Monthly Archives: August 2023

Why Infinity?

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First, right from the start: Infinity is NOT a number.

Lots of folks think of infinity as the largest number possible, greater than anything else. That’s understandable because infinity, denoted by the symbol \displaystyle \infty , is often used that way by those unlucky folks who don’t understand mathematics.

We’ll start with an example: Consider the fraction \displaystyle \frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}. This fraction has no value when x = 3 because there the denominator is zero. And you cannot divide by zero. Nothing personal, no one, no matter how smart, can divide by zero. Ever.  Permanently and forever not allowed. Don’t even think about it! (Actually, think about it; just don’t do it.)

What you should say in such cases is that the expression has no value, or is “undefined,” or “the limit does not exist,” abbreviated DNE.

In situations like the example we say, “the limit of the fraction as x approaches 3 equals infinity,” abbreviated  \displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=\infty . This means that the expression gets larger as x gets closer to three. The expression will be greater than any (large) number you want, if you are close enough to three.

You don’t believe me? Okay pick a large number, maybe \displaystyle {{10}^{8}}. I say pick any value for x between 2.9999 and 3.0001 (\displaystyle 3-{{10}^{{-4}}}<x<3+{{10}^{4}}) and the expression will be larger than \displaystyle {{10}^{8}}. Try it on your calculator.

How about \displaystyle {{10}^{{20}}}? Try a number between 2.9999999999 and 3.0000000001. I can play this game all day.

Try graphing the \displaystyle y=\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}} on your calculator. (Hint: Whenever you come across something like this, it is a great idea to graph the expression on your graphing calculator. Graphs can help you see what’s going on. Keep that in mind for the future.)

That’s the way to think about infinity: Infinity is what you say when you’re working with an expression that grows greater than any number you choose.

You may also use infinity to say what happens all the way to the left or right of the graph, its end behavior. The variable, x, may “approach infinity,” that is x moves further to the right (or is greater than any number you choose) the fraction above gets closer to zero: \displaystyle \underset{{x\to \infty }}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=0.

You may not do arithmetic with infinity.

\displaystyle \infty +\infty \ne 2\infty

\displaystyle \infty -\infty \ne 0

Arithmetic is for numbers.

You will see a number of expressions whose limit is equal to infinity, like \displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=\infty . Which really means, just what we saw above: that as you (not “you” but x) get closer to 3, the value of the expression will be greater than any number you pick. The \displaystyle \infty symbol is a shorthand way of saying this.

The opposite of infinity, \displaystyle -\infty , sometimes called “negative infinity,” means that the expression gets less than (i.e. more negative), than any negative number you choose.

Even though the expression has no limit, you are allowed to say the limit equals infinity. That’s funny when you think about it. It might be better if everyone said “undefined” or DNE, but they don’t. What can I say?

A word of warning: You may only say “equals infinity” is situations like the example above.

There are other similar expressions that have no limit where it is incorrect to say the limit equals infinity. For example,

  • \displaystyle \frac{{\left| x \right|}}{x} has no value, is “undefined,” when x = 0, but \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\left| x \right|}}{x}\ne \infty . (Hint: this is where you should look at a graph on your graphing calculator to see why.)
  • \displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{\left( {x-3} \right)}} does not exist. This is very similar to the first example but look at the graph and you’ll see a big difference.

So, good luck and enjoy your limitless journey through the infinite reaches of calculus. (Oh, wait! Can I say that?)

Finally,

Course and Exam Description Unit 1 topics 1.3, 1.14, 1.5 and others.

Why Limits?

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Welcome to AP Calculus.

Your journey into calculus starts with the topic of limits. Why? Because limits make calculus work. The two big things in calculus are called the derivative and the definite integral; both are limits.  

The first use of limits will be when you study continuity. A continuous function is one, roughly speaking, whose graph can be drawn without taking your pen off the paper. Limits will make this concept firm mathematically.

After that, if you flip through your book, you won’t see that many limits after the limit chapter. You will see derivatives and definite integrals – they are limits under a different name.

Notation: There is a new notation to learn for limits. It looks like this \displaystyle \underset{{x\to 6}}{\mathop{{\lim }}}\,\sin \left( {\tfrac{\pi }{x}} \right)=\tfrac{1}{2}. This is read, “The limit as x approaches 6 of the sine of π divided by 6 equals ½.”

