Graphing Integrals

The sixth in the Graphing Calculator / Technology series

The topic of integration is coming up soon. Here are some notes and ideas about the integration operation on graphing calculators. The entries are the same or very similar for all calculator brands.

The basic problem of evaluating a definite integral on a graphing calculator is done without finding an antiderivative; that is, the calculator uses a numerical algorithm to produce the result. The calculator provides a template,integral-templateand you fill in the 4 squares so that the expression looks exactly like what you write by hand. Then the calculator computes the result. (Older models require a one-line input requiring, in order, the integrand, the independent variable, the lower limit of integration and the upper limit in that order, separated by commas.) The first interesting thing is that the variable in the integrand does to have to be x. As the first figure illustrates using x or a or any other letter gives the same result.


This is because the variable of integration is just a place holder. Sometimes called a “dummy variable”, this letter can be anything at all, including x. On the home or calculation screen you might as well always use x, so the entry will look like what you have on your paper. As we will see, when graphing it may be less confusing to use a different letter.

The antiderivative, F, of any function, f, can be written as a function defined by an integral where there is a point on the antiderivative of f , which is with F ’ = f. The point (aF(a)) is the initial condition. (In the following we will use F(a) = 0 as the initial condition – the graph will contain the origin. As a further investigation, try changing the lower limit to different numbers and see how that changes the graph.)

The integration operation can be used to graph the antiderivative of a function without finding the antiderivative. You may graph the antiderivative when teaching antiderivatives. Have students look at the graph of the antiderivative and guess what that function is.

When graphing use x as the upper limit of integration and a different letter for the variable in the rest of the template. The calculator will use different values of x to calculate the points to be graphed.

The set-up shown in the next figure shows how to enter a function defined by an integral (blue line) and the same function obtained by antidifferentiating (red squares). As you can see the results are the same.


In this way, you can explore the functions and their indefinite integrals by graphing.

The Fundamental Theorem of Calculus

Another use of using the graph of an integral is to investigate both parts of the Fundamental Theorem of Calculus (FTC). Roughly speaking, the FTC says that the integral of a derivative of a function is that function, and the derivative of an integral is the integrand.

In the figure below, the same function is entered both ways; the graphs are the same. (Note the x’s in the second line must be x in all four places.)




The AP Calculus Concept Outline.

CED CoverThis is the first of four posts on the new AP® Calculus AB and AP® Calculus BC Course and Exam Description (CED). If you do not yet have a copy, click the link above to download the PDF version.

The CED contains general information about the AP Calculus program (format of the exams, number of questions, etc.). Probably the first thing you will want to look at is what topics will be covered on the exams. Turn to pages 11 – 23 for the Concept Outline for AB and BC calculus included in one document.

The topics are organized in what is known as the Understanding by Design model by Wiggens and McTighe. This is to help you see the overall structure of the courses.

The courses are first divided into four Big Ideas – limits, derivatives, integrals and the Fundamental Theorem of Calculus, and (for BC only) Series.

Under each Big Idea are listed:

  • Enduring Understandings (EU) These are the long-term takeaways; things students should understand about the Big Ideas.
  • Learning Objectives (LO) These are things that students will be able to do. The assessments are based on the LOs.
  • Essential Knowledge (EK) The EKs are things students should know and be able to recall. The EK lists consist of the topics that make up the course. The lists almost duplicate the Topical Outline in the previous course description. Tying them to the EUs and the LOs is to help teachers and students see the overall structure and organization of the course. (Yes, you can, and I would suggest you do, share them with your students at some point.)

Since all of the AB topics are also BC topics, there is only one list; BC topics are highlighted in blue and marked (BC).For clarification, the EKs list a few topics that are not tested on the exams such as the epsilon-delta definition of limit.

Overlying all of this are the Mathematical Practices (MPACs) that are the topic of my next post.

The CED contains almost 50 pages of sample exam questions, both multiple-choice and free-response. The LOs, EKs and MPACs that apply to each question are shown along with the question.

There are a few changes and additions to the topics in the courses. The previous Framework document listed the first two items below as changes to the old Topical Outline. A close reading of the new Concept Outline reveals a few other changes also included in the list below.

      • L’Hospital’s Rule has been added to the AB course. L’Hospital’s Rule was already a BC topic. (As a side note, I kind of wish they had not done this. I like the questions where the limit definition of the derivative was needed to find certain limits. Since these are always indeterminate forms of the type 0/0 they can now be done mechanically with L’Hospital’s Rule – but that’s just me).
      • The limit comparison test, absolute convergence, conditional convergence, and the alternating series error bound were added to the BC course.
      • Displacement – In rectangular motion situations, displacement is the definite integral of velocity over an interval of time. Displacement is the net directed distance from the initial position to the final position. Also for BC displacement and the magnitude of the displacement may be asked for in parametric/vector motion problems. (EK 3.4C1 and EK 3.4C2)
      • To the best of my recollection the word “displacement” has never been used on an AP Calculus exam, although the concept was there. Be sure students know the term and how to find the value.
      • Finding antiderivatives by algebraic manipulation. This includes long division and completing the square. Completing the square leads to antiderivatives that are inverse trig functions.

Let me end with examples from the last bullet point.

Long division of polynomials is the first step in this example.

\displaystyle \int{\frac{{{x}^{3}}-3x-4}{x+2}dx=\int{{{x}^{2}}-2x+1-\frac{6}{x+2}dx=\tfrac{1}{3}{{x}^{3}}-{{x}^{2}}+x-6\ln \left| x+2 \right|+C}}

Completing the square (the final step results from the u-substitution u=x-3):

\displaystyle \int{\frac{1}{{{x}^{2}}-6x+10}dx=\int{\frac{1}{\left( {{x}^{2}}-6x+9 \right)+1}dx=\int{\frac{1}{{{\left( x-3 \right)}^{2}}+1}dx=\arctan (x-3)+C}}}

And another completing the square example (just in case you need practice – I did):

\displaystyle \int{\frac{1}{\sqrt{8-{{x}^{2}}-2x}}dx}=\int{\frac{1}{\sqrt{9-\left( {{x}^{2}}+2x+1 \right)}}dx=}\int{\frac{1}{3\sqrt{1-{{\left( \frac{x+1}{3} \right)}^{2}}}}dx}

Then let u=\frac{x+1}{3} and du=\tfrac{1}{3}dx and the last integral above becomes

\displaystyle \int{\frac{1}{\sqrt{1-{{u}^{2}}}}du=\arcsin \left( u \right)+C}. Therefore,\displaystyle \int{\frac{1}{\sqrt{8-{{x}^{2}}-2x}}dx}=\arcsin \left( \frac{x+1}{3} \right)+C


Later this summer the College Board will publish a full practice exam in the new format. It will be secure (or as secure as they can be). It will be under the secure documents at your audit website. Be sure to look it over so you can see these new questions and how they are asked. Also look for some changes in style in some of the multiple-choice questions (see AB 26, 30, 79, 80, 83, and 86 and BC 22, 23, 29, 79, 82, and 87). I believe the document will also show a complete discussion of the multiple-choice question, which LOs, EKs, MPACs apply to each one, and what calculus mistakes lead to the wrong choices.

Another thing that affects the topics, style and format of questions on the new exams are the Mathematical Practices (MPACs). The MPACs will be the topic of my next post.

The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.












Mathematical Practices

This is the second of four posts on the new AP® Calculus AB and AP® Calculus BC Course and Exam Description (CED). If you do not yet have a copy, click the link above to download the PDF version.

What is really new, important, and nice in the new CED are the Mathematical Practices or MPACs.

I recently did some consulting for a company that was aligning calculus textbooks with the MPACs and Concept Outline. Publishers listed where their textbook dealt with each Learning Objective (LO), Essential Knowledge (EK), and MPAC statement from the CED. All of the books I reviewed met the LOs and the EKs at 95% or better. The MPACs not so much.

Part of the reason was that the publishers were looking for the MPACs in the text material. The text material did a good job of explaining the LOs and EKs and the reasoning behind them, applications, etc. The text is what the authors do. The MPACs are what the students do. At best, the texts occasionally provided exercise that addressed tangentially some of the things the MPACs require, but never in a purposeful manner. (The exception was the CPM text.)

