# 2019 CED Unit 1 – Limits and Continuity

This is the first of a series of blog posts that I plan to write over the next few months, staying a little ahead of where you are so you can use anything you find useful in your planning. Look for this series every 2 – 4 weeks.

Unit 1 contains topics on Limits and Continuity. (CED – 2019 p. 36 – 50). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions.

Logically, limits come before continuity since limit is used to define continuity. Practically and historically, continuity comes first. Newton and Leibnitz did not have the concept of limit the way we use it today. It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy and Weierstrass. But their formulation did not use the word “limit”, rather the use was part of their definition of continuity. Only later was it pulled out as a separate concept and then returned to the definition of continuity as a previously defined term.

Students should have plenty of experience in their math courses before calculus with functions that are and are not continuous. They should know the names of the types of discontinuities – jump, removable, infinite, etc. As you go through this unit, you may want to quickly review these terms and concepts as they come up.

(As a general technique, rather than starting the year with a week or three of review – which the students need but will immediately forget again – be ready to review topics as they come up during the year as they are needed – you will have to do that anyway. See Getting Started #2)

### Topics 1.1 – 1.9: Limits

Topic 1.1: Suggests an introduction to calculus to give students a hint of what’s coming. See Getting Started #3

Topic 21.: Proper notation and multiple-representations of limits.

There is an exclusion statement noting that the delta-epsilon definition of limit is not tested on the exams, but you may include it if you wish. The epsilon-delta definition is not tested probably because it is too difficult to write good questions. Specifically, (1) the relationship for a linear function is always  $\delta =\frac{\varepsilon }{{\left| m \right|}}$  where m is the slope and is too complicated to compute for other functions, and (2) for a multiple-choice question the smallest answer must be correct. (Why?)

Topic 1.3: One-sided limits.

Topic 1.4: Estimating limits numerically and from tables.

Topic 1.5: Algebraic properties of limits.

Topic 1.6: Simplifying expressions to find their limits. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.7: Selecting the proper procedure for finding a limit. The first step is always to substitute the value into the limit. If this comes out to be number than that is the limit. If not, then some manipulation may be required. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.8: The Squeeze Theorem is mainly used to determine $\underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\sin \left( x \right)}}{x}=1$ which in turn is used in finding the derivative of the sin(x). (See Why Radians?) Most of the other examples seem made up just for exercises and tests. (See 2019 AB 6(d)). Thus, important, but not too important.

Topic 1.9: Connecting multiple-representations of limit. This can and should be done along with learning the other concepts and procedures in this unit. Dominance, Topic 15, may be included here as well (EK LIM-2.D.5)

### Topics 1.10 – 1.16 Continuity

Topic 1.10: Here you can review the different types of discontinuities with examples and graphs.

Topic 1.11: The definition of continuity. The EK statement does not seem to use the three-hypotheses definition. However, for the limit to exist and for f(c) to exist, they must be real numbers (i.e. not infinite). This is tested often on the exams, so students should have practice with verifying that (all three parts of) the hypothesis are met and including this in their answers.

Topic 1.12: Continuity on an interval and which Elementary Functions are continuous for all real numbers.

Topic 1.13: Removable discontinuities and handing piecewise – defined functions

Topic 1.14: Vertical asymptotes and unbounded functions. Here be sure to explain the difference between limits “equal to infinity” and limits that do not exist (DNE). See Good Question 5: 1998 AB2/BC2.

Topic 1.15: Limits at infinity, or end behavior of a function. Horizontal asymptotes are the graphical manifestation of limits at infinity or negative infinity. Dominance is included here as well (EK LIM-2.D.5)

Topic 1.16: The Intermediate Value Theorem (IVT) is a major and important result of a function being continuous. This is perhaps the first Existence Theorem students encounter, so be sure to stop and explain what an existence theorem is.

The suggested number of 40 – 50 minute class periods is 22 – 23 for AB and 13 – 14 for BC. This includes time for testing etc. If time seems to be a problem you can probably combine topics 3 – 5, topics 6 -7, topics 11 – 12. Topics 6, 7, and 9 are used with all the limit work.

