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

Rate & Accumulation (Type 1)

Posted on March 11, 2022 by Lin McMullin

The Free-response Questions

There are ten general types of AP Calculus free-response questions. This and the next nine posts will discuss each of them.

NOTE: The numbers I’ve assigned to each type DO NOT correspond to the CED Unit numbers. Many AP Exam questions intentionally have parts from different Units. The CED Unit numbers will be referenced in each post.

AP Questions Type 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some quantities are changing. There are usually two rates acting in opposite ways (sometimes called an in-out question). Students are asked about the change that the rates produce over a time interval either separately or together.

The rates are often fairly complicated functions. If the question is on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing, integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}}$

where $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

Be ready to read and apply; often these problems contain a lot of words which need to be carefully read and understood.
Understand the question. It is often not necessary to do as much computation as it seems at first.
Recognize that rate = derivative.
Recognize a rate from the units given without the words “rate” or “derivative.”
Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)$

Find the final amount by adding the initial amount to the amount found by integrating the rate. If $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$ is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}}$ ,

Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included in the correct place. Think of the integral sign and the dt as parentheses around the integrand.
Find the average value of a function
Use FTC to differentiate a function defined by an integral.
Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) what the numerical argument means in the context of the question.
Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in the context of the question.
Store functions in their calculator recall them to do computations on their calculator.
If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids. Also, be ready to approximate a derivative using a quotient from the numbers in the table.
Do a max/min or increasing/decreasing analysis.

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.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the CED.

Typical free-response examples:

2013 AB1/BC1
2015 AB1/BC1
2018 AB1/BC1
2019 AB1/BC1
2022 AB1/BC1 – includes average value, inc/dec analysis, max/min analysis
2023 AB1/BC1 – Table stem, average value, MVT,
One of my favorites Good Question 6 (2002 AB 4)

Typical multiple-choice examples from non-secure exams:

2012 AB 8, 81, 89
2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021, March 11, 2022. February 17, 2024

AP Exam Review – General Suggestions

Posted on March 8, 2022 by Lin McMullin

It will soon be time to start reviewing for the AP Calculus Exams. So, it’s time to start planning your review. For the next weeks through the beginning of April I will be posting notes for reviewing. These are not new; versions have been posted for the last few years and these are only slightly revised and updated. The post will appear on Tuesdays and Fridays for the next few weeks. Posts are intentionally scheduled before you need them, so you can plan ahead. Most people start reviewing around the beginning or middle of April.

Student Goals

During the exam review period the students’ goal is to MAKE MISTAKES! This is how you and they can know what they don’t know and learn or relearn it. Encourage mistakes!

Ideas for reviewing for the AP Exam

Part of the purpose of reviewing for the AP calculus exams is to refresh your students’ memory of all the great things you’ve taught them during the rear. The other purpose is to inform them about the format of the exam, the style of the questions, the way they should present their answer, and how the exam is graded and scored.

Using AP questions all year is an effective way to accomplish this. Look through the released multiple-choice exams and pick questions related to whatever you are doing at the moment. Free-response questions are a little trickier since the parts of the questions come from different units. These may be adapted or used in part.

At the end of the year, I suggest you review the free-response questions by type – table questions, differential equations, area/volume, rate/accumulation, graph, etc. More detailed notes on what students needed to know about each of the ten types will be the topic of future posts on Tuesdays and Fridays over the next few weeks. Plan to spend a few days doing a selection of questions of one type so that student can see how that type of question is asked, the format of the question (i.e. does it start with an equation, a table, or a graph), and the various topics that are tested. Then go onto the next type. Many teachers keep a collection of past free-response questions filed by type rather than year. This makes it easy to study them by type. The “types” do not align exactly with the units of the Course and Exam Description (CED) since parts of each question often come from different units.

