave it totally unsimplified
and still get full
marks.
Q: Is it still acceptable, to leave an answer like, 3*2.1^2
+ 7*2.1 - 3 or
e^0 + sin(pi/2)
A: Answers do not have to be evaluated. If you have the
right answer on the
paper and evaluate it incorrectly you WILL lose credit.
Decimal answers must
be correct to three places after the decimal. This does not
mean they must
be rounded to three places or truncated to three places
(although either,
done correctly will get credit). You may leave the answer to
more than three
places as long as the first three are correct. The real
problem with
rounding is that, if it is done too early, it may adversely
affect the final
answer, if it does you will lose credit (mathematical
mistake - rounding too
soon). Lin McMullin
Q: Suppose the exact answer to a question is pi/7. Must
answers be "purely"
exact or "purely" approximate? What I'm getting at
is the treatment of a
"hybrid" answer like 0.143pi?
A: pi/7, (1/7)pi, 0.448, 0.449, 0.142pi, 0.143pi, 0.448799,
0.142857pi are
all acceptable. The best bet is to quit at pi / 7. There is
NO way to make
that answer better and pushing the wrong button could make
it a lot worse.
Lin McMullin
( vs.[
In denoting intervals where a function is increasing either
( or [ is
acceptable. I.e. The correct answer to, "Where is y =
x² increasing?" is
[0,infinity), but
(0,infinity) will be accepted.
The correct answer to,
"Where is y = x³/x increasing?" is (0,infinity).
If you are asked for the domain of a function and the
correct answer is
(-infinity, infinity), for example f(x) = 2x², you will lose
points for
answering
[-infinity,infinity].
Units
Q: Do units count?
I.e., if the answer is 500 meters and you write "500",
will you lose points?
A: Yes, unless the question is phrased "how many
meters...", i.e. the units
are given in the question.
Two Calculators?
During the administration of Section I, Part B and Section
II, students are
permitted to have two graphing calculators on their desks at
the same time.
The second calculator doesn't have to be just for backup
purposes; students
can use both calculators. AP staff
What's On The Exam?
<< Please help me in determining which of the
following if any are on the AB
or BC exam or both.
Inverse Trig BOTH
Trig substitutions NEITHER
Newton's Method NEITHER
hyperbolic functions >> NEITHER
Substitutions: Students need to know (for the Exam that is)
how to make
substitutions in integrals including changing the limits of
integration.
They
do no need to memorize specific substitutions such as the
Trig
Substitutions.
Lin McMullin
Are There Any Formulas Given?
Q: Are there any formulas given to students at the beginning
of the AP
Calculus exam like they do on the SATs?
A: No. Some formulas are given in problems where they may be
needed: things
like the Volume or surface area of a sphere. Calculus
formulae they are
supposed to know.
Lin McMullin
The formula for the volume of a cone was given with 1995/AB
5.
Is An Unneeded Absolute Value OK?
Q: Would you be marked down if you had an expression in
absolute value when
it wasn't necessary. For example, ln(abs(e^x + e^-x))? [e^x
+ e^-x is always
positive].
A: No. Correct answers are not marked wrong. Lin McMullin
Answer In Wrong Part
Q: How is a problem graded If a student needs a specific
answer to satisfy
the rubric in part a and the student does it all correctly but
puts the
required justification in part b instead of part a? Would it
make any
difference if the situation was reversed? What about
superfluous work that
makes it look like a student is on a fishing trip?
A: Generally a student must put the work and justification
in the part of
the answer sheet where it is needed. Thus the student who
immediately
differentiates the given function in part (a) where it is
not needed gets no
credit for it in part (b) where it is needed UNLESS he/she
specifically
refers back to the derivative in (a) - draws an arrow or
says "I found the
derivative back in part (a)" or whatever. The work must
be "on task" and
needed for the part of the problem being worked. We want to
know that the
student knows he/she needs the derivative in this part of
the problem - not
just that they can crank out the derivative. Lin McMullin
Formulas To Memorize
Q: Does anyone have suggestions for formulas that my
students should have
memorized for the AB exam? I expect my students to know:
Sin²x + Cos²x = 1
Sin ( A +- B ) = Sin A Cos B +- Sin B Cos A
Cos ( A +- B ) = Cos A Cos B -+ Sin A Sin B
Tan ( A +- B ) = ( Tan A +- Tan B ) / ( 1 -+ Tan A Tan B )
Product and quotient Rules of derivatives
generic conic section equations
Area/Circumference of circle
A: Quotient: tan x = sin x / cos x; cot x = cos x / sin x
Reciprocal: sin x = 1/csc x; tan x = 1/cot x; csc x = 1/sin
x; etc. (All 6)
Pythagorean: sin²x +cos²x = 1; tan²x + 1 = sec²x; cot²x + 1
= csc²x
Double-Angle: sin 2x=2sinx cosx; cos 2x = cos²x - sin²x
sin (-x) = -sin x, cos (-x) = cos x & cotrig(?/2-x) =
trig x.
