2021 Review Notes

Posted on March 16, 2021 by Lin McMullin

About this time of year, I have been posting notes on reviewing and on the ten types of problems that usually appear on the AP Calculus Exams AB and BC. Since the types do not change, I am posting all the links below. They are only slightly revised from last year. You can also find them under “AP Exam Review” on the black navigation bar above.

Each link provides a list of “What students should know” and links to other post and questions from past exams related to the type under consideration.

Note that the 10 Types are not the same as the 10 Units in the Fall 2020 Course and Exam Description. This is because many of the exam questions have parts from different units.

Here are the links to the various review posts:

AP Exam Review – General suggestions, hints, and information. Also, other resources for reviewing.
Type 1 questions – Rate and accumulation questions
Type 2 questions – Linear motion problems
Type 3 questions – Graph analysis problems
Type 4 questions – Area and volume problems
Type 5 questions – Table and Riemann sum questions
Type 6 questions – Differential equation questions
Type 7 questions – Miscellaneous
Type 8 questions – Parametric and vector questions (BC topic)
Type 9 questions – Polar equations (BC topic)
Type 10 questions – Sequences and Series (BC topic)

When assigning past exams questions for review (and you should assign past exam question), keep in mind that students can find the scoring standards online. Even though the AP program forbids this and makes every effort to prevent them from being posted, they are there. Students can “research” the solution. Keep this in mind when assigning questions from past exams. Here is a suggestion Practice Exams – A Modest Proposal

AP Calculus Exams Update

Posted on April 3, 2020 by Lin McMullin

Here is the latest information on the 2020 AP Calculus Exams as of April 3, 2020. Updated 4/29/2020

Update: A message from the AP Program 4/28/2020

Subject: How to Prepare Your Students for the 2020 AP Exams

Dear Colleagues,

Additional information is now available to help guide you and your students through the exam day experience.

New Resources

2020 AP Testing Guide (.pdf/10.9 MB): The guide, designed for educators to walk their students through test day, provides information about:

The AP Exam e-ticket

Five steps to take before test day

Understanding the test day experience

Exam scores, credit, and placement

2020 AP Exam Day Checklist (.pdf/526 KB): Teachers should have their students complete this checklist for each exam they take and keep it next to them while testing.

Explainer Videos: New videos are available to give students quick, easily accessible information about their test day experience, what they need to do to prepare, exam security, and more. Explore the playlist.

Other Reminders

AP Exam Demo (available May 4): AP students should use the clickable exam demo to practice the different ways to submit their exam responses. The demo will help students confirm that their testing device will be able to access and run the online exam. If they can’t access the demo, the final slide of the Testing Guide can help them troubleshoot. The sample content in the demo will be the same for all users and isn’t a practice exam. We’ll send educators and students an email to remind them when the demo is available. Please encourage your students to take this important preparation step.

Educator Webinars: Trevor Packer, the head of the AP Program, will walk participants through the 2020 AP Testing Guide. AP staff will answer questions during the presentation. This series of webinars includes:

Discipline-specific webinars for AP teachers, April 28–30

Updates for AP coordinators webinar, May 1

An exam day webinar for school counselors, May 4

Online AP Classes and Review Sessions prep week: The week before exams, online AP classes will focus on the exam day experience and sample exam questions.

Student Outreach: If any of your students are having difficulty receiving College Board emails, walk them through the steps on the “Confirm your email address” page.

Information for AP Chinese, French, German, Italian, Japanese, and Spanish Language and Culture Exams:

The above exams will be administered using a new dedicated app, the AP World Languages Exam App. Students taking these exams must use this app on smartphones or tablets. This free app will be available for download from the Apple App Store and Google Play Store the week of May 11. We’ll email students and their teachers to let them know when the app is available to download. Visit our site for more details.

A video walk-through of the test-taking experience will be available the week of May 4.

Details on accommodations for the above exams are also now available.

If your students are unsure about accessing the app, or if they don’t have a device, they can fill out this survey (or you can complete it on their behalf) as soon as possible so we can help support them (applicable to U.S. and U.S. territories).

Thank you for all that you’re doing for your students.

Sincerely,

Advanced Placement Program

General Information from the College Board

The previous announcement of March 20, 2020 from the College Board with details on the exam and what is and is not covered is here.

The College Board’s full email of April 3, 2020 is here.

A video of Trevor Packard’s online discussion on Thursday April 2, 2020 is here.

Video of the webinar for Math and Computer Science Teachers from April 14, 2020 is here.

The announcement regarding the exams published April 3, 2020 is here. Scroll down to the calculus sections for full exam details. Highlights are below.

The College Board’s Coronavirus Update page is here with information for teachers and students. This includes a FAQ page.

AP Calculus AB and BC

The AB exam will cover only Units 1 – 7 of the 2019 Course and Exam Description (NOT Unit 8)

The BC exam will cover only Units 1 – 8, and Unit 10 topics 2, 5, 7, 8, and 11 of the 2019 Course and Exam Description (NOT Unit 9 or Unit 10 topics 1, 3, 4, 6, 9, 10, 12, 13,14, and 15).

The format will be two free-response questions.

- The first multi-focus free-response question counts 60% and assess knowledge and skills from 2 or more units. Students will be allowed 25 minutes followed by 5 minutes to upload the answers. Once uploaded, students may not return to this question.
- The second multi-focus free-response question counts 40% assess knowledge and skills from 2 or more units. Students will be allowed 15 minutes followed by 5 minutes to upload the answer.
- Questions on the 2020 AP Calculus BC Exam are designed such that a graphing calculator or other calculator is not required. However, use of a calculator is allowed. Simple (“four-function”) calculators are freely available as apps for computers and phones (i.e. most or all internet-connected devices), and can be installed beforehand for use on the exam.
- No arithmetic or calculations will be required beyond what can readily be done with pencil and paper. As always, AP Calculus BC students are advised to submit “unsimplified” numeric answers, in order to avoid risking arithmetical errors not related to calculus.
- Accommodations for students who are entitled to them will be allowed. At the moment, I have no information on how this will work. I will edit this if/when I know.
- Video of the webinar for Math and Computer Science Teachers from April 14, 2020 is here.

