Category Archives: Reviewing

AP Exam Review 2024

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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 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 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 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

Why Review?

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The reason you review is TO MAKE MISTAKES!

When you’re reviewing for the AP Calculus exams your goal is to make mistakes. Why make mistakes? Easy: to find out what you’re doing wrong so you can fix it. And to find out what you don’t know so you can learn it.  

Your teacher will assign free-response questions, FRQs, from real AP Calculus exams from past years. Give yourself about 15 minutes and try to answer the question. (Fifteen minutes is about the time you have for each FRQ on the exam.) After fifteen minutes, stop. Check your work.

The questions, answers, solutions, and most importantly the scoring guidelines for FRQs are all online here for AB, and here for BC. Each FRQ is worth nine points. The scoring guidelines will show you what must be on your paper to earn each point.

Now you can copy the absolutely perfect answer for your FRQs and hand it in to your teacher. This won’t impress or fool your teacher because he or she has the guidelines too. More importantly this won’t help you. When reviewing mistakes are good. Study your mistakes and learn from them.

  • If you made a simple arithmetic or algebra mistake, learn to be more careful. One very common mistake is simplifying your answer incorrectly. Remember, you do not have to simplify numerical or algebraic answers. If you write “ 1 + 1 “ and the correct answer is 2, the “ 1 + 1 ” earns the point. But if you simplified it to 3, you lose the point you already earned. (The standards show simplified answers, so the readers will know what they are for (foolish?) students who chose to simplify.) simplifying also wastes time.
  • If you’re unsure how to write justifications, explanations, and other written answers, use the scoring guidelines as samples or templates. Learn to say what you need to say. Don’t say too much. You will not earn full credit for the correct answer and correct work if the question asks for a justification, and you don’t write one.
  • If you really don’t know how to answer a question you’ve made an important mistake. This is the thing you need to work on until you understand the concept or method. This requires more than just reading the solution on the guideline. Go back to your notes, ask your friends, ask your teacher, find out what you’re missing and learn it. Look at similar questions from other exams.
  •  For multiple-choice questions only the answers are available. Nevertheless, be sure you understand your mistakes.

There are 7 types of questions on the AB exam and an additional 3 on the BC exam. These are not the same as the ten units you’ve been studying, because AP exam questions often have parts from more than one unit. On March 5, I will post links to all the types. The discussion of each type will include a list of what you should know and be able to do for each type along with other hints.           

Now, when you actually take the AP Calculus exam your goal changes. Here you want to earn all the points you can. If you run across something you know you don’t know on the exam, leave it. Go onto something you do know. Don’t waste your time on something you’re not sure of. You can always come back if you have time.

Missteaks our heplfull.

AP Calculus Review 2023

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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.

I hope these will help as you review for this year’s exam.

General information and suggestions

  • 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 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.

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 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.

  • 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 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 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.

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 mor on Calc-medic see this post.

This year’s exam will be given on Monday May 8, 2023, at 8:00 am local time.

Update March 9, 2023 – Calc-medic

To dx or not to dx

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As exam time nears, teachers become concerned about exactly what to give credit for and what not to give credit for when grading their students’ work on past AP free-response questions.

Former Chief Reader Stephen Davis recently posted a note on the grading of a fictitious exam question showing how 2 points might have been awarded on a L’Hospital’s Rule question.  The note is interesting because it shows the details that exam leaders consider when deciding what to accept and what not to accept. It shows the details that readers must keep in mind while grading. This level of detail with examples is given to the readers in writing for each part of every question. With hundreds of thousands of exams each year and millions of questions to grade, this level of detail is necessary for fairness and consistency in scoring.

BUT as teachers preparing your students for the exam you really don’t need to be as concerned about all these fine points as readers are. Encourage your students to answer the question correctly and show the required work using correct notation. Even thought the guidelines show what is necessary to earn each and every point, there are still fine points and close calls that do not show up on the guidelines.

Don’t worry about the fine points. What if I say this, instead of that? What if I forget to write dx? Why show your students the minimum they can get away with? How does that help them? Do your students a favor: score the review problems more stringently than the readers. If a student’s answer is close but maybe not quite right, take off some credit and help them learn how to do better. It will help them in the long run. 

If your students try to answer and show their work but miss or overlook something, the readers will do their best to follow the student’s work and give her or him the points they have earned.