There are three other important limits that will be coming in later Units:

The definition of the derivative in Unit 2, topics 1 and 2

L’Hospital’s Rule in Unit 4, topic 7

The definition of the definite integral in Unit 6, topic 3.

Posts on Continuity

CONTINUITY To help understand limits it is a good idea to look at functions that are not continuous. Historically and practically, continuity should come before limits. On the other hand, the definition of continuity requires knowing about limits. So, I list continuity first. The modern definition of limit was part of Weierstrass’ definition of continuity.

Continuity (8-13-2012)

Continuity (8-21-2013) The definition of continuity.

Continuous Fun (10-13-2015) A fuller discussion of continuity and its definition

Right Answer – Wrong Question (9-4-2013) Is a function continuous even if it has a vertical asymptote?

Asymptotes (8-15-2012) The graphical manifestation of certain limits

Fun with Continuity (8-17-2012) the Diriclet function

Far Out! (10-31-2012) When the graph and dominance “disagree” From the Good Question series

Posts on Limits

Why Limits? (8-1-2012)

Deltas and Epsilons (8-3-2012) Why this topic is not tested on the AP Calculus Exams.

Finding Limits (8-4-2012) How to…

Limit of Composite Functions

Dominance (8-8-2012) See limits at infinity

Determining the Indeterminate (12-6-2015) Investigating an indeterminate form from a differential equation. From the Good Question series.

Locally Linear L’Hôpital (5-31-2013) Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

L’Hôpital’s Rules the Graph (6-5-2013)

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

# AP Calculus Prerequisites

College Board Prerequisites

Whenever I led a calculus workshop or APSI, I always spent a little time discussing the prerequisites for AP Calculus. Unfortunately, in some schools AP Calculus is a course for only the talented and little time is spent aligning the mathematics program and courses from 7th grade on so that more students will be able to take AP Calculus. But a program that includes the prerequisite for calculus will be a good program because of this. Such a program will also benefit students who do not take AP Calculus, but still need a good mathematics program for when they attend college.

Teachers in the earlier courses are usually appreciative of guidance from the AP Calculus teacher as to what should be included to prepare students for calculus. This is part of the rationale of the AP’s math Vertical Team program.

Below in blue is the entire prerequisite paragraph from the 2019 AP Calculus Course and Exam Description p. 7. I have separated the parts and commented on each.

Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students:

The four years is needed. Students should not be rushed.

In some respects, this is a political statement: four years means starting in 8th grade or earlier. While some of the most talented students can probably catch up by doing two years in one or three years in two, this is not the usual case. Learning math thoroughly takes four years.

Once in my district, our junior high decided to raise the standards for their “advanced” course that taught Algebra I in 8th grade. No one told us, so the next year we found only one class, instead of two, that could be ready for AP Calculus by the time they were seniors. We tried a three-years-in-two approach. It met with only limited success. Algebra I in 8th grade is required and really should be for everyone otherwise you are denying students the chance to even consider AP Calculus when they are seniors.

courses that should prepare them with a strong foundation in reasoning with algebraic symbols and working with algebraic structures.

Using and understanding the use of mathematical notation is a must. Throughout the four years, algebra and its structure should be emphasized.  So, it’s not just 4 years of math, but four years of a good algebra-based math program. But algebra is not the only thing:

Prospective calculus students should take courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions.

All these courses are related and lead to a fuller understanding of high school math topics.

These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions.

This is a list of the types of functions that should be included. They are the basic functions studied in the calculus. Linear and simple polynomial functions start in Algebra I and the others are added later. Piecewise-define functions also start early – the absolute value function is a piecewise-defined function.

In particular, before studying calculus, students must be familiar with the properties of functions, the composition of functions, the algebra of functions, and the graphs of functions.

