AP Calculus AB

The Adventure Continues ...

2006-06-29T15:49:00.000-05:00

Our adventures in blogging continue....

Watch for 3 new blogs going live September 6, 2006 ...

Pre-Cal 30S (Fall '06) (Grade 11)
Pre-Cal 40S (Fall '06) (Grade 12)
AP Calculus AB 2006-07 (Grade 12)

So Long ...

2006-06-28T18:15:00.000-05:00

We had our graduation exercises today. A gentle push into the world for all of you. I hope you're leaving with the keys to your future in your hand.

I'm so glad we've had this time together,

Just to have a laugh or learn some math,

Seems we've just got started and before you know it,

Comes the time we have to say, "So Long!"

So long everybody! Watch this space in the fall for pointers to new blogs for each of my classes.

Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)

Student Survey Results

2006-06-28T18:02:00.000-05:00

The exam is long over and we did a little survey in class. The results are below; 4 students participated. Here are the result. Please share your thoughts by commenting (anonymously if you wish) below .....

How prepared were you to write this exam? (Average score out of 100)

75%

How much effort did you put into preparing for this exam? (Average score out of 100)

67.5%

How good a job did your teacher do preparing you for this exam? (Average score out of 100)

92.5%

Did you have enough preparation using your calculator?

Yes 100% No 0%

Did you have enough preparation without using your calculator?

Yes 50% No 25% Middle 25%

Was I too hard or too easy on you??

Semester 1 was easy. Semester 2 was really hard for me. It killed me actually.
Just right.
Too easy.
Too hard.

What was the best learning experience you had in this class?

Blogging (2)
Mini exams (2)
Group Work
Wiki
Pre-Tests
Teaching and explanations were very clear and easy to understand

What was the worst learning experience you had in this class?

Homework (2)
None
Blogging On Blogging before tests

What suggestions can you share for next year?

More wiki assignments from past exams.
More exam-like long answer questions in class.
Have students doing boardwork in class.
The blog didn;t help me that much.

It's interesting to compare the items that were considered both the worst and best learning experiences. Also, take a look at the list of worst learning experiences compared to suggestions for next year. Help me do a better job next year by commenting on what you see here ....

The Places You'll Go!

2006-06-09T19:09:00.000-05:00

You have brains in your head.
You have feet in your shoes.
You can steer yourself
Any direction you choose. --Dr. Suess, Places You'll Go

It's almost graduation--what direction are you choosing?

Passing it forward--

2006-05-29T20:09:00.000-05:00

Some 43 years ago, when I was about your age, my father who used to travel a lot, wrote me a letter enclosing a small article clipped from a magazine. In that letter, he wished for me a life of purpose and joy. He shared with me the clipping which described such a life and quoted George Bernard Shaw whose words you can read above.

I carried that article with me for years; unfortunately, somewhere in the many moves, it was lost. But not the thought and the power of those words. They have become a part of who I am. I know that my career as a teacher, and the mentoring I do now for teachers seeking National Board Certification are but "drops in the ocean" in this often violent, globalized world of ours but my life has been filled with joy, happiness and love.

I'm sure that my father writing and sharing those thoughts adds to their meaning for me, especially now that Alzheimer's prevents him from recalling what occurred. But his belief in me, in mankind in general, shaped my world. I'm passing that forward --my belief in you, my belief in mankind, and my wish that you find the real joy in life!

Wishing you success ahead!

2006-05-26T18:58:00.000-05:00

I am thinking that these days each of you, your family, and your friends are looking to your future and wishing you success. I'd like to do that too!! To wish for you all that Emerson describes-- Would this meaning of success be one you'd be willing to adopt?

Two roads diverged in a wood---

2006-05-18T19:25:00.000-05:00

THE ROAD NOT TAKEN by Robert Frost

Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;
Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,
And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.
I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I-
I took the one less traveled by,
And that has made all the difference.

This has always been one of my favorites as I've found myself faced with those "two roads diverged" so many times in my life. I'm thinking you may now be viewing "two roads diverged" now as you are about to graduate and I know you will be many other times in your lives.

My choosing ( It wasn't necessarily an easy choice.) Earlham College was one of those times I "took the road less traveled" and it has made "all the difference" in my life. My years at Earlham have had a profound impact on who I am today, how I see the world, and what I believe.

