Scribe Post 12/7

For those of you who may not have been here to see it, we started off class today by looking at a video of a cute wombat. *Fun Fact* If you haven't already figured this out, OB's favorite animal is the wombat..

We were then subjected to a forty minute quiz on optimization problems, related rates, linearization, and differentials. Safe to say, it was a pretty tough quiz, seeing as there was minimal joking going on between us–something almost unheard of for period 4! While we were taking the quiz, an article of 15 Hidden Health Secrets of Lemons was on the Smartboard. Why, you may ask? Not sure, but if you want to know some more about lemons, feel free.

Once the quiz was over, O'Brien thought it'd be fun to compare our current knowledge of calculus to see how we would have done on the AP Exam that he took as a senior in 1990 (22 years ago..). According to OB, "back in the day" the exam was taken at the Congregational church (makes sense, since nothing goes more hand-in-hand than calculous and religion) and that you could not use a calculator on the free response questions. For this reason, back then, free response questions were more on "a type," whereas now, the concepts are more mixed and matched, making it important that we understand what actually going on in a problem.

So we looked at question 4, a related rate problem, which is about a sphere whose radius is increasing at a rate of 0.04 centimeters per second. Unfortunately, OB said that for this particular problem, a picture wouldn't really help you out that much. We should note, however, that as the rate of the radius increases, the volume will initially increase drastically and then begin to slow down over time. To help us visualize this, he used a balloon analogy: when you first blow air into a balloon, its volume increases rapidly, but as you continue to blow air into the balloon, you begin to notice the change in volume less and less. OB was immediately sassed out by Autumn for this analogy, perhaps rightfully so, seeing as this problem is CLEARLY about a sphere, not a balloon..

Okay, onto the problem. Seeing as you can all read what the question is asking for yourselves, I'm just going to dive in.

Part (a):
Given that:                                                                       Find:

The key to related rates problems, is to use a relationship that relates the variables that you're given. For this problem, that relationship would be the volume of a sphere:

The next step is to take the derivative of this relationship with respect to t, the independent variable. Don't forget to use the chain rule and implicit differentiation when you do so! Once you do that, sub in your given values and solve!

Autumn then makes the comment that, thus far, this AP question is easier than the quiz we just took. OB tells Autumn not to "whinge". 

Part (b):

Given that:                                                                             Find:

This problem wants us to find the cross-sectional circle within the sphere, or the largest circle possible within the sphere. To do this, we'll use the equation for the area of a circle:

Just like before, we're going to differentiate with respect to t:

Hold on though! While we know the rate at which the radius is increasing, we DON'T know what the radius is. This is where the given volume comes into play!

Now that we have a value for our radius, we can finish solving the problem:

Part (c):

This part sounds confusing, but it's actually quite simple. The volume and radius are increasing at the same numerical rate, but the rates themselves are different because volume and radius have different units. Here's how to solve it:

Answers and solutions to the rest of the 1990 AP exam are found here. 

Tips by OB on related rates problems:

1. Label everything nicely 

2. Draw a picture to help you visualize

3. Have a personal "cheering section" in your mind... (is this what you do in nordic too OB?)

Still need help with related rates problems? Check out the videos on Autumn's scribe post.

IW for next class:
pg. 256/ 14, 16, 25, 30, 31, 33, 39, 41

The next scribe, chosen at random out of a hat, will be...

If you can't scribe on Tuesday, try to find someone else to switch with you. Otherwise, good luck!







Update!!

Here's a nice little song for everyone to get you into the calculous mood.

Seeing as midterms are coming up, I feel like a good update would be a refresher on some difficult related rates problems, particularly (for me, anyway) those involving cones. A good example can be found here by PatrickJMT.