Master AP Calculus AB & BC



CHAPTER 1. All About the AP Calculus AB & BC Tests



• Frequently asked questions about the AP calculus tests

• Summing it up

Your goal and vision in any Advanced Placement class should be to take the AP test, pass it with a sufficiently high score, jump up and down like a lunatic when you receive your score, and attain credit for the class in the college or university of your choice. All AP tests are graded on a scale from 1 to 5, with 5 being the highest possible grade. Most colleges will accept a score of 3 or above and assign credit to you for the corresponding course. Some, however, require higher scores, so it’s important to know the policies of the schools to which you are applying or have been accepted. An AP course is a little different from a college course. In a college course, you need only pass the class to receive credit. In an AP course, you must score high enough on the corresponding AP test, which is administered worldwide in the month of May. So, it’s essential to know that all-important AP test inside and out.



Below are common questions that students pose about the AP Calculus tests. For now, this test is your foe, the only thing standing in your way to glorious (and inexpensive) college credit. Spend some time understanding the enemy’s battle plans so that you are prepared once you go to war.

What topics are included on the test?

The list of topics changes a little bit every couple of years. The College Board Web site ( always has the current course description. As your academic year draws to a close, use it as a checklist to make sure you understand everything.

What’s the difference between Calculus AB and BC?

The Calculus BC curriculum contains significantly more material than the AB curriculum. Completing Calculus BC is equivalent to completing college Calculus I and Calculus II courses, whereas AB covers all of college Calculus I and about half of Calculus II. The AB and BC curricula cover the same material with the same amount of rigor; BC simply covers additional topics. However, if you take the BC test, you will get both an AB and BC score (the AB score excludes all BC questions from the test).

Of the topics on the course description, which actually appear the most on the AP test?

This will vary, of course, but I asked my students to list the topics they saw the most. This is the list of topics they generated (BC topics are denoted with an asterisk): relative extrema (maximums and minimums), the relationships between derivatives of a function, the difference quotient, basic integration, integral functions with variables as limits of integration, volumes of solids with known cross-sections, motion (position, velocity, and acceleration functions), differential equations, area between curves, power series*, elementary series* (ex, cos x, sin x), Taylor polynomials*, radius of convergence*, and integration by parts*.

How is the test designed?

The test is split into two sections, each of which has a calculator-active and a calculator- inactive portion. Section I has 45 multiple-choice questions and lasts 105 minutes. Of that time, 55 minutes are spent on 28 non-calculator questions, and 50 minutes are dedicated to 17 calculator active questions. Section II has 6 free-response questions and lasts 90 minutes. Three of the free-response questions allow the use of a calculator, while 3 do not.

Should I guess on the multiple-choice questions?

You lose a fraction of a point for every multiple-choice question you answer incorrectly; this penalizes random guessing. If you can eliminate even one choice in a question, the odds are in your favor if you guess. If you cannot eliminate any choices, it is best to omit the question.

Should I have the unit circle memorized?

Oh, yes. The unit circle never dies—it lives to haunt your life.

Should my calculator be in degrees or radians mode?

Unless specifically instructed by the question, set your calculator for radians mode.

I have heard that the AP Calculus test is written by scientists living in the African rainforest, and that many tests are lost each year as couriers are attacked and infected by virulent monkeys. Is this true?


How many questions do I have to get right to get a 3?

There is no set answer to this, as the number varies every year based on student achievement. Unofficially, answering approximately 50 percent of the questions correctly usually results in a 3.

Why is the test so hard?

Many students are shocked that a 50 percent is “passing” on the AP test, but the exam is constructed to be a “super test” that tests not only your knowledge but also your ability to apply your knowledge under extreme pressure and in very difficult circumstances. No one expects you to get them all right. You’re shooting for better than 50 percent, so don’t panic.

How will I feel when the test is over?