Have you taken/will you be taking a road "less traveled by" and has it/will it made/make "all the difference"? Or does it matter?

Congratulations-- What's next?

2006-05-09T07:24:00.000-05:00

Congratulations on completing your AP Calculus exam!! That is quite an accomplishment!!! When you look back on the process, did you surprise yourself? Did you take some time to celebrate?

After celebrating, was your first thought "What's next?"

What will be the next immediate challenge? Isn't that the incredible part of living? That once you complete one challenge, another awaits. More hard work, more frustration, more hard work, always something.

Below is one of my favorite quotes (when I was teaching, we began each class with a quote; often ones that my students suggested; we felt each one captured something essential about living and life) ; Ara shared some of hers and since I've lived my life by a quote as you'll find in a future posting, I'll continue with this one here:

"Every day you may make progress. Every step may be fruitful. Yet there will stretch out before you an ever-lengthening, ever-ascending, ever-improving path. You know you will never get to the end of the journey. But this, so far from discouraging, only adds to the joy and glory of the climb." --Sir Winston Churchill

Your climb has just begun! Your AP exam was one great step!! What's next?

Kakuro Sunday

2006-05-07T11:12:00.000-05:00

'A Kakuro consists of a playing area of filled and empty cells similar to a crossword puzzle. Some black cells contain a diagonal slash from top left to bottom right with numbers in them, called “the clues”. A number in the top right corner relates to an “across” clue and one in the bottom left a “down” clue.

The object of a Kakuro is to insert digits from 1-9 into the white cells to total the clue associated with it. However no digit can be duplicated in an entry. For example the total 6 you could have 1 & 5, 2 & 4 but not 3 & 3. Sound simple? Be warned it gets hard and is as addictive as Sudoku.'

Click here for more Kakuros.

(Thanks again to Think Again!)

Running Up That Hill

2006-05-02T22:29:00.000-05:00

Take a look at this:

You're not the only ones using our blog to get ready for tomorrow. ;-)

If you're an AP Calculus student reading this now ... go to bed! You've got to be ready for tomorrow. Your brain is an organ; take good care of it. It needs sleep, a good breakfast tomorrow (like cheese, fruit, eggs and juice) and lots of water. Don't forget to do some exercise when they give you breaks so you can get the blood from your bottom into your brain. ;-)

Lastly, remember that luck has nothing to do with it. It's all about doing your best ... Learn Hard!

Answers to Mini-Exam #6

2006-05-02T09:45:00.000-05:00

(1) D
(2) D
(3) A
(4) B
(5) A

free response question

(a) Because g is the derivative of the function ƒ, ƒ will attain a relative minimum at ta point where g=0 and where g is negative to the left of that point and positive to the right of it. This occurs at x=6.

(b) Bcause g is the derivative of the function ƒ, ƒ will attain a relative amximum at a point where g=0 and where g is positive to the left of that point and negative to the right of it. This occurs at x=3.

(c) We are trying to find the area between the graph and the x-axis from x=-3 to x=6. From x=-3 to x=3, the region is a semicircle of radius 3, so the area is 9π/2.
From x=3 to x=6, the region is a semicircle of radius 3/2, so the area is 9π/8.
We substract the latter region from the former to obtain: (9π/2) - (9π/8) = (27π/8)

(d) Because ƒ''(x) = g'(x), we are looking for points where the derivative of g is zero. This occurs at the horizontal tangent lines at x=0, x=4.5, and x=7.5.

Just do it!!!!

2006-05-01T18:35:00.000-05:00

On the day that I was to take my "big" test, I was just about to leave for the testing center when I asked my husband to wish me luck. "No," he said, "I won't do that." I was crestfallen. I felt like I needed one last boost before the "big" one.

Then he continued, "You don't need luck. You're smart. You're prepared. You're good. I believe in you. Go out there and just do it!!!! I'll be here when you get back."

It's time for me to pass that forward to you! You don't need luck. You're smart. You're good. You're well prepared (thanks to your hard work and Mr. K). I believe in you. Go out there and just do it!!!!

Your second wind--

2006-04-30T18:20:00.000-05:00

Most people never run far enough on their first wind to find out they've got a second. Give your dreams all you've got and you'll be amazed at the energy that comes out of you. --William James

I wonder, would this apply to calculus too?