What that means is that as an AP Calculus teacher you have to provide opportunities for your students to practice the MPACs. In preparing the questions for the AP Calculus exams the College Board is aware of the MPACs; all the sample questions (p. 46 – 94), list which MPACs apply to each question. While you probably won’t be teaching individual lessons on each MPAC, a number of them will naturally be used in each lesson. Make your students aware of them as you use them.

The MPACs are designed to “capture important aspects of the work that mathematicians engage in, at the level of competence expected of AP Calculus students…. [They enable] students to establish mathematical lines of reasoning and use them to apply mathematical concepts and tools to solve problems…. [T]hey are highly integrated tools that should be utilized frequently and in diverse contexts.” (p. 8)

There are 6 MPACs, with lists of what the students should be able to do. They each start with “Students can:” Here are summaries of the Mathematical Practices for AP Calculus (paraphrased from p 9 – 10).

MPAC 1: Reasoning with Definitions and Theorems. Students can use theorems and definition in justifying answers, proving results, confirming hypotheses, interpreting quantifiers, developing conjectures (possibly from using technology), and solving problems. They can give examples and counterexamples, understand converses, and test conjectures.

MPAC 2: Connecting Concepts. Students can relate the concept of limit to other parts of the calculus. They can use the connection between concepts (such as rate of change and accumulation, or processes such as differentiation and antidifferentiation) to solve problems. They can connect visual representations of concepts with and without technology and identify common structures in different contexts.

MPAC 3: Implementing algebraic/computational processes. Students can select appropriate algebraic and computational strategies. They can sequence and complete them correctly. They can apply technology, attend to precision analytically, numerically, graphically and verbally, and specify units of measure. They should be able to connect these processes to the question they are working on.

MPAC 4: Connecting multiple representations. This refers to what is known as the Rule of Four: considering mathematics analytically, verbally, graphically, and numerically. Students can associate the various representations of functions, develop concepts, identify characteristics of function, extract and interpret mathematical content, relate and construct one representation from another, and select or construct a useful representation for solving a problem.

MPAC 5: Building notational fluency. Students know and can use a variety of notations, connect the notations to their definitions and to different representations of functions (Rule of Four). They should be able to interpret notation accurately and understand its meaning in different contexts.

MPAC 6: Communicating. Students can clearly present methods, reasoning, justifications and conclusions in accurate and precise notation and language. They can explain the meaning of results, notation, expressions, and units of measure in context. They can explain connections between concepts, and accurately report and critically interpret information provided by technology. Students can analyze, evaluate, and compare the reasoning of others.

The MPACs are written with reference to learning and using the calculus. The MPACs should, I think, become part of mathematics learning and instructions at all levels K – 12 leading up to calculus and then beyond it into higher levels of mathematics. When the College Board announced that starting in 1995 graphing calculators would be required for use on the AP Calculus exams, graphing calculators and other technology quickly worked their way down through the secondary math curriculum in most high schools. I think that was a good thing. I hope the same thing happens with the MPACs. The MPACs should have a big impact.

The third post in this series will look at some of the other items in the CED to help you organize and teach a better course.

The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.


Instructional Approaches for AP Calculus

This is the third of four posts on the new AP® Calculus AB and AP® Calculus BC Course and Exam Description (CED). If you do not yet have a copy, click the link above to download the PDF version.

In previous posts I discussed the Concept Outline from the new CED, the Big Ideas and the related Enduring Understandings (EU), Learning Objectives (LO), and Essential Knowledge (EK). The second concerned the Mathematical Practices for AP Calculus (MPACs).  The CED contains a section called AP Calculus AB and AP Calculus BC Instructional Approaches.

The instructional approaches begins with recommendations on organizing the course from three perspectives: inquiry, applications, and technology. The second part of this section gives hints on linking the Learning Objectives to the MPACs and scaffolding them.

The third section called Teaching the Broader Skills talks about justification, reasoning, modeling, interpretation, drawing conclusions, building arguments, and applications. A succinct table related these to the MPACs, suggests question techniques and other strategies for each.