Simulated Exam

Plan to give a simulated (mock) exam. Full exams from past years are available. The free-response questions through 2021 are available here for AB and here for BC. The secure 2014 – 2019 exams are available through your audit website. If possible, find a time when your students can take an entire exam in one sitting (3.25 hours). Teachers often do this on a weekend day or in the evening. This will give your students a feel for what it is like to work calculus problems under test conditions. If you cannot get 3.25 hours to do this, give the sections in class using the prescribed time. Some teachers schedule several simulated exams. Of course, you need to correct them and go over the most common mistakes.

Be aware that all the exams (yes, including the secure exams unfortunately) are available online. Students can find them easily. Here is a modest proposal for how to deal with this:

Don’t grade the practice exam or count it as part of the students’ averages.

Athletes are not graded on their practices, only the game counts. Athletes practice to maintain their skills and improve on their weakness. Make it that way with your practice tests.

Calculus students are intelligent. Explain to them why you are asking them to take a practice exam. Explain how making mistakes is a good thing because it helps them find their weaknesses so they can eliminate them. Use the simulated exam to maintain their skills and find their weakness. This will help them do better on the real exam. By taking the pressure of a grade away, students can focus on improvement.

Make it an incentive not to be concerned about a grade.

Directions

Print a copy of the directions for both parts of the exam and go over them with your students. For the free-response questions emphasize the need to show their work, explain that they do not have to simplify arithmetic or algebraic expressions, and explain the three-decimal place consideration. Be sure they know what is expected of them. The directions are here can be found on any free response released exams. Yes, this is boiler plate stuff, but take a few minutes to go over it with your students. They should not have to see the directions for the first time on the day of the exam. This and other information is explained in detail in this article: How, not only to survive, but to prevail. Copy this article for your students!

Explain the scoring

There are 108 points available on the exam; each half (free-response and multiple-choice) is worth the same – 54 points. The number of points required for each score is set after the exams are graded and changes slightly every year.

For the AB exam, the minimum points required for each score out of 108 points are, very approximately:

for a 5 – 69 points,
for a 4 – 52 points,
for a 3 – 40 points,
for a 2 – 28 points.

The numbers are similar for the BC exams are again very approximately:

for a 5 – 68 points,
for a 4 – 58 points,
for a 3 – 42 points,
for a 2 – 34 points.

The actual numbers vary from year to year, but that is not important. What is important for students to know is that they may omit or get a number of questions wrong and still earn a good score. Students may not be used to this (since they skip or get so few questions wrong on your tests!). They should not panic or feel they are doing poorly if they miss a few questions. If they understand and accept this in advance they will calm down and do better on the exams. Help them understand they should gather as many points as they can, and not be too concerned if they cannot get them all. Doing only the first 2 parts of a free-response question will probably put them at the mean for that question. Remind them not to spend time on something that’s not working out, or that they don’t feel they know how to do.

Resources for reviewing

How, Not Only to Survive, but to Prevail… – Notes and advice for your students. You may copy and duplicate this for your class.

Calculator Use on the AP Exams – hints and instruction.

Ted Gott’s Multiple-choice Index – an excel spreadsheet searchable by topic, and referenced to the CED by Learning Objectives (LO) and Essential Knowledge (EK)

Type Analysis 1998 – 2023 a listing of the questions on both free-response and multiple-choice questions by type, so you can find them easily.

NEW: Searchable index to all FR questions since 1998 is here https://www.calc-medic.com/ap-calc-free-response-questions

Revised March 8,2022, March 12, 2023, February 17, 2024

Adapting 2021 AB 4 / BC 4

Posted on July 13, 2021 by Lin McMullin

Four of nine. Continuing the series started in the last three posts, this post looks at the AP Calculus 2021 exam question AB 4 / BC 4. The series considers each question with the aim of showing ways to use the question with your class as is, or by adapting and expanding it. Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

2021 AB 4 / BC 4

This is a Graph Analysis Problem (type 3) and contains topics from Units 2, 4, and 6 of the current Course and Exam Description. The things that are asked in these questions should be easy for the students, however each year the scores are low. This may be because some textbooks simply do not give students problems like this. Therefore, supplementing with graph analysis questions from past exams is necessary.