How To Justify That You Have Found The Requested Min, Max Or
POI
Many students develop the bad habit of thinking that y' = 0
guarantees an
extreme value and that y" = 0 guarantees a point of
inflection (POI). A POI
requires a change in sign of y". Don't forget that an
extremum also requires
a change in sign. The "standard" AP method of
showing this is to label a
horizontal number line "sign of y'" or "sign
of y" versus x" and putting "+"
and "-" signs over the line to indicate where y'
or y" is positive or
negative. You write a zero over the line where y' or y"
is zero. This is a
quick and acceptable way to "justify" that you
have found the requested min,
max or POI without writing a paragraph.
Q: Is the shells method included in the AP topics?
A: No. But you can use it if you want to Lin McMullin
Q: Will points be taken off for failure to write the
increment (dx or dy,
for example) in their integrals?
A: We are not usually that picky. Lin McMullin
Q: Are the two multiple choice parts equally weighted? There
are fewer
calculator-active multiple choice questions.
A: Each of the 45 MC questions is worth the same. The score
is Number
correct minus 1/4 of the number wrong (blanks are not
counted as "wrong").
The multiple choice and free response sections are equally
weighted The
multiple choice score is multiplied by 54/45 = 1.2 to make
it of equal
weight with Part II [free response] (6 questions @ 9 points each).
Lin
McMullin
Basic Functions To Understand
Q: Students need to have a, "knowledge of derivatives
of basic functions
including....logarithmic functions." What is considered
a basic function?
While I'm sure that ln x and e^x are, what about a^x and log
base a of x.?
A: Derivatives: power, exponential, logarithmic,
trigonometric and inverse
trig along with sums, products, quotient, compositions
(Chain Rule) and
implicit differentiation plus for BC only parametric, polar
and vector.
Integration: those that follow directly from the derivatives
above plus for
BC only how to substitute, parts, partial fractions. As for
a^x, I make sure
they know a = e^(ln a) so that a^x = e^((ln a)x) - e to a
constant times x)
which does not require any other new formula past the chain
rule. For
log(base a) (x) = (ln x)/(ln a) the "Change of Base
Rule" from precalculus
makes this simply a constant (1/ (ln a)) times ln x. Lin McMullin
BC (and not AB) students are responsible for integration by
parts and
therefore should be able to do (but not necessarily
memorize) integrals with
logarithms and inverse trig functions. Lin McMullin
***********
Q: Do AB students need to memorize, derive, or be able to
use inverse trig,
exponential, and logarithmic derivatives?
A: Memorize: not necessarily, since they can always derive
them, but then
the more you can memorize the faster things go.
Derive: Yes they should know the procedure for finding
derivatives of the
elementary functions and their inverses (e.g. y = arcsin x,
then x = sin y
and dx = cos y dy so dy/dx = 1/ cos y = 1/ SQRT(1 - sin^2y)
= 1/SQRT( 1 -
x^2) ) I doubt they would be asked to do such a derivation
for the
elementary functions, but they should know it and be able to
use it so that
if other functions arise they can handle them.
Use: certainly. Lin
McMullin
******************
Q: If a student works a free response problem two different
ways, how will
it be graded?
A: Both solutions are graded separately and the scores are
averaged and
rounded DOWN. Thus for a 4 point part of a question, if the
student has a
solution worth 1 point and another worth 4 points, the
student gets 2
points. Lin McMullin
Q In a free response question, students are asked to use the
Maclaurin
series for cos x to derive a Taylor series for f(x) centered
at x = 0. If a
student gives only this answer: -x/2! + x^3/4! - x^5/6! + .
. . + (-1)^(n)*
x^(2n - 1)/(2n)! + . . . I.e. neglects to give the sigma
notation for the
series, would they receive full credit for this response,
since the prompt
did not specifically ask for the notation?
