Scribe Post November 30, 2010

Here is the google doc to my scribe post.
Click here

(Update in google docs)

Leap Frog Game

Leap Frog Game 

• Have students arrange their game cards on their desks 
• After the first part of a statement is read using the Power Point presentation, students complete the statement by holding up the most complete correct answer 
• Students who answer incorrectly must remain in their seats 
• Students who answer correctly will proceed to the next available seat, “leap frogging” over the students who were incorrect 
• Play continues until a student returns to his/her original seat 

Rules: 

• All functions are continuous and twice differentiable. 
• Once a student holds up a card, they have committed and cannot change cards 

Relationships Game—Part One (Basic Relationships) 

• Uses the verbal cards (with “changes signs”) 
• Bonus questions have two correct answers and allow the students to leap frog two desks. A student must hold up both correct cards to be eligible to leap. 

Relationships Game—Part Two (Graphical and Numerical) 

• Uses the numerical cards 
• The functions represented by numerical tables are monotonic between data points 
• Game card values represent x values 
• Students hold up two cards to represent an interval 

Relationships Game—Part Three (Linear Motion) 

• Uses the verbal cards (with “equals zero”)

Scribe Post Tuesday 11/23/10

Click here for the scribe post from 11/23/10

Scribe Post Friday November 19, 2010

We started class with working on a problem that Mr. O'Brien had on his AP exam all those years ago. Back in that day they didn't use graphing calculators, so the problems were a little different. After working on this question for a few minutes Mr. O'Brien told us some about how our exam will be in May. Then, he continued by going over the problem on the board.

Steps to solve related rates word problems:
1) Find our what it is about, and draw a picture
2) Identifying variables and constants
3) Write equation(s) relating the variables
4) Differentiate LHS and RHS with respect to t
5) Solve by adding in variables

This struck the end of related rates and lead us into our next set of word problems.

The next set of word problems are for extreme values of functions. So, we started learning the Calculus we will need to do to solve this problems by doing a few example problems...

f(x)=-x^3+6x^2
on [-1,5]

f'(x) = -3x^2+12x
Where f'(x) = 0

0=-3x^2-12x

=-3x(x-4)

x = 0,4

Now we can plug these numbers back into the f(x) function and we get
f(0) = 0

f(4) = 32

So, The minimum is 0 because it is the lower of the two and the maximum is 32 because it is the higher.

We graphed this as well... however my computer is having problems posting images. I'll work on it and try to get these up for everyone soon.

Next, we did the same problem however with a limit of [-1,7].
What was interesting about this was that when we graphed this we could see that the graph starts to have a steady negative slope between the x values of 5 and 7, and this made it so the the absolute minimum changed quite a lot. This is easy to get a visual on geogebra, but again it won't let me upload the images onto the blog and therefore I will have to add this images later.

However, we can solve the minimum without a graph by plugging in 7 as x in the function. And.... f(7) = -49. Since this value is the lesser now it is apparent that -49 is the minimum.

When looking for extreme values of a function:
1) Locate places where the derivative of the function is 0 or undefined, and these are the critical points
2) Evaluate the function at these critical points
3) Evaluate the function at the end points
4) Pick out biggest and smallest and call these the extreme values.

An important note is that if derivatives change from (+) to (-) it is a max. And, if the derivative changes from a (-) to a (+) then it is a min. Remember this can be seen by the slope of the graph. This is called the first derivative test... however there is a problem with it that you must always have in the back of your mind. It is only true with continuous functions as Robin "Beans" Wilder pointed out to us.

Next we went on with more examples...

Find the extreme values of f(theta) = tan(theta)
for [-pi/3, pi/4]

F'(theta) = sec^2 = 1/cos^2

There is no points where its zero or undefined so, there are no critical points. Therefore plug is -pi/3 and pi/4.

tan(-pi/3) = -sqrt(3)<------ This is the minimum

tan(pi/4 = 1 <----------- This is the maximum

Here are some links:

http://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx

http://www.youtube.com/watch?v=r56mq7iBH4o

http://www.themathpage.com/acalc/max.htm ----- UPDATE

http://library.thinkquest.org/3616/Calc/S2/FMME.html ----- UPDATE

Related Rates Warm up

Mr. O'Brien graduated from Camden-Rockport High School in 1990. How would you do on his AP Calculus exam? Let's try Question 4!

November 17th Blog Post

Click here to see my link.

Related Rates Warm-up and Assignment

Related Rates Warm-up and Assignment

November 15th - Scribe Post

I hope this works, I wasn't sure how to go about using google docs with a scribe post, but i tried it.

here's the post


https://docs.google.com/document/d/14BA-d-Tc76M-fJO0gbp9wxdNk4sOr7cDDRpqgUQ8doQ/edit?hl=en&authkey=CIH9muwK try this one

Novembah 12th's Blog Post

Our class today started off by our class splitting up into little groups to answer the first 3 questions of a worksheet with a mysterious method on it: "Newton's Method". We spent a few moments trying to work it out in small groups. I personally had no idea even where to begin, which is great considering I am writing this blog post.

It all really centers around roots. The root can be thought of as any x-value in a function that will produce a result of zero.

But what is really important about today is the concept of Newton's Method, or the method of approximating the roots in a function.

The process of the method is as follows:

1. Start with an initial guess that is close to the actual root.

2. Find the function's tangent line. Treat this as an approximation of the function itself.

3. Find the x-intercept of the tangent line. This will (generally!) be a better approximation than the one you guessed in STEP 1.

4. Rinse and repeat to get even better approximations.

Here is a nice little visual of these steps. It also happens to be in German. But just in case you're really bad at translation (REALLY BAD), "Funktion" refers to the blue function, and "Tangente" refers to the red tangent lines. It also happened to be created by Ralf Pfeifer, who may or may not be a talented Ju-Jitsu Fighter.

Finally, we found each of the equations for finding the roots.

We started with wanting to find in terms of .

Now, since the slope of a function (in this case f(x)) is its derivative, we can call the slope of the function f'(). And since is the point we are trying to find, we can define the point as (x, f()).

By inputting this into point-slope form and by plugging in our x-intercept at (, 0) into the equation of the tangent line, I can solve the equation for .

Now inputting the x-intercept:

To find , do the same process.

A very cool visual on using Newton's Method is found here.

THIS LINK is actually very cool. It essentially completes Newton's method for you. Just type in your equation, and then your guess for the first root of the function, and it will instantly find x1, x2, x3, and all the way up to x9!

AND FINALLY Oh Lawdy! In case you still don't quite really understand this stuff, this guy is THE MAN. He really clearly explains WHAT Newton's Method is, its point, and why it exists! This video in fact really helped me write this blog post! His voice is really high, and he uses excessive amounts of sound effects. But hey, he's trying!

IN FACT!!! (For review on ANYTHING in CALCULUS we've done so far) please take a look at this link. It is done by the same guy, but reviews ALL of essentially our first semester in under 20 minutes. And it's incredibly entertaining. It really allows you to understand the connections between everything, and it clears up everything!

Happy Trails!

Newton's Method

Today's investigation

november 2, 2010 :)

https://docs1.google.com/document/edit?id=12FYDYFD9MDjscRCxVyqxPkPYuzInMDFP9JTulYUW2fg&hl=en#