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

 

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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

 

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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

 

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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.

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