# AP Calculus Review 2023

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 dx

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.

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)

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 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)

### 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

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)

### 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

# Other Problems (Type 7)

### AP Questions Type 7: Other topics

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. The topics discussed here are not asked often enough to be classified as a type of their own. The topics listed here have been the subject of full free-response questions or major parts of them. Other topics occasionally asked are mentioned in the question list at the end of the post.

Implicitly defined relations and implicit differentiation

These questions may ask students to find the first and/or second derivative of an implicitly defined relation. Often the derivative is given, and students are required to show that it is correct. (This is because without the correct derivative the rest of the question cannot be done.) The follow-up is to answer questions about the function such as finding an extreme value, second derivative test, or find where the tangent is horizontal or vertical.

What students should know how to do

• Know how to find the first derivative of an implicit relation using the product rule, quotient rule, chain rule, etc.
• Know how to find the second derivative, including substituting for the first derivative.
• Know how to evaluate the first and second derivative by substituting both coordinates of a given point. (Note: If all that is needed is the numerical value of the derivative then the substitution is often easier done before solving for dy/dx or d2y/dx2, and as usual the arithmetic need not be done.)
• Analyze the derivative to determine where the relation has horizontal and/or vertical tangents.
• Write and work with lines tangent to the relation.
• Find extreme values. It may also be necessary to show that the point where the derivative is zero is actually on the graph and to justify the answer.

Simpler questions about implicit differentiation may appear on the multiple-choice sections of the exam.

Example:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

2021 AB 5 (a) Implicit differentiation.

2023 AB 6 implicit differentiation, find horizontal and vertical tangent line location,                   related rate

BC classes see Implicit differentiation of parametric equations, and A Vector’s Derivative

Related Rates

Derivatives are rates and when more than one variable is changing over time the relationships among the rates can be found by differentiating with respect to time. The time variable may not appear in the equations. These questions appear occasionally on the free-response sections; if not there, then a simpler version may appear in the multiple-choice sections. In the free-response sections they may be an entire problem, but more often appear as one or two parts of a longer question.

What students should know how to do

• Set up and solve related rate problems.
• Be familiar with the standard type of related rate situations, but also be able to adapt to different contexts.
• Know how to differentiate with respect to time. That is, find dy/dt even if there is no time variable in the given equations using any of the differentiation techniques.
• Interpret the answer in the context of the problem.
• Unit analysis.

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

For previous posts on related rates see Related Rate Problems I and Related Rate Problems II.

Examples

2014 AB4/BC4,

2016 AB5/BC5

2019 AB 4 Related Rate

2019 AB 6

2022 AB2 (d), AB4/BC4 (d) Good example that requires using product and evaluation of an expression that include dr/dt and dh/dt.

2023 AB 6 implicit differentiation, find horizontal and vertical tangent line location,                   related rate

Good Question 9

Family of Functions

A “family of functions” is defined by an equation with a parameter (sort of an extra variable). Changing the parameter gives a different but similar curve. Questions should be answered in general, that is, in terms of the parameter not a specific value of the parameter. These questions appeared on some exams long ago, may be making a comeback.

Examples:

1995 BC 5

1996 AB4/BC4

Good Question 5: 1998 AB2/BC2

2019 BC 5

Other Topics

Free response questions (many of the BC questions are suitable for AB)

• Finding derivatives using the chain rule, the quotient rule, etc. from tables of values: 2016 AB 6 and 2015 AB 6, 2023 AB 5
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 4(c), 2023 BC 4(c)
• Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
• Arc length (BC Topic) 2014 BC 5
• Partial fractions (BC Topic) 2015 BC 5
• Improper integrals (BC topic): 2017 BC 5, 2022 BC5 (c), 2023 BC 5 (b)

Multiple-choice questions from non-secure exams:

• 2012 AB 27 (implicit differentiation), 77 (IVT), 88 (related rate)
• 2012 BC 4 (Curve length), 7 (Implicit differentiation), 11 (continuity/differentiability), 12 (Implicit differentiation), 77 (dominance), 82 (average value), 85 (related rate) , 92 (compositions)

These questions may come from any of the Units in the CED.

Updated March 12, 2021, April 1, and May 14, 2022, June 4, 2023

# Linear Motion (Type 2)

### 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 positions(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 xa 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