We began class with a lovely 40-minute quiz covering IWs 1-7. Then Mr. O’Brien casually mentioned that the scribe was LEXI, but LEXI REFUSED. Thus, a new scribe had to be chosen–i.e. Alex W., who was absolutely thrilled to take the job. O’b mentioned that luckily it was an "easy" class because he had decided we would not go into the next topic (inverse trig function identities), so Alex W's immense burden was somewhat tempered.
After the quiz and the sudden shift in responsibility, we looked at the IW #8 sheet, which showed us how to find the derivative of a negative integer:
becomes
which makes "– n" a positive integer, and then we can use the power rule as we use it on positive numbers.
And, just like that, we realized that with the aid of the Powerful Power Rule, we could find the derivative of ANY rational function. That’s right: linear functions, exponential functions, sums, differences, products, quotients, trig functions, multiple functions chained together... you name it, we derive it.
But... aren’t we missing something? We can find the tangent lines of trig functions easy peasy. But what about the inverse of those functions??
We took a quick break to talk about the Exploration 3-9. Mr. O’Brien said that answers will be posted, but not until next class (as he would like us to Explore more thoroughly before discussion ensues).
After the update, we tackled question #5 of the IW.
Geometrically, why would
(a sphere) have a derivative of
(the surface area)?
To better understand this, we can think of the surface area of the sphere as a super duper thin shell around the sphere. It only make sense, then, that the shell would grow with the volume. The change is instantaneous, and thus can be expressed as a derivative!
We moved on to #7, which looked at co-function identities...
CO-FUNCTION IDENTITIES! That's our new topic!
Buuuuut... let's lay off for a bit, chill-out, slow ride, take it eeeeasy...
Instead, since we were all clearly overwhelmed by the vast number of functions we were able to derive, we decide to look at inverse function (and how to derive those).
First of all, what is the inverse of a function? From Algebra 1 and 2 we know that to find the inverse, we solve for the "other" variable. For instance in the function .gif)
we would merely state "x" in terms of "y":
or in function notation, .gif)
Note that the "y" value has now
become an "x". The variable really doesn't matter. Heck, it could even be "w" for all we care. Ha! What a ridiculous letter...
Anyway, to really truly get an accurate idea of what an inverse means, we must perform that strange, ancient ritual known as "graphing".
The graph shows us that a function and its inverse have symmetry over the y=x axis. This means that a point on the function, (1, 2) for instance, will swap its "x" and "y" values to be the symmetrical point on the inverse function [making our example point (2,1)].
But that is algebra, and we're in Calculus. AP Calculus. We are the ELITE! So let's apply this to calc.
Mr. O'b started us out by asking this question: How does the tangent line of f(x) compare to the inverse of that tangent line?
Crockett piped up, telling us that since tangent is y/x, or rise over run, the inverse is x/y, or run over rise. Which is legit, but not complicated enough.
So, Mr. O'Brien wrote it "More Formally":
"If f and g are inverses, then f(g(x)) = x"
This is because inverses have a tendency to cancel each other out, seeing as they are reciprocals:
Since by their natures, they undo each other, the answer is just whatever they are multiplied by [in the case of f(g(x)), that would be "x"].
During this segment came the comparison to Dr. Seuss' "Sneetches", courtesy of Will.
Sneetch Video
Start at 4:15 to see the machine, and start at 7:00 if you just want to see the inverse. However, I recommend watching the whole thing because there is a valuable lesson to be learned about prejudice.
"Star on, star off", so to speak.
Getting back to our More Formal Function, f(g(x))=x, we must find the derivatives of both sides in order to evaluate the derivative of the whole. The derivative of the right side here is easy, as the derivative of x is 1. However, that f(g(x)) is pretty daunting if we're trying to take the derivative... Or is it? Two words: Chain. Rule. BAM!
Using the chain rule, we can find that .gif)
(note that the right side of this equation is the reciprocal of g'(x), as f' simply indicates the function must be reciprocated.)
If we go back to the very first inverse function example of 
which has a reciprocal of 
. This time, we want to evaluate the function at (1,2), meaning the derivative at (2,1). We can convert that to an exponential form and take the derivative. But are you looking at that thing? That's got a cube root on it! We need something extra potent to conquer this function. What about two extra powerful derivative rules working TOGETHER? Could it be? The Powerful Power Rule AND Chain Rule?! Yes!!!
That's hot. Let's plug something in and rev this baby up!
I'm drooling.
But another issue is arising... what happens if getting the inverse isn't so easy?
The Bluester Cluester gave us the following problem:
Given: .gif)
Task: find .gif)
at x=3
Sadly, solving for "x", the customary way to find the inverse, is impossible (or at least terribly taxing. Look at all the exponents!)
Therefore, we must look to the other piece of information we were given. What did we set out to do in the first place? Why, find the inverse of the derivative of the inverse at x=3! Let's swap that x=3 and make it y=3 because of the nature of inverses. If we know y=3, the entire problem simply becomes a polynomial to simplify:
If we bring across the 3 from the left to the right, and then use the polynomial rule we learned about last year, we can guess and check with just a few factors to find that x=1.
Now lets take the derivative of .gif)
, which is .gif)
and then substitute in 3 (because we're finding the function at x=3) to get the answer of 1/4.
That was all.
I haven't found much in the way of helpful links as of yet, but if you want to hear a boring-sounding lady explain a simplified version of how to get the derivative of an inverse function, click here.
40-Minutes
After we read the scribe posts and determined that Alex C's font was, in scientific terms, microscopic, Mr. O'b made a confession.
The "proof" he had written for the first part of problem 2 of IW# 8 proved NOTHING. It seemed to magically convert the given form to the desired one:
He explained that because he is such a rock star/god-figure, we sometimes fail to question his infinite wisdom. However, this was one case in which the working was not quite enough to prove anything.
Here's a blurry picture of the real thing!
The scribe for Friday will be Lexi Doudera.
UPDATE:
Another thought about the chain rule: if the "chain" consists of just a function and its inverse, you needn't do any work, since the two functions cancel each other out (leaving only the affected expression behind.)
Examples:
No matter how complicated that inside expression is, it will always just be itself once the function does and undoes it.
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