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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Monday, October 17, 2005

Derivative: a CoNtrovErsY ... bRACKET or pARENTHESIS?

Second Derivative - the derivative of the derivative of a function.

>> sounds confusing right?

Well me too, I'm confused. But to fully understand what a second derivative is and how to make it, I'm gonna help you guys, and myself too by recalling the things we've learned in today's discussion.

First up, we started the day with the usual practice activity on derivatives. Mr. K. asked us to

  • write equations for the line tangent to a function at a given point

- to do this, use any formula for getting an equation that is applicable to the given problem. In our case this morning, since the points and the tangent (which is also the slope) are provided, we used the POINT - SLOPE FORMULA.

  • name the interval where f' is positive, f'' (or second derivative) is positive and determine the critical numbers of f, given the graph of f (...too much f's :))

- this is the part where the controversy began. it is pretty much easy to find the interval where f' is positive because you only have to know where the graph of f is increasing, but to write it down, there we've got a problem. Some of us may probably use "[ ]" than "( )" in giving intervals or points because we consider those "points" part of the increasing graph. But according to Mr.K, it is more preferable to use "( )" for the fact that at those "points", the graph is stationary, that is neither increasing nor decreasing.

- to get the points where f'' is positive, we first have to know the graph of f'. then, looking at f', try to plot its derivative and that becomes f''. So to get its positive points, take a look at the point where the graph of f' is increasing and wWALAHH! there you have the points where f'' is positive.

- lastly for this portion, critical numbers are the points where the tangent or slope is equal to zero

*** the second derivative is just like considering the first derivative your mother or parent function and plotting a derivative on it.

In addition to this, here are some important notes that we took up today:

  • the vertical position of a point has nothing to do with the derivative
  • where the graph of the parent function is concave down, the graph of the second derivative is below x-axis, and vice versa
  • the parent function's increase/decrease tells you about the first derivative, while its concavity tells you about the second derivative
  • to get the maximum and the minimum points, you don't only have to look for the point where f' equals zero, but also where f' is undefined or does not exist

There you go. that's all i can remember from this morning's class.

There'll be more on derivatives tomorrow, as MaryAnn update you guys.

cYa! :)

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  • At 10:33 PM CDT, Blogger Mr. Kuropatwa said…

    An excellent post Ara! You did a good ob summarizing a difficult concept.

    You wrote: critical numbers are the points where the tangent or slope is equal to zero

    Tangents cannot equal zero but the slope of the tangent line (i.e. the derivative) can.

    Also, don't forget that critical numbers can also be where the derivative (slope of the tangent line) does not exit.

    Sarah wrote about this at her new blog. She's got a cool new chatbox too. ;-)


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