Thursday, September 28, 2006

2.4: The Precise Definition of a Limit

Hello class.

Lately we've been defining limits in a vague way. Now, with Mr. French's help, we've learned to be more precise in our definitions. I will attempt to review the notes for 2.4 forthwith.

The Book's "precise definition" of a limit is as follows:

"Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write

lim (x-->a) f(x) = L

if for every number e (epsilon) > 0 there is a number d (delta) > 0 such that

|f(x) - L| < e whenever 0 < |x-a| < style="font-style: italic;">a because a is sometimes discontinuous.

By making an interval around the x value, also called d, we may find the corresponding interval around the y value, or e. Sometimes, if we are finding, say, the d interval using the e interval, it will not be evenly distributed about the x value. In this case, we must choose the lesser distance, because that is the distance that does not surpass the interval in either direction on the x-axis.

When using one-sided limits, we restrict the interval to lie on one side of the x interval, left or right for left and right hand limits, respectively.

For example, let's take y = e ^ x.

Now, we know that y = e ^ x is an exponential function, and therefore has a constantly changing slope. Thus, we can assume that the interval we are finding is also uneven. Here is the problem.

Given lim (x-->2) e^x, find delta if epsilon equals .05

Since epsilon is our interval around y, or e ^ 2, we know that our interval ranges from (e ^ 2) + .05 to (e ^ 2) - .05. So, to find our x interval, we must first find the x values that correspond to our y values. The simplest way to do this is to graph y = e ^ x, y= (e ^ 2) + .05, and y= (e ^ 2) - .05, and find where the curve intersects with both lines. We can also use logs to solve this algebraically, plugging (e ^ 2) + .05 and (e ^ 2) - .05 into the y value of y = e ^ x and solving for x. Either way, we get our corresponding x values to be 1.993 and 2.006.

Now we must consider the distances from the ends of our x interval, which ranges from 1.993 to 2.006, to 2, our original x value. With some nifty subtraction, we come up with .007 and .006. Therefore our delta value is .006, as .007 would pass the x interval on the right side, whereas .006 stay within our limits.

There you go. I feel so much more precise. How about you? Don't worry. If you don't, here is a link to another website to make you feel better. It always makes me happy when I'm feelin' blue.

The Precise Definition of a Limit

A Reminder. John, you're posting next.

Here's part of a poem by Emily Dickenson that I like that has the number one in it, a number widely accepted in most states.

To make a prairie it takes a clover and one bee,
One clover, and a bee,
And revery.
The revery alone will do
If bees are few.

To lose one's faith surpasses
The loss of an estate,
Because estates can be
Replenished, faith cannot.

Inherited with life,
Belief but once can be;
Annihilate a single clause,
And Being 's beggary.

Afraid? Of whom am I afraid?
Not death; for who is he?
The porter of my father's lodge
As much abasheth me.

Of life? T'were odd I fear a thing
That comprehendeth me
In one or more existences
At Deity's decree.

Of resurrection? Is the east
Afraid to trust the morn
With her fastidious forehead?
As soon impeach my crown!

Have a nice day, class.

Wednesday, September 27, 2006

Quiz 2.1-4 Topics

Here’s a list of topics that will be covered on this Friday’s 2.1-4 Quiz. I’ve tried to indicate where a similar homework problem would be helpful.

Calculate the slope of a secant line to a given level of accuracy. (2.1, #9)
Find an average rate of change from a data table. (2.1, #1)
Determine limits from a graph (2.2, #5,7)
Determine infinite limits given a function (2.2, #25)
Determine a value “a” to create a limit in a rational function (2.3, #59)
Apply the limit laws to determine a limit. (2.3, #1)
Determine delta given x, f(x), L and a (algebraically) (2.4, #1)
Determine delta given x, f(x), L and a (graphically) (2.4, #5)

That's it! I'll be in early on Friday. Don't forget the makeup test for chapter 1 has to be completed before the weekend.

Tuesday, September 26, 2006

Section 2.3: Calculating Limits Using the Limit Laws

Hey guys! So bear with me, I’m not sure how great this will be…

Verbally, it goes something like this:

1. The limit of a sum is the sum of the limits (Sum Law)
2. The limit of a difference is the difference of the limits (Difference Law)
3. The limit of a constant ( c ) times a function is the constant times the limit of the function (Constant Multiple Law)
4. The limit of a product is the product of the limits (Product Law)
5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not equal to zero) (Quotient Law)

(Ok, so that all makes sense, right? Don’t worry, because in the next section we will prove Law Numero Uno. The other proofs are in an appendix somewhere. For fun with proofs on this stuff, as well as more info on this leson, go to chapter 2 on this page)

But wait, there’s more!

So far, all of the limits make sense, right? It’s all pretty intuitive.

Oh man. Ok, the book tells me that that’s it for the laws. However, we are not yet done. We have miles to go before we sleep (ok, maybe not miles, just hours)

Ok, there you go! Now, before my brain shuts off entirely from exhaustion, allow me to present a sample problem.


