AP Calculus

Every single AP Calculus AB and BC past exam paper including full papers, mark schemes and written mark schemes/model solutions for the multiple choice and free response questions from 1969 onwards can be found below. 

 

    AB and BC Options

There are two AP calculus options - AB and BC. How can students decide between taking AB and BC?  Precalculus is the main prerequisite for both Calculus AB and Calculus BC. If you did well in precalculus, especially if you found this course quite easy then you should consider going taking BC.  Also check which course your univiersity requires. 

 

AB Topics

The main topics in AP calc are:

• Graphing rational functions and finding asymptotes
• Limits - defining limits, determining limits and using limits to analyse asymptotes
• Continuity - defining continuity, confirming continuity, ensuring continuity, connecting limits, types of discontinuities and intermediate value theorem
• Differentiation + applications -  first principles, average rate of change versus instantaneous rate of change at a point, connecting differentiability and continuity, basic rule plus chain rule (power, exponential, ln and trig) combined with product and quotient rule, implicit differentiation, related rates, stationary points (max and min),  optimisation, increasing/decreasing, points of inflection, concavity, analysing derivate graphs and Mean Value Theorem
• Integration + applications -  antiderivatives and indefinite integrals, definite integrals, fundamental theorem of calculus, accumulation of change, integration by substitution, area under a curve, volume of revolution, volumes of known cross sections, integral graphs, differential equations (solving basic equations and slope fields), Riemann sums (summation and integral notation), Mean Value Theorem for integrals
• Kinematics

Pre-requisite topics that are required from pre calc are:

• Solving Quadratics and Inequalities
• Algebraic fractions
• Logs
• Indices
• Trigonometry
• Polynomial Division
• Functions - composite, inverse, domain and range, modulus functions and asymptotes

You can still get away with doing very well in the exam without being “super strong” at these pre-requisite topics.  Make sure you are strong with using your graphing calculator though.

 

    

Differences Between AB and BC  

The major differences between Calculus AB and BC are the range of topics rather than difficulty. BC has all the same topics as AB plus some more topics. In addition to topics from Calculus AB, the AP Calculus BC exam covers the following:

• Polar Coordinates
• Parametric Equations
• Vector Calculus (vector functions) - motion in plane
• Improper Integrals
• Partial Fractions
• Euler’s Method
• Integration by parts
• Lagrange error bound
• Logistical growth models, and more
• Analysis of Series (Geometric, Harmonic, Taylor and Power series)

Students taking the Calculus BC exam will earn a score for both Calculus AB and Calculus BC (in other words students will get an AB subscore in addition to their BC exam).  This is because everything from the Calculus AB exam is also covered on the Calculus BC exam. The AB subscore shows how one did on the part of the BC exam devoted to Calculus AB topics, which is approximately 60% of the exam. The AB sub-score is based on the correct number of answers pertaining to AB material only.

 

Exam Format

 

 

The structure of the AB and BC exams are identical.  Both the AP calculus AB and BC exam are three hours and fifteen minutes long.  There are 2 sections (multiple choice and free response) each worth 50% of the exam.  Both AB and BC have the same number of multiple choice and free response problems (45 multiple choice and 6 free response).

Students are given a 10 minute break between section I multiple choice and section II free response.

Section I consists of 45 multiple choice questions and lasts for 105 minutes. The multiple choice is separated into 2 parts

• Part A: 30 questions, 60 minutes (non calculator). This counts as 33.3% of the total exam score. 

• Part B: 15 questions, 45 minutes (calculator).  This counts as 16.7% of the total exam score. 

Each part in section I is timed separately, and you may work on each part only during the time allotted for it. Once you have finished part A after 60 minutes you cannot go back to it anymore. You must put your calculators under your desk for part A and may not revisit part A after starting part B. Each multiple choice question has four possible answer choices (A, B, C, or D).  You will be given a multiple choice booklet and an answer sheet in the exam.  You must complete the answer sheet using a No. 2 pencil only.  The answer sheet has circles marked A–E for each question.   For Calculus [AB/BC], you will use only the circles marked A–D. Completely fill in the circles. No credit will be given for anything written in the exam booklet. Scratch paper is not allowed, but you may use the margins or any blank space in the exam booklet for scratch work.

Test writers know all the common errors that students make, so they incorporate those missteps into the wrong answer multiple choice answers on the exam. 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.

