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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Friday, March 31, 2006

Scribe for last Thursday

Differebtial Equations-intro Euler's Method


Consider the differential equation dy/dx=y-x
Complete the table below, starting at the point (0,0) to generate a numeric solution to be the differential equation on interval (0,1) in 4 steps (so n=4 and △ x=0.25).

Nest scribe is Chris



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Thursday, March 30, 2006

You guys need to see this ....



This is a video recording of a presentation called: "From Smart Toilets to Smart Schools." It was given by a world class educational technologist named Alan November. He travels all over the world teaching teachers about powerful uses of technology for their classrooms. He's been talking about us all over the world. In the video he talks about our whole class, and Ara, he mentions you by name. (How cool is that?) This particular presentation was given in Ohio on February 14th. If you didn't really believe me when I said you had a world wide audience reading and learning from your work, well, ..... watch the video.

Start watching the video at the 38 minute mark. I think it will go offline on April 14th so don't put off watching it. You'll need to have realPlayer media player installed on your computer (it's free). Just click on the link to [RealPlayer - Free]. Here's the video! You should be very proud of yourselves ... I'm proud of you. ;-)




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Wednesday, March 29, 2006

Urgent, Important? First things first!


I can certainly relate to Sarah’s feeling that there are too many big rocks for the bucket of our life, as she mentioned in a comment. In early 2003, my bucket seemed to overflow. I felt then that I had to find a way to make it all work. Achieving my goals was too important to me and my test (which if I passed would lead to a national teacher certification; only 40% of test takers passed) was only one of many goals. I was really trying to be superwoman—just as I imagine many of you work at being superteens. I thought I was prioritizing but ooohh---a to-do list with 20 items all the time!!!

What I struck me one day when I found myself in tears because I had not met a school deadline, was that I was not always prioritizing by what would help me meet my goals. Urgent “stuff” kept happening and I was always responding to that. Does that sound familiar?

That night I sat down and broke my test review into manageable chunks (I had 2 months until the test; I divided the topics into sessions for those two months) and put them into my planner. I did that first because the goal to pass the test was so important. And I planned to turn off instant message, not answer the phone, or have the TV on during the review sessions. I had tried to plan before but was always interrupted by the phone or my students on instant message with questions about our studies and I found that I didn’t accomplish half as much then.

Then I looked at my other rocks, and categorized them: vital, important, or nice. Then I took the vital rocks and categorized them again: vital, important, or nice. So my house wasn’t very clean during the process, and we didn’t have gourmet dinners. But the laundry was done and we had quickie suppers. My student’s work was graded but I didn’t plan any big field trips or projects during that time. I set aside a time every second night to evaluate their work. As I look back now, I prioritized, and then prioritized again. I did a mental daily check of my goals and made every effort not to be dragged down by urgent if it didn’t help me achieve my goals.

Of course, I had to be flexible at times. I couldn’t always follow the plan exactly. But since I knew where I was going and I had planned for time to get there, my review was accomplished by “the day”.

I truly believe, that with good preparation and putting first things first, you'll too feel that great rush of a job well done, and a goal achieved when you learn your scores. I share these experiences, knowing that you are planning and reviewing, but wondering is there one little piece here you could use to help you on your way? Or can you point us to some tips that are really helping you manage the rocks and put first things first?



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Monday, March 27, 2006

New yet Old

�One of the greatest pains to human nature is the pain of a new idea.�

Or is it? Sometimes new beginnings are just too hard to take. But some do give you something to think about. It's the same feeling as the day you packed up all your stuff to move in to a new house. Boy you really wished you never went to that neighborhood. But eventually you learned to like it. Or that time when babies had to stop drinking from their milk bottles and start acting like "grown ups." You see them whine and do every little thing they can to persuade you into giving back their tiny, precious milk containers. And you always wonder "What's the big difference?" Well, tell you my friend, no matter how alike two things are, you'll always see yourself looking for that same old thing you've grown to love. It's never the same.


But then, the case here is quite clashing to what you've just read. (Some introduction huh? ;P) I just put that up to make it a little bit more interesting. We do have a new topic (if we all would consider that) but it's not that heart-breaking or agonizing to take. It's pretty cool actually.

This day, we learned some new terms of those we've known in the past chapters. NEW WORDS, SAME CONCEPT.
  • Differential Equation - equations that relate derivatives to other functions
  • Order - refers to the highest-order derivative of the differential equation
  • General Solution - corresponds to a family of solution curves
  • Particular Solution - exact solution for the differential equation
  • Initial Condition - The values assumed by the variables in a component, system or model at the beginning of some specified duration of time
  • Initial Value Problem - the combination of an initial condition and a differential equation
These are just some and I know there's a bunch to come.

