Scribe Post 10/4

Hello Everybody!


UPDATE!!!!

We talked about three different ways to write the derivative:

We also discussed how to type a cube root into your calculator:
cbrt(x) in Geogebra, also you can raise it to the 1/3rd power.

We also checked out the function:

and talked about how there are two different slopes: 1 and -1.  This forms a corner not a cusp.

We also looked at a few equations to prove that non-differentiable (when there is no derivative) at cusps, corners, vertical tangents, and discontinuities.  

We decided to use a function with a cube root to show that if f is differentiable, f is continuous.  (Thus, proving yet another theorem).


p. 124/1, 5, 11, 25, 29, 32, 37, 39, 53, 54
p. 126/1, 2, 4

We began our electrifying class with the 40 minute SuperCorrection test created by the masterful Mr. O'Brien.  It (hopefully) jogged our memory on how to solve interesting and exciting limit problems!

We discussed the very thrilling math meet on Wednesday!  The mathletes came in so very close behind Lincoln Academy.  Many of us made varsity, including Duncan, Crockett, Eddie, and Cal.  Wednesday the 31st is our Halloween Math Meet in Boothbay! We will be in costume!  Think about joining us and talk to O'B!

Then we got down to business. 

THEOREMS AND PROOFS

We first went over question 3 from the IW.  There were two equations:

First Equation

Second Equation

O'B showed us that for these problems, we could use the derivative equation (which should be memorized by now):

Derivative Equation

Then, since  

we can efficiently substitute k in for f(x).  That means, since x is approaching c, we can also substitute k in for f(c).  Doing this, we come up with the final result of this equation:

which ends up equaling 0 when the two k's cancel.  

Then, we tackled part two of question three.  The equation was:

We took the same approach, by using the derivative formula and then substituting in mx+b whenever there was an f(x) or f(c).

In the end we got this:

Then, that simplified as well:

Victory!  Then we went over question 10.  Oh, wait! We were almost starting to discuss when Rebecca had to ask a question about #3...

"Can you cancel instead of factoring m out?"

O'Brien retorts, "its all sort of the same idea." In principle, the limit of x-c/x-c is 1, and the limit properties allow us to extract the m. 

If you still need more help understanding how to utilize the limit properties, check out this link from Scotty's scribe post. 
O'B reminds us all of the handy second form of the derivative equation.  

Derivative Equation Take 2

To tackle question 10, O'Brien wittily decides to alter the equation from:

into

This is because we will be working with h as a component of the derivative equation. The variable "s" was chosen to stand as the function letter because it sounds like sum and there are two functions being added together in the "s" function.

Now we will prove the Sum or Difference theorem when it comes to derivatives! Using the derivative equation above, we changed all the f's to s's.  Then we started to expand everything.

THEOREM:

We determined that since s was the sum of f and g functions, s(x+h) would really be f(x+h) added to g(x+h) and so on and so forth.  We applied these applications to get the following monster:

AHHHH, that is scary looking! Luckily O'Brien knew what he was doing, and was using this as a part of his master plan of genius.
"Split the big thing into two different limit pieces!" -O'B (Using the Limit Properties again!)
And thus, it was cleft in twain:

since each limit piece is the derivative equation itself, we can conclude that

But of course, Rebecca had to ask yet another question:

"What's with the big thing over h, how did it become the two separate limits?"

O'Brien sneakily explained how he first used the distributive property:

then, he used the commutative property to rearrange the order of things:

Finally, limit properties were used again to separate the two limits.  

Hope that clears everything up!
If you want to see a real live math student perform the above actions, check out this Youtube Video! He's a real bro and he does pretty much exactly what we just did, except with j and k instead of f and g.

Next order of business!

"Let's prove the power rule!" -OB 

The power rule (also known as the infamous "Damian Rule") is that clever trick that allows us to quickly find the derivative by bringing down the exponent and multiplying it by the coefficient to get a new coefficient.  Then, you reduce the top exponent by one.  See the theorem below:

                     THEOREM:        

Then, we can apply the Constant Multiple rule to come up with the official Damian rule below:

To be warned: this proof will only concern natural numbers, proving irrationals and other unnatural numbers will come later.  
For now however, to acknowledge that we will prove it, let it be known that:

Starting out we plunked down the beginnings of a proof:

Then, using substitution, we ended up with this:

Then it got tricky, so we put it on the back burner, and referred to question 11 from the IW to help with this long and complicated proof.  

