Category Archives: AP Calculus Exams

AP Exam Review

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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. There are not new; versions have been posted for the last few years and these are only slightly revised and updated. A schedule for the dates of the posts appears at the end of this post. My posts are intentionally scheduled before you will probably be needing them, so you can plan ahead. Most people start reviewing around the beginning or middle of April.

Exams for AP Calculus are scheduled for Tuesday May 14, 2019 at 08:00 local time.

Ideas for reviewing for the AP Exam

Part of the purpose of reviewing for the AP calculus exams is to refresh your students’ memory on 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 a good way to accomplish some of 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. For a list of the types see the posting schedule at the end of this post. Plan to spend a few days doing a selection of questions of one type so that student can see how that type question can be used to test a variety of topics. 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.

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. 

Simulated Exam

Plan to give a simulated (mock) exam. Each year the College Board makes a full exam available. The exams for 1998, 2003, 2008, and 2012 are available at AP Central  and the secure 2013 – 2017 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 avail online. Students can find them easily. For suggestions on how to handle this see Practice Exams – A Modest Proposal. 

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.

For the AB exam, the minimum points required for each score out of 108 point 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 are not what is important. What is important is that students to know is that they can omit or get wrong many questions 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 number of 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.

Directions

Print a copy of the directions for both parts of the exam and go over them with your students. Especially, for the free-response questions explain 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: AB Directions and BC Directions. 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. Emphasize  the need to clearly show their work and justify their answers, and the three-decimal accuracy rule. This rule and lots of other information is explained in detail in this article: How, not only to survive, but to prevail. Copy this article for you students!

Schedule of future posts for reviewing for the 2019 Exams

Exams for AP Calculus are Tuesday May 14, 2019 at 08:00 local time

  • Tuesday February 26, 2019 – Resources for reviewing
  • Tuesday March 5, 2019 – Type 1 questions – Rate and accumulation questions
  • Friday March 8, 2019 – Type 2 questions – Linear motion problems
  • Tuesday March 12, 2019 – Type 3 questions – Graph analysis problems
  • Friday 15, 2019 – Type 4 questions – Area and volume problems
  • Tuesday Match 19, 2019 –  Type 5 questions – Table and Riemann sum questions
  • Friday March 22, 2019 – Type 6 questions – Differential equation questions
  • Tuesday March 26, 2019 – Type 7 questions – miscellaneous
  • Friday March 29, 2019 – Type 8 questions – Parametric and vector questions (BC topic)
  • Tuesday April 1, 2019 – Type 9 questions – Polar equations
  • Friday April 5 – Type 10 questions – Sequences and Series

 

 

 

 

Revised for 2019

Integration

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Integration – DON’T PANIC

As I’ve mentioned before, I try to stay a few weeks ahead of where I figure you are in the curriculum. So here. early in November, I start with integration. You probably don’t start integration until after Thanksgiving in early December. That’s about the midpoint of the year. Don’t wait too much longer. True, your kids are not differentiation experts (yet); there will be plenty of differentiation work while your teaching and learning integration. Spending too much time on differentiation will give you less time for integration and there is as much integration on the test as differentiation.

The first thing to decide is when to teach antidifferentiation (finding the function whose derivative you are given). Many books do this at the end of the last differentiation chapter or the first thing in the first integration chapter. Some teachers, myself included, prefer to wait until after presenting the Fundamental Theorem of Calculus (FTC). Still others wait until after teaching all the applications. The reasons  for this are discussed in more detail in the first post below, Integration Itinerary.

Integration itinerary – a discussion of when to teach antidifferentiation.

The following posts are on different antidifferentiation techniques.

Antidifferentiation u-substitution

Why Muss with the “+C”?

Good Question 12 – Parts with a Constant?

Arbitrary Ranges  Integrating inverse trigonometric functions.

Integration by Parts I (BC only)

Good Question 12 – Parts with a Constant  How come you don’t need the “+C”?

