Blog Guide

As I am not doing many new posts these days, I want to call your attention to the “Blog Guide” tab above. This tab will guide you to the information on the blog. It will help sort through the approximately five hundred posts and find those that concern the topic you are interested in.

The “Before Calculus” section discusses things usually taught before calculus.

The “Pedagogy” sections had notes on pacing, teaching, testing, grading, and scoring.

The “Graphing Calculator Use” page contains links to what students should know and be able to do on the AP Calculus Exam with their graphing calculator. There are also links to how to use a graphing calculator to teach some of the topics in the course.

The “AP Exam Review” has links to the ten common type questions on the exams with notes on what students should know about each of them. Good for review and as you teach each topic during the year.

Then there are links to the ten units, different from the type questions, in the current Course and Exam Description for AP Calculus AB and BC.

The “Good Questions” links are to specific questions, mostly from past AP exams, which are discussed in detail. They explore the richness of the question.

“Odds and Ends” has links to, well, odds and ends – other posts you may find interesting and helpful.

Meanwhile: An interesting article “How I Rewired My Brain to Become Fluent in Math”

Today and Tomorrow

Today is this blog’s tenth anniversary!

 My first post was on July 15, 2012. At the time I was working with the Arkansas Advanced Initiative for Math and Science. I was thinking of a series of emails with teaching hints for the calculus teachers I was working with. It occurred to me that a blog format would be more useful to them and to others who stumbled across it. So, that’s how all this all got started.

This is my 492nd post in addition to the 98 pages available from the menu bar. As of this morning, the blog has had, 956,803 visitors and 1,628,857 page views – and counting.

Teaching mathematics is more than just proving the theorems and doing the standard examples.  I certainly have not posted about everything there is to know about calculus – which would be difficult, since I don’t know everything. It was never my intent to write an online calculus book or even cover all the topics in the course description. Textbooks do that well enough. I hoped to provide some insight and ideas to help teachers explain things.

But I seem to have little more to add. I have found little new to write about recently. For the past few years, as you’ve probably noticed, many of my post were lists of links to past posts of actual calculus content.

So, I’ve spent some time this month looking at all my past posts and sorting out the ones with real content from those linking to the content posts. I’ve added a new drop-down menu to the navigation bar at the top of the screen called Blog Guide. Here you will find all the content posts organized in a way that I hope you will find useful. (The “link” posts are not there but are still available if you’ve bookmarked any of them.)

Please take a minute to look at the Blog Guide. I hope its organization will help you find your way around. (The “Search, “Posts by Topics,” and the “archives” on the sidebar will also help.)  

From now on, the blog will be on autopilot, so-to-speak. There will be few new posts. If I get an interesting idea, I will share it, but will not be posting regularly.

Some of my best inspiration comes from readers. So, if you have a calculus topic you would like me to discuss or expand on, please email me here and I’ll see what I can do. (The address is also on the navigation bar under “About.”), Also, I would appreciate you letting me know of any typos or broken links.

If you click on the “Follow” link in the sidebar, you will receive an email whenever a new post appears.

I hope to have helped you at least a little and hope to continue to do so. Thanks for reading and supporting

Enjoy your summer and have a good school year.

Starting the Year

As you get ready to start school, here are some thoughts on the first week in AP Calculus. I looked back recently at some of the “first week of school” advice I offered in the past. Here’s a quick (actually, a bit longer then I planned) summary with some new ideas.