Let's travel to Wolfram Alpha Land!
Question 11 looked kind of like this


11. Next class, we will prove the wonderful Power Rule. To prepare us for this event, please visit 
wolframalpha.com and calculate (a + b) raised to the power of n for some natural number values of n. What do you observe about the 

powers? How are the coefficients related to Pascal’s Triangle?

So we set off to experiment with different natural number values of n.

Our exploration proved fruitful and left us with the following chart (in mid-creation in the shot below):

                                                 

                                                                            and so it goes...

There was something clearly up, though there usually is when O'Brien is this deep in a complicated math exploration.  Then, all of a sudden, SURPRISE! The ball was dropped... it was PASCAL'S TRIANGLE!

Pascal's Triangle of Awesomeness

Remember this crazy thing? We all learned about it in our math childhood.  It's that cool repeating pattern where a number is just the sum of the two numbers top left and top right of it.  This all fits in, because you can see thats whats happening with the coefficients as n gets larger with our Wolfram Alpha equations above! The exponents of a just decrease by one each term and increase by one for b.  Math is truly wonderful.

Then, O'B relates and asks whether our old teachers ever told us about the connection to combinations. Scotty retaliates "Fitz didn't."

Well combinations are pretty cool, basically you can plug them in on your calculator and answer cool probability questions with ease (think SAT?) Here's a FUN link about Combinations and Permutations! If you thoroughly scour that site, you'll see lots of cool connections with Pascal, including the fact that with combination nCr, n represents the number of rows down in the triangle, while r represents how many spots in.
Basically a combination has a few parts, in nCr, n stands for the number of items total, of which you have to choose r items not paying attention to order.
Example: 5 elephants, you have to pick 2, order doesn't matter.  n=5, r=2.  There are 10 combinations of elephants.
This is the factorial explanation, used by Patrick JMT in a video a little ways down the post:

Here's how you do it on a calculator:

n value -> Math -> Prb -> nCr -> r value

Here's me typing it in on my TI-83:

Since we know that Pascal, which can be represented combinations, determines coefficients, and that exponents decrease each term by 1 from n for a, and increase by 1 for each term from 0 for b, we can deduce the following sequence to represent 

etc...

Patrick JMT, in this helpful video, explains the above definition, also known as the BINOMIAL THEOREM.
However, note, he doesn't use the term "C" as combination but rather uses the definition of a combination as 

Anyhoo, we've got this crazy thing: etc... and we ended up sticking that onto our proof right after that step way back when we had 

Luckily, things simplify a lot because stuff cancels and something as confusing as that first term:

 can reduce all the way down to x raised to the nth.  

Yippee, now we can look at something as simple as this:                                  

Yeah, just kidding, thats not simple in the slightest, but O'Brien points out that since we know the big repeating thing ends in a -x to the nth, we can cancel a lot more terms.  Everything cancels with its opposite except for the golden term, 

At this point, a frazzled O'Brien, hair tossed, glasses askew, compares himself to a mad professor.

The scribbles on the board look like the crazed etchings of a delusional scientist.

The proof is crazy, everybody hopes there are ways to look at the power rule which are more intuitive.

A polite insult to Ms. Damian: "Math is too hard for you."

Yes, this proof is difficult, but maybe this guy can help.  It really starts getting interesting about halfway through, as he'll show you a way to get the same answer by step by step substitution. Thanks calculussuccess!

WHAT DO I ACTUALLY NEED TO KNOW? <- you are probably asking yourself this

You need to know two different forms, recognize the derivative, and should know that all these cool results that we are proving come from these definitions.

Work on these problems still! They are tough! He's not gonna post the answers for a little while because he is maniacal, devious, tricky, and trying to help!

Scribe for Next Class: The MAGNIFICENT Duncan Hall!!!!!!