The next three posts discuss the tabular method in more detail. This is used when integration by parts must be used more than once. If memory serves, using integration by parts twice on the same function has never shown up on the AP exams. Just sayin’.

Integration by Parts II (BC only) The Tabular method.

Parts and More Parts   (BC only) More on the tabular method and on reduction formulas

Modified Tabular Integration  (BC only) With this you don’t need to make a table; it’s quicker than the tabular method and just as easy.

 

 

 

 

Revised and updated November 4, 2018

On Grading

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I got to thinking about grading the other day after seeing a question on Facebook. We’ll get to that question in a minute, but first I want to try to outline a grading scheme I used towards the end of my teaching career. It is based on how free-response questions on AP Calculus exams are graded, but the ideas are usable in any course. Here are some suggestions and examples of how to do that. There are also some suggestions for grading multiple-choice and True-False questions.

Free-response questions

The AP Calculus scoring standards are considered as a guide for awarding partial credit. Partial credit is earned for taking correct steps on the way to the solution. Points are earned, not deducted. Examples that follow will expand on these principles:

  • Each step is worth 1 or 2 points. For 2-point steps, it must be possible to earn only one point.
  • Students earn the point(s) for showing they are doing a good thing.
  • Once earned, the point cannot be lost by some later mistake. (It’s “in the bank,” as readers say.)
  • Since a mistake will affect the final answer, the student may earn later points, including the answer point, for continuing correctly. However, some mistakes are so bad that earning the rest of the points is not possible. Mistakes must not simplify the remaining work.
  • The standard must allow for different methods of solution.

Example 1: Consider a typical volume problem worth four points. Students are required to write a definite integral. By the washer method the work should look like

\pi {{\int_{a}^{b}{{\left( {f(x)} \right)}}}^{2}}-{{\left( {g\left( x \right)} \right)}^{2}}dx= a numerical answer.

  • 1-point is earned for the constant and both limits of integration.
  • 2-points are earned for the integrand. If the integrand is of the form something squared minus something else squared, they earn 1-point; if both correct quantities are squared they earn the second point. If the integrand is something squared plus something else squared this is considered a calculus (major) mistake they earn 0 points and are not eligible for the answer point. (No deduction for a missing dx)
  • 1-point for the answer from their calculator. Saying that the correct answer is equal to an incorrect integral such as \pi {{\int_{a}^{b}{{\left( {g(x)} \right)}}}^{2}}-{{\left( {f\left( x \right)} \right)}^{2}}dx= the correct answer is a mistake (negative = positive) and does not earn the answer point. However, the reversed integrand and the correct answer not connected by an equal sign recoups the integrand point and earns the answer point (i.e. full credit). (Subtracting in the wrong order and taking the opposite of your (negative) answer is a correct algorithm, even if inefficient and “ugly.”)

You can see how much consideration goes into setting the grading standards.

There is no reason you must use the exact AP exam standard. In your class you may want to be more specific in hopes of helping your students be more precise. You may make this question worth more points. So, you could use this standard:

  • 1-point for the \pi .
  • 2-points for the limits of integration (one each)
  • 1-point for the form of the integrand (square minus square)
  • 1-point for the first squared quantity
  • 1-point for the second squared quantity
  • 1-point for dx
  • 1-point for the answer from their calculator

Example 2: An example from Algebra 1. Find the solution of 4x-\left( {x-3} \right)=x+7. The expected solution is

4x-x+3=x+7

3x+3=x+7

2x=4

x=2

You could count this as 3-points:

  • 1-point for removing parentheses
  • 1-point for collecting like terms
  • 1-point for the answer

Or you could count it as

  • 1-point for knowing to remove parentheses
  • 1-point for removing parentheses correctly
  • 1-point for collecting the x-terms
  • 1-point for collecting the constant terms
  • 1-point for the answer – any arithmetic mistakes in collecting terms fails to earn the answer point

Example 3: from Algebra 1: Solve {{x}^{2}}-8x=9

Expected solution:

{{x}^{2}}-8x-9=0

\left( {x-9} \right)\left( {x+1} \right)=0

x=9\text{ or }x=-1

As a 3-point standard

  • 1-point for setting equal to zero
  • 1-point for factoring
  • 1-point for answers

Or a 5-point standard

  • 1-point for setting equal to zero
  • 2-points, one for each correct factor
  • 2-points, one for each answer.

