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!!!!!!

Scribe Post 10/2

     We started class with a quick warmup that focused mostly on using our calculators to find derivatives numerically and graphically(to do it numerically math 8, function, x, value of x, precision(optional)). It is really hard to explain without a visual so here is a youtube video from DrPhilClark

   
     Just for review, in order to find a symmetric difference quotient use the equation   .  Mr. O'Brien told us that the calculator does not have mystical elves working inside to find the derivative for us, but instead it just uses this equation.  In order to graph a derivative on your calculator use the syntax illustrated by the picture below:

     The function we looked at in the warmup is the Y1 function in the picture above, and the picture below shows the graph of the function  :

     When we looked at the graph we found that when the slope was positive, the door was opening, and when the slope was negative, the door was closing.

      At this point Mr. O'Brien made a graph FURRY and I noticed some people wanted to know how to make their own furry graphs. For those of you who dont know how to do that here is a little video I made.  Go to the y= window and press the left key a bunch of times. Once you have the line next to Y1 alternating from / to _ just press enter: 

      Mr. O'Brien showed us something strange: when we graph the original and derivative together, the vertical scale on the graph is used for both y and y'.  On the original function, the points were in the form (time, º), but the derivative showed (time, º/second). In other words, the original shows (x,y), and the derivative shows (x, slope). The original is in blue and the derivative is in red

     After further investigation we found that there was a zero for the graph of the derivative.  We knew this because the graph started above the x axis with a y intercept of 200, and dropped below with a minimum value at about (2.885, -27.067).  According to the Intermediate Value Theorem* because 0 is between 200 and -27.067, at some point with an x value between 0 and 2.885, the function will cross the x axis.  The x value of zero for the derivative was the x value of the Maximum of the original function!  At that point, the door is neither opening or closing.
*for review of the Intermediate Value theorem check Sarah's blog post

    In order to use solver on the calculator press the MATH key then scroll down to SOLVER. Enter your equation, then press ALPHA, SOLVE(ENTER). For a more in depth explanation watch this very basic video on Solver by learning4mastery.  It involves a quadratic, but the basic concepts are applicable to derivatives.  You may not even need to watch much if you're just looking for keystrokes.

     If you have already entered a function into the y= window and you are feeling a little lazy, you can copy that function by pressing the VARS key then the right arrow to get to Y-VARS, press ENTER on the "Function" option and then select the equation you want.

     For more calculator help check out this website.

    While working on the warmup we learned new vocabulary: a Point of Inflection is the point of a graph at which the slopes curvature changes signs, or in our example below, the point at d(2.885). Flecto in Latin means to bend.  Inflection is the point at which the function changes from bending down to bending up! It goes from concave to convex! Check out wolframs definition here.

(side note: O'Brien reminded us that sinø/ø is not 1, but the lim of sin/ as x-> =1)
     A question was raised about #2 on the test, which related to #3 of the practice test(IW#8). On #3, it looked like the lines only crossed 2 times, but we have to remember that an exponential equation does shoot up eventually.

Question 3

At how many points do the graphs of the functions   (in red) and

(in blue) intersect?

At first it appears that there are only 2 intersections at points C and D, however, if we zoom WAY out we find that there is a third intersection at point E below. 

    In the same way #2 on the test at first looked like this:

How many zeroes does the function g(x)=sin(ln x) have for 0< x ≥1?

when we graph this function on a simple window we get this:

Okay, so now you might say "easy 2 intersections." Instead when you zoom in you can see 3 below, and if you zoom in more, and more, you will find more and more zeroes. The sin made the ln(x) oscillate as it got close to 0.  We usually wont be able to see these types of things on graphing calculators, but instead we need to see things like the fact that as we plug numbers close to 0 into ln(x) we get infinitely small numbers.  Then when you put all of those numbers into sine, you get numbers oscillating between -1 and 1 infinitely.  Finally by using the intermediate value theorem we see that if there are infinite oscillations over the x axis, then there is an infinite amount of 0s :

Now we checked out the Power Rule, Mr. O'Brien showed us how the rule worked with the following example:

This table shows a function of a certain power and it's derivatives.  Notice it works for negative and fractional exponents too!

The Power Rule and the Damian Trick are the same thing! when the function is , the derivative is .  What a POWERful function!

We went to the thatquiz links posted by O'Brien to work on using this rule and class ended.

     If there is a coefficient on the X, ignore the coefficient at first: k*f '(x). It is the same thing just more mathematical than the explanation offered by physics. You can prove the mathematical way by using limits to find the derivative, you take the k out to find the limit, then multiply by k.


FORESHADOWING:
Sum/Difference Rule:
[f(x)+g(x)]=f'(x)+g'(x)
[f(x)-g(x)]=f'(x)-g'(x)

The ln(3x)= ln(3)+ln(x)
       ln(20)=ln(4)+ln(5)
       ln(20)=ln(2)+ln(10)
       ln(20)=ln(1)+ln(20)
So: horizontal dilations of a derivative would be simply a translation for all log functions!

UPDATE:

I mentioned the sum/difference rule above, but what about the product and quotient rules?

Product rule:

So with this rule, you multiply f(x) with the derivative of g(x) and vise-versa, then simply add the products and you have the derivative of a product!

The Quotient rule is similar, but with a few crazy differences:

Notice that in the numerator it is the same as the product rule with one key difference: SUBTRACTION.  Also, notice the funky denominator: g(x) squared.


Cal

 NEXT SCRIBE: jk, side deals going down, Will is the next scribe again…

   

Links for 10/2

Warm up 
ThatQuiz problems first set 
ThatQuiz problems second set 
IW #2