Section II consists of 6 free response questions and lasts for 90 minutes.  

The free response is separated into 2 parts.

• Part A: 2 problems, 30 minutes (calculator). This counts as 16.7% of the total exam score
• Part B: 4 problems, 60 minutes (non calculator). This counts as 33.3% of the total exam score

For Section II, if students finish Part A before the end of the timed 30 minutes for Part A, they cannot begin working on Part B and hence must wait until the beginning of the timed 1 hour for Part B.  However, during the timed portion for Part B, students may work on the questions in Part A without the use of a calculator.

Free response means that you will not be given any answer choices.  Instead, you will have to work out the problem, showing all steps, and come with a final answer yourself. Since there are only 6 questions in total you can expect these problems to be time consuming.  Therefore the free response questions will take a lot longer to work out than the multiple choice.  A typical free response problem states a scenario and has 3-4 questions/parts about that scenario.  Sometimes the various parts of the question rely on previous answers, but not always. You must show your work. For this section of the exam you will use a pen with black or dark blue ink or a No. 2 pencil to write your responses. You may make notes in the question booklet, but no credit will be given for any work written in this booklet. You must write your responses in the free-response answer booklet. If you use a pencil, be sure that your writing is dark enough to be easily read. 

 

Pacing Yourself

Multiple Choice:

Students have 2 minutes per problem in the non calculator section and 3 minutes per problem on the calculator section. So if you find yourself spending 4 minutes or more on any given question, it’s time to move on.  Every multiple choice question can be described as a "stand alone" question.  Where each question occurs in the test makes no difference because there is no patterned order of difficulty. This means tough quesitons are scattered between easy and moderately difficult questions. Each question covers a particular topic and the question that follows covers a different topic, so the stand alone questions look like a bunch of disconnected calculus questions one after the other. Because they aren't connected to eachother, there is no reason you have to answer the questions in a sequential order.  You may wish to do the questions that you find easiest first in each part. Skip questions that stump you to put "fresh eyes" on them later. Often, when a student is struggling , whatever they were missing on the first view will be caught when they return 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.

Free Response:

Students have to spend an average of 15 minutes per problems or less. Teachers often say to move on if you are stuck on a question and come back to it, but for these free response I recommend sticking with a free response problem until you can complete all parts.  However, if you to want to allow youself some time to revisit a particular question then try to spend no more than 10 minutes per free response question.

In contrast to the multiple choice questions, I recommend working out a free response question from beginning to end, even if it takes you more than 15 minutes. A complete solution to one question looks much better (and may be worth more points) than partial work in multiple problems. The graders are looking for significant progress towards the final answer, rather than a few stabs in the dark. Therefore, it is better to completely work out the problems that you know how to answer before trying any that you might have trouble with. 

With at least two minutes per question (and 15 minutes per free-response question), timing should not be a major concern. If you have space time take the opportunity to do problems two different ways (not the same way twice where you might just repeat the same mistake).  This will allow you to nearly guarantee that you’ve gotten a question correct.

 

Scoring

Multiple Choice: 

Your score on the multiple-choice section will be based solely on the number of questions answered correctly. Equal weight is given to each question. A correct answer receives 1 point.  Omitted and incorrect answers receive 0 points and hence there is no penalty for guessing so you should use a process of elimination if all else fails and maybe you can get it down to a choice between 2 or 3 answers.  In the worst case scenario, if you have no idea what a question is asking, take your chances on a random selection. The total score is multiplied by 1.2 to calculate the adjusted multiple choice score.  So, there are 45 questions and 1 point for each correct anwer and 0 for each incorrect answer.  Therefore the raw score of 45 is scaled up to 54 points maximum (45 x 1.2).

The multiple choice questions are marked by a computer.  

Free Response:

Your points are based on how much you answer from each question. Notice that the number of problems is vastly smaller in this section as well. You only have to work on 6 problems total! Each question can score from 0 to 9 points.  The graders are looking for specific steps in each part of a free response question (there is a detailed graded rubric that the markers follow). You must show your work. Marks are awarded for showing the specific steps in a logical order that allows you to arrive at your final answers. Even if you used a calculator function such as numerical integration, you must at least write down the equation or formula as well as an indication of which calculator feature you received. If you don't know how to do a problem, write as much as you know to get a change of earning at least part of the available 9 points.  Marks can be awarded for doing correct work on an incorrect function. This means even if you did an early calculation wrong and got a wrong number, but you carried on using this wrong number with a correct method for the next part(s) you will get what is known as follow through marks for having the correct process. So, even if you have no idea how to do the first part write down something reasonable anyway and move on to the next parts. 