However, whatever method you use should allow for a solution by quadratic formula, or completing the square, or even by graphing. (Unless the direction specifically read “Solve by factoring.”)

Example 4: This is the question that got me started on this post. It is from a September 20, 2018 post on the AP Calc TEACHERS – AB/BC Facebook page. The teacher’s question is at the top.

The teacher is right about being concerned with proper notation and right about requiring students to use it.

On an AP exam this limit would be a multiple-choice question (see below) and so notation does not enter in. Even on a free-response question – judging from past exams – only the answer would be required. Just because it’s an AP class, does not mean that you must do things only as they are done on the exams.

To provide for notation, this could be scored as a 3- or 4-point question:

  • 1 point for knowing what algebra to use to find the limit
  • 1-point for doing the algebra correctly (For a 3-point value, this could be included in the answer point, but there is a fair amount to do and it’s not straightforward, so an additional point here is reasonable.)
  • 1-point for the answer (If the student does not earn both of the first two points, then the answer should agree with their work.)
  • 1-point for correct use of limit notation throughout the problem.

The student in the example does not earn the last point. The solution shown earns 3 of 4 points (or 2 of 3).

But there is more here. Suppose the student did not write “lim” in the second through fifth lines of the solution; there is no reason the must since they are just doing some algebra. Also, the lines are not connected with equal signs. Then, he or she has not misused the notation and should earn full credit.

Some comments on the Facebook post were also concerned about dividing out the x’s and not mentioning that x\ne 0. If that is a concern for you, then another point could be included for that.

So, the idea is to be very precise about what earns a point and what does not. Seeing a “+3” instead of a “-1” next to their work encourages students. Noticing that a number of your students are not earning the same point will help you see where the class is confused. One of the reasons you give tests is to help you see where your class as a whole is missing some idea. Consider that when giving multiple-choice and True-False questions.

 Multiple-choice questions – forget scan sheets.

As a teacher, you need to see the students work, so you can find their mistakes and help them do better (a/k/a formative assessment). When giving a multiple-choice question, require students to show their work and award partial credit for incorrect answers. Two- or three-points seem to work well – one-point for knowing what to do, one-point for doing it, and one-point for the answer.

Examples:

  • Find where a function is increasing: one-point for knowing to examine the derivative, one-point for finding the derivative, one point for the answer.
  • Find the acceleration: one-point for finding the derivative of velocity, one-point for the answer.
  • Questions with statements I, II, and III: one-point for each statement identified correctly as true or false. (Think of the answer “I only” as T,F,F etc.)
  • Set up the integral: one-point for limits of integration, two-points for integrand (Algebra/notation mistake loses one point, calculus mistake loses both points).

True or False questions

For many years I used a textbook (I think it was Larson and Hostetler 2nd edition – I’m showing my age) that had True-False questions for each set of exercises. I really liked them. Newer textbooks rarely have them. One exception is the new Calculus for AP by Steward and Kokoska that starts almost every exercise set with a few True-False questions.

For two-points, have students say if the statement is true or false, AND require them to explain why the statement is true or false: what theorem or idea is illustrated; what hypothesis is not met, give counterexamples, etc.

Another approach to True-False questions is to change them to Always, Sometimes, or Never True questions.

For example: Is the statement “If f ”(a) =0, the (a, f(a)) is a point of inflection,” sometimes, always, or never true?

Answer: Sometimes. If f(x) = x3, then (0,0) is a point of inflection, but if f(x) = x4, then (0,0) is not a point of inflection.

Another answer: Sometimes: If f ”(x) changes sign at (a, f(a)), then (a, f(a))  is a point of inflection, if f ”(x) does not change sign at (a, f(a)) , then it is not a point of inflection.