So as this point, saying DERIVATIVE or SECOND DERIVATIVE is prohibited. We all have to get used to the new ones. Because if not, it's never going to be a habit. And if it's not a habit, it's never going to run successfully in our system. And if it doesn't, then there was no learning.

�The greatest pleasure in life is doing what people say you cannot do.�

Are you ready for this? :)



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Friday, March 24, 2006

Invading your space?

Greetings AP Calculus!
I do hope that you don’t feel that I’ve invaded your space! That certainly is not my intent! But after reading your blog, I felt that we might have something in common—facing a major testing challenge. I made it through my testing successfully, thanks to hard work, mentors, and lots of encouragement-- and I hoped I could help to “pass it forward” so to speak. Thinking we all might learn from each other if we had an opportunity for conversations about dealing with challenging and stressful testing (and mine was more than stressful!), I just jumped right in without asking your permission (I had Mr. K’s of course). Let me know if you’d rather I’d not post up front. However, I’m not sure you can get me to go away completely, because it’s my sense that such a talented group as you should have a good cheerleading section on your way to May 3!

I asked you a question in my last posting and since I’ve had no takers with an answer in the comments, I’m going to answer it myself, given the time to May 3 is steadily dwindling! If you haven’t read that posting yet, do it now before you read the answer!










The answer is “Habit”.

I know from lots of experience that habits can ensure that I reach my goals or hold me back. Given that, there have always been 7 habits that if I worked at them have helped me be effective! Have you seen these 3? Do you think these habits can be of value as you look to “the test”?
--------------------------------------------
I think this short autobiography is a good introduction to habit 1.
There's A Hole In My Sidewalk: AUTOBIOGRAPHY IN FIVE SHORT CHAPTERS
by Portia Nelson
Chapter 1
I walk down the street.
There is a deep hole in the sidewalk.
I fall in.
I am lost.... I am hopeless.
It isn't my fault.
It takes forever to find a way out.
Chapter 2
I walk down the same street.
There is a deep hole in the sidewalk.
I pretend I don't see it.
I fall in again.
I can't believe I am in the same place.
But, it isn't my fault.
It still takes a long time to get out.
Chapter 3
I walk down the same street.
There is a deep hole in the sidewalk.
I see it is there.
I still fall in.... It's a habit.
My eyes are open.
I know where I am.
It is my fault.
I get out immediately.
Chapter 4
I walk down the same street.
There is a deep hole in the sidewalk.
I walk around it.
Chapter 5
I walk down another street.


Habit 1: Be Proactive®
Take responsibility for your life.


--------------------------------------------------------------
Does this cartoon say it all for Habit 2?

Habit 2: Begin with the end in mind.®
Define your mission and goals in life..


-----------------------------------------
Finally, Habit 3: Rocks, Pebbles, Sand, Water—And Calculus is which?

A time management specialist was asked to give a presentation on her specialty. She decided to do a demonstration. First she asked her assistants to bring a big bucket and put it on the table in front of the audience. Then she asked for large, grapefruit-sized rocks and filled the bucket with them.
"Is the bucket full," she asked?
"Yes!" said the crowd, but she asked for more to put in anyway. This time her assistants brought in pebbles. She poured the pebbles in the bucket and it held a surprising number in the space between the big rocks.
"Now is the bucket full?" she asked.
"Yes!" "No!" "Yes!" "No!" said various persons in the crowd. Some people were uncertain; some were getting suspicious. The time management specialist asked for more. This time the assistants brought her sand. She poured sand in the bucket and it filled the spaces between the pebbles.
"Now is the bucket full?" she asked.
"No!" they answered. By now, everyone was suspicious. So she asked for water and poured in quite a lot. Now no one could think of anything else that could fit in that bucket.
"What does this process demonstrate?" asked the time management specialist.
One member of the audience spoke up: "No matter how busy you are, you can always fit in one more thing."
"I can see how you might think that was my point, but it is not," said the specialist. "I was trying to show you that if you don't put the big rocks in first, you'll never get them in at all!"

Habit 3: Put First Things First®
Prioritize and do the most important things first.


Best,
Lani



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Thursday, March 23, 2006

Who Am I ?

I am your constant companion. I am your greatest helper, or your heaviest burden. I will push you onward, or drag you down to failure. I am at your command. Half of the task you do, you may just as well turn over to me. I will do them quickly and correctly.