There are 6 questions and 9 points maximum for each question and hence 54 raw points are available. This raw score total is not scaled up/adjusted like the multiple choice. The raw score from the free response is just added to the adjusted multiple choice score to receive a maximum composite score of 108 points.  Notice how there were 54 points for both tthe multiple choice section and free response section which should make sense as each section is worth 50%. This composite score is compared to a composite-score scale (known as a curve) for that partifcular year's exam. The scaling/curve varies from year to year as it depends on the difficulty of the exam and the performance of the students who took it (how hard students found the exam). The raw score conversion process is kept confidential by the College Board and you will never see your composite scaled score. The scaled score is converted into an AP score of 1- 5 which is what you receive in July.  See grade boundaries below for an explanation of this. 

Use mark schemes of past papers to study how the points are allocated. Look out for the kinds of things that always lead to point deduction, like leaving off units. 

The free response section is hand-graded by trained maths teachers. 

 

Grade Boundaries   

This is not like a classroom exam where 90 is an A, 80 is a B, 70 is a C and so on.  For AP, grades range from 5 to 1, with 5 being the highest. 

The equivalences are as follows:

• 5 = extremely well qualified and is the equivalent of an A+ or A. 
• 4 = very well qualified and is the equivalent of an A, B+ or B.
• 3 = qualified and is the equivalent of a B-, C+ or C. 
• 2 = fail and is the equivalent of a D. This means possibly qualified and that one fails to qualify for any college credit.  It is usually not accepted by colleges unless there are extenuating circumstances.
• 1 = fail and is the equivalent of an F. This means no recommendation.

Students only need a 3 to pass which is roughly 50%. Grade boundaries depend on how well everyone did overall and the curve that the AP use. The boundaries vary from year to year. You can get quite a lot of questions wrong and still get a 5.  Over 50% of students usually score a 3 or higher.

Percentage-wise the grade boundaries are roughly:

Percentage	AP Exam Grade
70% - 100%	5
60% - 69%	4
50% - 59%	3 (pass)
25% - 49%	2 (fail)
0% - 24%	1 (fail)

Note: A 65% often gives a 5 and this has been as low as 60% in the past.  A 4 has been as low as a 51% in the past.  Very roughly speaking, on average around 20% of students get each of the grades above.

The percentages are not strictly what your score comes out to. AP Calculus is not technically graded this way but the percentages are about what it is, technically speaking. This is because the AP score is scaled from the composite score which is made up of the adjusted multiple choice score and raw free response score, not the raw multiple choice score.

Composite score-wise the grade boundaries are roughly:

Composite Score Range	AP Exam Grade
64-108	5
50-63	4
38-49	3
26-37	2
0-25	1

A good rule of thumb is to expect a 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 25 to receive a 1.

Does getting a score with the score range guarantee that you will achieve the respective AP exam grade? The answer is no! The boundaries vary from year to year due to the scaling and can be as high as 70 for a 5.  In order to be safe to achieve a 5 one should get a 70 or above.  This is not difficult - it means you can get 31 multiple choice questions wrong (remember each one counts as 1.2 marks) or drop 38 points in the free response or some combination of multiple choice and free response.

Grade Boundaries to ensure to be safe of achieving your tagret grade:

Composite Score Range	AP Exam Grade
70-108	5
57-69	4
45-56	3
37-44	2
0-36	1

BC scales/curves tend to be slightly more forgiving and therefore the boundaries are lower for BC and a larger amount of students who take the AP calculus BC exam earn a score of 5 (roughtly 50% of students get a 5). This is more a testament to the hard work and preparation of the students that take this course and mathematically strong students.

 

Accuracy Of Answers

Students must present all numerical answers to 3 decimal places unless stated otherwise.

 

Calculator Usage

Always look for opportunities to use the calculator anyway, especially on questions about functions that can be graphed. Some questions purely rely on the use of a calculator and can't be done without. 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 always ask yourself can I use a calculator?