You can then have the class discuss and criticize each other’s answers. These become good writing questions and good preparation for the “Justify your answer” and “Explain your reasoning” questions on the AP exams. For you, they help you see what the student is thinking and, if wrong, help them correct it (a/k/a formative assessment again).

Next Friday some thoughts On Scaling – and all tests are scaled.

Get(ting) ready

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August and the school year is about to start!

As I did last year, this year my weekly posts will point you to previous posts on topics that will be coming up a week or two later. I try to stay a little ahead of you so you’ll have time to read them and incorporate what you feel is helpful into your plans. I will occasionally write some new posts as ideas come to me. (You could help them come by sending them. Send your questions and suggestions to LinMcMullin2@gmail.com)

Resources

First, here are some suggestions on pacing.

The Course and Exam Description

  • The Course and Exam Description  (CED)This is the official course description from the College Board. The individual list of topics that are tested on the exams (the Concept Outline) begins on page 11 and are listed in the Essential Knowledge (EK) column along with its Big Idea (BI), and Learning Outcome (LO) . Also, you will find the Mathematical Practices (MPACs) starting on page 8. These apply to all the topics.

 

 

  • To help you organize all this see my post on Getting Organized using Trello boards. A board listing all the Essential Knowledge and MPAC items are included.

Exam Questions

AP Calculus teachers should have a collection of the past AP Exams handy. Use them for homework, quizzes, and test through the year. Study them yourself to understand the content and style of the questions. Here are some places to find them:

  • The College Board has “home pages” for each course with links to past exams and other good information. AB Home Page and BC Home Page.

 

  • Another good reference is Ted Gott’s free-response question index and his MC unsecure Index by topic 1998 to 2018 The indices reference all the released free-response and multiple-choice questions. They are Excel spreadsheets. Each question is referenced to its Key Idea, LO and EK and includes a direct link to the text of the question. Click on the drop-down arrow at the top of each column and choose questions exactly on the EK you want to see. Ted plans to update this after the new multiple-choice questions are released. I will let you know when and where it is available. Thank you again, Ted!

 

  • I have an index of a different sort. It lists the ten Type Problems and which question, multiple-choice and free-response, that are of each type. You can find it here. This will be updated when the 2018 exams become available.

 

  • Past free-response questions that have been released along with commentary, actual student samples, and data can be found at AB FRQ on AP Central and here BC FRQ on AP Central. Be aware that these are available to anyone including your students.

 

  • Multiple-choice questions from actual exams are also available. The 2012 exam in the blue box on the course home pages (see above). This is open to anyone including students. More recent exams can be found at your audit website under “secure document” on the lower left side. This must be kept confidential because teachers use them for practice exams – they may not be posted on-line, on your school website or elsewhere, or even allowed out of your classroom on paper. Unfortunately, some teachers have not obeyed these rules and the exams can be found online by students with very little effort. Be aware that, nevertheless, your students may have access to the secure questions. For my suggestion on how to handle that see A Modest Proposal.

The AP Calculus Community

  • Finally, if you are not already a member, I suggest you join the AP Calculus Community. We are fast approaching 17,000 members all interested in AP Calculus. The community has an active bulletin board where you can ask and answer questions about the courses. Teachers and the College Board also post resources for you to use. College Board official announcements are also posted here. I am the moderator of the community and I hope to see you there!

Have a great year!

PS: Here is a link to some precalculus topics that come up in calculus

Good Question 18: 2018 BC 2(b)

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In this post we look at another part of the AP Calculus BC exam. Good Question 15 discussed the unusual units in 2018 BC 2(a). In this post we look at 2018 BC 2(b) where units help us find the correct integral to answer the question.

How do you answer a question of a type you’ve never seen before? I expect that’s what many of the students taking the 2018 AP Calculus exam were asking when they got to BC 5. If you’ve never done a density question how do you handle this one? 

