Sunday, February 21, 2010

Scribe Post Feb 19th

Related Rates!!

Followed by last scribe post, in this post we will talk about the extension of Porky Problem.

This post will follow the exploration done in class: Section 4.9 A: Related Rates Warm Up



Review!
Calculus is the mathematics of change, and everything changes with respect to time.

In algebra we study the relationship between variables.

In calculus we study the relationship between the rate of change of variables.


There are infinite amount of scenarios that can be produced.
- A bathtub is filling at the rate of 2gal/min
- A balloon is deflating at a rate of 3 cubic centimeters/second
- A football is kicked with an initial upward velocity of 30yd/s

Lets translate these scenarios into mathematical models!
all theses scenarios are rates volume changing respect to time.





Now let's play with some real related rates problems!
Make sure you have the exploration in front of you!

Rippling Water Problem

- This problem deals with the radius of the circle, and the area.
Therefore we should write out the equation involving two different variables.

Let A be Area

Now we are using Implicit Differentiation
to express the rate of change of the area of the circle with respect to time t.

Process starts with stating that this equation will be derived with respect to variable.


Use chain rule to implicitly differentiate the equation.


This is the hardest part where everyone is having hard time, or at least some of us!!
It is simple, let's look at again.

- This is our original equation

each of variables, in this case A and r, is a function composed with t.
Since the letter r itself is a function, outside function is squared, and inside function is r

Using chain rule,
derive the outside, leave inside alone

- We get the result.

now we have to multiply the derivative of inside function, which is r'
However, since the function r is the function of t, we can also write out the derivative of r in different form.


It's the change in numerator over change in denominator.
This is why there is additional fraction representing derivative after each implicit differentiations.

This is how to do the implicit differentiation relating two different rates.
Rest of the question and answers are posted in Bru's post of Friday's class.
If there is any additional question, please let me know.

-Hyunhwa-


:0 Our next Scribe Post will be by Skirdude
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8 comments:

  1. dammitimmadFebruary 21, 2010 at 5:31 PM

    Hey Hyunhwa, great post. Good use of colors and of the equation editor. I think the one bump in this post is when you explain the implicit differentiation of the equation for area of a circle. You say that both variables are composed with t, and although I know what you are trying to say because I was in class, I think the whole explanation comes off a bit unclear. Were you trying to say something more along the lines of 'you are deriving both variables with respect to time?' Also, I think it could have been helpful to juxtapose an example of deriving 'A' with respect to 'r' in addition to this one which is with respect to 't.' But other than this that, great job!

    ReplyDelete
  2. MarleyFebruary 21, 2010 at 6:23 PM

    Hey Hyunhwa! I enjoyed reading your scribe post, and I understand what you're saying throughout the post. Yes you explained 'the hardest part of this exploration,' but considering someone who didn't see this in class, I think it would be better to explain the steps before the final answer is given.

    Oh! also, I like the picture you used as your introduction. Moreover, it's always good to insert a review part in the post.
    Great job!

    ReplyDelete
  3. JoanFebruary 21, 2010 at 6:41 PM

    awesome post! it was really helpful for me that you put a review first thing in your post. I am not the elephant Bru wishes i was, so it is good to remember the main themes of the class before i do my homework. It was also great that you had a clear set of answers from the exploration. A little more explanation about the steps to each answer would be nice, but otherwise you have a solid post. Bravo!

    ReplyDelete
  4. YDplusSBFebruary 21, 2010 at 8:35 PM

    I think I have nothing to say. This post is so awesome. So many details I have never thought about. A lot of colors highlight the most important things. Nice organization and clear explanation. Great job! I tried so hard to find a flaw or a little unclear part, however, I failed. Nice job! I think you provided not only us but also yourself a great chance of reviewing and catching up. Thank you.

    ReplyDelete
  5. blitzenFebruary 21, 2010 at 10:42 PM

    Nice post, really easy to understand it's concise, but has adequate information to recall what happened in class. I like how you paid special attention to deriving implicit functions since it's a really confusing subject. It might have been helpful to include another example, maybe one from the homework packet or the book just to solidify the information again.

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  6. flying slugFebruary 22, 2010 at 7:59 AM

    That was a great post, I really enjoyed reading it. Although I had already done the problem by the time that I read it, it still clarified some tough topics. You possibly could have chosen a different problem than the example that we did, but i still think that you did a great job. And because you chose the problem we did it was easier to understand where you were coming from in your directions. This was a very thorough, and helpful scribe post. Good job.

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  7. BestonFebruary 22, 2010 at 9:15 PM

    Wow, two great posts in one week. Something is up here. I want to point out that you and I like the same kind of formatting, although everything that I made a mistake on you got right! I have to tell you that my favorite part about your post was the last sentance of your paragraph, that you are so willing to help is great and I know you will help if you can. This post was a job well done and the only thing I can add is that from your post you sound very much in a rush, don't forget to stop and smell the roses.

    Have a wonderful night!


    Beston

    ReplyDelete
  8. SKSFebruary 26, 2010 at 12:39 PM

    Hyunhwa,

    Your post is very clearly organized. I particularly like the use of color to highlight important things.

    In order to show how both the radius and the area are functions of time, it might help to include multiple representations of this idea. You could include a data table (numerical) or a picture/diagram (visual) that could help the reader understand that both the radius and the area are changing with respect to time.

    This might support your verbal and algebraic explanations of this challenging idea.

    Great work!

    SKS

    ReplyDelete

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