I am easily managed, but you must be firm with me. Show me exactly how you want something done. After a few lessons I will do it automatically. I am the servant of all great people; of all great failures as well. Those who are great, I have made great. Those who are failures, I have made failures.

I am not a machine, though I work with all the precision of a machine, plus the intelligence of a person. You may run me for profit, or run me for ruin--it makes no difference to me.

Take me -- train me -- be firm with me, and I will place the world at your feet. Be easy with me, and I will destroy you!

Who Am I ?
And what can this possibly have to do with calculus?



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Sunday, March 19, 2006

Box Up Sunday!

This is a clever little game. You've got to get the small blue box inside the large red box. You can only push a box from the inside. The black boxes, if used cleverly, can help you get the blue box inside the red one. But sometimes they're just in the way. I made it to level 4 pretty quickly, but then it starts getting tough. How far can you go? ;-)

Have fun with this!




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Tuesday, March 14, 2006

Seek and ye shall find ...



The Coin Hunt has officially begun as of 12:30 this afternoon. The race is on! Who will be the first to find the coin? Will the students find their coin before the teachers find theirs? Who will win the pizza party? Which charity will benefit from this year's hunt?

Check the walls of the building as you walk into school in the morning for hints to figuring out the puzzles.

Happy π Day!!
Have fun with it. ;-)



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Sunday, March 12, 2006

When test time comes, be ready

It doesn't matter what you do in before the test, what matters is what you do on your own time. This unit was fairly short, section 1 was net distance which is the integral of the velocity function and total distance traveled which is the absolute value of the velocity function then take the integral of that. Section 2 was volumes by slicing, which involves revolving the graph of a function around the x-axis. You can find the volume of that solid by finding the integral from (a,b) of the function A(x). A(x)=(pi)(x)², x represents the radius of the circle, which is the function value. If it is a washer then in place of x, subtract the smaller function from the larger function. In section 3 there was the shell method for solving a problem if a function was revolved around the y-axis. The volume of that solid can be found by taking the integral from (a,b) for 2(pi)*x*f(x). Section 4 was the average value over an interval. The average value of a function is represented by the integral from (a,b) integrate f(x) and multiply it by the constant (1/(b-a)). Section 5 was the hard part of the unit, WORK, and DENSITY. Work = (Force)*(Distance) and Density= mass/volume. The best way to attempt these problems is to break them up in pieces and figure out what each part of the equation is before attempting to find the final solution. For example, if you have a WORK problem, find the FORCE and then find the DISTANCE before combining them in an integral. The problems from this section of the textbook are the hardest we've faced thus far, so good luck.



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Sunday Knight



How far can you go? Play here! ;-)



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Thursday, March 09, 2006

The Mystery Coin Hunt!

π Day is around the corner .. it's five days away! Soon, soon, the hunt will be on!




Somewhere on the property of DMCI a coin will be hidden. Hidden so carefully and cleverly that it cannot be discovered by chance or simply by looking for it. On March 14, π Day, the coin's location will be revealed buried in a series of riddles and puzzles. Until it is discovered the coin's location will remain a mystery....





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Wednesday, March 08, 2006

The amount of WORK

Yes calculus takes a lot amount of work. Imagine our brain cells, constantly moving the data through the brain. Our brain only has a particular volume to it, but it takes a lot of force to move all the data swimming inside it to a particular destination.

*hehe* Just kidding guys.

But really the topic we covered on Tuesday was on WORK. To those who took Physics, you may already have conjured up the formula for work:

WORK = (FORCE) x (DISTANCE)

Applying Calculus…

Overlooking the Problem from Tuesday…

We are given a rectangular prism tank 6 ft long, width of 5 ft, and height of 4 ft. On the top of it is a spout 2 ft. tall. The prism is filled with a liquid whose density is 40 lbs/ft³. Write the integral on amount of force it takes to move the water out of the tank.

When a picture is not provided, it is always good to draw one to understand the problem more.

To start off the problem:
1) Draw a picture, unless it is provided.
2) Write down what you are given.
3) Recall a formula.
For this problem, it’s the WORK formula. Work = F x D

In the problem we are not given the force straight off the bat. We are looking for the force of gravity, which is measured in lbs. We are given that the density in the units lbs/ft³. What do we multiply the density by to leave us off with just lbs.?

Lbs/ft³ x _____ = lbs.

The answer is: multiply it by ft³. In other words we multiply by the volume.

To find the volume of a rectangular prism: (length) (width) (height)

We are to move all of the water out of the tank. Taking an arbitrary slice, the height of the slice will be a very small itty bitty change in y.

We all know that it’s dy.

