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Thursday, December 4, 2008

Big and Tall...Oops, I mean Small :P

Today's class started off with Mr. K giving us a small chat over the tags on our blog. Since we had so many due to most of us tagging our work wrong, he went over what the right tags to use for our posts were. Here are the main three tags we are to use every time we post something onto the blog:

  • Name
  • Type of Post (Scribe Post, BOB, On My Mind, or Links for Learning - Explained in some detail later)
  • Title of Current Unit (They are shown in the book, or, you can see the slides for them)

Now to get into more detail about the Types of Post, especially the last two which are indeed new to all of us.

  • Scribe Post: Self-explanatory, this is what we use if we're posting the scribe for the day
  • BOB: Use this to tag Pre-Test thoughts and such, we should know how to use this by now
  • On My Mind: When posting something of interest that isn't a scribe post or BOB, use this tag. A good example of this tag would be one of Benchmen's standalone posts that wasn't a BOB or scribe post (See post after the last test date where he was asking for opinions on the test)
  • Links for Learning: Mr. K will mostly be using this tag to post links that will help us with current or past units (Will have the unit tag along with it). I believe students may use this as well.

We then moved on to talking about the video assignment, "What is a Derivative?", that was previously talked about on the day Mr. K came back. The details are as followed:

  • Create a commercial describing what a derivative is
  • Must describe using little to no algebra
  • Cannot use derivative rules
  • Must be 30 seconds MAX, there is a 10% deduction for every second over the max time limit
  • Must be posted as a video response to Mr. K's video which will be posted on youtube (Will be posted on blog with the tag "Calculus Commercials"
  • This will not only be open to us, but to the whole youtube community. Who knows? We might even go viral!
  • No specific due date but must be done by the holidays!

After that, we then moved onto a few questions that was review for yesterday's class on limits.

Like yesterday's questions, first start off by expanding the numerator and denominator. Once done with that, you multiply the numerator and denominator by 1/x2 / 1/x2 since that is the highest degree in this question. This, as a reminder, reduces the highest degree to 0, leaving us only with 6/2, along with the remaining parts of the numerator and denominator divided by a degree of x. Those go away as the limit reaches infinity and we are now left with the horizontal limit equaling 3.

An easier way, as shown boxed in red would be to just take the coefficients of the leading terms and dividing them since all we care about are the coefficients. It saves us time and thus makes a seemingly long question a lot shorter.

Now onto a quick review of the 1st and 2nd Derivative Tests.
  • 1st Derivative Test: Finds critical numbers, as well as looking for where the parent graph is increasing or decreasing
  • 2nd Derivative Test: Determines the concavity of parent graph along with finding inflection points
In reality, these two tests both do the same job. That same job being the fact that they both can be used to max or the mins of the parent graph. Although, they do require different procedures to go about it.

From there, we then started the new stuff. Optimization Problems. These are basically problems where we are looking for where a function is big and small (max and mins). An example would be owning a business and trying to make as much money as possible while keeping production costs at a minimum.

There are six steps to solving Optimization Problems. They are as follows (Click to enlarge image for easy viewing):

We then went on to using these steps to solve a problem.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square the will produce a box having the largest possible volume.

First use Step 1 to find what we are trying to maximize or minimize, in this case it is the volume.

We then follow Step 2 and write an equation for what we found in Step 1, which is the volume. So...

I've included a diagram so you can get a better grasp of the situation, plus they look pretty :). We then move onto Step 3, which is take the optimization equation (which was made in Step 2) and turn it into a two variable equation so we can easily get the derivative. So just substitute in the values from the diagram into the optimization equation and you'll get this:

We need the domain of c first so we can then use the quadratic formula on the derivative function and then take the value within the domain. That will be the size of the corner of the square which will provide a maximized volume, thus answering our question. If we continue on and do the quadratic formula, we would then find out that the only value that fits into our domain is 3. Thus, the size of the square that will provide us the maximized volume is a 3x3 one. Tada, we're done!

Well that was fun, the next scribe shall be, following Francis' lead on taking the next person on the list, Joseph. I refuse to call you Master Joseph :D. Oh yeah, homework is Chapter 5.4 Questions 1-6. Alrighty then *puts in PROPER tags*...*leaves*

1 comment:

Lani said...

Hi Christina,

For anyone who missed class or has an interest in these topics, you've described and annotated the concepts well with illustrations and colors to guide your reader. Thanks for taking time to share so completely.

I'm looking forward to the commercials!!! How is yours coming along?