The algebra of functions means learning how to add, subtract, multiply, divide, and compose functions and how doing so affects the properties and graphs of the resulting functions. The graphs of these functions and how doing algebra, composition, and transformations affects the graph is important.

Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and descriptors such as increasing and decreasing).

The list of the language functions is too short. Some terms such as increasing, decreasing, maximum and minimum values, concavity and others often considered the province of calculus all come up in the study of functions and can and should be discussed when they arise using the correct terminology and notation. There is no need to wait for calculus to use them to describe functions, graphs and transformations. An informal use and understanding of continuity and limits should be included. Asymptotes should not be overlooked (they are the graphical manifestation of limits and continuity or the lack of same). The more students learn before calculus, the less you’ll have to do in calculus.

Students should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions at the numbers $0,\tfrac{\pi }{6},\tfrac{\pi }{4},\tfrac{\pi }{3},\tfrac{\pi }{2}$ and their multiples.

Yes, with all the technology available these basic trig facts should be learned (learned, not just memorized); they are always tested on the AP Exams.

Students who take AP Calculus BC should have basic familiarity with sequences and series, as well as some exposure to parametric and polar equations.

Here I disagree. Parametric equations, vector equations and polar equations should be a part of the curriculum for all students. Students who do not take BC calculus, may well take more math courses in college and should understand these ways of working with the plane and with functions defined in different ways.

This list does not define the entire high school math program. There are other topics that can and probably should be included – statistics, systems of equations, linear algebra and matrices, proofs, probability to name a few. What it does define is what should be included so that students will be ready for calculus.

What I think is missing here is the use of technology. In the world today mathematics is done with technology. The proper use of technology should be an integral part of the program from before Algebra I.

AP Statistics is a great course. Students who have completed Algebra II should consider this course. However, AP Statistics it is not an algebra-based course. About three-quarters of the course and its exam is writing; there is very little algebra involved. Therefore, students should not be taking AP Statistics instead of AP Calculus, or if they are not taking calculus, instead of a third year of Algebra. The AP Statistic prerequisites state:

Students who wish to leave open the option of taking calculus in college should include precalculus [i.e. a third year of algebra] in their high school program and perhaps take AP Statistics concurrently with precalculus.

Students with the appropriate mathematical background are encouraged to take both AP Statistics and AP Calculus in high school.

AP Statistics 2019 Course and Exam Description p. 7, emphasis added.

The point is that students should not have a year in high school without an algebra course. A year in which to forget their algebra before going to college where they may need it again is not a good idea.

I like to think of all the mathematics courses before calculus as “precalculus.” In many schools, “precalculus” is the name of the last course before calculus. That’s okay, I guess. What I disagree with is that often the precalculus teacher, with the good intention of preparing their students for calculus, teaches them “derivatives.” By which they mean the rules for computing derivatives. This really does not help the students or the calculus teacher.

Derivatives are limits and derivatives are slopes; computing derivatives is the least of your worries. If students have learned all the other precalculus topics (including parametric, vector, and polar equations) well and there is time left, consider delving further into limits and continuity. Limits seem to be more difficult to understand and some repeating of the topic when students arrive in calculus will do no harm. Leave the calculus for the calculus class. (The exception is when the precalculus class is intentionally meant to get an early start on the calculus; when it is taught by the calculus teacher or a teacher who is aware of the Essential Knowledge and Learning objective of the AP Calculus course.)  – Just my opinion.

High School Prerequisites

Some high schools add their own prerequisites to enter AP Calculus courses. This usually means students have to earn a significantly higher score than just a passing grade in the precalculus course(s). I do not agree with such a policy.  It excludes students who may benefit. If your student passed the precalculus course, even with a low grade, how can you say they are not ready for calculus? What will make them more ready? True, they may have to struggle, but that won’t hurt them. You may want to council them (and their parents) and explain, without discouraging them, the amount of work and time required in a college level course like AP. Explain the amount of time and work they will have to spend once they get to college in a course that meets far fewer times then AP Calculus to cover the same material. Even if they end up without earning a qualifying score on the AP Exam, they will still benefit by putting in the time and effort. If they want to try, encourage them.

