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Topic: [ap-calculus] Concavity
Replies: 2   Last Post: Nov 12, 2008 11:45 PM

 Messages: [ Previous | Next ]
 BCrombie@AOL.COM Posts: 108 Registered: 12/8/04
Re: [ap-calculus] Concavity
Posted: Nov 12, 2008 11:45 PM

There is an alternative type of definition that helps in the interpretation
of the second order derivative.

The important idea is that we can use the features of polynomials as a
standard to measure the features of other functions. For example the tangent line
can be considered the closest line to a graph. The closest line to the graph
is the best linear approximation to the graph at the point of tangency. Being
the closest line the direction of the tangent line is also the direction of
the graph. The slope of the tangent line will then tell us whether the graph
is increasing or decreasing at the point of tangency.

While direction characterizes lines, parabolas can be characterized by their
concavity. The tangent parabola is the closest parabola to the graph at a
given point or, again, the best quadratic approximation to the graph. The second
derivative is, within a factor, the coefficient of the x^2 term of the
tangent parabola. That coefficient determines whether the parabola is concave up
or concave down.

More than two hundred year ago Roger Joseph Boscovich made an interesting
observation on this topic.

"We fashion our geometry on the properties of a straight line because that
seems the simplest of all. But in reality all lines that are continuous and of
a uniform nature are just as simple as one another. Another kind of mind
which might form an equally clear mental perception of some property of any one
of these curves, as we do the congruence of a straight line, might believe
these curves to be the simplest of all, and from that property of these curves
build up the elements of a very different geometry, referring all other curves
to that one, just as we compare them to a straight line. Indeed, these
minds, if they noticed and formed an extremely clear perception of some property
of say, the parabola, would not seek, as our geometers do, to rectify the
parabola, they would endeavor, if one may coin the expression, to parabolify the
straight line."

The second derivative "parabolifies" a graph.

Bill Crombie

In a message dated 11/12/2008 9:59:54 A.M. Eastern Standard Time,
jgrunloh@d127.org writes:

I understand why the first derivative test tells you where the function is
increasing or decreasing but why do we use the second derivative test to tell
us where the function is concave up or concave down?

Jill Grunloh

Grayslake North H.S - Mathematics Department
847.986.3100 (ext. 5616)
_jgrunloh@d127.org_ (mailto:jgrunloh@d127.org)
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