• Using a calculator does not mean you should show no work.  Whilst you don't have to do the calculations by hand, always write the equation your'e solving or the value you are evaluating (for example the integral and limits of integration when finding a deifinite integral on the calculator). 
• Make sure your calculator is always in radian mode.
• Save functions in your calculator instead of re-typing them each time, especialy if you are using the same function multiple times like kin free response questions.
• Don't round intermediate values. Only round your answers at the end to 3 decimal places.
• Be fluent with using a calculator for things such as storing functions, storing values, finding zeros of a function, finding intersections of curves or solving equations, graphing derivatives, calculating derivatives numerically and evaluating definite integrals.

 

Formulae

There are certain formulas for AP Calculus AB that you should have down pat as there is no formula sheet given on the AP exam.  You'll have to memorize the formulas you'll need! Click AP Calculus AB Formula Sheet to see my formula sheet to review before the exam.

 

Free Response Questions

Many argue that the multiple choice section of the exam is easier. It is true that it is easier to guess on a multiple chocie question than to guess the correct answer to an open-ended free response question. However, free response papers are very similar across papers and doing lot of papers can help you prepare well. Most students struggle to get used to the style at first, but once you get used to them are easy! You just need to practice, practice, practice these papers! The calculator section of the AP Calculus free response section relies heavily on the use of a calculator. Typical types of questions include:

Rates/Flows:

These are usually given in a table. These questions always get combined with forming an equation for the total amount and then differentiating this to find the max/min amount using the fundamental theorem of calculus. The questions often requires you to use mean value theorem/intermediate value theorem, but these are phrased in a “hidden” way that doesn't explicity say to use these theorems and students therefore struggle with this. Also be aware of the differences between average rate of change, instantaneous rate of change and average value. This question often involve using Riemann sums and trapezium rule. 

Areas/Volumes of Revolution/Volumes Of Known Cross Sections: 

These questions are pretty straight forward.  Watch out when you finding the area about the y axis (you need to swap letters) and also watch out for when you are revolving the area between curves about another line (washer method).  

f, f, f’’ Graphs:

These questions are also pretty straight forward.  Students often find the questions where you have to go backwards (f' to f or f'' to f') the hardest. You have to understand the difference between what an integral of an f graph represents versus what an integral of an f’ graph represents.

Related Rates:

These give you the rate of one quantity changing and ask you to fin the rate of something else that is changing as a result.  It is usually helpful to draw a picture and write out everything you know and what you want. You are usually given an equation or you might be given a scenario where you have to build an equation.  These questions involve differentiating with respect to time.  Be sure to understand dx/dt means x is changing at a rate of ... Sometimes these questions come up at the end of the table rate/flow questions.

Differential Equations

These are pretty standard questions. The worst type that could come is not being given the differential equation. Instead you have to form your own equation based on a given scanrio in words and solve it.

 

Form A versus Form B

Form 2002-2011 you will notice that the free response past papers have 2 forms (form A or form B). These forms represent different time zones. Form B is primarily given in Europe. It's to prevent students from finishing the exam prior to other students starting and relaying information.

Note: There is a late-testing date a few weeks after the normal exam for people that couldn't take a test for some reason (testing conflict etc), but this is a form that is never released and posted.

 

Link To University Courses 

AP Calculus AB and BC are similar to university courses.  AP Calculus AB is similar to a university Calculus I course. AP Calculus BC is equivalent to taking both Calculus I and II.

 

Link To A Levels

A Level Maths is definitely a lot harder that AP calculus.  Some might try and tell you otherwise, but this is simply not true.  IB higher courses are the hardest of all courses.  AP calculus is a 1 year course and offers 2 levels of difficulty (AB and BC).  A Level Maths is a 2 year course and also 2 levels of difficulty (regular A levels and Further A Levels). One cannot take Further maths without taking regular maths unlike AP Calculus where you either take one or the other. You receive a sub grade for AB if taking BC.  In UK students have a choice of several exam boards (5) for A Level maths and some are harder than others.  Singapore only offer 2 exam boards which are either Edexcel IAL or Cambridge CAIE.  Cambridge CAIE is the hardest exam board that there is and can also be sat in the UK. The IB higher level maths courses are harder than regular A Level maths and AP Calculus (but comparable with A Level Further Maths). 

AP Calculus AB is quite a condensed course and easy to do well in due to the similarity of papers and predictability of the free response questions.  The questions (like flow/rate table quesitons) are often phrased in a 'disguised' way and students struggle to know which topic is being tested and what they are meant to do. Therefore the papers may seem hard at first, but once you have done a few papers and get the technique down, the papers are easy.  AP consists of NARROW range of topics, but in a good level of DETAIL (limits, continuity including intermediate value theorem, differentiation (first principles,  averages and rates versus instantaneous rate of change, techniques and rules combined with product rule quotient rule and implicit diff, related rates, stationary points,  increasing/decreasing, points of inflection, concavity, analysing derivate graphs, mean value theorem) plus integration (technique and rules, area under a curve, volume of revolution, integral graphs differential equations, and Riemann sums) and kinematics.  If you learn these topics well then you are always fine for the exam.  Of course there are also pre-requisite topics from previous courses such as pre calculus like logs, indices, trig, solving quadratics, polynomial division solving inequalities, functions inc. domain and range etc, but you can still get away with doing very well in the exam without being “super strong” at these topics. 

AP Calculus BC quite a bit is harder than Calculus AB due to the inclusion of extra topics (it includes all of AB plus more). In the main these extra topics are polar coordinates, parametric equations, vector functions and motion in plane, improper integrals, partial fractions, Euler’s method, series, integration by parts and logistic model.  A Level further maths is much much harder than Calculus BC. A Level Further maths prepares a student well for their first year studying a maths degree at uni whereas AP Calculus BC does not.  Further maths involves at least a couple of the first few months of a math degree. Non maths university courses will offer similar calculus courses to AB and BC calc though (normally called calculus I and Calculus II), so these are great courses for a student to take if they want to study Econ at uni for example with some maths courses, but will definitely not prepare a student well for a maths degree. 

A Level Maths (I will call it regular here to distinguish the difference between regular and Further maths) consists of all these topics above (apart from limits, continuity, intermediate value theorem and mean value theorem which are not considered hard topics anyway) plus A LOT MORE.  A Level exams are definitely a lot less predictable and require more problem solving skills.  Differentiation and integration is the main chunk of AP calculus AB and A level regular maths includes basically all of the differentiation and integration topics (apart from volume of revolution and harder inverse trig integrals not in regular maths, but these are in Further Maths and much harder there).  Volume of revolution used to be in regular A Level maths, but got moved to further in 2017). Some might try to argue that AP calculus has a non calculator section whereas A level maths is all calculator which makes AP harder, but this is not true in the slightest.  

A Level regular maths covers pure, mechanics (the maths parts of physics) and statistics (data, regression, sampling, probability, binomial and normal distribution and hypothesis testing). Studying all pure, stats and mechanics is mandatory whereas AP calculus only covers the pure side and a very very tiny part of mechanics (kinematics) whereas the mechanics in A Level covers SUVAT, forces and connected particles such as pulleys, moments and vectors.  The mechanics and statisticss side is equally as big as the pure side. For pure, A Level maths also covers binomial expansion, geometric and arithmetic series, geometry such as straight line graphs and circles, goes into much greater detail with trigonometry, numerical methods such as iteration, vectors etc.  Also, the pre-requisites topic mentioned above are not tested too heavily for AP calculus and one can still get a 5 without being strong in them and knowing which parts they need to know well, whereas A Levels requires a very strong grasp in all areas of these topics. You have core pure 1 and 2 for Further maths which is so much more advanced than calculus and then get to choose what area you specialise in for the Further maths topics which is nice. 

Regular A Level Maths has some of the topics of BC like parametric equations, integration by parts, partial fractions.  The only topics not included in regular A Levels from AP Calculus AB are volume of revolution, harder integration techniques (which is minor) and polar coordinates but these are included in Further A level maths and then so much more. 

The take away from this is as follows:

AP calculus relies on more memorisation and computation and does not require such a deep level of understanding.  A Level regular maths consists of all the topics from AP Calc AB (apart from limits, continuity, intermediate value theorem and mean value theorem which are not considered hard of big topics anyway) plus A LOT more. Differentiation and Integration are the main topics from AP calculus and covered pretty much the same in A level. Also level regular maths doesn’t have the midpoint Riemann sum and go into the same level of detail detail for Riemann sums (e.g. right versus left), but it does with trapezium rule and they are basically just the same concept anyway. A Level regular maths does not include all of the topics from BC (it has the 3 topics parametric equations, integration by parts, partial fractions but not the 5 topics  motion in plane, improper integrals, Euler’s method, series and logistic model).  However, A level further maths has everything from AP calculus AB and BC plus way way more.