# Mathematical Practices

In March, I attending a training session given by the College Board on the new 2019 AP Calculus Course and Exam Description (2019 CED). I was impressed by the copious other materials the College Board had prepared for the roll-out that will be available at summer institutes. Among these was Mathematical Practices. The MPACs (Mathematical Practices) from the 2016 CED have been revised and condensed from six down to four. In both forms they summarize how mathematicians work, think, and communicate. Therefore, they outline what students need to learn and do when learning mathematics.

The Practices are summarized on page 13 – 14 of the 2019 CED and discussed in detail in the “Developing the Mathematical Practices” chapter (p. 214 – 220) where, included with each of the skills, are Key Questions, Sample Activities, and Sample Instructional Strategies. Each unit in the 2019 CED starts with a short discussion of the Mathematical Practices that apply to that unit.

While the Practices are listed with examples specifically for the AP Calculus courses, they really apply to the entirety of a student’s mathematical learning and thinking from grade school on. If your school district has a Math Vertical Team, an ongoing discussion of the Practices is certainly an appropriate topic. Otherwise, share them with the teachers from the lower grades and sending schools. They are relevant at all grade levels.

One thing you can do to help students with the Practices is to make and keep them aware of them. Put them on a poster in the room. Make a handout of pages 13 and 14 for the front of their notebook. Refer to them whenever you use one of the items on the list.

The practices are these. (I have slightly edited them to remove the numbering and the calculus-specific examples.) My thoughts and comments are below the quotes.

Practice 1: Implementing Mathematical Processes – Determine expressions and values using mathematical procedures.

• Identify the question to be answered or problem to be solved.
• Identify key and relevant information to answer a question or solve a problem.
• Identify an appropriate mathematical rule or procedure based on the classification of a given expression.
• Identify an appropriate mathematical rule or procedure based on the relationship between concepts or processes to solve problems.
• Apply appropriate mathematical rules or procedures, with and without technology.
• Explain how an approximated value relates to the actual value.

The first Practice really describes the problem-solving process. This Practice is applicable throughout a student’s study of mathematics from grade school on.

The first two bullets while marked as “not assessed [on the AP Calculus exams]” are the beginning of the problem-solving process. The next two are how you start the work of problem solving, and the fifth applies to carrying out the rules and procedure you’ve decided upon. The last needs to be considered whenever your answer is not exact – which may be most of the time.

Practice 2: Connecting Representations – Translate mathematical information from a single representation or across multiple representations.

• Identify common underlying structures in problems involving different contextual situations.
• Identify mathematical information from graphical, numerical, analytical, and/or verbal representations.
• Identify a re-expression of mathematical information presented in a given representation.
• Identify how mathematical characteristics or properties of functions are related in different representations.
• Describe the relationships among different representations of functions ….

Multiple representations, often called the “Rule of Four”, help one see and delve deeper into mathematical situations. Graphs, tables, and symbolic expressions representing the same thing show different ways of expressing and understanding mathematical ideas. Expressing the relationships in words by writing, talking, discussing, and arguing about them helps students understand and internalize the mathematics (see Practice 4). Technology is invaluable in doing this.

All four should be considered in every situation and for every concept. Sometimes one is more informative and useful than the others, other times a different perspective sheds additional light on the concept. And, once again, this should be done from the beginning of a student’s mathematical career.

Practice 3: Justification – Justify reasoning and solutions

• Apply technology to develop claims and conjectures.
• Identify an appropriate mathematical definition, theorem, or test to apply.
• Confirm whether hypotheses or conditions of a selected definition, theorem, or test have been satisfied.
• Apply an appropriate mathematical definition, theorem, or test.
• Provide reasons or rationales for solutions and conclusions.
• Explain the meaning of mathematical solutions in context.
• Confirm that solutions are accurate and appropriate.