A: Unless the question specifically asked for sigma
notation, then what you
have would get credit (assuming it's correct -- I didn't
check it). Usually
the
question is worded something like "Give the first four
terms and the general
term of the series for ...." your answer has only 3
terms and for that you
would lose credit. Lin McMullin
Crib Sheets & Programs
Q: Do you suggest to your students that they should keep a
crib sheet in
their calculator memories? Of course they will need to know
things for the
non-calculator portions of the exam, but it seems obvious
that having notes
in their calculator will help them to remember formulas and
even procedures.
Do you recommend any actual programs to be stored as well?
A: For some students, organizing the information to store in
the calculator
memory is one of the best possible ways for them to study.
Copying a program
from another student who has done this, of course, is worse
than useless
(they'll only waste time on the test trying to find things).
Joshua Zucker
Q: It Is OK to Bring a Calculator to the AP with Programs
Stored in It
A: Students may have programs etc in their calculator memory
at the AP
calculus exam. They will NOT be cleared before or after the
exam. The catch
is they may not use them: that is, they must show the set up
and work for
everything they do except the 4 allowed operations
(graphing, equation
solving, definite integrals and value of y ' at a point).
Lin McMullin
Q: If a question on the calculator-ok portion of the free
response section
asks for a max or a min value, are you allowed to use the
max or min feature
of their calculators?
A: They must take the derivative, set equal to zero, solve,
and evaluate
etc. Except for the 4 allowed uses of the calculator work
should be shown.
Do not try to justify with the calculator results. Also do
not try to get
around using a table by doing a calculator regression, etc.,
etc. The
general directions of the Free Response sections say,
"Show all of your
work. Indicate clearly the methods you use, because you will
be graded on
the correctness of your methods as well as the accuracy of
your answers.
Correct work without supporting work may not receive credit.
Lin McMullin
Q: Do you recommend any programs to be stored on the
graphers?
A: Programs may be stored in calculators but may not be used
on the exam.
For example a program for Euler's Method is helpful for BC
students BUT THEY
MUST SHOW ALL THE WORK ON THEIR PAPER. If it's not there the
correct answer
will not get credit (See 1998 BC 4 (b)) - so what good is
having it for the
exam? Only the four things allowed to be done on calculators
may be done
without showing work. So after writing down the integral
they need, they may
write the answer next to it with no intermediate work. They
may write down
the equation they need to solve and write the solution down
with no
intermediate work. But to say that the Trapezoidal rule
gives this answer
without the intervening arithmetic would not get credit. As
for the Multiple
choice section where calculators are allowed I doubt there
is anything there
that a program will help with. The folks who write the exam
are very aware
of what calculators can do and take that into account when
writing the exam.
Lin McMullin
*************************
After talking with ETS and reading the Acorn book, I put
together the
following information about using calculators on the AP
examinations for my
students. Earl Mitchelle
1. To achieve an equitable level of technology, the AP
Calculus Development
Committee develops examinations based on the assumption that
the only
requirements regarding a calculator are that it has the
following
capabilities:
* Produce the graph of a function within an arbitrary
viewing window.
* Find the zeros of a function.
* Compute the derivative of a function numerically.
* Compute definite integrals numerically.
2. The memory of a calculator will not be cleared before the
examination
begins. This means that anything may be stored in the memory
of a calculator
such as formulas, rules, programs, etc.
3. No material from the examination may be stored in the
memory of a
calculator and taken out of the examination room. A
student's score will be
invalidated if he attempts to remove any test material from
the testing room
by any method.
4.A student may bring no more than two approved calculators
to the
examination.
5. For solutions obtained using one of the four required
calculator
capabilities, students are required to write only the setup,
e. g., a
definite integral, an equation, or a derivative, that leads
to the solution
along with the result produced by the calculator.
6. For solutions obtained using a calculator capability
other than one of
the four required ones, students must also show the
mathematical steps that
lead to the answer. An unsupported calculator result is not
sufficient.
7. When a student is asked to justify an answer, the
justification must
include mathematical reasons, not just calculator results.
8. Any decimal answer must be correct to three decimal
places unless
otherwise indicated. Do not round values before the final
answer is
determined. Carry as many decimals as possible until the
final answer is
determined.
9. Students should be aware that there are limitations
inherent in graphing
calculator, e. g., answers obtained by tracing along a graph
to find roots
or points of intersection may not produce the required
accuracy. Unless
there is a very good reason to obtain a decimal
approximation, do not do it.
Leave the answer in exact form.