From ZZZZZ's to A's

2006-04-28T18:38:00.000-05:00

The bottom line: Teens need 9.25 hours of sleep per night

from Do Teens Get Enough Sleep?

In experiments done at Harvard Medical School and Trent University in Canada, students go through a battery of tests and then sleep various lengths of time to determine how sleep affects learning. What these tests show is that the brain consolidates and practices what is learned during the day after the students (or adults, for that matter) go to sleep. Parents always intuitively knew that sleep helped learning, but few knew that learning actually continues to take place while a person is asleep. That means sleep after a lesson is learned is as important as getting a good night's rest before a test or exam.
from Adolescents and Sleep

At the risk of sounding "mom-ish", have you taken this into consideration in your preparation for your upcoming test? I saw in a scribe that Mr. K had mentioned it!

Asking only because, when I was sleep deprived, I know I wasn't fully aware of how much more difficult problem solving and remembering was. I never fully realized how sleep deprivation changed my abilities and me until after I started getting adequate sleep.

Another factor in your preparation to be your very best for your test??

Your self portrait!

2006-04-26T18:46:00.000-05:00

How is your self portrait coming?

Visualizing excellence in calculus!

2006-04-24T19:48:00.000-05:00

"Almost all of the world-class athletes and other peak performers are visualizers. They see it; they feel it; they experience it before they actually do it. They began with the end in mind. You can do it in every area of your life. Before performance, a sales presentation, a difficult confrontation, or the daily challenge of meeting a goal, see it clearly, vividly, relentlessly, over and over again. Create an internal "comfort zone." Then, when you get into the situation, it isn't foreign." --Steven Covey

Are you an athlete? Do you visualize already? Maybe you've already thought about this before--

I wanted to share because I think that with your success on your mini exam (saw that in one of Sarah's comments! Congratulations to you all!) and all that you are sharing, and reading, and problem solving on this blog, you really can visualize excellence for May 3! You are doing everything the quote from Stephen Covey suggests! Now with visualization for May 3, you'll be in your comfort zone and on your way to a peak performance.

I had mentioned earlier in an earlier post that I had been faced with a major 3 hour assessment. I sense that I prepped in some of the same ways that you are. I "relentlessly" researched and reviewed all I could find that could help me with the six test questions. The organization provided generic test questions and scoring guides for each question. I practiced answering the question within the half hour framework. I practiced with the software that I would be using in the testing center. I visualized how each question might be phrased and how I would respond. I can't honestly say I was in a comfort zone when I entered the testing center, but I know that when I took a deep breath and began, all that I had visualized and practiced seemed to flow from my brain, through my fingers and into the testing software.

Do you think visualizing could be helpful to you too?

Ara's Blog Assignment

2006-04-24T10:15:00.000-05:00

An object in motion along the x-axis has velocity v(t) = (t + e^t) sin (t^2) for the interval 1, 3.

a. How many times is the object at rest?

Defining an object at rest means the slope is zero, that is, the same way as saying the derivative of the parent function is equal to zero. Therefore, since the velocity function or the first derivative is given, we only have to look for the points where v(t) is zero.

0 = (t + e^2) sin (t^2)

Plotting it into the graphing calculator, we obtain the roots at x = 1.7725 and x = 2.5066
Having these points, we say that at x = 1.7725 and x = 2.5066, the object is at rest.

b. When is the object moving to the left? Justify your answer.

A movement to the left, in my understanding, is similar to moving backward as opposed to forward. Therefore, the object is moving towards left on the interval where the velocity function is negative.

'velocity less than zero'

Looking at the graph, the points where the velocity is negative are pretty obvious. Having said that, on the interval [1.7725 , 2.5066] the object is moving to the left.

c. During the interval found in part b, when is the speed of the object increasing?

In this question, the acceleration or the second derivative is needed since the speed of the object is being asked.

a(t) = (1 + e^t) (sin (t^2)) + (t + e^t) (2t) (- cos (t^2))

*it is actually much easier if you guys just plug the function into the calculator at Y2 using nDeriv.

Finding the interval where the speed of the object is increasing also means looking for the positive values of a(t).

The roots of a(t) are located at x = 1.377 , x = 2.2161 and x = 2.8307.

