AP Calculus Exam: Tips from tutors that can help you prepare
Although calculus has a reputation for being a grueling, demanding course, those who take the class often realize that it is one of the most exciting courses offered in high school. For many students, a calculus course is the first time that math feels like something more than a byzantine academic requirement as they can apply the subject to many real-life situations. In one class they might be working out the extent of an oil spill at an offshore rig; in the next class they might be mapping the volume of an underground cave. From physics to biology to finance, calculus opens up the door to so many interesting pursuits.
There are two Advanced Placement Calculus courses for high school students, AB and BC, and most schools offer both. This year the AP Calculus exams are being offered on paper and digitally in three separate administrations. Schools will decide which sitting to offer their students, but the paper exams and the digital ones are very similar, and they all follow the format of the pre-2020 AP Calculus exams. The only major differences are that the digital exams will be constructed such that students’ responses to Free-Response questions will be simple to type using the exam app and that no question in the traditional no-calculator sections of the digital exam will be meaningfully aided by the use of a calculator. Nevertheless, students are allowed to use a calculator on all sections of the digital AP exam. Also, like all digital APs this year, students will not be able to return to questions from earlier in a section.
The Nuts and Bolts of the AP Calculus Exams
Before you begin studying for an AP Calculus exam, let’s look at the format of the tests. Both the AB and BC AP Calculus exams are structured the same way, and though these structures should hold whether the test is on paper or digital, the no-calculator sections will not have questions on the digital version where a calculator would be of significant use.
- A 30-question no-calculator multiple-choice section given over 60 minutes
- A 15-question calculator-allowed multiple-choice section given over 45 minutes
- A 2-question calculator-allowed free-response section given over 30 minutes
- A 4-question no-calculator free-response section given over 60 minutes, where you are allowed to go back to the 2 calculator-allowed free-response questions but without use of the calculator (on paper versions)
- Equal weighting given to the multiple-choice and free-response sections: 54 points each for a total of 108 points
- Multiple-choice questions are given 1.2 points if correct, and no points are given or deducted for wrong answers. Free-response questions are scored by trained math teachers, and each question can score between 0 and 9 points depending on a pre-set rubric.
Though scaling of the AP Calculus exams varies from administration to administration and the BC scales tend to be slightly more forgiving at the top end, a good rule of thumb is to expect any raw score (out of 108) that is above 65 to receive a 5, any score in the 50s or low 60s to receive a 4, any score in the 40s to receive a 3, any score in the 30s to receive a 2, and any score below 30 to receive a 1. There may, however, be more variance in scaling on the 2021 exams because of all of the educational disruptions.
It is interesting to note that the distributions of scores on these tests are not similar. Where the AB exam generally has a uniform distribution – in the past, approximately 20% of students have garnered each of the five possible scores – the BC exam is traditionally one of the most top-heavy of all APs. Over 40% of students score a 5 most years on the BC exam, and only about 5% score a 1. This may be because the BC exam is taken by fewer than half the number of students that take the AB exam, and the BC cohort boasts only the most dedicated math students in each school.
Content Covered on the AP Calculus Exams
The AB Calculus exam covers the concepts of limits and continuity, derivatives and their applications, integration (including u-substitution but not parts or partial fractions) and its applications, slope fields, and simple differential equations. The BC exam covers everything you might see in the AB exam plus Taylor series, convergence tests for series, parametric and polar functions, integration by parts, integration by partial fraction decomposition, improper integrals, and differential equations for logistic growth.
Expert Tips: How to Succeed on the AP Calculus Exams
Since you only need to get about 60% of available points to score a 5 on either AP Calculus exam, and since you have ample time on all sections, you can strategize the exam differently than you would almost any classroom test. Many tutors will tell you, correctly, that because of the generous scale, students with limited study time can focus their attention on either just the multiple-choice half or just the free-response half of the exam. If you start your preparation early enough, however, you don’t have to choose and can organize a more flexible attack.
Tip #1: Whenever possible, do a problem multiple ways
With at least two minutes per question (and 15 minutes per free-response question), timing should not be a major concern. Though it is a luxury many students can’t afford on some other tests, the opportunity to do problems two different ways – not the same way twice where you might just repeat the same mistake – will be there for many questions on the AP Calculus exam, and it will allow you to nearly guarantee that you’ve gotten a question correct. It is practically impossible to make two separate mistakes that would get you to the same wrong answer, so when you get the same answer doing a problem two different ways, you can bank the question as automatic points. If, on the other hand, the two methods lead to different answers, then at least you know that you’ve made an error somewhere and can check your work on both methods to try to find where one of them went wrong.
