jedi (implicit)

Yoda

SCRIBE EPISODE 4:Return of the Jedi

SCRIBE EPISODE 4:Return of the Jedi

So, Some people (those who are not in our class, or weren't there yesterday) reading this Scribe post may be a

little confused by the title.

The title is as such because Mr. K. actually started off the class with a couple "Jedi mind tricks"!!! Isn't that

awesome?!?!

I won't go into detail, but let's just say:

THERE ARE NO GREY ELEPHANTS IN DENMARK!!!

And so I begin the Calculus Related part of this Scribe post...

SCRIBE EPISODE 4:Return of the Jedi

Well, Mr. K. then shoed us a couple pictures (man I wish the slides were up) with questions pertaining the existence of somethings that were not shown in the pictures, but were

eluded to by certain hints in these pictures.

i.e. You see a shadow of person on the ground, but have not turned around to see this person yet. Is there someone behind you???

When we gave an answer he asked us why? We said because it is implied. There aren't any visuals of the objects in question,

but certain evidence allows us to imply that they exist.

This leads to the topic of the day:IMPLICIT FUNCTIONS!!!

SCRIBE EPISODE 4:Return of the Jedi

So we moved on to a quick review of composite functions and the chain rule, because we would have to use it later

on in the lesson.

In one example the answer was:

SCRIBE EPISODE 4:Return of the Jedi

Then we were asked to identify THIS function:

SCRIBE EPISODE 4:Return of the Jedi

It is, of course, a SEMI CIRCLE

SCRIBE EPISODE 4:Return of the Jedi

Now what's the equation of the full circle???

SCRIBE EPISODE 4:Return of the Jedi

SCRIBE EPISODE 4:Return of the Jedi

SCRIBE EPISODE 4:Return of the Jedi

And that can be drawn as...

SCRIBE EPISODE 4:Return of the Jedi

2-2

2

-2

SCRIBE EPISODE 4:Return of the Jedi

However, a circle is not a function. But, is there a function (besides √4-x2 and its opposite: -√4-x2) that is a

part of the circle???

SCRIBE EPISODE 4:Return of the Jedi

Sure there is: There's this one...

2-2

2

-2

SCRIBE EPISODE 4:Return of the Jedi

2-2

2

-2

And this one...

SCRIBE EPISODE 4:Return of the Jedi

And this one...

2-2

2

-2

SCRIBE EPISODE 4:Return of the Jedi

Do you understand that there are an infinite amount of functions with x as the

variable?

All of these functions hidden in the main graph of a circle, are

IMPLICIT FUNCTIONS!!!

SCRIBE EPISODE 4:Return of the Jedi

Now think of a relation such as the circle given previously, and think how you would find the derivative

of any given point in the domain that relation?

There are infinite implicit functions, so who knows which one you should find the derivative of because there are

many of these functions that go through the same points!

SCRIBE EPISODE 4:Return of the Jedi

The only way to find the derivative of a circle at any given point is by finding the

derivative of the relation using the following derivation:

SCRIBE EPISODE 4:Return of the Jedi

First of all, let's use the following equation of a circle:

SCRIBE EPISODE 4:Return of the Jedi

Now let's find the derivative:

SCRIBE EPISODE 4:Return of the Jedi

First rewrite it as such:

SCRIBE EPISODE 4:Return of the Jedi

Now we can recognize that the derivative of 25 is always 0.

SCRIBE EPISODE 4:Return of the Jedi

Next, use the chain rule on the left side of the equation...

remember it is:

SCRIBE EPISODE 4:Return of the Jedi

Identify ƒ(x), g(x), and their derivatives:

The reason why the inner function "g(x)" is simply "y" is because the inner function of y^2 is an implicit function. We do not know exactly what it is, but we know it is there.

SCRIBE EPISODE 4:Return of the Jedi

Did everyone see the red text on the previous slide? It is very important!

The reason why the inner function "g(x)" is simply "y" is because the inner function of y^2 is an implicit function. We do not know exactly what it is, but we know it is there.

SCRIBE EPISODE 4:Return of the Jedi

Anyway, now using the outer and inner functions, complete the chain rule:

SCRIBE EPISODE 4:Return of the Jedi

Now combine it with the derivative of x2

(2x) and you have the derivative of the left side of the equation...

SCRIBE EPISODE 4:Return of the Jedi

therefore...

SCRIBE EPISODE 4:Return of the Jedi

Now solve for " y' "...

SCRIBE EPISODE 4:Return of the Jedi

So now we know that the derivative (slope of the tangent line) at any given point on

the circle x2 + y2 = 25 is given by the following formula

SCRIBE EPISODE 4:Return of the Jedi

So let's test it out.

Let's say we need to find the derivative of the previous function at x=3.

SCRIBE EPISODE 4:Return of the Jedi

First, find the output of when x=3:

SCRIBE EPISODE 4:Return of the Jedi

The next step is simple. To find the derivative at that point, plug the coordinates into the

equation of the derivative of the circle:

SCRIBE EPISODE 4:Return of the Jedi

The derivative (slope of the tangent) at x = 3 is (-3)/4... Let's see if that is true.

SCRIBE EPISODE 4:Return of the Jedi

So yes, the derivative (slope of the tangent) at any point of any circle with its center at the

origin (because the derivative of the constant will always be zero) is given by:

SCRIBE EPISODE 4:Return of the Jedi

From there, Mr. K. gave us some examples to try and find the

derivative functions of relations (which could have been seen on

the slides if they were up).

SCRIBE EPISODE 4:Return of the Jedi

Then one final point was made:

When the numerator of the derivative function of the relation

is 0, the tangent is a horizontal line.

When the denominator is 0, then the tangent is a vertical line.

(as seen on the next slide)

SCRIBE EPISODE 4:Return of the Jedi

SCRIBE EPISODE 4:Return of the Jedi

Yoda

About it, that is...

For reading, thank you...

Grey-M, the next scribe is...