For several years now, I’ve been posting a series of notes on reviewing for the AP Calculus Exams. The questions on the AP Calculus exams, both multiple-choice and free response, fall into ten types. I’ve published posts on each. The ten types have not changed over the years, so there is not much to add. They are updated from time to time. The posts may be found under the “Blog Guide” tab above: click on AP Exam Review. The same links are below with a brief explanation of each.
This year’s AP Calculus exams will be given on Monday May 13, 2024, at 8:00 am local time.
I hope these will help as you, teachers and students, to review for this year’s exam.
General information and suggestions for teachers
- AP Exam Review – Suggestions, hints, information, and other resources for reviewing. How to get started. What to tell your students. Simulated (mock) exams.
- To dx or not to dx – Yes, use past exams and the scoring guideline to review, but don’t worry about the fine points of scoring; be more stringent than the readers.
- Practice Exams – A Modest Proposal Like it or not (and the AP folks certainly do not) the answers are all online. What to do about that. Don’t overlook the replies at the end of this post.
General Information for students
Why Review? To make mistakes!
How, not only to survive, but to Prevail… Things students should know to do well on the exams. Copy this article for your students or share the link with them.
The Ten Type Questions.
Other than simply finding a limit, a derivative, an antiderivative, or evaluating a definite integral, the AP Calculus exam questions fall into these ten types. These are different from the ten units in the CED. Students are often expected to use knowledge from more than one AP Calculus unit in a single question.
These posts outline what each type of question covers and what students should be able to do. They include references to good questions, free-response and multiple-choice, and links to other posts on the topic.
These ten types appear in multiple-choice and free response questions. This type analysis provides an index to the questions by type. In addition the multiple-choice questions include straightforward questions (find a limit, compute a derivative, etc.)
Type 1 questions – Rate and accumulation questions. Contextual questions about things that are changing. Careful reading is the first step. Good graphing calculator skills are essential since this is usually a calculator active question.
- Type 2 questions – Linear motion problems. Motion on a line in a context or not. Students must analyze the motion: position, velocity, and acceleration. Again, often a calculator active question.
- Type 3 questions – Graph analysis. Given the derivative often as a graph, students must answer questions about the function – extreme values, increasing, decreasing, concavity, etc.
- Type 4 questions – Area and volume problems. Student must find the area of a region enclosed by one or more curves, find the volume of a solid with regular cross-sections, and/or find the volume of a solid of revolution (which is, of course, a regular cross-section).
- Type 5 questions – Riemann Sum & Table Problems. Starting with a short table of values of a function, and/or its derivative students are required to find Riemann sums and other information about the function often in a contextual situation.
- Type 6 questions – Differential equations. Students are asked to solve a first-order separable differential equation, work with a slope field, or other related ideas. BC students may be asked to use Euler’s Method to approximate a value and discuss the logistic equation.
- Type 7 questions – Miscellaneous. These include finding the first and second derivative of implicitly defined relation, solving a related rate problem or other topics not included in the other types.
- Type 8 questions – Parametric and vector questions. (BC topic only) Students are asked to analyze motion given in parametric or vector form and find velocity and acceleration vectors.
- Type 9 questions – Polar equations (BC topic) Students are asked to analyze polar equations and find areas enclosed by polar curves.
- Type 10 questions – Sequences and Series (BC topic) Questions ask student to determine the convergence of series using various convergence tests and to write and work with a Taylor and Maclaurin series, find its radius and interval of convergence.
The notes always available from the menu line at the top of the page: click on Blog Guide > AP Exam Review
Also, Calc-Medic has posted a searchable database of all the AP Calculus Free-response questions from 1998 on. The link is here. While you’re there take a look at their website which has lots of resources and free lesson plans. For more on Calc-medic see this post.
Updates: March 9, 2023 – Calc-medic, March 2, 2024
Teaching Calculus 
Off topic, but could you post about what you think of the AP Precalculus course? The content itself and more importantly the decision to have a lower level “Advanced Placement” course itself. If you have already done so, please link. I searched, but did not find it. (Thanks!)
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First, keep in mind that the course description for AP Precalculus mathematics is developed by a committee of college mathematics teachers from around the USA. It is their “content” that the colleges have agreed on. Of course, not everyone or every college would choose the same topics. You may have other good ideas yourself.
My own opinion is that it’s a good course. I would like to see some more time spent on analytic geometry (especially writing the equations of various loci). In the parametric/vector/polar units I would also like to see more application to the uses of these alternate and important ways of coordinatizing the plane and writing the equations of moving objects (e.g. cycloids). While matrices are an important part of working with vectors, they are not anything one is going to see in AB or BC Calculus.
Since this is a full-year course, there should be time to introduce some of these things. I a little concerned that the applications are not listed in the course description. After we see a few exams, we’ll know better what to expect and what applications to teach.
If you mean in terms of difficulty, then I think most of the AP courses are “lower level” compared to AP Calculus. That’s not to put them down. AP Statistic, a course I think everyone in high school would benefit from, has little mathematics in it. It is a writing and analyzing course. So, the math is easier but that doesn’t make the course “lower level.”
Just my opinions.
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Thank you for your opinion. Wanted to hear it without prejudicing you with mine. Really respect you and like to hear what you have to say…despite being much more educationally conservative than you!
I think of “pre-calculus” as remedial at the college level. I graduated high school in the mid 80s and even at that time, the norm for freshman year STEM majors at colleges (and not CIT/MIT, but even second tier state schools) was to take calculus in first semester. (Advanced kids placed out of THAT.) Precalculus was defacto remedial. Yes, I realize there was a time (maybe WW2ish or the 30s when my dad went to school, where high schools just went through geometry and trig/analytic geometry/algebra 2/etc. were college material. But that ship seemed to have sailed a long time ago. Like well before I graduated high school.
If anything, I wish AP had courses for 3rd semester calculus and for “fourth” (ODEs). And not some new fad content like the TI-time. But classic Thomas Finney and Spiegel type stuff. Even in the 80s, there were starting to be kids getting AP BC done in junior year or earlier. And this has only accellerated. There would surely be the demand for it. And taking such courses in HS and with a good AP exam at the end is way better scaffolding than having to waste time at night school at a community college.
I get the impression College Board is a little playing in the game of junior year high school “honors courses” as IB does in a sense, as opposed to “advanced placement”. Moving to a precursor course as opposed to a more advanced one shows this trend.
This was always my impression (even in the 80s) of Physics B (i.e. non calculus physics, i.e. high school physics, essentially the same level as PSSC!) Similarly APUSH. Great course really…but more of a high school honors course in terms of impact. US History is an iconic junior year class…absolutely a core part of a high school education, but very uncommon to be required at college level (world history still is at some schools). You could probably also throw AP Government in that bag…iconic required senior year high school course (“civics”)…but incredibly unimportant and unrequired at the college level.
I’m sure they got a committee of college teachers, but not that impressed. Can get whatever you want for whatever you want to drive. Just look at NTCM and the “technology” push.
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About the lower level thing, I think we’re in agreement. I would say that AP Chem, Bio, and English were not lower level versus AP BC (seemed similar…real college course), at least when I took them.
AP Physics C seemed even a little more advanced than BC, at least when I took it, especially the electrostatics part, which required some aspects of 3rd semester calc. In any case, PSSC sure did not prepare me for real college physics. Ended up getting a 4 on mechanics (didn’t know the rotational stuff, but it’s all similar to translational…and if you know calculus, kinematics is just intuitive.) But a 2 on E&M. The little DC circuits stuff from PSSC was nothing like a real calculus-based intro E&M course.
(No huge point, just sharing.)
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