The question concerns density. Density gives you how much of something exists in a certain length, area, or volume.  Density questions have appeared on the exam now and then, most recently 2008 AB 92 (which really isn’t recent, but then there are a lot of questions we never see). I have a blog post about the density here with several examples. In that post the alternate solution to example 3 explained how I used a unit analysis to find the answer; I used a similar approach here.

2018 BC 2 (b)

The stem of the question tells us that at a depth of h meters, 0 < h < 30, the number of plankton in a cubic meter of sea water is modeled by p\left( h \right)=0.2{{h}^{2}}{{e}^{{-0.0025{{h}^{2}}}}} million cells per cubic meter. Part (b) asks for the number of million of plankton in a column of water whose horizontal cross sections have a constant area of 3 square meters.

If the density were constant, then it is just a matter of multiplying the volume of the column times the constant density. Alas, the density is not constant; it varies with the depth. What to do?

Since an amount is asked for, you usually look around for a rate to integrate. Density is a kind of rate: the units are millions of cells per cubic meter. You need to integrate something concerning the density so that you end up with millions of cells; something that will “cancel” the cubic meters.

Consider a horizontal slice thru the column at depth h meters. While I’m not sure plankton is a good topping for pizza, you could picture this as a rather large pizza box whose sides are \sqrt{3} meters long and whose height is  \Delta h meters. This box has a volume of 3 \Delta h cubic meters. For small values of \Delta h the number of million plankton in the box is nearly constant, so at depth hi , there are p(hi) million plankton per cubic meter or {3p\left( {{{h}_{i}}} \right)\Delta h} million plankton in the box.

Notice how the units of the individual quantities combine to assure you the final quantity has the correct units:

\displaystyle (3\text{ square meters)}\cdot \left( {p\left( {{{h}_{i}}} \right)\text{ }\frac{{\text{million plankton}}}{{\text{cubic meters}}}} \right)\left( {\Delta h\text{ meters}} \right)=3p\left( {{{h}_{i}}} \right)\Delta h\text{ million plankton}

Now to find the amount in the column of water we can add up a stack of “pizza boxes.” The sum is \sum\limits_{{i=1}}^{n}{{3p\left( {{{h}_{i}}} \right)\Delta h}}. Now, if we take thinner boxes by letting \Delta h\to 0, we are looking at a Riemann sum. And calculus gives us the answer.

\displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{n}{{3p\left( {{{h}_{i}}} \right)\Delta h}}=\int_{0}^{{30}}{{3p\left( h \right)dh}}\approx 1,675 million plankton in the column of water (rounded to the nearest million as directed in the question.)

Previous Good Questions can be found under the “Thru the Year” tab on the black navigation bar at the top of the page, or here.

Renumbered 3-14-24 Was Question 16.

Good Question 15: 2018 BC 2(a)

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My choices for the Good Question series are somewhat eclectic. Some are chosen because they are good, some because they are bad, some because I learned something from them, some because they can be extended, and some because they can illustrate some point of mathematics. This question and the next, Good Question 16, are in the latter group. They both concern units. They both are taken from this year’s AP calculus BC exam; both are suitable for AB classes. In this question 2018 BC 2(a) has some unusual units and in the next 2018 BC 2(b) the units help you figure out what to do. Part (c) concerns an improper integral and pard (d) is about parametric equation, neither of these are AB topics. 

2018 BC 2(a)

2018 BC 2 gave an equation that modeled the density p(h) of plankton in a sea in units of millions of cells per cubic meter, as a function of the depth, h, in meters.  Specifically, p\left( h \right)=0.2{{h}^{2}}{{e}^{{-0.0025{{h}^{2}}}}} for 0\le h\le 30. Part (a) asked for the value of {p}'\left( {25} \right) and also asked students “Using correct units, [to] interpret the meaning of {p}'\left( {25} \right) in the context of the problem.”

Plankton

This was a calculator active question, so the computation is easy enough: {p}'\left( {25} \right)=-1.17906

Now units of the derivative are always very easy to determine; this should be automatic. The derivative is the limit of a difference quotient, so its units are the units of the numerator divided by the units of the denominator. In this case that’s millions of cells per cubic meter per meter of depth.