  1. The last time I taught AP Calculus during review time a student asked if there was a list of what’s on the exam. Duh! Why didn’t I think of that? So, I made copies of the list (from the old Acorn Book) and gave it to everyone. I should have done that on Day 1. So, my first suggestion is to make a copy of the “Mathematical Practices” and the “Course at a Glance” from the 2019 AP Calculus Course and Exam Description (p. 14 and p. 20 – 23) and give them to your students. Check off the topics as you do them during the year. 
  1. DON’T REVIEW! Yes, students have forgotten everything they ever learned in mathematics, but if you reteach it now, they will forget it again by the time they need it next week or next January. So, don’t waste the time, rather, plan to review material from Kindergarten thru pre-calculus when the topics come up during the year. Include short reviews in your lesson plans. For instance, when you study limits you will need to simplify rational expressions – that’s when you review rational expressions. When you look at the graphs of the trigonometric functions, that’s when to review the graphs of the parent functions, a lot of the terminology related to graphs, discontinuities, asymptotes, and even the values of the trigonometric functions of the special angles. Months from now you’ll be looking at inverse functions, that’s when you review inverses.
  1. In keeping with Unit 1 Topic 1, you may want to start with a brief introduction to calculus. Several years ago, when I first started this blog, Paul A. Foerster, was nice enough to share some preview problems. They give a taste of derivatives and integrals in the first week of school and get the kids into calculus right off the bat. Here is an updated version. Paul, who retired a few years ago after 50 (!) years of teaching, is Teacher Emeritus of Mathematics of Alamo High Heights School in San Antonio, Texas. He is the author of several textbooks including Calculus: Concepts and Applications. More information about the text and accompanying explorations can be found on the first page of the explorations. Thank you, Paul!
  1. If you are not already a member, I suggest you join the AP Calculus Community. We have over 18,000 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!
  1.  Here are some links to places on this blog that you may find helpful:
    1. Pacing– organizing your year.
    2. Check the Resource page from this blog.
    3. Calculator information:
    4. Miscellany: These posts discuss basic ideas that I always hoped students knew about mathematics before starting calculus

Adapting 2021 BC 6 the last in the series on adapting questions from the 2021 exam will appear in two weeks on Auguest 31, 2021

Revised August 16, 2021

Summer … At last!

Summer … At last!

I hope you have all either completed your year or are close to it. Take some time to relax.

I am working on a series of nine summer post which will be begin on Tuesdays starting June 22. Each will look at one of the nine 2021 free-response questions. I will not be presenting their solutions; you can find them online. Rather, I will try to suggest ways that you can adapt the questions for use during the year. This may include ways to slightly change the question, ask additional questions from the same stem, and use the question to explore the topic further and deeper. I hope you’ll find them useful.

Please join me then and enjoy your summer!

2021 Review Notes

About this time of year, I have been posting notes on reviewing and on the ten types of problems that usually appear on the AP Calculus Exams AB and BC. Since the types do not change, I am posting all the links below. They are only slightly revised from last year. You can also find them under “AP Exam Review” on the black navigation bar above.  

Each link provides a list of “What students should know” and links to other post and questions from past exams related to the type under consideration.

Note that the 10 Types are not the same as the 10 Units in the Fall 2020 Course and Exam Description. This is because many of the exam questions have parts from different units.

Here are the links to the various review posts:

When assigning past exams questions for review (and you should assign past exam question), keep in mind that students can find the scoring standards online. Even though the AP program forbids this and makes every effort to prevent them from being posted, they are there. Students can “research” the solution. Keep this in mind when assigning questions from past exams. Here is a suggestion Practice Exams – A Modest Proposal


Analytical Applications of Differentiation – Unit 5

Unit 5 covers the application of derivatives to the analysis of functions and graphs. Reasoning and justification of results are also important themes in this unit. (CED – 2019 p. 92 – 107). These topics account for about 15 – 18% of questions on the AB exam and 8 – 11% of the BC questions.

You may want to consider teaching Unit 4 after Unit 5. Notes on Unit 4 are here.

Reasoning and writing justification of results are mentioned and stressed in the introduction to the topic (p. 93) and for most of the individual topics. See Learning Objective FUN-A.4 “Justify conclusions about the behavior of a function based on the behavior of its derivatives,” and likewise in FUN-1.C for the Extreme value theorem, and FUN-4.E for implicitly defined functions. Be sure to include writing justifications as you go through this topic. Use past free-response questions as exercises and also as guide as to what constitutes a good justification. Links in the margins of the CED are also helpful and give hints on writing justifications and what is required to earn credit. See the presentation  Writing on the AP Calculus Exams and its handout

Topics 5.1

Topic 5.1 Using the Mean Value Theorem While not specifically named in the CED, Rolle’s Theorem is a lemma for the Mean Value Theorem (MVT). The MVT states that for a function that is continuous on the closed interval and differentiable over the corresponding open interval, there is at least one place in the open interval where the average rate of change equals the instantaneous rate of change (derivative). This is a very important existence theorem that is used to prove other important ideas in calculus. Students often confuse the average rate of change, the mean value, and the average value of a function – See What’s a Mean Old Average Anyway?

