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

Differential Equations (Type 6)

Posted on March 20, 2020 by Lin McMullin

AP Questions Type 6: Differential Equations

Differential equations are tested almost every year. The actual solving of the differential equation is usually the main part of the problem, but it is accompanied by a related question such as a slope field or a tangent line approximation. BC students may also be asked to approximate using Euler’s Method. Large parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam. What students should be able to do

Find the general solution of a differential equation using the method of separation of variables (this is the only method tested).
Find a particular solution using the initial condition to evaluate the constant of integration – initial value problem (IVP).
NEW Determine the domain restrictions on the solution of a differential equation. See this post for more on this.
Understand that proposed solution of a differential equation is a function (not a number) and if it and its derivative are substituted into the given differential equation the resulting equation is true. This may be part of doing the problem even if solving the differential equation is not required (see 2002 BC 5 – parts a, b and d are suitable for AB)
Growth-decay problems.
Draw a slope field by hand.
Sketch a particular solution on a given slope field.
Interpret a slope field.
Multiple-choice: Given a differential equation, identify is slope field.
Multiple-choice: Given a slope field identify its differential equation.
Use the given derivative to analyze a function such as finding extreme values
For BC only: Use Euler’s Method to approximate a solution.
For BC only: use the method of partial fractions to find the antiderivative after separating the variables.
For BC only: understand the logistic growth model, its asymptotes, meaning, etc. The exams so far, have never asked students to actually solve a logistic equation IVP

Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:

one point for separating the variables
one point each for finding the antiderivatives
one point for including the constant of integration and using the initial condition – that is, for writing “+ C” on the paper with one of the antiderivatives and substituting the initial condition; finding the value of C is included in the “answer point.” and
one point for solving for y: the “answer point”, for the correct answer. This point includes all the algebra and arithmetic in the problem including solving for C..

In the past, the domain of the solution is often included on the scoring standard, but unless it is specifically asked for in the question students do not need to include it. However, the 9 CED. lists “EK 3.5A3 Solutions to differential equations may be subject to domain restrictions.” Perhaps this will be asked in the future. For more on domain restrictions with examples see this post. 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. For some previous posts on differential equations see January 5, 2015 and for post on related subjects see November 26, 2012, January 21, 2013 February 1, 6, 2013 The Differential Equation question covers topics in Unit 7 of the 2019 CED.

Free-response examples:

2019 There was no DE question in the free-response. You may assume the topic was tested in the multiple-choice sections.
2017 AB4/BC4,
2016 AB 4, BC 4, (different questions)
2015 AB4/BC4,
2013 BC 5
and a favorite Good Question 2 and Good Question 2 Continued

Multiple-choice examples from non-secure exams:

2012 AB 23, 25
2012 BC: 12, 14, 16, 23

Previous posts on these topics for both AB and BC include:

Differential Equations A summary of the terms and techniques of differential equation and the method of separation of variables

Domain of a Differential Equation – On domain restrictions.

Accumulation and Differential Equations

Slope Fields

An Exploration in Differential Equations An exploration illustrating many of the ideas of differential equations. The exploration is here in PDF form and the solution is here. The ideas include: finding the general solution of the differential equation by separating the variables, checking the solution by substitution, using a graphing utility to explore the solutions for all values of the constant of integration, finding the solutions’ horizontal and vertical asymptotes, finding several particular solutions, finding the domains of the particular solutions, finding the extreme value of all solutions in terms of C, finding the second derivative (implicit differentiation), considering concavity, and investigating a special case or two.

Previous Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation

Logistic Growth – Real and Simulated

Revised 2/20/2021

Riemann Sum & Table Problems (Type 5)

Posted on March 17, 2020 by Lin McMullin

AP Questions Type 5: Riemann Sum & Table Problems

Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested. This question often includes an equation in a latter part of the problem that refers to the same situation.

What students should be able to do:

Find the average rate of change over an interval
Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating. This amounts to finding the slope or rate of change. Show the quotient even if you can do the arithmetic in your head and even if the denominator is 1.
Use a left-, right-, or midpoint- Riemann sums or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible 9.)
These questions are usually presented in some context and answers should be in that context. The context may be something growing (changing over time) or linear motion.
Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem.
One of the parts of this question asks a related question based on a function given by an equation.
Unit analysis.

Do’s and Don’ts

Do: Remember that you do not know what happens between the values in the table unless some other information is given. For example, do not assume that the largest number in the table is the maximum value of the function, or that the function is decreasing between two values just because a value is less than the preceding value.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient even if the denominator is 1.

Do Not do arithmetic: A long expression consisting entirely of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you a simplify correct answer incorrectly, you will lose credit.

Do Not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

This question typically covers topics from Unit 6 of the 2019 CED but may include topics from Units 2, 3, and 4 as well.

Free-response examples:

2007 AB 3 (4 simple parts on various theorems, yet the mean score was 0.96 out of 9),
2017 AB 1/BC 1, and AB 6,
2016 AB 1/BC 1
2018 AB 4

Multiple-choice questions from non-secure exams:

2012 AB 8, 86, 91
2012 BC 8, 81, 86 (81 and 86 are the same on both the AB and BC exams)

Revised March 12, 2021

Teaching Calculus

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

Category Archives: AP Calculus Exams

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)

Differential Equations (Type 6)

AP Questions Type 6: Differential Equations

Previous posts on these topics for both AB and BC include:

Previous Posts on BC Only Topics

Riemann Sum & Table Problems (Type 5)

AP Questions Type 5: Riemann Sum & Table Problems

Share this:

Share this:

AP Questions Type 9: Polar Equations (BC Only)

Share this:

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

Share this:

AP Questions Type 7: Other topics

Share this:

AP Questions Type 6: Differential Equations

Previous posts on these topics for both AB and BC include:

Previous Posts on BC Only Topics

Share this:

AP Questions Type 5: Riemann Sum & Table Problems

Share this: