(Di)Lemma's, Proofs, and Other Things.

By now it should be no mystery to everyone that I, (Hi I'm Justus, glad to meet you) am the scribe for today. Now I regret to inform you that this scribe post might now be wholly as spectacular or colorful as the one I last left you with, and this is due to the fact that well, I just don't have that much time on my hands today ;p As a result you get the scribe post you may or may not be about to read. I apologize in advance for any trouble this may cause.

So without further ado, onward with the show!

Where to begin...where to begin...

This is indeed a hard question, since we really didn't start with anything in particular, but instead jumped all over the place across a few topics we all felt would be good to talk about. Well I'll just start with what I have first in my notes here.

All right, so to kick things off we started with some more proofs (Or rather, logical explanations) for the differentiation rules (Which Jamie so graciously typed up for us in a lovely format.)

The first (according to my notes) was;



Now Mr.K explained to us that it is quite possible to go through the algebraic explanation of this differentiation rule, but also said that it is much simpler, so just remember the graphs of sine(x) and cosine(x)


Sine Graph



Cosine Graph



Looking at the sine graph as the graph of some function we want to differentiate, we may pick out the important points/bits of information found within the graph itself. By doing a quick glance you can find the local minimums and maximums, as well as the inflection points for the graph. These will be used to construct the derivative graph of sine(x).

Our experience with the graphs of functions and the graphs of their derivatives tells us that the red points are the local minima/maxima and that the blue points are inflection points. We know that because the red dots are local extremum, they will become roots for the derivative graph of f. We also know that the blue point(s) is a maximum because the slope at the inflection point is positive.




At this stage it should be easy to see, that connecting these dots gives you a derivative graph identical to cos(x). Therefore;



We also dealt with the derivative of cos(x) being -Sin(x) following basically the exact same steps. Those can be seen in the following image.






Moving along nicely then :]

The next topic of discussion, is kind of a twisty windy, jump off into several lemmas, resulting in an organization dilemma, and me having to sort out some thoughts kinda topic. For the most part, it involves this;



and the question, how would you find the derivative of that logarithm? This is indeed a tough question, as we didn't (at that point in time) have any differentiation rules for logarithms of any base (which in reality, is what we were trying to find at the time.) However, by using the change of base law, and some knowledge about differentiation rules with did know at the time, Mr.K showed us the way.

LEMMA TIME

(something about lemmas imagery here)

Change of Base Law, how it works, and what it looks like on the inside.

So, say you have a logarithm base "a", that equals "n", that you would like to change to base "c."



You must first start by re-writing the logarithm as a power (In this case, a to the n equals b.)



The next step is to take the log of both sides giving you the following;



After you have completed that, you would use the power law to bring down the "n" exponent on the left side leaving you with something you can work with.



After that, all you have to do, is isolate N. Looking back at the beginning we see that in the very definition of what N was, lies the logarithm we wanted to change the base of. What we're left with, is N (aka. our logarithm) equals a new logarithm, now with a different (hopefully more useful) base.



I hope that made sense ^_^;, just in case, here's all the steps together, so you can see it as one fluid thing, instead of a bunch of seperate steps.

1.

2.

3.

4.

5.

Back To Our Regularly Scheduled Programming.

So we have this logarithmic function



which we would like to find the derivative of. The first step would be to apply our newly a wholly understood change of base law to end up with this;



From here, we will pull out 1/ln2 so that we have;



the reason for this lies in the fact that although ln2 is a logarithm, it is still just a number, and therefore, a constant. By pulling it out of the whole thing, we get the nice and easy to differentiate, lnx by way of the constant multiple rule. For those of you who may be having trouble remembering exactly what that is, it basically means that;

"The derivative of a constant times a function is the constant times the derivative of the function."

As a result of this differentiation rule, all that remains is to find the derivative of lnx which we know to be 1/x

*Note* We did a Lemma for the differentiation rule for lnx but when I looked at my notes I couldnt really makes heads or tails of it. I'll need to check it over with Mr.K and get that up here once I understand it. Sorry guys D:

The result, is the derivative, which in this case, happens to be



Now if we think back to the beginning, the primary goal was to see if we could find a rule for all logarithmic functions, and to this end I believe we achieved our goal. By simply substituting a variable (lets say a) into the spot of 2, you get the rule for differentiating logarithms.



With that stuff out of the way we moved onto the proof for d/dx of tan x, which happens to go something like this.




After converting tan into something more friendly, you simply apply the quotient rule.







Tadaa! If you wanted to find the derivative of say Cotangent, you would do the exact same thing. Turn cotan into Cos x over sin x, and apply the quotient rule from there.

Another neat little tidbit of information for you. According to Mr.K, the derivative of all "co" trig functions is negative. Just thought you'd like to know ;p

Now, after alllll that stuff behind us, we got back to antiderivatives. Remember that an antiderivative lives on the idea that anything that can be done, can be undone. However, with derivatives, because of how we figure them out, there is a small kink. Any constant in a parent function, is lost when you take the derivative of that function. Essentially, you may know a functions derivative, but you may NOT know its vertical positioning by the same method. In effect, when you differentiate a funtion, you are finding a family of functions, not just one. Now there is a way to figure out what the constant was, if you have a point from the graph you'd like to single out. These are called intial value problems. Lets see an example.




Because we know an antiderivative gets rid of the constant, we need to add that back in. We do so with the value "c", seen above.

Know this is called the inital value theorem because we have a value at the beginning (initial) of the problem. In this case its the value of f at 1



From this point we substitute the initial value into the antiderivative and solve for c.





And thats really all there is to it :] In case that didnt make much sense, here's the slide that explains it better then me I think D:



Okay guys, I think that about wraps it all up for tonight. Not gonna go into a big spiel this time, its late, and I'm tired, and I have basketball tomorrow, and the day after, and the day after lol.

SO with that said, lawrence shall be scribe for the next day (which is technically today.) Alright? alright. Night all :]