Oh, and by the by, Taylor, you're on deck...

Sorry about the huge gaps, you guys...I didn't crop it properly when I re-formatted the equations! My apologies!

Monday, September 25, 2006

Section 2.2 and such

Hello, this is Evan, the Lord Protector of Math.

Today I'm going to be reviewing limits.

Our Calculus book defines limits like this:

"We write
and say 'the limit of f(x), as x approaches a, equals L'

if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a."

So, to put this a little less pretentiously,
L is the value that f(x) is approaching as x approaches a.

One of the main things to remember when dealing with limits is:


Still having trouble remembering? Maybe this creepy comic book guy will help you remember.

It is also helpful to ask yourself this question:
What value is y approaching as x approaches the limit?

There are two main ways that we have learned to estimate the value of limits.

The first method is numerically.

With this method, we plug in values that are close to the limit, and see if we can spot a trend.

The second method with which we can solve limits is algebraically.
To use this method, we plug the value that x is approaching into the equation.
Look at this picture:

We can see here that this equation has been solved algebraically, and a graph has been drawn to make the concept a bit clearer.

With the graph, we can see that as x approaches -1, y is approaching 4. When -1 is plugged into the equation for x, 4 is the number we obtain.

And remember, even when plugging in values gives you undefined, the equation can still be solved! If you want to solve it algebraically, try factoring. Or, you could always try to spot the numeric trend.

Try this example problem:

Guess the value of

Now, the equation is not defined when x = 1, but it doesn't matter because the concept of a limit tells us that we are considering values close to
1 but not equal to it.

Let's try plugging some values into the equation.
x = 0.5, f(x) = .6666666667
x = 0.9, f(x) = .526316
x = 0.99, f(x) = .502513
x = 0.999, f(x) = .500250

Are we noticing a trend here?
We can safely guess that
= 0.5

Here's a link if you want further clarification of the concept:

Reminder: Kate, it's your turn tomorrow!

See you all tomorrow!

Sunday, September 24, 2006

Review: 2.1 The Tangent and Velocity Problems


  • tangent line: a tangent to a curve is a line just touching one point on the graph of a function; finding the slope of the tangent line finds the slope at that particular point and the limit as that point is approached; it should go in the same direction as the curve at the point of contact
  • secant line: these go through the point of the tangent line and another nearby point so that we are able to find a slope and eventually a limit
  • limits: secant lines approaching tangent lines illustrate the idea of these; they arise when we attempt to find the tangent to a curve or the velocity of an object
  • instantaneous velocity: the limiting value of average velocities over shorter and shorter time periods


  • lim x--> a f(x) - f(a) / x - a
If f(x)=x2a , and we wanted the equation of the tangent line through P (1,1), we could solve for x increasingly closer to 1 until we find that we are clearly approaching 2. So, lim x--> 1 x2-1 / x-1 = 2 We can check with x=1.001, 1.0012-1 / 1.001-1 = 2 approximately
  • average velocity = distance traveled / time elapsed (by using a very small interval of time, we can guess the instantaneous velocity)

Sample problem: A ball is dropped from 450 m above ground. What is the velocity of the ball after 5 seconds?

We must know Galileo’s law that tells us s(t)=4.9t2

Because we are dealing with an instant, there is no time interval and an average for a short interval must be used.

First, we will solve for the velocity between 5 and 5.1 seconds

=s(5.1)-s(5) / 0.1 = 4.9(5.1)2 - 4.9(5)2 / 0.1 = 49.49 m/s

Now, we do this again and again with smaller intervals (making a table is recommended) until about 5.001, at which point we can tell the average velocity is getting very close to 49m/s.

Time interval (s)

Average velocity (m/s)

5 to 5.1


5 to 5.01


5 to 5.001


Thus the instantaneous velocity at 5 is v=49m/s.


  • (this one goes into section 2.2, sorry Evan!)

Reminder: Evan, I do believe Monday’s notes on 2.2 are yours to post.

Personal Touches:

  • This stuff has a clear real-world application! We can find instantaneous velocity.
  • Why don’t we use point C to find a limit? Because c can’t line. (Yes, it is a pitiful joke)
  • a limerick I found on the web that does not have anything to do with the lesson:
        Integral z-squared dz
               from 1 to the square root of 3
               times the cosine
               of three pi over 9
               equals log of the cube root of 'e'.

Tuesday, September 19, 2006

Test 1 Topics

Yes, it's confirmed - we will meet in the library tomorrow to work on our Calculus Movie Project! See you there...

Here’s a list of topics that will be covered on this Thursday’s Chapter 1 Test. I’ve tried to indicate where a similar homework problem would be helpful.