Revised February 17, 2024

Sequences and Series (Type 10)

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AP Questions Type 10: Sequences and Series (BC Only)

The last BC question on the exams usually concerns sequences and series. The question may ask students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in multiple-choice questions. Questions about the convergence of sequences may appear as multiple-choice questions. With about 8 multiple-choice questions and a full free-response question this is one of the major topics on the BC exams.

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a convergence test chart students should be familiar with; this list is also on the resource page.

Students should be familiar with and able to write a few terms and the general term of a Taylor or Maclaurin series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it, or integrating it.

The general form of a Taylor series is \displaystyle \sum\limits_{{n=0}}^{\infty }{{\frac{{{{f}^{{\left( n \right)}}}\left( a \right)}}{{n!}}{{{\left( {x-a} \right)}}^{n}}}}; if a = 0, the series is called a Maclaurin series.

What Students Should be Able to Do 

  • Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.  and the posts “What Convergence Test Should I use?” Part 1 and Part 2. In 2022 BC 6 (a) students were asked to state the condition (hypotheses) of the convergence test they were asked to use.
  • Understand absolute and conditional convergence. If the series of the absolute values of the terms of a series converges, then the original series is said to be absolutely convergent (or converges absolutely). If a series is absolutely convergent, then it is convergent. If the series of absolute values diverges, then the original series may or may not converge; if it converges it is said to be conditionally convergent.
  • Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
  • Distinguish between the Taylor series for a function and the function. DO NOT say that the Taylor polynomial is equal to the function (this will lose a point); say it is approximately equal.
  • Determine a specific coefficient without writing all the previous coefficients.
  • Write a series by substituting into a known series, by differentiating or integrating a known series, or by some other algebraic manipulation of a series.
  • Know (from memory) the Maclaurin series for sin(x), cos(x), ex and \displaystyle \frac{1}{{1-x}}and be able to find other series by substituting into one of these.
  • Find the radius and interval of convergence. This is usually done by using the Ratio test to find the radius and then checking the endpoints. for a geometric series, the interval of convergences is the open interval \displaystyle -1<r<1 where r is the common ration of the series.
  • Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, \displaystyle {{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}}. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
  • Be familiar with the harmonic and alternating harmonic series. These are often useful series for comparison.
  • Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
  • Determine the error bound for a convergent series (Alternating Series Error Bound or Lagrange error bound). See my posts on Error Bounds and the Lagrange Highway
  • Use the coefficients (the derivatives) to determine information about the function (e.g., extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the free-response section is usually quite straightforward. Topics and convergence tests may appear on the multiple-choice section. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked, how it is a sked, and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series or any other topic you are interested in.

Free-response questions:

  • 2004 BC 6 (An alternate approach, not tried by anyone, is to start with \displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin \left( {5x} \right)\cos \left( {\tfrac{\pi }{4}} \right)+\cos \left( {5x} \right)\sin \left( {\tfrac{\pi }{4}} \right)). See Good Question 16
  • 2011 BC 6 (Lagrange error bound)
  • 2016 BC 6
  • 2017 BC 6
  • 2019 BC 6
  • 2021 BC 5 (a)
  • 2021 BC 6 – note that in (a) students were required to state the conditions of the convergence test they were asked to use.
  • 2022 BC 6 – Ratio test, interval of conversion with endpoint analysis, Alternating series error bound, series for derivative, geometric series.
  • 2023 BC 6 – Taylor polynomials, Lagrange error bound

Multiple-choice questions from non-secure exams:

  • 2008 BC 4, 12, 16, 20, 23, 79, 82, 84
  • 2012 BC 5, 9, 13, 17, 22, 27, 79, 90

These questions come from Unit 10 of the CED.

Revised March 12, 2021, April 12, 16, and May 14, 2022, June 4, 2023

Polar Equation Questions (Type 9)

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AP Questions Type 9: Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graphs may be given or easy to find.

