Implicit Differentiables

Okay, late as usual, it is I Not Paul, posting his not scribe post.

Today we welcomed back Mr. K again for our official first full day of class that isnt interrupted by assemblies. First thing Mr. K did was show us what he had posted last night on the blog, which was the Wordle. The Wordle was an image that graphically depicted what most of our BOBs had in common and visually highlighted commonly used words such as "understanding" and "Mr.K" by making them large compared the less frequently used words which were small. In this way we all got to see what everyone on the class had in common in regards to the class.

After the wordle we discussed some of the things we thought we needed to know, and Mr. K stressed the importance of understanding both Integrals and Differentiables. He proceeded to prove the Chain Rule ([f o g](x)) = f'(g(x)) * g'(x)) with a circular function. His proof went something along these lines:

We have a equation: 1 = x^2 + y^2

Now while not explicit in this equation, we actually have two other implicit functions in it. We can extract the top half of the circle: y = (4-x)^(1/2) and the bottom half: -y = (4+x)^(1/2)

Now, we'll find the derivative of a circle, or the tangent of any point on a circle:

a = x^2 + y^2 = 1

a' = 2x + 2y * y' = 0

2x = -2yy'

2x/-2y = y'

y' = -x/y

This is true for any function we imply from the original equation.

We then went onto a question about derivatives. The questions is as follows:

Gas is pumped into a spherical balloon at 5 cubic ft/min. Find the rate the radius is changing if the diameter is 18.

So we volume of the sphere, as given by the equation [4(pi)r^3]/3

The radius is given as 9 inches or 3/4 feet.

Since we know dv/dt = 5, and we are looking for dr/dt, we need to find dr/dv because dv/dt * dr/dv = dr/dt.

So we find the derivative of [4(pi)r^3]/3 because that is dr/dv.

= [12(pi)r^2]

So our solution is 12(pi)r^2 * 5 which is 60(pi)r^2.

Sorry if I missed anything, and also sorry this is so late. Also, I'm iffy on whether or not I am correct with the solution to the problem here because we didnt actually finish this question in class today, and I didnt take notes for the entire class like I should have.

Good night/morning, the next scribe is Francis.