The length and width is given in the problem 6 ft. and 5 ft.

Good Job. You have found the weight of the substance. Now for distance:

Looking at the prism, imagine an arbitrary slice on the surface. That slice has 2 ft of distance to move out of the tank. Now imagine an arbitrary slice on the very bottom. The height of the prism add the height of the spout gives you 6 ft of distance to cover to get out of the tank. What equation would give you the possible heights?

Answer: 6-y

One thing left to figure out, is the limits of integration. The problem had said the prism is filled with a liquid substance. The height of the prism is 4 ft. So therefore, the limit of integration is 0 to 4.

Now to write the integral it is:


On Tuesday we were given another similar problem but this time it was a cylinder. The only thing that changed was how to find a volume of a cylinder. I guess this is where our bank of volume formulas comes to play once again. It is always good to have any of our formula sheets around.

So it was another day in calculus. I believe we have only 18 classes to go? Or maybe even less before the AP exam. Not to scare anyone away, its one tough exam. Everything Mr. K says is true. It’s nothing like any of the exams we had ever written. Good practice, a good understanding of the concepts is needed, and a great memory too. Take advantage of all resources available to us. We’ll need it.

Well see you all tomorrow =)


Reply to COMMENT: I find the quizzes a good review of the topics we've already covered. It also sets us ready for the time given in the exam to answer multiple choice questions. It also helps us not to be so dependant on the calculator. So overall, its helps us get ready for that exam on the beginning of May =)



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Sunday, March 05, 2006

Sunday Gridlock



In this game you have to move the blocks (vertical blocks move only vertically and horizontal blocks move only horizontally) out of the way so that the blue block can slide out the "door" on the right. Although this game can sometimes get frustrating there is always a way out. Remember Sisyphus!

So far I've made it to level 6, how far can you get?



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Friday, March 03, 2006

Solid by Revolution

In today's class we started with a quiz of course, just like every day. We then went on to practice integrals by solving problems using disks and cylindrical shells. One such problem is what would be the volume of a solid obtained by revolving the region bounded by the lines y=9-x^2 and y=-7 around the line y=-7. The points of intersection of the two functions are at x=-4 and x=4.

Using the disk method is the easier method for this particular problem. Because a disk can be thought of as a cylinder and the equation for a cylinder is V=(pi)hr^2, if we were to think of the radius of the disk as the value of the function +7 and if the height of each disk was an infinitely small amount, we could piece together a function for each individual disk as V=(16-x^2)^2 dx.
Now if we were to take the integral of that function with the limits of integration as the points of intersection of the functions (-4 to 4), it would give us the volume of the solid.

The cylindrical shell method is a little more difficult than that however. To get a cylindrical shell, we would have to evaluate the integral as a function of y not a function of x. Therefore we should rearrange our parabola for y, which is y=sqrt(9-y). So now that we have that we need to think of the equation of the volume of a cylindrical shell, the volume being the circumference multiplied by the height and width. Because the distance between the large and small radii will be so insignificant, we can think of the width as dy. The equation for circumference is 2(pi)r, and because r is the distance between y=-7 and the function value, r can be thought of as (y+7). The height of each individual shell will be the value of x at each y value multiplied by two because it is a parabola and it's simetrical, so h=2sqrt(9-y).
So now if we put the two together, we get the equation V=2(pi)(y+7)2sqrt(9-y) dy
Now we integrate the function with the limits of integration being all the y-values between the line y=-7 and the maximum function value (-7 to 9) and that will give us the volume of the solid.

Closer to the end of class Mr. K gave us a stencil with similar problems on it.
(Sorry about my lack of the proper characters, the system I'm on at the moment has very limited resources.)



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Thursday, March 02, 2006

It's Coming ...

Wednesday, March 01, 2006

Scribe for Wednesday

We worked on solving for the volume of a function that is revolved around the y-axis. The integral for the interval [a,b], the inner function is the length times the width times the height.

V= (2(3.141...)x)(f(x))

The height is represented by f(x), the width is dx, the length is 2(pi)x.
In class, one of the examples was f(x)=√(-x+3)

V=2(pi) √(-x^3 +3x²)

When the graph is revolved around the line x=5

V=2(pi) (5-x)√(-x+3)

The second part of class was about the average value of a function. Which can be defined as:

[f(x1)+f(x2)+f(x3)+...+f(xn)]/n

dx= (b-a)/n
n=(b-a)/dx

1/(b-a) f(x)
The Average Value of a Function is defined by the previous expression. This expression can be used to find the mean value or f(c).
The next scribe is Chris



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