There are many additional questions that can be asked based on this stem and the stems of similar problems. Usually, the graph of the derivative is given, and students are asked questions about the graph of the function. See Reading the Derivative’s Graph.

Some years this question is given a context, such as the graph is the velocity of a moving particle. Occasionally there is no graph and an expression for the derivative or function is given.

Here is the 2021 AB 4 / BC 4 stem:

The first thing students should do when they see $G\left( x \right)=\int_{0}^{x}{{f\left( t \right)}}dt$ is to write prominently on their answer page ${G}'\left( x \right)=f\left( x \right)$ and $\displaystyle {G}''\left( x \right)={f}'\left( t \right)$ . While they may understand and use this, they must say it.

Part (a): Students were asked for the open intervals where the graph is concave up and to give a reason for their answer. (Asking for an open interval is to remove any concern about the endpoints being included or excluded, a place where textbooks differ. See Going Up.)

Discussion and ideas for adapting this question:

Using this or similar graphs go through each of these with your class until the answers and reasons become automatic. There are quite a few other things that may be asked here based on the derivative.
- Where is the function increasing?
- Decreasing?
- Concave down, concave up?
- Where are the local extreme values?
- What are the local extreme values?
- Where are the absolute extreme values?
- What are the absolute extreme values?
There are also integration questions that may be asked, such as finding the value of the functions at various points, such as G(1) = 2 found by using the areas of the regions. Also, questions about the local extreme values and the absolute extreme value including their values. These questions are answered by finding the areas of the regions enclosed by the derivative’s graph and the x-axis. Parts (b) and (c) do some of this.
Choose different graphs, including one that has the derivative’s extreme value on the x-axis. Ask what happens there.

Part (b): A new function is defined as the product of G(x) and f(x) and its derivative is to be found at a certain value of x. To use the product rule students must calculate the value of G(x) by using the area between f(x) and the x-axis and the value of ${f}'\left( x \right)$ by reading the slope of f(x) from the graph.

Discussion and ideas for adapting this question:

This is really practice using the product rule. Adapt the problem by making up functions using the quotient rule, the chain rule etc. Any combination of $\displaystyle G,{G}',{G}'',f,{f}',\text{ or }{f}''$ may be used. Before assigning your own problem, check that all the values can be found from the given graph.
Different values of x may be used.

Part (c): Students are asked to find a limit. The approach is to use L’Hospital’s Rule.

Discussion and ideas for adapting this question:

To use L’Hospital’s Rule, students must first show clearly on their paper that the limit of the numerator and denominator are both zero or +/- infinity. Saying the limit is equal to 0/0 is considered bad mathematics and will not earn this point. Each limit should be shown separately on the paper, before applying L’Hospital’s Rule.
Variations include a limit where L’Hospital’s Rule does not apply. The limit is found by substituting the values from the graph.
Another variation is to use a different expression where L’Hospital’s Rule applies, but still needs values read from the graph.

Part (d): The question asked to find the average rate of change (slope between the endpoints) on an interval and then determine if the Mean Value Theorem guarantees a place where $\displaystyle {G}'$ equals this value. Students also must justify their answer.

Discussion and ideas for adapting this question:

To justify their answer students must check that the hypotheses of the MVT are met and say so in their answer.
Adapt by using a different interval where the MVT applies.
Adapt by using an interval where the MVT does not apply and (1) the conclusion is still true, or (b) where the conclusion is false.

Next week 2021 AB 5.

I would be happy to hear your ideas for other ways to use this questions. Please use the reply box below to share your ideas.

Adapting 2021 AB 3 / BC 3

Posted on July 6, 2021 by Lin McMullin

Three of nine. Continuing the series started in the last two posts, this post looks at the AB Calculus 2021 exam question AB 3 / BC 3. The series considers each question with the aim of showing ways to use the question in with your class as is, or by adapting and expanding it. Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

2021 AB 3 / BC 3

This question is an Area and Volume question (Type 4) and includes topics from Unit 8 of the current Course and Exam Description. Typically, students are given a region bounded by a curve and an line and asked to find its area and its volume when revolved around a line. But there is an added concept here that we will look at first.