What students should know how to do:

  • Calculate the coordinates of a point on the graph,
  • Find the intersection of two graphs (e.g. to use as limits of integration).
  • Find the area enclosed by a graph or graphs: \displaystyle A=\frac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{{\left( {r\left( \theta \right)} \right)}}^{2}}d\theta }}
  • Use the formulas \displaystyle x\left( \theta \right)=r\left( \theta \right)\cos \left( \theta \right)\text{ and }y\left( \theta \right)=r\left( \theta \right)\sin \left( \theta \right)  to convert from polar to parametric form,
  • Calculate \displaystyle \frac{{dy}}{{d\theta }} and \displaystyle \frac{{dx}}{{d\theta }} (Hint: use the product rule on the equations in the previous bullet).
  • Discuss the motion of a particle moving on the graph by discussing the meaning of \displaystyle \frac{{dr}}{{d\theta }} (motion towards or away from the pole), \displaystyle \frac{{dy}}{{d\theta }} (motion in the vertical direction), and/or \displaystyle \frac{{dx}}{{d\theta }} (motion in the horizontal direction).
  • Find the slope at a point on the graph, \displaystyle \frac{{dx}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

This question typically covers topics from Unit 9 of the CED.

Free-response questions:

  • 2013 BC 2
  • 2014 BC 2
  • 2017 BC 2
  • 2018 BC 5
  • 2019 AB 2

Multiple-choice questions from non-secure exams:

  • 2008 BC 26
  • 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)

         Visualizing Unit 9 Desmos Demonstrations for Polar, Vector and Parametric Curves

Revised March 12, 2021, April 8, 2022

Parametric and Vector Equations (Type 8)

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AP Questions Type 8: Parametric and Vector Equations (BC Only)

The parametric/vector equation questions only concern motion in a plane. Other topics, such as dot product and cross product, are not tested.

In the plane, the position of a moving object as a function of time, t, can be specified by a pair of parametric equations \displaystyle x=x\left( t \right)\text{ and }y=y\left( t \right) or the equivalent vector \displaystyle \left\langle {x\left( t \right),y\left( t \right)} \right\rangle . The path is the curve traced by the parametric equations or the tips of the position vector. .

The velocity of the movement in the x- and y-direction is given by the vector \displaystyle \left\langle {{x}'\left( t \right),{y}'\left( t \right)} \right\rangle . The vector sum of the components gives the direction of motion. Attached to the tip of the position vector this vector is tangent to the path pointing in the direction of motion.

The length of this vector is the speed of the moving object. Speed = \displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}. (Notice that this is the same as the speed of a particle moving on the number line with one less parameter: On the number line speed \displaystyle =\left| {v\left( t \right)} \right|=\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}}}.)

The acceleration is given by the vector \displaystyle \left\langle {{x}''\left( t \right),{y}''\left( t \right)} \right\rangle .

What students should know how to do:

  • Vectors may be written using parentheses, ( ), or pointed brackets, \displaystyle \left\langle {} \right\rangle , or even \displaystyle \vec{i},\vec{j} form. The pointed brackets seem to be the most popular right now, but all common notations are allowed and will be recognized by readers.
  • Find the speed at time t: Speed = \displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}.
  • Use the definite integral for arc length to find the distance traveled \displaystyle \int_{a}^{b}{{\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}}}. Notice that this is the integral of the speed (rate times time = distance).
  • The slope of the path is \displaystyle \frac{{dy}}{{dx}}=\frac{{{y}'\left( t \right)}}{{{x}'\left( t \right)}}. See this post for more on finding the first and second derivatives with respect to x.
  • Determine when the particle is moving left or right,
  • Determine when the particle is moving up or down,
  • Find the extreme position (farthest left, right, up, down, or distance from the origin).
  • Given the position find the velocity by differentiating.
  • Given the velocity, find the acceleration by differentiating.
  • Given the acceleration and the velocity at some point find the velocity by integrating.
  • Given the velocity and the position at some point find the position by integrating. These are just initial value differential equation problems (IVP).
  • Dot product and cross product are not tested on the BC exam, nor are other aspects.

When this topic appears on the free-response section of the exam there is no polar equation free-response question and vice versa. When not on the free-response section there are one or more multiple-choice questions on parametric equations.

Free-response questions:

  • 2012 BC 2
  • 2016 BC 2
  • 2021 BC 2
  • 2022 BC2 – slope of tangent line, speed, position, total distance traveled
  • 2023 BC 2 – acceleration vector, speed, tangent line total distance traveled.

Multiple-choice questions from non-secure exams

  • 2003 BC 4, 7, 17, 84
  • 2008 BC 1, 5, 28
  • 2012 BC 2

Visualizing Unit 9 Desmos Demonstrations for Polar, Vector and Parametric Curves

This question typically covers topics from Unit 9 of the CED.

Revised March 12, 2021, April 5, and May 14, 2022, March 6, 2023, June 6, 2023