AP Calculus Review 2023

Posted on March 7, 2023 by Lin McMullin

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.

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 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 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.

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.

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 dx

Posted on April 23, 2022 by Lin McMullin

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; it shows the details that readers must keep in mind while grading. This type 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, 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 concerned about all these fine points as readers do. Encourage your students to answer the question correctly and show the required work using correct notation. This is shown on the scoring standard for each question (on Stephen’s sample it is in the ruled area directly below the question). Don’t worry about the fine points – what if I say this, instead of that. 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.

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 their answer is not quite right, take off some credit and help them learn how to do better. It will help them in the long run.

Sequences and Series (Type 10)

Posted on April 12, 2022 by Lin McMullin

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), e^x 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)

Posted on April 8, 2022 by Lin McMullin

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), $\displaystyle \frac{{dr}}{{d\theta }}$ (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

Parametric and Vector Equations (Type 8)

Posted on April 5, 2022 by Lin McMullin

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

Linear Motion (Type 2)

Posted on March 15, 2022 by Lin McMullin

AP Questions Type 2: Linear Motion

We continue the discussion of the various type questions on the AP Calculus Exams with linear motion questions.

“A particle (or car, person, or bicycle) moves on a number line ….”

These questions may give the position equation, the velocity equation (most often), or the acceleration equation of something that is moving on the x– or y-axis as a function of time, along with an initial condition. The questions ask for information about the motion of the particle: its direction, when it changes direction, its maximum position in one direction (farthest left or right), its speed, etc.

The particle may be a “particle,” a person, car, a rocket, etc. Particles don’t really move in this way, so the equation or graph should be considered a model. The question is a versatile way to test a variety of calculus concepts since the position, velocity, or acceleration may be given as an equation, a graph, or a table; be sure to use examples of all three forms during the review.

Many of the concepts related to motion problems are the same as those related to function and graph analysis (Type 3). Stress the similarities and show students how the same concepts go by different names. For example, finding when a particle is “farthest right” is the same as finding when a function reaches its “absolute maximum value.” See my post for Motion Problems: Same Thing, Different Context for a list of these corresponding terms. There is usually one free-response question and three or more multiple-choice questions on this topic.

The position, s(t), is a function of time. The relationships are:

The velocity is the derivative of the position $\displaystyle {s}'\left( t \right)=v\left( t \right)$ . Velocity has direction (indicated by its sign) and magnitude. Technically, velocity is a vector; the term “vector” will not appear on the AB exam.
Speed is the absolute value of velocity; it is a number, not a vector. See my post for Speed.
Acceleration is the derivative of velocity and the second derivative of position, $\displaystyle {{s}'}'\left( t \right)={v}'\left( t \right)=a\left( t \right)$ It, too, has direction and magnitude and is a vector.
Velocity is the antiderivative of acceleration.
Position is the antiderivative of velocity.

What students should be able to do:

Understand and use the relationships above.
Distinguish between position at some time and the total distance traveled during the time period.
The total distance traveled is the definite integral of the speed (absolute value of velocity) $\displaystyle \int_{a}^{b}{{\left| {v\left( t \right)} \right|dt}}$ .
Be sure your students understand the term displacement; it is the net distance traveled or distance between the initial position and the final position. Displacement is the definite integral of the velocity (rate of change): $\displaystyle \int_{a}^{b}{{v\left( t \right)dt}}$
The final position is the initial position plus the displacement (definite integral of the rate of change from x= a to x = t): $\displaystyle s\left( t \right)=s\left( a \right)+\int_{a}^{t}{{v\left( x \right)dx}}$ Notice that this is an accumulation function equation (Type 1).
Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. These are easily handled with the accumulation equation in the bullet above but may also be handled as an initial value problem.
Find the speed at a given time. Speed is the absolute value of velocity.
Find average speed, velocity, or acceleration
Determine if the speed is increasing or decreasing.
- When the velocity and acceleration have the same sign, the speed increases. When they have different signs, the speed decreases.
- If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See the post on Speed.
- There is also a worksheet on speed here
- The analytic approach to speed: A Note on Speed
Use a difference quotient to approximate the derivative (velocity or acceleration) from a table. Be sure the work shows a quotient.
Riemann sum approximations.
Units of measure.
Interpret meaning of a derivative or a definite integral in context of the problem

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

This may be an AB or BC question. The BC topic of motion in a plane, (Type 8: parametric equations and vectors) will be discussed in a later post.

The Linear Motion problem may cover topics primarily from primarily from Unit 4, and also from Unit 3, Unit 5, Unit 6, and Unit 8 (for BC) of the CED.

Free-response examples:

2017 AB 5, Equation stem
2009 AB1/BC1, Graph stem:
2019 AB2 Table stem
2021 AB 2 Equation stem
2022 AB6 Equation stem – velocity, acceleration, position, max/min
2023 AB 2 Equation stem – velocity (given), change of direction, acceleration, speeding up or slowing down, position, total distance.

Multiple-choice examples from non-secure exams:

2012 AB 6, 16, 28, 79, 83, 89
2012 BC 2, 89

Updated: March 15, and May 11, 2022, June 4, 2023

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Category Archives: Reviewing

AP Calculus Review 2023

General information and suggestions

To dx or not dx

Sequences and Series (Type 10)

Polar Equation Questions (Type 9)

AP Questions Type 9: Polar Equations (BC Only)

Parametric and Vector Equations (Type 8)

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

Linear Motion (Type 2)

AP Questions Type 2: Linear Motion

General information and suggestions

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AP Questions Type 9: Polar Equations (BC Only)

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

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AP Questions Type 7: Other topics

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AP Questions Type 2: Linear Motion

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