This is followed by an excellent section on representative instructional strategies. A table all lists these strategies:

  • ask the expert,
  • construct and argument,
  • create a plan,
  • create representations,
  • critique reasoning,
  • debriefing,
  • discussion groups,
  • error analysis,
  • graph and switch,
  • graphic organizers,
  • guess and check,
  • identify a subtask,
  • look for a pattern,
  • marking the text,
  • model questions,
  • notation read aloud,
  • note-taking,
  • paraphrasing,
  • predict and confirm,
  • quickwrite,
  • sharing and responding,
  • simplify the problem,
  • think aloud,
  • think-pair-share,
  • use manipulates, and
  • work backwards.

A brief definition of the strategy, its purpose, and an example are included for each. Quite a list! Of course no one can do all of every day or even every month, but the list provides a succinct reference; it’s a place to find new things to try with your class. No one will do all of them, but maybe you can find some new ones you like.

The strategies are followed by a brief discussion on the importance of students being able (that is, taught) to communicate their solution and their reasoning well.

This is followed by section on using formative assessment to monitor student learning and provide ongoing feedback. It suggests ways to help you understand what students know and addressing what they do not (yet) understand. Relating these to the instructional strategies, ways to assess learning while teaching, and ways to provide good feedback to students round out this chapter.

There are further short comments on vertical teaming, using graphing calculators and other technology, and resources for strengthening teacher practice complete this chapter.

The final part of the book gives information about the format and timing of the exam. A selection of sample exam questions for AB and BC calculus is included. The applicable LOs, EKs, and MPACs are included for each question. This will help you see how the LOs, EKs, and MPACs applied in preparing the test questions. A secure practice exam is also scheduled for release later this summer and will be available at your audit website, and at College Board sponsored workshops and summer institutes. The questions are also linked to the appropriate LOs, EKs, and MPACs.

Taken together the CED gives a complete look at the new program and many resources for AP Calculus AB and BC teachers. The CED gives a good basic outline of what an AP Calculus teacher need to know. It is a good read, maybe not for the beach, but for AP teachers.

The next and last post in this series will be a little different. In it I’ll show you a good tool for organizing all this and arranging it for teaching.

The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.

Getting Organized

There is a lot to put together in the new AP® Calculus AB and AP® Calculus BC Course and Exam Description (CED). Today I’m going to show you a way to organized all of this and arrange it to plan you lessons and your year. Don’t worry: I’ve done the tedious part for you and we’ll get to that in a few minutes.

The key to this is an app called Trello. Go to and sign up (free) for your own account. Trello is an electronic cork board. You put your ideas and notes on “cards” and tack them on the board to make lists. It was a great help recently when we moved from Texas to New York: we made “To Do” lists for my wife, and me, and for both of us. As we completed the items we moved them to a “Done” list – except they’re not all done yet!.

Bookmark Trello on your computer. While you’re there look at their tutorials. Play with a little and you’ll catch on quickly. Everything you do will be saved and available on all your devices. There are Trello apps for iPads and other tablets and phones.

While it may look a little overwhelming at first, remember that there are only three objects to consider:  boards that contain lists, lists that contain cards, and cards.

Once you have the idea go to . This will open a board called “AP Calculus CED” in your Trello account. I suggest you immediately copy the board so you have your own copy to rearrange for you own use. To do this on the upper right side click on “… Show Menu,” then click on “more,” and finally click on “copy board.” Give it a new name when prompted. Your new board will be ready to help you plan your year.

Trello 1

Take a look at the AP Calculus CED board. The first 6 lists after the notes are the MPACs. The next groups of lists are the 4 Big Ideas (BI). For each BI there are lists for the Enduring Understands (EU), the Learning Objectives (LO), and several Essential Knowledge (EK) lists.On each list, each card contains one item numbered as in the CED. The color coding refers to the appropriate LO for that Big Idea. (There are only so many colors, so the colors are reused in each LO list.) Hover over the colored bar to see what it refers to.The final list on the right lists the Instructional Strategies.

Here is how you can use the AP Calculus CED board to organize and plan your lessons and your year.