Hopefully, you will still be able to function. Most students are completely exhausted and drained at the end of the ordeal. Students who are well prepared (like those who buy my book) experience less depression than others. In general, students have a vague positive feeling when they exit the test if they dedicated themselves to studying all year long. I have found that the way you feel when exiting the test is independent of how you will actually perform on the test. Feeling bad in no way implies that you will score badly.

Are there any Web sites on the Internet that could help me prepare for the AP test?

There are a few good sites on the Web that are free. Among them, one stands clearly above the rest. It offers a new AB and BC problem each week, timed to coordinate with a year-long curriculum to help you prepare for the test. Furthermore, each problem is solved in detail the following week. Every problem ever posted is listed in an archive, so it’s a very valuable studying tool for practicing specific skills and reviewing for tests throughout the year. To top it off, the site is funny, and the author is extremely talented. The site? Kelley’s AP Calculus Web Page, written and shamefully advertised here by yours truly. You can log on at Enjoy the same hickory-smoked flavor of this book on line for free each and every week. Another good problem of the week site is the Alvirne Problems of the Week (

ALERT! Tie World Wide Web is constantly in a state of flux. Web sites come and go, and their addresses change all the time. All of these links were active at publication time.

How are the free-response questions graded?

Each free-response question is worth up to nine points. Free response questions usually have multiple parts, typically two or three, and the available points are dispersed among them. Many points are awarded for knowing how to set up a problem; points are not only given for correct answers. It is best to show all of the setup and steps in your solution in an orderly fashion to get the maximum amount of credit you can. The College Board has examples of excellent, good, and poor free response answers given by actual test takers on their Web site ( In addition, they have the most recent free-response questions with their grading rubrics. You should try these problems, grade yourself according to the rubrics, and see how you stack up to the national averages.

Are free-response questions the same as essays?

No. Free-response questions are most similar to questions you have on a typical classroom quiz or test. They require you to solve a problem logically, with supporting work shown. There’s no guessing possible like on the multiple-choice questions. There’s really no need to write an essay—it just slows you down, and you need every last second on the free response—it’s really quite meaty.

TIP. The past free-response questions, course outline, and grade rubrics are all available free of charge on line. The College Board Web site is updated far more frequently than printed material. Check there for breaking news and policies.

Should I show my work?

Yes, indeedy. The AP graders (called readers) cannot assign you partial credit if you don’t give them the opportunity. On the other hand, if you have no idea what the problem is asking, don’t write a detailed explanation of what you would do, and don’t write equations all over the paper. Pick a method and stick to it—the readers can definitely tell if you are trying to bluff your way through a problem you don’t understand, so don’t pull a Copperfield and try to work magic through smoke and mirrors. Also, keep in mind that any work erased or crossed off is not graded, even if it is completely right.

What if the problem has numerous parts and I can’t get the first part?

You should do your best to answer the first part anyway. You may not get any points at all, but it is still worth it. If the second part requires the correct completion of the first, your incorrect answer will not be penalized again. If you complete the correct sequence of steps on an incorrect solution from a previous answer, you can still receive full credit for the subsequent parts. This means you are not doomed, so don’t give up.

TIP. Cross out any of your errant work instead of erasing it. Erasing takes more time, and time is money on the AP test.

Is it true that a genetically engineered chicken once scored a 4 on the AB test, but the government covered it up to avoid scandal?

I am not allowed to comment on that for national security reasons. However, I can say that free-range poultry typically score better on free-response questions.

Should I simplify my answers to lowest terms?

Actually, no! The AP readers will accept an answer of 39/3 as readily as an answer of 13. Some free-response questions can get a little messy, and you’re not expected to make the answers pretty and presentable. However, you still need to be able to simplify for the multiple-choice questions. For example, if you reach a solution of ln 1/3 but that is not listed among the choices, you should be able to recognize that —ln 3 has the same value, if you apply logarithmic properties.

NOTE.  according to the log property that states 

How accurate should my answers be?

