Thursday, October 20, 2005

Finishing up Derivatives

So in class you guys told me how you liked those tutorials. You liked the fact that there was someone talking and that it used movement in the pictures. So I went home, after dealing with my first priorities (*ahem*babysitting*ahem*), I went ahead and googled some flash tutorials. I came up with none. The site where those flash tutorials came from does have a chapter 2, but we have no use for it YET. The ones we really need still says (COMING SOON!). I guess we’ll see it someday, and unfortunately that someday isn’t today. But, the first tutorial in chapter 2 is somewhat useful if you want a more clearer picture of secant lines and tangent lines. It’s on “The Difference Quotient”. You guys can take a look at that.

So I guess pictures will have to do for now. Sorry guys. I hope it helps though. Even though it’s boring.


I’ll start with interpreting derivative graphs:

Parent Function -> First Derivative -> Second Derivative

So looking back, first derivative. The first derivative tells us three things about the parent function. For the following graphs consider the lime green graph the parent function and the red graph the first derivative.

1) Where the parent function is increasing and decreasing.
-------- > How?
Where the first derivative is positively valued (above the x-axis), the parent function is increasing (slope of tangent lines are positive). Where the first derivative is negatively valued (under the x-axis), the parent function is decreasing (slope of tangent lines are negative).


2)Where the parent function has a minimum.
-------- > How?
Where the first derivative has a root (equal to 0), the parent function has a CANDIDATE for a minimum OR a maximum.
-------- > How is the candidate a minimum NOT a maximum?
When it’s negatively valued (under the x-axis) on the left side of the root, and positively valued (above the x-axis) on the right side of the root, it is a minimum.

3) Where the parent function has a maximum.
-------- > How?

Where the first derivative has a root (equal to 0), the parent function has a CANDIDATE for a maximum OR a minimum.
-------- > How is the candidate a maximum NOT a minimum?
When it’s positively valued (above the x-axis) on the left side of the root, and negatively valued (under the x-axis) on the right side of the root, it is a maximum.


So now the second derivative. The derivative of the first derivative. Confusing rights? Hmm not necessarily. If you look at this calculus stuff and not knowing what question you have in mind, maybe then it’ll confuse you. But if you know what you are being asked, you’ll know which part of the “BIG” picture to look at. Let’s get something straight first.

Behind the first derivative and the second derivative is some PARENT function. It’s what you first started out with.

The first derivative tells us where the parent function is increasing or decreasing and where it has local minima and local maxima.

The second derivative tells us where the parent function is concave up or concave down and where it has a points of inflection.

Remember the second derivative is the derivative of the first derivative? Meaning the second derivative also tells us where the first derivative is increasing or decreasing and where it has local minima and maxima.


Now onto the how can you tells.

The Second Derivative

The second derivative also tells us three things about the parent function. (The next following graphs, red graph is the parent function, the lime green graph is the first derivative, and the blue graph is the second derivative.) They are:

1) Where the parent function has a point of inflection.
-------- > How?
Where the second derivative has a root, the parent function has a point of inflection.








2) Where the parent function is concave up.
-------- > How?
Where the second derivative is positively valued (above the x-axis), the parent function is concave up.
3) Where the parent function is concave down.
-------- > How?
Where the second derivative is negatively valued (under the x-axis), the parent function is concave down.





So that pretty much conclude my post on derivatives. I'm not sure if I should post up more on limits and continuity. Tell me if I should. So the pre-test is on Monday and the test is on Tuesday. Don't forget to do your test blogs guys. Wouldn't want to lose a mark on that. I'm thinking of posting up another blog tomorrow. I'm thinking of applying the stuff on derivatives in 2-3 questions. If you have any questions in mind, put in your comments, or just tell me in the shoutbox. Hope those pictures helped and you understand the written stuff up there. Now just apply it to questions. Till tomorrow.


P.S. October 20, 2004 ---> The day I transferred into Calculus last
year. Yups I was 2 chapters late when I got into Calculus. I didn't understand
what the whole derivative stuff were. Much less the pre-cal stuff. I was also
taking Pre-Cal 40 at that time with Ms. Antymniuk. But you know what? I didn`t
let that stop me. Mr. K explained the concepts to me, even though it was
somewhat brief. That whole week, I made notes from the textbook. I learned half
of Pre-Cal 40S from the Cal Textbook in 2 days. The Calculus stuff it took me a
longer time to analyze it all. But I understood LITTLE parts of it. I wrote
the Chapter 2 test the same time the class wrote it. What I got in it, I was
proud of. 63%. You might ask why I would be proud of such a mark?
Considering I started the class when there were already in Chapter 2.6, and
nearing a test. The fact that I ACTUALLY passed, was good enough for me at that
time.


Unlike Sysiphus we are NOT set up TO FAIL, we are set up TO SUCCEED.
It all depends on the effort you put in, and how hard you try.

1 Comments:

At 8:16 PM, October 21, 2005, Blogger Mr. Kuropatwa said...

Awesome post Sarah! You did such a good job of explaining a difficult set of concepts -- and the personal touch of explaining "where you were this time last year" is both courageous on your part and motivational for your classmates.

You're really showing us what learning is all about. I'll bet the process of writing your posts has helped you understand the material much better in your own head.

I also really appreciated the mention of Sysiphus. ;-) Keep up the good work!!

 

Post a Comment

<< Home

site hit counter
Stats