The stem is:

First, let’s consider the c. This is a family of functions question. Family of function questions appear now and then. They are discussed in the post on Other Problems (Type 7) and topics from Unit 8 of the current Course and Exam Description. My favorite example is 1998 AB 2, BC 2. Also see Good Question 2 and its continuation.

If we consider the function with c = 1 to be the parent function $\displaystyle P\left( x \right)=x\sqrt{{4-{{x}^{2}}}}$ then the other members of the family are all of the form $\displaystyle c\cdot P\left( x \right)$ . The c has the same effect as the amplitude of a sine or cosine function:

The x-axis intercepts are unchanged.
If |c| > 1, the graph is stretched away from the x-axis.
If 0 < |c| < 1, the graph is compressed towards the x-axis.
And if c < 0, the graph is reflected over the x-axis.

All of this should be familiar to the students from their work in trigonometry. This is a good place to review those ideas. Some suggestions on how to expand on this will be given below.

Part (a): Students were asked to find the area of the region enclosed by the graph and the x-axis for a particular value of c. Substitute that value and you have a straightforward area problem.

Discussion and ideas for adapting this question:

The integration requires a simple u-substitution: good practice.
You can change the value of c > 0 and find the resulting area.
You can change the value of c < 0 and find the resulting area. This uses the upper-curve-minus-the-lower-curve idea with the upper curve being the x-axis (y = 0).
Ask students to find a general expression for the area in terms of c and the area of P(x).
Another thing you can do is ask the students to find the vertical line that cuts the region in half. (Sometimes asked on exam questions).
Also, you could ask for the equation of the horizontal line that cuts the region in half. This is the average value of the function on the interval. See these post 1, 2, 3, and this activity 4.

Part (b): This question gave the derivative of y(x) and the radius of the largest cross-sectional circular slice. Students were asked for the corresponding value of c. This is really an extreme value problem. Setting the derivative equal to zero and solving the equation gives the x-value for the location of the maximum. Substituting this value into y(x) and putting this equal to the given maximum value, and you can solve for the value of c.

(Calculating the derivative is not being tested here. The derivative is given so that a student who does not calculate the derivative correctly, can earn the points for this part. An incorrect derivative could make the rest much more difficult.)

Discussion and ideas for adapting this question:

This is a good problem for helping students plan their work, before they do it.
Changing the maximum value is another adaption. This may require calculator work; the numbers in the question were chosen carefully so that the computation could be done by hand. Nevertheless, doing so makes for good calculator practice.

Part (c): Students were asked for the value of c that produces a volume of 2π. This may be done by setting up the volume by disks integral in terms of c, integrating, setting the result equal to 2π, and solving for c.

Discussion and ideas for adapting this question:

Another place to practice planning the work.
The integration requires integrating a polynomial function. Not difficult, but along with the u-substitution in part (a), you have an example to show people that students still must do algebra and find antiderivatives.
Ask students to find a general expression for the volume in terms of c and the volume of P(x).
Changing the given volume does not make the problem more difficult.

Next week 2021 AB 3/ BC 3.

I would be happy to hear your ideas for other ways to use this questions. Please use the reply box below to share your ideas.

Common Mistakes that Make Readers Pull Their Hair Out

Posted on April 20, 2021 by Lin McMullin

For years, there have been various lists of the most common mistakes on the AP Calculus exams. This one was originally drawn up by the late Ben Cornelius. Larry Peterson added some comments and posted it to the AP Calculus Community bulletin board a few days ago. Both Ben and Larry are long time AP Calculus teachers, exam readers, table leaders, and question leaders. I have revised it slightly and added some thoughts of my own starting with the first one.

This is for students; so, be sure to share it with them.

“Common Mistakes that Make Readers Pull Their Hair Out.”