Now, to the left of x = 1.377 the graph is positive and to its right negative. So on the interval 1 , 1.377 the speed is increasing.

at x = 2.2161, the graph is negative to its left and positive to its right. on the other hand, at x = 2.8307, the graph has its positive values on its left.

But, since the interval is specified, we only have to consider the roots between x = 1.7725 and x = 2.5066. Therefore, the object's speed is increasing on the interval 2.2161 , 2.5066

Steve's Blog Assignment

2006-04-23T17:03:00.000-05:00

The town of Calcuville has a water tower whose tank is a circular ellipsoid, formed by rotating an ellipse around its minor axis. The tank is 20 feet tall and 50 feet wide.
a) If there are 7.46 gallons of water per cubic foot, what is the capacity of the tank to the nearest 1000 gallons?
b)Calcuville imposes water rationing regulations whenever the tank is only one quarter full. How deep is the water in the tank when water rationing becomes necessary?
c) During peak usage, 5000 gallons of water are used per hour. How fast is the water level dropping if it is 12 feet deep when peak usage begins?

Equation for an ellipse ((x-h)²)/a² + ((y-k)²)/b² = 1

width is related to the x axis, and height is related to the y axis

50 = 2a 20 = 2b
a = 25 b = 10

Substitute the values of a and b into the equation and the coordinate (0,0) because no specific coordinate was mentioned and it is also easier to work with.

((x-0)²)/25² + ((y-0)²)/10² = 1
x²/625 + y²/100 = 1

Find volume by using the technique of slicing. Solve for x because ellipse is rotated around the minor axis which happens to be the y axis.

x² + 625y²/100 = 625
x² = 625 - 625y²/100
x = (+/-)√(625-(25y²/4))
You need both the positive and the negative of the square root because it is an ellipse and it exists in both the first and fourth quadrants.

V(y) = πΣ(0,10) √(625-(25y²/4))²dy
Σ takes the place of the integral sign and the numbers after it are the (a,b) of the integral.
That part covers only the top half or the first quadrant of the ellipse, but the bottom or fourth quadrant MUST be included in order to find the volume of the whole ellipse.
V(y) = πΣ(0,10)(√(625-(25y²/4)))² + πΣ(-10,0)(-√(625-(25y²/4)))²
Because the inner function is squared, it becomes positive which means that it becomes the same as the positive one and it can become one integral.
V(y) = πΣ(-10,10)(√(625-(25y²/4)))Â²
The square root and the squared cancel each other out. Leaving a parabola.
V(y) = πΣ(-10,10)(625-(25y²/4))
Antidifferentiate then evaluate
= π(625y-(25y^3/12) from -10 to 10
= π{[625(10) -(25(10^3)/12)]-[625(-10)-(25(-10^3)/12)]}
= π[(4166.6667)-(-4166.6667)]
= π(8333.3333)
= 26179.9388 ft^3
Use the constant of 7.46 gallons of water per cubic foot to calculate how many gallons of water there are.
V= (26179.9388 ft^3)(7.46 gal/ft^3)
V= 195302.3433 gallons
Answer to Part a) 196000 gallons of water
Part b) Requires changing volume from gallons to cubic feet and take one quarter of that amount.
0.25V= (1/4)(195302.3433 gallons/7.46 gallons/ft^3)
0.25V= 6544.9847 ft^3
Now take the integral from the bottom of the tank to y and set it equal to 0.25V
πΣ(-10,y)(625-(25y²/4)) = 6544.9847
Divide by π
Σ(-10,y)(625-(25y²/4)) = 2083.3333
Antidifferentiate the left side and evaluate it while still equaling 2083.3333.
625y-(25y^3/12) from -10 to y
[625y-(25y^3)/12]-[625(-10)-(25(-10^3)/12)] = 2083.3333
625y-((25y^3)/12)+4166.6667 = 2083.3333
625y-(25y^3)/12) = -2083.3333
Solve for y to find the depth.
Multiply by 12.
7500y -25y^3 = -25000
0 = 25y^3 -7500y -25000
0 = 25(y^3 -300y -1000)
Using a calculator, I found y to equal -3.472964, in terms of depth, -3.472964 is not the depth, but it is the y coordinate on the graph of the ellipse. For example, when y=-10, depth is 0 ft. If y =0, depth is 10 ft, and finally if y=10, depth is 20 ft, the max.
depth = 10-3.472964
Answer Part b) depth = 6.5270 ft
Part c) Given dV/dt = -5000 gallons/hour and the depth of the water is 12 feet, I can find the y value needed to find the rate of how fast the water level is dropping. y=2.
V = 7.46π[(625y-(25y^3)/12)-4166.6667]
Differentiate implicitly
dV/dt = 7.46π[625y'-(25y²y'/4) -0]
dV/dt = 7.46π[625y'-(25/4)y²y']
Substitute the values for dV/dt, and y to solve for y'.
-5000 = 7.46π[625y'-(25/4)(2²)y']
-213.3444 = 625y' -25y'
-213.3444 = 600y'
-.3556 = y'
Answer Part c) y' = -0.3556 feet/hour, the depth is decreasing at a rate of 0.3556 feet per hour when the depth is 12 feet.