Secondary methods can include traditional standardized test strategies like working backwards from the answer choices or plugging in numbers for variables. Also, since in many cases a derivative can be thought of as the reverse of an integral, you can do a problem forward and backward as your two ways. Additionally, visual estimations – like sketching a graph to see the relative size of the area under a curve to check an integral, or the apparent steepness of a slope to check a derivative – can be excellent secondary assessments.
Tip #2: Move around each section nimbly, skipping questions that stump you to put “fresh eyes” on them later (only on paper version since there is no way to return to questions on the digital version)
Generally, when a student is having trouble with a math question, continuing to stare at it in the hope that the answer will manifest itself is not the best strategy. In many cases, whatever the student was missing on the first view will be caught when the student returns to the question, even if the second look is just a minute later. For that reason, make a point of allowing yourself to skip questions with the intention of returning to any that you’ve skipped when you reach the end of the page or the end of the section.
On some tests with less generous timing, you might fear that you won’t have a chance to return to skipped questions, but the AP Calculus exams are more leisurely paced, with very few questions per page, both of which make bouncing around a section fairly comfortable. Then, when you feel that you’ve completed the section, be sure to check your answer sheet to see that you’ve answered every question, and don’t forget that you lose no points for wrong answers.
Tip #3: Have a plan for the Multiple Choice section that recognizes the purpose of wrong answer choices.
Students often ask me “who thinks of all the wrong answer choices for multiple-choice questions?” I have some insight into this because I write a lot of math tests and materials. We math teachers know all of the common errors that students make. We see them all the time, and, frankly, we make them ourselves sometimes too. So when it comes time to write up the wrong answer choices for a multiple-choice problem, we have a fairly good idea what kinds of things to write. For example, calculus students are probably pretty familiar with making the mistake of using the power rule when taking the derivative of an exponential function or forgetting to change the limits on an integral when doing a u-substitution. Test writers are familiar with those things too, so we incorporate those missteps into the wrong answer choices we write.
If you take the time to examine all the choices, even after you’ve arrived at your answer, you may notice that each of the wrong answer choices represent a path that is born out of a common error. You might even realize that you have fallen into one of those errors, and that will give you the chance to fix it. Let the choices speak to you. They hold many secrets, and they are never random.
Tip #4: Have a plan for the Free Response section that maximizes point allocation
Though there aren’t that many actual full AP Calculus exams online, you can find free-response questions, and solutions with proper point allocations, from every exam given. This will give you a good sense of what is always tested in those sections. While you are preparing, study how the points are allocated and, if detailed rubrics are included, look for the kinds of things that always lead to point deduction, like leaving off units when they are required or failing to write an improper integral to infinity using limit notation.
Write more when you know more. Don’t ever feel like brevity is a virtue on a free-response section. Your answer should be clear and correct, but elaborating and using full sentences to explain certain steps can’t really harm you if you know your stuff.
Tip #5: Value diagrams and graphs, whether they are provided or you have to draw them yourself
Geometry and function analysis are at the core of calculus, so make use of the visual any time you get the chance. Again, time should not be a major factor, so if a sketch of a graph or diagram will help you round out your understanding of a problem, don’t skimp on your picture. A question involving Reimann sums is always clearer when you have the graph and some of the rectangles or trapezoids drawn out, even if you’ve memorized formulas. Related rates problems are inherently visual, and relationships between dimensions on the shapes involved in a related rates problem will be instantly recognizable once your diagram is drawn.
Tip #6: Don’t neglect your calculator
On the AP Calculus exams, there are a total of 17 questions where you’ll be allowed your calculator, two of which are free responses worth nine points each. Usually more than half of the multiple-choice questions on calculator-allowed sections don’t really require a calculator to get to the answer, but you should look for opportunities to use the calculator anyway, especially on questions about functions that can be graphed. Having the calculator will also allow you to expand your use of plugging in numbers for variables as a secondary check mechanism.
On the free-response calculator-allowed section, use your calculator’s graphing options in conjunction with the calculus-specific functions like nDeriv and fnInt to guide sketches that will visually enhance your answer and possibly tease out an extra point or two. Though you may know how to find relevant functions on a TI-84 calculator using the MATH menu, you should also familiarize yourself with the various shortcuts and pop-up menus available by hitting the alpha button followed by the different top-line buttons below the screen.
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