BAD ALGEBRA. Yes, calculus students should know how to do algebra. The AP exams are calculus exams and the necessity for doing algebra is kept to a minimum. Algebraic simplification is not required. Once you find a derivative using the quotient rule, stop. Don’t take the chance of simplifying a correct answer and making an algebra mistake. Questions do not require lines and lines of algebra. If you are doing a lot of algebra, you have made a mistake; start over.
BAD ARITHMETIC. There is no need to simplify arithmetic. It won’t make the answer any more correct (even a long Riemann sum). If you get 2 + 3 or sin(π/6) stop – that’s good enough.
Crossed out work. Don’t cross out your work, unless you know you can do better.
Incorrect or missing units. Units are required only if specifically asked for.
Go for partial credit.: If you are worried that your result in part a) is incorrect, use it anyway to finish the problem.
Go for partial credit. When asked to write an integral, start with the limits and any constants of multiplication; that’s worth a point, even if you are not sure of the integrand.
When using a calculator, show the mathematical set-up (e.g., the definite integral); describe what you are doing clearly in mathematical terms, not in calculator speak. Calculator work is limited to the four required functionalities: graphing, roots, numerical derivative, and numerical integration. You will not be required to do anything else with your calculator and no question will be asked where using an additional feature would give an advantage (e.g., curve fitting). Anything else you do will not earn credit.
Writing bad math. For example, “slope of the derivative.” or “6.2368 = 6.237″ or (since you know the answer should be positive) “–17.241 = 17.241″)
Rounding or truncating mistakes. Remember: three or more decimal places, rounded or truncated. Do not round or truncate before the final answer (early rounding may affect the accuracy of the final answer.)
Say what you mean. Don’t write f(x) = 2(1.5) + 3 if you mean f(1.5) = 2(1.5) + 3. Don’t give a Taylor series when asked for a Taylor polynomial.
Pronouns need an antecedent. Name the function you are referring to. Do not say, “It is increasing because it is positive”, say “The function, f, is increasing, because the derivative of f is positive.” Refer to the function by name, especially if there is more than one function.
Know the difference between increasing and positive: f is increasing when f ’ is positive.
Know the difference between local and global extrema.
“Value of a function” means the y-value. Do not give an ordered pair, if only the value is asked for. Know the difference between the extreme value (y-coordinate) and the location of the extreme value (x– and y-coordinates).
Write justifications and reason in complete sentences (without pronouns). You may use mathematical symbols.
Make sure the equations flow correctly from one line to the next. Do not use stream of consciousness. Do not connect equations with equal signs, unless you are sure they are equal. (e.g., 3 x+ 12 = 0 = x + 4 = –4 makes no sense). Rather work vertically:

3x + 12 = 0

x + 4 = 0

x = –4

Here is even more advice: “How, not only to Survive, but to Prevail” is the introduction I wrote for to Multiple-choice & Free-response in Preparation for the AP Calculus (AB) Examination and Multiple-choice & Free-response in Preparation for the AP Calculus (BC) Examination published by D&S Marketing Systems, Inc. More advice and information for students about to take the AP Calculus Exams.

And good luck on this year’s exams.

Teaching Calculus

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

Category Archives: Reviewing

Linear Motion (Type 2)

AP Questions Type 2: Linear Motion

Rate & Accumulation (Type 1)

The Free-response Questions

AP Questions Type 1: Rate and Accumulation

AP Exam Review – General Suggestions

Adapting 2021 AB 4 / BC 4

2021 AB 4 / BC 4

Adapting 2021 AB 3 / BC 3

2021 AB 3 / BC 3

Common Mistakes that Make Readers Pull Their Hair Out

AP Questions Type 7: Other topics

Share this:

AP Questions Type 2: Linear Motion

Share this:

The Free-response Questions

AP Questions Type 1: Rate and Accumulation

Share this:

Share this:

2021 AB 4 / BC 4

Share this:

2021 AB 3 / BC 3

Share this:

Share this: