Monthly Archives: August 2018

Local Linearity

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If you use your calculator or graphing program and zoom-in of the graph of a function (with equal zoom factors in both directions), the graph eventually looks like a line: the graph appears to be straight. This property is called Local Linearity. The slope of this line is the number called the derivative. (There are exceptions: if the graph never appears linear, then no derivative exists at that point.) Local Linearity is the graphical manifestation of differentiability. 

To find this slope, we need to zoom-in numerically. Zooming-in numerically is accomplished by finding the slope of a secant line, a line that intersects the graph twice near the point we are interested in. Then finding the limit of that slope as the two points come closer to our point. This limit is the derivative. It is also the slope of the line tangent to the function at the point. 

While limit is what makes all of the calculus work, people usually think of calculus as starting with the derivative. The first problem in calculus is finding the slope of a line tangent to a graph at a point and then writing the equation of that tangent line. The slope is called the derivative and a function whose derivative exists is said to be differentiable. 

This week’s posts start with local linearity and tangent lines. They lead to the difference quotient and the equation of the tangent line.

Local Linearity I

Local Linearity II      Working up to difference quotient. The next post explains this in more detail.

Tangent Lines approaching difference quotients on calculator by graphing tan line.

Next week: Difference Quotients.

 

 

 

Revised from a post of August 29, 2017

 

Continuity

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Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. Included in that definition was the epsilon-delta definition of limit. This definition has been pulled out, so to speak, and now is usually presented on its own. So, which came first – continuity or limit? The ideas and situations that required continuity could only be formalized with the concept of limit. So, looking at functions that are or are not continuous helps us understand what limits are and why we first need them.

In the ideal world, students would have plenty of work with continuous and not continuous functions before starting the calculus. The vocabulary and notation, if not the formal definitions, would be used as early as possible. Then when students got to calculus, they would know the ideas and be ready to formalize the ideas.

The Intermediate Value Theorem (IVT) is an important property of continuous functions.

Using the definition of continuity to show that a function is or is not continuous at a point is a common question of the AP exams, as is the IVT.

Continuity The definition of continuity.

Continuity Should continuity come before limits?

From One Side or the Other One-sided limits and one-sided differentiability

How to Tell Your Asymptote from a Hole in the Graph  From the technology series. Showing holes and asymptotes on a graphing calculator.

Fun with Continuity Defined everywhere and continuous nowhere. Continuous only at a single point.

Theorems The Intermediate Value Theorem (IVT) and suggestions on teaching theorems.

Intermediate Weather  Using the IVT

Right Answer – Wrong Question Continuity or continuity “on its domain”?

 

 

 

 

 

Revised from a post of August 22, 2017

 

Limits – They Make the Calculus Work.

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In an ideal world, I would like to have all students study limits in their pre-calculus course and know all about them when they get to calculus. Certainly, this would be better than teaching how to calculate derivatives in pre-calculus (after all derivatives are calculus, not pre-calculus).

Limits are the foundation of the calculus. Continuity, an important property of functions, depends on limits. All derivatives and all definite integrals are limits. For AP Calculus students need a good intuitive understanding of limits, what they mean, and how to find them The formal (delta-epsilon) definition is not tested and need not be taught, however, do not feel that you have to avoid it. If your students can handle it, let them try.

Here are a few of my previous posts on limits.

Why Limits?

Finding Limits  How to … and the use of “infinity” vs “DNE”

Dominance  Finding limits the easy way.

Deltas and Epsilons Not tested on the AP Exams; here’s why.

Asymptotes The graphical manifestation of limits at or equal to infinity.

Next Week: Continuity

Update of my post of August 15, 2017.

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