The Da Vinci Code Quest Sunday

2006-04-23T15:34:00.000-05:00

It started last week. Google releases one puzzle each day for 24 days until the movie "The Da Vinci Code" is released in May. So far 7 puzzles have been released. You have to solve the puzzle to reveal a clue. Then you have to answer the clue question(s) to advance to the next puzzle. You can win a prize for solving all 24 puzzles. Now I realize this is all about marketing and they're really just trying to get as many of us as possible to go see the movie but the puzzles are really cool! Google searching often helps to find the answers. One of the puzzle questions can be answered using The Fundamental Principle of Counting and the very first (sudoku-like) puzzle uses a couple of mathematical symbols.

Challenge 1: What is the question that can be solved using The Fundamental Principle of Counting and how do you use the counting principle to find the answer?

Challenge 2: What mathematical symbol is used in the very first puzzle and what number does it represent? (Not the "delta," in a later puzzle it has a different meaning.)

You have to sign up for a Google Homepage in order to play, but that's a free and very useful service. After that you can begin the game. Click on the US button to start 24 days of fun! (Actually, 17 because you could work through the first eight today.) Don't forget to also find the answers to the Challenge Questions above!. ;-)

Blog Assignment 1

2006-04-23T12:46:00.000-05:00

A differentiable function f defined on -7 < 0< 7 has f(0)=0 and f'(x) = 2x sin x- e^(-x^2) +1
a) describe the symmetry of f.
b) On what intervals is f decreasing?
c) For what values of x does f have a relative maximum? Justify your answer.
d) How many points of inflection does f have? Justify your answer.

a) f is an even function.

b) f'(x) = 2x sinx - e^(-x^2) + 1

2x sinx- e^(-x^2) +1 =0

x=-6.2024, -3.294, 0, 3.294, 6.2024

+ - + + - +

--------1----------1----------1-----------1-----------1-----------

-6.2024 , -3.294 , 0 , 3.294 , 6.2024

If f'(x) less than zero,f(x) is decreasing, therefore, f(x) is decreasing at (-6.2024, -3.294) and (3.294, 6.0224).'

c) If f'(x) larger than zero, f(x) is increasing; If f'(x) less than zero, f is decreasing.

from increasing to decreasing, f reaches a maximum. f has relative maximum at x= -6.2024, and x= 6.2024

d) f"(x)=2x cos x+ 2 sinx + 2x e^(-x^2)
2x cosx+2 sin x +2x e^(-x^2) = 0
x= -4.9136, -2.0405, 0, 2.0405, -4,9136
f has inflection points when f"(x)= o, so, f has 5 inflection points

Sarah's Blog Assignment

2006-04-20T20:32:00.000-05:00

Question:
Let C represent the piece of the curve y = (64-16x^2) ^ (1/3) that lies in the first quadrant.
Let S be the region bounded by C and the coordinate axes.
a) Find the slop of the tangent line to C at y=1.
b) Find the area of S.
c) Find the volume when S is rotated arount the x-axis.
d) Find the volume when S is rotated arouind the line x=-2.

I have tried to find an equation editor that will show the following math signs properfly but the files seem to be given out incomplete. As for point, it really is hard to use and I have no time to fiddle with it, so bare with how my answers will be, and sorry in advance.