Unless specified otherwise, the answer must be correct to at least three decimal places. You may truncate (cut off) the decimal there or round the decimal there. For example, a solution of x = 4.5376219 maybe recorded as 4.537 (truncated) or 4.538 (rounded). If you want to write the entire decimal, that is okay, too, but remember that time is money.

Should I include units in my answer?

If the problem indicates units, you need to include the appropriate units in your final answer. For example, if the problem involves the motion of a boat and phrases the question in terms of feet and minutes, velocity is in ft/min, and the acceleration will be in ft/min2.

TIP. Never, never, never round a number in a problem unless you are giving the answer. If you get a value of 3.5812399862 midway through a problem, use the entire decimal as you complete the problem. Rounding or truncating during calculations almost always results in inaccurate final answers.

When can I use the calculator to answer questions?

You may use the calculator only on calculator-active questions, but you probably figured that out, Mr. or Ms. Smarty Pants. Occasionally, you may use a calculator to completely answer a question and show no work at all. You can do this only in the following circumstances: graphing a function, calculating a numerical derivative, calculating a definite integral, or finding an x-intercept. In fact, your calculator is expected to have these capabilities, and you are expected to know how to use them. Therefore, in these four cases, you need only show the setup of the problem and jump right to the solution. For example, you may write

without actually integrating by hand at all or showing any work. In all other circumstances, you must show supporting work for your solutions.

How should I write an answer if I used my calculator?

As in the above example is all you should write; the readers understood and expected you to use your calculator. Never write “from the calculator” as a justification to an answer. Also, never write calculator language in your answer. For example, a free-response answer of  cannot get a point for the correct setup, though it may get points for the correct answer.

TIP. Because you can use your calculator to find x-intercepts, you can also use it to solve any equation without explaining how. See the Technology section in Chapter 2 for a more detailed explanation.

What calculators can I use on the AP test?

The most current list of calculators can be found on the College Board Web site. Most favored among the calculators are the Texas Instruments 83 and 83 + (and probably the TI 89 before too long). It’s a matter of preference. Some people live and die by HP calculators and will jump down your throat in the blink of an eye if you suggest that the TI calculators are better. Calculators like the TI-92 cannot be used because they have QWERTY keyboards. Make sure to check the Web site to see if your calculator is acceptable.

I recently made a calculator out of tinfoil, cat food, and toenail clippings. Are you telling me I can’t use it on the AP test?

Sorry, but you can’t. By the way, I shudder to think about how the toenail clippings were put to use.

Can I have programs stored in my calculator’s memory?

Yes. Programs are not cleared from the calculator’s memory before the test begins. Many of my students have stored various programs, but I don’t think a single student has ever used a program on the test. The test writers are very careful to construct the calculator portions of the test so that no calculator has an advantage over another. It’s really not worth your time to load up your calculator.

NOTE. Only in the four listed circumstances can you use the calculators reach an answer. For instance, most calculators can find the maximum or minimum value of a function based on the graph, but you cannot use a calculator as your justification on a problem such as this.

If I can store programs in the calculator memory, can’t I store formulas and notes? Why do I need to memorize formulas?

Technically, you can enter formulas in the calculator as programs, but the test writers also know you can do this, so it is highly unlikely that such a practice could ever be useful to you. Remember that more than half of the testis now calculator inactive! Don’t become so calculator dependent that you can’t do basic things without it.

NOTE. A QWERTY keyboard, for those not in the know, has keys in the order of those on a computer keyboard.



• All AP tests are graded on a scale from 1 to 5, with 5 being the highest possible grade. Most colleges will accept a score of 3 or above and assign credit to you for the corresponding course (see the Appendix at the back of this book).

• Completing Calculus BC is equivalent to completing college Calculus I and Calculus II courses. AB covers all of college Calculus I and about half of Calculus II.

• The test is in two sections: Section I has 45 multiple-choice questions and lasts 105 minutes; Section II has 6 ftee-response questions and lasts 90 minutes.

• The most current list of calculators can be found on the College Board Web site.