Technologies (in the broad sense of anything other than paper and pencil: blocks, beads on wires, and other manipulatives in grade school, to computer programs, spreadsheets, CAS, and Oh BTW, graphing calculators) are an increasingly important tool for mathematicians. Technology should be incorporated at all grades and levels. Students should learn how to use them no only to do and check their work, but also to explore mathematics and discover mathematical ideas (even if these are already known to more advanced students).

Definitions and theorems formalize the results of mathematical exploration and point the way to other discoveries. Students should become familiar, not just with a few theorems and definitions, but with the structure of them and relationships between them (converses, inverses, and contrapositives). They need to know that if the hypotheses are true, then the conclusion is true. They need to be able to show (confirm) that the hypotheses are true before they apply a theorem or definition to a given situation.

In early grades, stating theorem formally is not always necessary or desirable. Still, students should be aware that there are certain rules (which after all are theorems) and they may be used only when appropriate. I’ve often told students that in real life you can do whatever you want unless there is a law saying you can’t, but in mathematics you can’t do anything unless there is a law that say you can.

Part of the problem-solving process in Practice 1 should include making sure your result makes sense in context. That means student mathematicians need to understand the meaning of their results and be able to confirm that the work and the solution are accurate and appropriate. Explaining this verbally to other and in writing, a communication skill from Practice 4, is a way to do this. This can be does at all grade levels.

The previous MPACs from the 2016 CED list “Students can … analyze, evaluate, and compare the reasoning of others.” (MPAC 6f.) At all levels, this is one way to have students confirm and explain their results and understanding.

Practice 4: Communication and Notation – Use correct notation, language, and mathematical conventions to communicate results or solutions.

• Use precise mathematical language.
• Use appropriate units of measure.
• Use appropriate mathematical symbols and notation
• Use appropriate graphing techniques.
• Apply appropriate rounding procedures.

As we’ve all learned early in our teaching careers, after teaching a topic two or three times we understand it much better. We see the fine points and appreciate the connections. It was that communication, the teaching of it, that helped us understand it. Activities where students communicate help them understand as well.

The items under Practice 4, are important because communication with others orally and in writing will help your students learn and understand mathematics. To use the language of mathematics, students need to know the structure of mathematical reasoning (return to Practice 3 – theorems and definitions), and the tools for doing so (notation, units, etc.). At all grade levels, students should practice in communicating and using the language and notation – this will help them learn.

Take a good look at the Mathematical Practices and incorporate them into your thinking and teaching. Help your students look at what they are doing, to look at the big picture. It will help with the details.

# Pacing for AP Calculus

Some thoughts on pacing and planning your year’s work for AP Calculus AB or BC.  The ideas are my own and are only suggestions for you to consider.

Almost all textbooks provide an AP pacing guide among their ancillary material. You can consult the guide for your book for specific suggestions for the number of days on each topic or section.

Keep a copy of the latest Course and Exam Description handy. Changes in the exam are announced in this book; to keep up to date be sure you always read the following year’s edition which is available at AP Central shortly after the exam is given in May. The book contains the “Topical Outline” for the AB and BC courses. The topics listed here are what may be tested on the exams. What is not listed will not be tested. For example, calculating volumes by the method of Cylindrical Shells is not listed; any volume problem on the exam can be done by other methods. This does not mean you may not or should not teach the topics that are not listed if you believe your students will benefit from them. If you wish to teach them you may still do so. Students may use these methods on the exam; they will not be penalized for correct mathematics. Many teachers teach these topics in the time after the exam.

Get out your school calendar. The AP Calculus exams are usually given during the first week in May; the exact date will be at AP Central.