Other information

Most exams will have one or two free-response questions, and each question will be timed separately. Students will need to write and submit their responses within the allotted time for each question.

- Students will be able to take exams on any device they have access to—computer, tablet, or smartphone. They’ll be able to type and upload their responses or write responses by hand and submit a photo via their cell phones.
- For most subjects, the exams will be 45 minutes long, plus an additional 5 minutes for uploading. Students will need to access the online testing system 30 minutes early to get set up.
- Again, The announcement regarding the exams published April 3, 2020 is here. Scroll down to the calculus sections for full exam details.

Exam Dates

The AP Calculus Exams AB and BC will be administered online on Tuesday May 12, 2020 simultaneously worldwide, specifically:

- Eastern time zone at 2:00 p.m.
- Central time zone at 1:00 p.m.
- Mountain time zone at 12:00 noon
- Pacific time zone at 11:00 a.m.
- Alaska time zone at 10:00 a.m.
- Hawai’i time zone at 8:00 a.m.
- Greenwich Mean Time (GMT) 6:00 p.m. (18:00)

Make Up Exams for Calculus will be Tuesday June 2, 2020 at 20:00 GMT (8 p.m.) That’s

- 4:00 p.m. Eastern,
- 3:00 p.m. Central,
- 2:00 p.m. Mountain,
- 1:00 p.m. Pacific,
- 12:00 noon Alaska
- 10:00 a.m. Hawai’i

Review Links

Links to my review blogs are below. The “type” numbers are not the same as the CED unit numbers. One type may and probably does require knowledge from several of the CED Units.

AP Exam Review 2020 https://wp.me/p2zQso-286
Reviewing Resources 2020 https://wp.me/p2zQso-28f
Rate and accumulation questions (Type 1) https://wp.me/p2zQso-265
Linear motion problems (Type 2) https://wp.me/p2zQso-269
Graph analysis problems (Type 3) https://wp.me/p2zQso-26e
Area and volume problems (Type 4) https://wp.me/p2zQso-26m
Table and Riemann sum questions (Type 5) https://wp.me/p2zQso-27K
Differential equation questions (Type 6) https://wp.me/p2zQso-27U
Other questions (Type 7) https://wp.me/p2zQso-28q
Sequences and Series questions (Type 10) BC Topic https://wp.me/p2zQso-298

Revised 4/9/2020: Additions and corrections.

Sequences and Series (Type 10)

Posted on March 27, 2020 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 largest 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 several 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
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 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 $\displaystyle \tfrac{1}{1-x}$ 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.
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 and 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 test 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 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 $\displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin (5x)\cos \left( {\tfrac{\pi }{4}} \right)+\cos (5x)\sin \left( {\tfrac{\pi }{4}} \right)$ )
2011 BC 6 (Lagrange error bound)
2016 BC 6
2017 BC 6
2019 BC 6

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 question come from Unit 10 of the 2019 CED.

Revised March 12, 2021

Polar Equation Questions (Type 9)

Posted on March 26, 2020 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 graph 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 (to use as limits of integration).
Find the area enclosed by a graph or graphs: Area = $\displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}$ θ $\displaystyle ){{)}^{2}}d$ θ
Use the formulas $x\left( \theta \right)\text{ }=~r\left( \theta \right)\text{cos}\left( \theta \right)~~\text{and}~y\left( \theta \right)\text{ }=~r(\theta )\text{sin}\left( \theta \right)~$ 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, $\displaystyle \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$ .

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.

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 March 25, 2020 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 $x=x\left( t \right)\text{ and }y=y\left( t \right)$ or the equivalent vector $\left\langle x\left( t \right),y\left( t \right) \right\rangle$ . 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 $\left\langle {x}'\left( t \right),{y}'\left( t \right) \right\rangle$ . 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. $\text{Speed }=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}$ . (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 $\text{Speed}=\left| v \right|=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}}$ .)

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

What students should know how to do:

Vectors may be written using parentheses, ( ), or pointed brackets, $\left\langle {} \right\rangle$ , or even $\vec{i},\vec{j}$ 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: $\text{Speed }=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}$
Use the definite integral for arc length to find the distance traveled $\displaystyle \int_{a}^{b}{\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}}dt$ . 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

Multiple-choice questions from non-secure exams

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

This question typically covers topics from Unit 9 of the 2019 CED .

Revised March 12, 2021

Linear Motion (Type 2)

Posted on March 6, 2020 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 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 to be 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 the 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, ${s}'\left( t \right)=v\left( t \right)$ . Velocity is 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, $\displaystyle a\left( t \right)={v}'\left( t \right)={{s}''}\left( t \right)$ . It, too, has direction and magnitude and is a vector.
Velocity is the antiderivative of the 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. The speed is the absolute value of the velocity.
Find average speed, velocity, or acceleration
Determine if the speed is increasing or decreasing.
- If at some time, the velocity and acceleration have the same sign then the speed is increasing.If they have different signs the speed is decreasing.
- 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 2019 CED.

Free-response examples:

Equation stem 2017 AB 5,
Graph stem: 2009 AB1/BC1,
Table stem 2019 AB2

Multiple-choice examples from non-secure exams:

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

Teaching Calculus

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

Category Archives: Reviewing

2021 Review Notes

AP Calculus Exams Update

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

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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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