Topics 5.2 – 5.9

Topic 5.2 Extreme Value Theorem, Global Verses Local Extrema, and Critical Points An existence theorem for continuous functions on closed intervals

Topic 5.3 Determining Intervals on Which a Function is Increasing or Decreasing Using the first derivative to determine where a function is increasing and decreasing.

Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema Using the first derivative to determine local extreme values of a function

Topic 5.5 Using the Candidates’ Test to Determine Absolute (Global) Extrema The Candidates’ test can be used to find all extreme values of a function on a closed interval

Topic 5.6 Determining Concavity of Functions on Their Domains FUN-4.A.4 defines (at least for AP Calculus) When a function is concave up and down based on the behavior of the first derivative. (Some textbooks may use different equivalent definitions.) Points of inflection are also included under this topic.

Topic 5.7 Using the Second Derivative Test to Determine Extrema Using the Second Derivative Test to determine if a critical point is a maximum or minimum point. If a continuous function has only one critical point on an interval then it is the absolute (global) maximum or minimum for the function on that interval.

Topic 5.8 Sketching Graphs of Functions and Their Derivatives First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.

Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.

Topics 5.10 – 5.11

Optimization is important application of derivatives. Optimization problems as presented in most text books, begin with writing the model or equation that describes the situation to be optimized. This proves difficult for students, and is not “calculus” per se. Therefore, writing the equation has not be asked on AP exams in recent years (since 1983). Questions give the expression to be optimized and students do the “calculus” to find the maximum or minimum values. To save time, my suggestion is to not spend too much time writing the equations; rather concentrate on finding the extreme values.

Topic 5.10 Introduction to Optimization Problems 

Topic 5.11 Solving Optimization Problems

Topics 5.12

Topic 5.12 Exploring Behaviors of Implicit Relations Critical points of implicitly defined relations can be found using the technique of implicit differentiation. This is an AB and BC topic. For BC students the techniques are applied later to parametric and vector functions.


Topic 5.1 is important and may take more than one day. Topics 5.2 – 5.9 flow together and for graphing they are used together; after presenting topics 5.2 – 5.7 spend the time in topics 5.8 and 5.9 spiraling and connecting the previous topics. Topics 5.10 and 5.11 – see note above and spend minimum time here. Topic 5.12 may take 2 days.

The suggested time for Unit 5 is 15 – 16 classes for AB and 10 – 11 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Finally, were I still teaching, I would teach this unit before Unit 4. The linear motion topic (in Unit 4) are a special case of the graphing ideas in Unit 5, so it seems reasonable to teach this unit first. See Motion Problems: Same thing, Different Context

This is a re-post and update of the third in a series of posts from last year. It contains links to posts on this blog about the differentiation of composite, implicit, and inverse functions for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get to the topic. 

Previous posts on these topics include:

Then There Is This – Existence Theorems

What’s a Mean Old Average Anyway

Did He, or Didn’t He?   History: how to find extreme values without calculus

Mean Value Theorem

Foreshadowing the MVT

Fermat’s Penultimate Theorem

Rolle’s theorem

The Mean Value Theorem I

The Mean Value Theorem II


Concepts Related to Graphs

The Shapes of a Graph

Joining the Pieces of a Graph

Extreme Values

Extremes without Calculus


Reading the Derivative’s Graph

        Other Asymptotes

Real “Real-life” Graph Reading

Far Out! An exploration

Open or Closed  Should intervals of increasing, decreasing, or concavity be open or closed?


Lin McMullin’s Theorem and More Gold  The Golden Ratio in polynomials

Soda Cans  Optimization video

Optimization – Reflections   

Curves with Extrema?

Good Question 10 – The Cone Problem

Implicit Differentiation of Parametric Equations    BC Topic

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

Limits and Continuity – Unit 1  (8-11-2020)

Definition of t he Derivative – Unit 2  (8-25-2020)

Differentiation: Composite, Implicit, and Inverse Function – Unit 3  (9-8-2020)

Contextual Applications of the Derivative – Unit 4   (9-22-2002)   Consider teaching Unit 5 before Unit 4

Analytical Applications of Differentiation – Unit 5  (9-29-2020) Consider teaching Unit 5 before Unit 4 THIS POST

LAST YEAR’S POSTS – These will be updated in coming weeks

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