Make a list for your units

  • Make your own list for each unit or week of your course – whatever works for you. To do this scroll to the far right. You will see “Add a list …” Click on this, give the list a name when prompted, then click save.
  • Let’s say your list is “Unit 4: Applications of the derivative.” Drag the list and place it near the Essential Knowledge lists for derivatives. Choose which EK you want in this unit and drag the card from the EK list to your new list. Arrange them in the order you want.
  • If you want to add a topic that is not on the EK list, just click on “Add a card” at the bottom of your list and type what you want, save it, and move it to the position in the list where you want it.
  • As you go through your planning the original lists will empty. As the list becomes empty you will see what you have yet to include. For example, by the time you get to derivative applications most of your Limit lists will be empty except for L’Hospital’s Rule; drag this card to the Applications of derivatives list.

Copying Cards

You will probably use the EK cards only once. The MPAC and Instructional Strategies are used across entire course. The way to handle this is to copy the card and move the copy. To do this left click on the card and the “back” of the card will open. I’ll discuss the “back” of the card next, but for now just find and click “copy” on the right side. Then click “create card” at the bottom. This will place a duplicate card in the same list. Move it to where you want it.

The “back” of a card

If you left click on any card its “back” will appear.

Trello 2

This is a nice feature of Trello. On the back of the card you can do several things. Some of these will help you this year and others will help you in the future.

  • You may add comments. You may want to include:
    • the assignments and homework for the EK on the card,
    • any activities for this lesson,
    • student misconceptions, and
    • reflections on what went well and what did not – so you can address them next year.
  • Click “Attachment” on the right and attach things like the following. Then they will be right where you can find them next year without having to search your computer. Attach things like
    • Tests
    • Quizzes
    • Work sheets
    • Readings
    • Activities
  • Labels – click this to color code a card or remove the code.
  • Checklist – add a checklist that you can check off things as you do them
  • Other things you can do are listed here as well. The “Archive” tab will hide, but not delete the card (unless you click: delete”), To un-archive something click on “… show menu” at the top right and then on “Archived items.”
  • The “Activity” list at the bottom of the card automatically notes the additions and changes you made to the card. This is mostly for teams that contribute to a board – so everyone can see who made changes and what the changes are.

Making your own board

The original board is quite wide and when you add your own lists it gets wider. To alleviate this crowding you can make new boards. Once you’ve planned a unit you can move that list to a new board. After creating a new board (explained above), click on the “ …” at the top right of a list. Click on “Move list …” Select the destination board and click and “move.” If you want to keep the list and move it, make a copy and move the copy.


If you are fortunate enough to work with another teacher or two, form a team by (1) clicking on the “+” sign in the upper right, then (2) clicking on “Private” in the upper right and choosing “team” and then (3) clicking on “…show menu” and add members.

And that’s it. I hope this helps. Now I’m going to use Trello to plan my APSI.

The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.

May 2016

starting overThe AP Calculus year is almost over. The exam is a few days away. I hope you kids do well.

They say, that being a teacher, no matter how well you do your job you have to start over again next fall. I guess I can’t argue with that. AP Calculus teachers are no different, but next year will be a little different from this year. The College Board has published a new AP Calculus AB and BC Course and Exam Description (CED) for the 2016 – 17 school year. (A previous publication called the Framework has been available for some time; the CED includes the Framework and additional information.)

I’m planning four blog posts on the CED that will be posted later this month (you’re entitled to a few days to relax after the exam).

The first will discuss the new Concept Outline from the CED. This is the list of topics tested on the AB and BC exams. This is what you should teach and includes the (very) few new topics that have been added to the previous  course description.

The next will be about the Mathematical Practices for AP Calculus (MPACs). These are not topics to be included, but rather the MPACs “capture important aspects of the work that mathematicians engage in, at the level of competence expected of AP Calculus students” (p. 8). These apply to many if not all of the topics in the course and with some modifications to doing mathematics at every level.

The third post will look at the Instructional Approaches: the broader skills you can use, representative instructional strategies, and other suggestions suitable for AP Calculus courses.

The final post will give you a way to organize and plan all this. This is new and I think you’ll like what I’ve put together to help you.

What is a Solution?