  • Interpolation – given data or a graph, estimate/determine new points in the relationship/function. (1.1 - #17, 1.2 - #11,19)

  • Extrapolation – given data or a graph, estimate/determine new points in the relationship/function. (1.1 - #17, 1.2 - #11,19)

  • How does a graph change when certain values within the function vary? (1.4 - #31,34)

  • Given f(x) and g(x) (algebraically, numerically or graphically), determine values for f(g(x)) given x. (1.3 - #55)

  • Determine f(x) given a graph. (1.5 - #17)

  • Determine the range of a function. (1.1 - #23)

  • Determine the domain of a function. (1.1 - #23)

  • Determine an appropriate viewing window for a given function. (1.4 - #7,13)

  • Determine g(x) as a result of transformations of f(x). (1.3 - #3)

  • Work with inverse functions. (1.6 - #19)

  • Sketch a graph illustrating a given functional relationship. (1.1 - #11)

  • Sketch a graph given a transformed function. (1.3 - #3)

Sunday, September 10, 2006

Blog Postings

As a reminder, the requirements for each blog posting will consist of:
  • A review of the main point of the class lecture/demonstration.  This summary should highlight any relevant formulas and/or graphs and communicate your interpretation of the concept covered in class. (15 pts)  For this part of the posting, I am looking for quality, not quantity.

  • An example problem, including a statement of the problem, the answer, and the solution method.  (For your first post, using an example covered in class is acceptable, for additional postings, original examples will be required.) (10 pts)

  • A link to an additional Internet resource supporting the Topic of The Day. (5 pts)

  • A reminder to the next BlogMaster of their responsibility to post. (5 pts)

  • A “personalization” of your posting.  This personalization can be a comment about the day’s class, an image, a quotation, a question posed for discussion, a joke, or something else that reflects you as a student.  These personalizations must be in good taste!  (5 pts)

In addition to your posting, you will be expected to comment on a minimum of two (2) of your classmates postings during each quarter.  These comments must either further enhance your classmates’ understanding of the posted topic or further a discussion question posed in the original posting.

Additional Notes:
  • Postings will be due within 24 hours of class.  I will post a schedule of class scribes for the first quarter once everyone has joined the class blog.

  • For help with posting equations and graphs, please feel free to come ask me for assistance.

  • Initially, the blogs will be hosted on  As the year progresses, we hope to migrate to an internal website.

  • While we are on, there is some software available through the website that allows creation/editing of posts via Microsoft Word.

Tuesday, September 05, 2006

Blog Policies

There are some things I want you to remember about blogging. Many of things have been discussed by other teachers and classes, so I will paraphrase them here and try to give them proper credit:

First of all, our class will not be the only people to view our postings. The Internet is accessible almost everywhere these days, and even if a post is deleted, there’s no guarantee that the posting hasn’t been copied and propagated to other sites or linked to from those sites. This has a couple of implications:

First, privacy. We will only be using first names on the site. If I post pictures or video, no one will be identified, other than “Mr. French’s class”. Do not use pictures of yourself for your profile here. If you want a graphic image associated with your profile, use an “avatar” – a picture of something that represents you but is not you. Here’s a link to a fun image creator.

Second, etiquette, appearance and common sense. Bud the Teacher has these suggestions, among others:

  1. Students using blogs are expected to treat blogspaces as classroom spaces. Speech that is inappropriate for class is not appropriate for our blog. While we encourage you to engage in debate and conversation with other bloggers, we also expect that you will conduct yourself in a manner reflective of a representative of this school.

  2. Never EVER EVER give out or record personal information on our blog. Our blog exists as a public space on the Internet. Don’t share anything that you don’t want the world to know. For your safety, be careful what you say, too. Don’t give out your phone number or home address. This is particularly important to remember if you have a personal online journal or blog elsewhere.

  3. Again, your blog is a public space. And if you put it on the Internet, odds are really good that it will stay on the Internet. Always. That means ten years from now when you are looking for a job, it might be possible for an employer to discover some really hateful and immature things you said when you were younger and more prone to foolish things. Be sure that anything you write you are proud of. It can come back to haunt you if you don’t.

  4. Never link to something you haven’t read. While it isn’t your job to police the Internet, when you link to something, you should make sure it is something that you really want to be associated with. If a link contains material that might be creepy or make some people uncomfortable, you should probably try a different source.
Are there other considerations we should take into account? Use the comment feature to add any others or to clarify/expand on one of the above.

Monday, September 04, 2006


Congratulations! You found our class blog! This is where we as a team will hopefully create a resource to help us conquer any issues that arise during our class this year. This is the place to talk about what’s happening in class; to ask a question you didn’t get to ask in class; to share your knowledge with fellow classmates and any other Internet users who choose to read our notes;…and most importantly it’s a place to reflect on what we’re learning.

A large part of retaining knowledge requires reviewing and discussing new information on a regular basis. This blog is intended to help each of you do just that. Between creating your own posts and commenting on your classmates’ posts, you will have the opportunity to explore each of the topics we cover this year in greater depth. I hope you will use this forum to help yourself and your classmates in whatever ways you can think of.

Blogging Prompt

Occasionally I will include a posting of my own, either to clarify a concept or to generate some further discussion. These postings will have a title similar to the one above this paragraph.

To get things rolling, here’s a question for you to think about and respond: Is God a mathematician? Why or why not?

Don’t forget to email me with the information I requested in class so I can include you on the team!