• Count back about 2 school weeks from the exam date (don’t count your spring break week). Allow an extra week if you are prone to many snow days. This time will be used for review. (This brings you to a week or so into April.)
• Count back two more weeks. I’ll discuss what this should time should be used for later. (Mid-march) This is when you should aim to be done the material and ready to begin review. Finishing by the beginning of March is even better.
• Count the number of weeks between the beginning of school and the week above. (About 26 – 27 weeks if your start just after Labor Day; 28-30 weeks if you start in mid-August). This is the number of week you have to teach the material. Don’t panic: the AB course is taught typically in college in 30 – 35 classes in one semester. You do have time, but by the same token, you still need to stick with the calendar and keep you students on it as well.
• Take half of this number and find the middle week of the year. This is sometime in early to mid-December. To allow equal time for derivatives and integrals, this is when you should finish derivatives and start integration. Don’t delay starting integration beyond the first class of the New Year.
• Now plan your work so that you can do it in the time allowed. You all want your students to do well. It is not unknown for teachers to spend a few extra days now and then to give extra work on derivative. But this time adds up. Remember half the exam is integration; you need to cover that too. Don’t get behind.
• If you are in an area where there are closings due to weather or other reasons, plan for them. You usually get some short warning that snow is coming. Be ready on short notice to post an assignment, a video to watch, or some other useful work on your website. If it looks like several days off, tell the students you will post the assignment daily and make them responsible for finding them and doing them.

Look over past exams. Learn what is tested and how it is tested and plan your time accordingly. Here are some hints as to where you can shave some time.

STARTING THE YEAR

• Summer assignments: Personally, I do not see the use in summer assignments. What is their purpose? To keep the material fresh in the kids’ minds, I suppose. But the good students will do it right away and then forget anyway over the summer, and the others, will forget “everything” over the summer and try the assignment at the end of the summer and get nowhere.
• If you want to keep their minds on mathematics over the summer, assign a good book to read. Maybe they will spread that out over the summer. Reading suggestion: Is God a Mathematician? by Mario Livio.
• Ideally, limits and continuity should be taught in pre-calculus. Work with your pre-calculus teachers and help them arrange their curriculum so that the things students need to know coming into calculus are taught in pre-calculus. This is one of the things vertical teaming can accomplish. (Incidentally, be sure they do not start learning about derivatives and the slope of tangent lines in pre-calculus as some textbooks do; the time is better spent elsewhere.) Remember the delta-epsilon definition is not tested and is optional.
• DO NOT begin the year with a week or two (or even a day or two) of review of mathematics up to calculus. It won’t help. Later in the year when you get to one of those topics students “should” know, they will have forgotten it all over again. So instead of a week or two (or more) of review at the beginning of the year, plan 10 – 15 minutes of review when these topics come up during the year. (You’ll have to do this anyway.)
• If the first chapter of your textbook is review, as most are, skip this chapter. Make your first night’s assignment to read this chapter and ask about anything they don’t remember. This chapter can be used for reference when necessary later in the year.
• Do begin the year with derivatives (or limits and continuity if students have not studied this before). The very fact that this is new will help get and retain the students’ interest.

DERIVATIVES

Here are some places you may shave a few days off while teaching derivatives:

• Computing derivatives is important. Product rule, Quotient rule, Chain rule are all tested on the exam. But look at some past exams: the questions are not that complicated. It is rare to find “monster” problems involving all three rules together along with radicals and trig functions. Sure, give one or two of those, but the basics are what are tested. Furthermore, you can and should include these all thru the year, so students stay in practice.
• Optimization problems: Building a cheaper box or fencing in the largest field with a given amount of fence are great problems. They do not appear on the AP exams (at least not since 1982). They do not appear because the hard part is writing the model (the equation); if a student misses this they cannot earn anymore points in the problem. If these problems were on the exam, missing the equation means the student could not go on and cost the student all 9 points on a free-response question. Finding maximums and minimum, which require the same calculus thinking and techniques, are tested in other ways. On the multiple-choice section, optimization questions, if any, are of the easiest sort. The model may even be given, and there will  be no more than one such question. Spend only a day or two on the modeling.
• Related Rate problems: These questions do appear on the exams. A multiple-choice question on related rates may appear. As with any multiple-choice question it cannot be too difficult. Every few years a related rate question shows as part of a free-response question. You cannot cut this out completely, but you can shave some time off here if you are short of time.
• Practice the differentiation skills, and later the antidifferentiation skills, and the concepts associated with derivatives by including them on all your tests. Make all tests cumulative from the beginning of the year; just a random question or two will keep them on their toes.
• Look for and assign differentiation problems based on graphs and tables of values in addition to the usual analytic (equation) questions. Use your textbook; however, some textbooks are rather thin on questions with tables and graphs in the stem. Use released exams or a review book for sources.

