Prep Session: AP Calculus Exam Tips

Philosophy (AP College Board)

Calculus AB and Calculus BC are primarily concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications. The courses emphasize a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important. Calculus BC is an extension of Calculus AB rather than an enhancement; common topics require a similar depth of understanding. Both courses are intended to be challenging and demanding. Broad concepts and widely applicable methods are emphasized. The focus of the courses is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems, or problem types. Thus, although facility with manipulation and computational competence are important outcomes, they are not the core of these courses. Technology should be used regularly by students and teachers to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Through the use of the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling, the course becomes a cohesive whole rather than a collection of unrelated topics. These themes are developed using all the functions listed in the prerequisites.

Goals (AP College Board)

• Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.• Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems.• Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.• Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

• Students should be able to communicate mathematics and explain solutions to problems both verbally and in written sentences.• Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.• Students should be able to use technology to help solve problems, experiment, interpret results, and support conclusions.• Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

AP Exam Format (AP College Board)

Section I Part I (55 minutes): 28 multiple choice non-calculator questions Section I Part II (50 minutes): 17 multiple choice calculator-active questions Section II Part I (45 minutes): 3 free response calculator-active questionsSection II Part II (45 minutes): 3 free response non-calculator questions

Students may go back and work on problems in Section II Part I during the last time period but without a calculator.

AP Exam Scoring (AP College Board)

The scores for Sections I and II are given equal weight. The raw multiple choice score is calculated by taking the number of questions correct and subtracting one-fourth the number of questions missed. Questions left blank score no points. This score is then multiplied by a scale of factor of 1.2 so that the highest multiple choice score a student can earn is 54. Each free response question is worth 9 points so that the highest free response score a student can earn is 54. However, students are not expected to answer every question. The multiple choice and free response scores are added to give a composite score. The composite score is then converted to a grade on AP’s 5 point scale.

5 Extremely well qualified4 Well qualified3 Qualified2 Possibly qualified

1 No recommendation

AP Exam grades of 5 are equivalent to A grades in the corresponding college course. AP Exam grades of 4 are equivalent to grades of A-, B+, and B in college. AP Exam grades of 3 are equivalent to grades of B-, C+, and C in college.

2008 AP Calculus AB Free Response Scoring Statistics (AP College Board)

Question Number

Mean Score

Points Possible

1 4.8 92 3.36 93 2.45 94 2.60 95 3.70 96 3.06 9

Calculator Do’s

Do plot graphs of a function within an arbitrary viewing window. Do find zeros of functions (solving equations numerically) Do calculate numerical derivatives of functions Do calculate numerical values of a definite integral. Do put your calculator in radian mode.

Calculator Don’ts

Don’t trace to find an intersection. Don’t calculate a definite integral from the graph window. Don’t use a regression to curve fit a function from a table. Don’t round a number in the middle of a problem. Store values

in memory so the entire number can be used in calculations. Don’t use a program or built-in utility to do the work for a

problem that is not one of the above mentioned (e.g. if a Trapezoidal Rule problem is on the free response section, you must show the original equation and a solution).

Don’t forget to put fresh batteries in before the AP exam. Don’t forget to take your calculator to the AP exam.

Multiple Choice Hints (Lin McMullin and Mark Howell)

Expect to leave a few blank. Don’t guess randomly. If you can eliminate two or more

choices, it is to your advantage to guess. Don’t go back and change an answer unless you found an error

in your work. Your first instinct is usually correct. There are probably no “none of the above” answers. Some questions can be worked backwards by plugging in

answer choices to the problem. Answer choices will include good distracters that use

predictable mistakes. Be careful! Some questions ask only for a set-up. Look at answer choices so

you don’t work too far. You can expect I, II, or III questions. These are three true/false

questions in a multiple choice format. On the calculator active section, you do not need a calculator for

every question. If all the answer choices include numbers with three decimal places, you will definitely need your calculator.

Don’t get stuck on a question. Move on and come back to it. If you can eliminate some incorrect answers in the multiple-

choice section, it is advantageous to guess. Otherwise it is not. Wrong answers can often be eliminated by estimation, or by thinking graphically.

Free Response Hints (Lin McMullin and Mark Howell, Dan Kennedy)

Write neatly and clearly. Someone who is not your teacher will be scoring your exam.

Try all questions. The questions are not necessarily arranged easiest to hardest.

Do not let the points at the beginning keep you from getting the points at the end. If you can do part (c) without doing (a) and (b), do it. If you need to import an answer from part (a), make a credible attempt at part (a) so that you can import the (possibly wrong) answer and get your part (c) points.

Do not waste time erasing bad solutions. Cross out the bad solution after you have written the good one because crossed-out work will not be graded. If you have no better solution, leave the old one there. If two solutions are shown, they will be graded separately and you will be given the average of the two scores even if one of them is worked perfectly.

If you are doing a long computation, you are probably doing it the hard way or the wrong way. The exam is not testing computation as much as it is conceptual understanding.

Write your answers to each part in the answer booklet designated for that part. Graders do not hunt and search for answers in different parts of the booklet.

Do not round or truncate until the final answer. Store intermediate values in your calculator so you can use them in further calculations.

Three decimal places rounded or truncated in final answers! You do not have to simplify answers. You can leave equations of lines

in point-slope form and you do not need to simplify trigonometric or logarithmic expressions completely. You can leave answers in terms of π and e.

Functions that are expressed in terms of a family of function (e.g. y = sin bx) are not to be solved with a specific value of b. Solve them in general terms.

Do not use calculator notation when you are writing equations. Use standard derivative and integral notations.

Be sure you have answered the question. If asked for the value of a function, write the y-value only. If asked for a point, then write the (x, y) ordered pair. Answer only what is being asked. Answer yes or no questions with a yes or a no.

Write limits of integration. Always consider endpoints in absolute extrema or interval problems

by writing about them. If you use your calculator to solve an equation, write the equation

first. An answer without an equation might not get full credit, even if it is correct.

If you use your calculator to find a definite integral, write the integral first. An answer without an integral will not get full credit, even if it is correct.

Expect to write justifications. This is a calculus exam so use the language of calculus in your justifications.

No pronouns in your justifications! You can show a number line as part of your analysis but label the

number line as f or f’. A number line alone, though, is not enough Know the names of basic theorems (MVT, IVT, FTC) because you can

use them by names in your justifications. Be sure to express your answer in correct units if units are given. When in doubt, take a derivative and set it equal to zero.

Free Response Types

Area and Volume (every year since 1969) Particle Motion Interpretation of graphs- f, f’, and f” Rates and Accumulation Differential Equations and Slope Fields Related Rates

Implicit Differentiation

Top Ten Student Errors (Dan Kennedy)

1. is a point of inflection.2. If is a maximum or a minimum at x, then . If ,

then is a maximum or a minimum at x .

3. The average rate of change of f on [a, b] is .

4. Volume by washers:

5. Separable differential equations can be solved without separating the variables.

6. Omitting the constant of integration, especially in initial value problems.

7. Graders will assume the correct antecedents for all pronouns used in justifications.

8. If the correct answer came from your calculator, the grader will assume your setup is correct.

9. Logarithmic antidifferentiation:

10. Chain Rule errors:

2008 AB Free Response Questions