While “millions of cells per meter to the fourth power” is technically correct and will probably earn credit, what is a meter to the fourth power?

It is similar to the better-known situation with velocity and acceleration. I never liked the idea of saying the acceleration is so many meters per square second. What’s a square second? Are there round seconds? Acceleration is the change in velocity in meters per second per second; that is, at a particular time the velocity is changing at the rate of so-many meters per second each second. 

Returning to the question, a cubic meter (volume) and a meter of depth (linear) are not things that you should combine. The notational convenience of writing meters to the fourth power hides the true meaning. So, a better interpretation is “At depth of  25 meters, the number cells is decreasing at the rate of 1.179 million cells per cubic meter per meter of depth.” or “The number of cells changing at the rate of -1.179 million cells per cubic meter per meter of depth.”

Had the model been given using volume units such as millions of cells per liter, then the units of the derivative would be millions of cells per liter per meter. That makes more sense.

But what does it mean?

Let’s look at the graph of the derivative. The window is 0 < h < 30 and –2.5 < p(x) < 2.5

It means, that as we pass thru that thin (thickness \Delta h\to 0) film of water 25 meters down, there are approximately 1.179 million cells per cubic meter less than in the thin film right above it and more than in the thin film right below it.

For reference, p\left( {25} \right)\approx 26.2014 million cells per cubic meter. Of course, that thin (thickness \Delta h\to 0) film of water has very little volume; it is kind of difficult to think of a cubic meter exactly 25 meters below the surface (maybe a cube extending from 24.5 meters to 25.5 meters?). As \Delta h\to 0 does a cubic meter approach a square meter?

The cubic meter above h = 25 has \displaystyle \int_{{24}}^{{25}}{{p\left( h \right)(1)dh}}=26.763 cells and the cubic meter below has 25.586 million cells. This is a decrease of 1.1767 million cells. So, the derivative is reasonable.

(To make the units of \displaystyle \int_{{24}}^{{25}}{{p\left( h \right)(1)dh}} correct, I included a factor of 1 square meter, this multiplied by p(h) million cells per cubic meter and by dh in meters give a result of millions of cells. More on why this is necessary in Good Question 16 on density.)

Previous Good Questions can be found under the “Thru the Year” tab on the black navigation bar at the top of the page, or here.

NCTM Calculus Panel Discussion

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The annual AP Calculus Panel Discussion at the NCTM Annual meeting was held on Saturday April 28, 2018. The principal speaker was Stephen Davis, the chief reader for calculus. Stephen has made his slides available for anyone who is interested. The slides are here http://www.ncaapmt.org/archive/crTalks/nctm-apr2018-notes.pdf .

The items highlighted in blue are the ones Stephen discussed in detail. Thank you to Stephen for making them available to everyone. The last slide for each of the 9 questions contains comments on the scoring of the question.

Here are a few notes I took at the meeting about specific problems from the 2017 operational (Main US) exam:

  1. AB3/BC3 (d) Avoid words like “pointy” it is better to discuss the one-sided limits.
  2. AB4/BC4 (b) Students need to be able to jump into the middle of the problem. Some students solved the differential equation, then differentiated the answer to get to the equation that was given.
  3. AB5 Sign charts appear on the standards. This is not a change; sign charts are excellent ways to organize information. However, sign charts should not be used as justifications; readers want students to write about what they the sign chart tells them.
  4. AB 6 (d) Justify by showing (saying) that the hypotheses of the theorem or definition are met.
  5. BC 2 (c) Students had trouble understanding that w(theta) = g(theta) – f(theta) was. They seemed not to understand what g and w represented graphically.
  6. BC 5 (d) Either the integral test or the limit comparison test may be used. Students need to state the conditions of whichever test they use.
  7. Communication is becoming more important in all questions.

Teachers should look at and study the “Chief Reader Report” that is available for each exam on the same page as the questions and scoring standards at AP Central. The sample student responses are also helpful in understanding what is and is not a good response.