Answer:
a) slope @ y=1

1 = cube root ( 64 - 16x^2 )

cube both sides

1 = 64 - 16x^2

-63 = - 16x^2

63 / 16 = x^2

square root both sides

square root ( 63 / 16 ) = x

x = 1.9843

( 1.9843 , 1 )

y = ( 64 - 16x^2 )^(1/3)

y' = (1/3)( 64 - 16x^2 )^(-2/3) * ( -32x)

y' = (-32x) / [( 3 ) ( cube root [ ( 64 - 16 x^2)^2])]

therefore, y' (1.9843) = - 21.1538

(**NOTE: It is always good to store exact values into your calculator. If you are using programs like the Riesum programs, be careful where you store it, the alpha letter you may be using might be used by the program as well. **)

b) Area of S.

x - intercept ( 2 , 0 )

c) Volume of S when rotated around the x-axis.

Taking a slice, it would look like a disc. A circle. Area of a circle is the pi times the radius squared. Pi becomes a constant. The radius is measured by the function itself so therefore,

d) volume when S is rotated around the line x= -2.

When revolving it around the vertical line, we get a cylinder formed. Method that we use is cylindrical shells. Thinking of a piece of paper and just wrapping it around some center, How do we find the volume of that piece of paper? L * W * H. Height in this case is still the function value. Width is the itty bitty x difference, dx. Lenght though is a the circumference of the circle, 2 pi radius or 2 pi x. Therefore we have the integrand,

P.S. To that post, what else makes me focus? Other than candies, if I'm not at home its the music, not just any particular ones right now are korean r&b and pop. As long as it sounds good to me and something that I could never sing or know what its about. =D

See you all Monday.

The Main Thing

2006-04-19T19:40:00.000-05:00

Spring has sprung in Chardon, Ohio! Yes! We get lots of snow and winter seems so long that when the daffies bloom and the forsythia bursts forth with yellow, I'm estatic. (Last year this time, a late storm dumped 15 inches of snow on us) Is Winnipeg the same?

With spring, I'm finding that it is more difficult to stay with the course I'm designing for superintendents and principals now. I'd much rather be walking in the park, or out in the garden-- And so I've been using the thought above to help keep me focused.

What about you? What helps keep you focused?

endings-- beginnings--

2006-04-13T18:44:00.000-05:00

Ara's post really hit a spot with me. This entire notion of an end to something or not---
Do you think this quote applies as you work toward to your graduation?

today's last.. doesn't seem like it

2006-04-13T05:30:00.000-05:00

"Give a man a fish and you feed him for a day, teach a man to fish and you feed him for a lifetime"

Wow. It surprises me that I lasted this long in this course. I really had no idea that I am close to finishing it. I'm telling you guys, it was pretty tough along the way... the thought of quitting had crossed my mind hundreds of times already. I felt like through the entire span of this class I was always hanging on to the ropes. But Mr. K, he stretched his arm, reached for me and pulled me through. I wouldn't be surviving this class without you.

Although, this is actually my last post on blogging on blogging, it seems to me... it's not. As that Chinese proverb says, learning will never end. In every step I take and in every move I make (sounds familiar huH? ;D ), there will always be an instance where I'll remember and apply (well, somehow) some of the lessons we took in this course: the optimization, the related rates, the volumes by slicing (ooh... I hated that!), the slope fields and all. Could I be telling my friends the best time they should drink their hot cocoa? Or would I tell my mom the right dimensions in wrapping christmas gifts? Or maybe, just maybe, I can estimate the rate of how horrifyingly fast or sadly slow the airplane flies when I spend my holiday back to the Philippines... Hmm... I might consider that one! ;)

This could be the last, but it's never going to be the last. There are a lot more things to learn in this course and I am actually looking forward to struggling through them... *smiles*

A New Feed Window

2006-04-09T23:08:00.000-05:00

There is another Winnipeg AP Calculus class sharing their learning on a blog. You can peek in on what they're learning by checking out the new feed window way down there at the bottom of the side bar underneath the del.icio.us box. Take a look at their blog. Are they publishing anything that you find helpful? If so share it with us in the comments to this post.

I'll bet you like getting comments on our blog. Be a good netizen; drop in on them and leave them a positive comment too. ;-)