You will start by finding the limits of various functions. Sometimes the limit is easy to find – just substitute the number x is approaching into the (that’s what happened above); what you get is the limit. If a limit exists, it is a number. If you get a number by substituting, usually that’s the limit – end of problem.

But other times you get expressions that are not numbers when you substitute (like maybe you end up trying to divide by zero). In that case, you will need to do some sort of algebraic or trigonometric simplification. Your teacher will help you learn the “tricks” involved. Derivatives and the definite integral are both limits involving dividing by zero.

Some limits may not exist at all; in this case, you say, “Does not exist” or “DNE.” Others do not exist, but we say they are “equal to infinity.” Infinity will be the subject of the next post in this series.

As you learn to find limits, look for patterns. The limits of similar looking expressions are often found in similar ways.

One good way to see what a limit is, or is not, is to graph the expression. Use your graphing calculator.

Your calculator may “misinform” you sometimes! But even that is a help. (Hint: when your calculator does misinform you, about limits or anything else, that’s a time to look deeper into the situation: something interesting is going on.)

Producing a table of values (on your calculator) can often help you see what’s happening, as well.  (Hint: while tables are useful, what happens between the values in the table is not always clear; that’s where the trouble may be.)

The first thing you will use limits for is to investigate continuity. When limits do not exist, continuity is usually the problem. Continuity will be the subject of a later post.

So, get ready for your trip through calculus: it’s an unlimited journey.

The next post in this series, “Why Infinity?”, will appear on Friday August 25, 2023.

Course and Exam Description: Unit 1 all topics

The Why Series

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Since 2012 this blog has been written with teachers in mind, hence the title “Teaching Calculus.” Students may read it too; I hope they do and find it helpful. Please share any of the posts that you think will be helpful with them.

This year I plan to write a series of posts for students. The first post will appear next Tuesday August 22, 2023. It’s called “Why Limits?” and will discuss briefly why we use limits and how they fit into calculus. Following weeks will see post on Infinity, continuity, and then derivatives.

My idea is to introduce the topics, to help students sort through what they are about to learn, and why. I will not be providing detailed notes; that’s your job. I hope I can provide a thorough line so students can get an idea of where they are going in calculus and why.

If you find the posts helpful, please share the link with your students. Ask them to “Like” the post (If they like the post) or add comments, suggestions, and especially their questions using the “Leave a Reply” link at the very end of each post.

Following the timing suggested in the Course and Exam Description, the posts will be timed to appear at least a week before you get to the topic (even longer for schools starting in September). This is so you may read them in advance and decide which to share. Give the links to your class when they fit your schedule.

As always, I am happy to have your suggestions for posts and your students’.

Starting the Year

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As you get ready to start school, here are some thoughts on the first week in AP Calculus. I looked back recently at several of the “first week of school” advice posts I offered in the past. Here’s a summary with some new ideas.

  • 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 through pre-calculus when the topics come up during the year. Plan for short reviews. For instance, when you study limits, you will need to simplify rational expressions – that’s when you review rational expressions. Months from now you’ll be looking at inverse functions, that’s when you review inverses.
  • Make a copy of the “Mathematical Practices” and the “Course at a Glance” from the Current 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. Or give them the more detailed Unit Guide (e.g., p. 32 – 33 for Unit 1 Limits) as you start each unit. Either way, have your students check off the topics (1) as you teach them and (2) when they understand them.
  • 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!
  • If you are not already a member, I suggest you join the AP Calculus Community. We have over 23,000 members all interested in AP Calculus. The community has an active bulletin board where you can ask and answer questions about the courses. Questions often ask how to better teach a topic – get hints and share your ideas. Teachers and the College Board post resources for you. College Board official announcements are also posted here. I moderate of the community, and I hope to see you there!
  • If you haven’t done so already, read your Instructional Planning Report (IPR). Especially helpful will by the comparison of your classes mean scores with the state and global mean scores for each question. If yours are higher great; if lower, that’s where you might need to do something different.

This blog has been written with teachers in mind. Students are always welcome to read it. You may give links to any of the posts you think your class may be interested in.

This year I plan to write a new series of posts especially for students. It will be called “The Why Series.” The inspiration was Why Radians, one of my most read posts. The posts will be short pieces introducing units and parts of units that (I hope) will explain why the topic is important, where the topic is leading. The first post “Why Limits” will appear soon.