INTEGRALS

• As with derivatives, the finding of antiderivatives is important, but the antiderivatives, definite and indefinite integrals are not very difficult. There are no trig substitution integrals, and nothing too monstrous. Integration by Parts is only on the BC exam.  Give students lots of practice spread over the second half of the year.
• Trapezoidal Rule is not really tested on the exams. Students do not need to know the formula or the error bound formula for the Trapezoidal Rule. Questions do ask for a “trapezoidal approximation.” Like the left-, right; and midpoint-Riemann sums approximations, these questions can be answered by actually drawing a small number of trapezoids and computing their areas. This should be done from equations, graphs and tables. This tests the concept and often the graphical interpretation, not the mindless use of a formula. Error analysis is tested based on whether the approximating rectangles or trapezoids lie above or below the graph. Simpson’s Rule is not tested.
• Look for and assign integration problems based on graphs and tables of values in addition to the usual analytic (equation) questions. Use your textbook, released exams or a review book for sources.

THOSE TWO WEEKS BEFORE THE REVIEW STARTS

The free-response and the multiple-choice sections of the exam contain some questions very similar to questions that are in textbooks and in contiguous sections of the textbook. These include:

The free-response and the multiple-choice sections contain some questions that are very different from questions that are in textbooks. This is because these questions are on topics from different parts of the year (limit, differentiation and integration concepts in the same question), and these questions are just not asked in the same way in textbooks. These include:

• Rate/accumulation questions
• Graph Analysis Differentiation and integration questions about a function given the graph of its derivative and functions defined by integrals
• Motion on a line (AB), or motion in a plane (BC – parametric and vector equations)
• Polar Equations (BC only)
• Questions, both differentiation and integration, given a table of values.
• Overlapping topics in the same question such as a particle motion question based on a graph or table stem, or a question about an important theorem based on value in a table.

The topics in this latter list pull the entire year’s work together. At first students find this disconcerting since they have rarely seen questions like these; so be sure they do see them before the test. Use these two weeks to pull these topics together and get your students thinking more broadly. This will lead naturally into the full-scale review; in fact, some of this work may profitably spill over into the review time.  Spend 2 – 3 days on each type using actual AP questions for each so the students can see the different variations on the same idea, and the different ways the same idea can be tested. (This is preferable starting the review with one complete free-response exam with 6 different type questions to do. However, later in the review you should do this.)

Another way to approach these problems is to include parts of them throughout the year as the students learn the topics tested in each part. Released multiple-choice problems can be used for this purpose as well.

THE REVIEW TIME

Once the students are familiar with the style of questions, give them a mock exam. For the multiple-choice questions use one of the released exams or one of the genuine-fake exams in a good review book. Give the free-response questions from a recent year. If possible, give the mock exam under the same conditions and timing as the exam. This can be done on a Saturday. If you cannot get 3.25 hours in a row, then give the parts with their proper timing during class periods. Grade the exam according to the standards which are available at AP Central.  Teach them some good test taking strategies.

Spend a fair amount of time doing multiple-choice questions. The released exams from 1998, 2003 and 2008, 2012 and 2013 (and soon 2014) are available. You can also use questions from a good review book (AB or BC). Pay attention to the style and wording, as well as the concepts tested.

Make your calendar up in advance and stick to it. You won’t help the students by getting behind; in college they will have to go a lot faster than in high school. Help them get used to it.

I hope this helps you get started and keep a proper pace through the year.

Revised and updated June 6, 2021