“How does one solve x + ln(x) = c algebraically?” A teacher asked that on the old calculus electronic discussion group (EDG) and it got me to thinking about what a “solution” really is.

We start by teaching students how to solve linear equations; the idea is to do some arithmetic and/or algebraic operations on an equation so that you end up with x = some number. And that’s what students learn: do some operations on an equation so that you end up with x = some number.

Next come quadratic equations. You can perform a series of operations (completing the square) and end up with x = two numbers. And then you learn two short cuts – factoring and the quadratic formula. (Okay, maybe you learn factoring first.) These solutions often involve radicals, and therefore, do not necessarily give a recognizable number.

Next come cubics and higher degree polynomials. Sometimes you can factor, and there is a cubic formula. There is also the rational root theorem and synthetic division that can help you find rational solutions to polynomial equations; but if the solutions are not rational, you may be out of luck.

Equations involving trigonometric functions always have solutions that are the so-called “special angles.” Exponential and logarithmic equations always involve convenient bases. So, everything can be solved that way – or not.

These kinds of answers are usually called “closed form” expressions. That is, they are given in a notation that indicates what arithmetic must be done to get a decimal answer. That is, 5192/13 means to get “the answer” divide 5192 by 13. Since decimals are not always “exact,” the closed-form answers are preferred – they are “exact.”

We rarely consider solving by graphing by-hand, graphing calculators, computer algebra systems (CAS), or searching the internet. Why not? Because by-hand graphing is not very accurate, you only get numerical answers, graphing is “just for checking,” and of course technology is not allowed on the state exams.


So, returning to the EDG here are some of the answers that were offered to the question “How does one solve x + ln(x) = c algebraically?”

Then folks started complaining.

Someone wrote “’Naming’ a previously unknown function isn’t ‘solving’ a problem.”

To which someone else replied

  • Before Joe Arcsine invented his function, one could not “solve” sin(x) = 0.123
  • Before Betty Logarithm invented her function, one could not “solve” {{e}^{x}}=0.123
  • Before Johann Lambert invented his function, one could not “solve” x{{e}^{x}}=0.123

And he is correct – well okay, his idea is correct, even if he may not have ascribed the solutions to the correct mathematicians. We actually do it all the time. Functions like the arcsin(x), ln(x), and the like are simply names given to functions for which we may not have a series of arithmetical operations leading to a closed-form solution. (Taylor series, being infinite in length, are not closed form.)


Then there are the answers themselves. Simplifying makes things easier.

  • x = 3, x = 0.125, x=3.7\overline{53} are fine, but x = 4.567… not so much.
  • x = ½ and x = \displaystyle \frac{5192}{13} are great, but x = \displaystyle \frac{5192}{33} is not.
  • We like x = \sqrt{6}, we like \sqrt{7}, we’re not too sure about x = \sqrt{8}, x = \sqrt{9} is just plain wrong, but x = \sqrt{10} is okay.
  • x = \displaystyle \frac{2}{\sqrt{3}-1} cannot be correct because we have to rationalize the denominator; try that with x\displaystyle \frac{1}{\pi }
  • Likewise, x = \displaystyle \arcsin \left( \tfrac{7}{8} \right) is okay, but x\displaystyle \arcsin \left( \tfrac{1}{2} \right) is not.
  • Which of x = \displaystyle \frac{\ln \left( 2 \right)}{\ln \left( 7 \right)-\ln \left( 3 \right)} and the equivalent x = \displaystyle \ln \left( {{2}^{\frac{1}{\ln \left( \frac{7}{3} \right)}}} \right)  is simplified?
  • When we differentiate tan(x) using the quotient rule we get \displaystyle \frac{1}{{{\cos }^{2}}\left( x \right)}, but that has to be changed to {{\sec }^{2}}\left( x \right) even though there is no secant button on a calculator.
  • And then there is LambertW(e4)

So, what have we learned?

  • Each kind of equation has a different process for finding its solution.
  • We are allowed to make up new functions to solve equations.
  • To the outsider (read: student) this looks like a hodgepodge – and for good reason.
  • Most of our “solutions” are really directions for finding the solution.

Keep calm