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 £© The Geometry of Graphs
In Section we discussed the graph of a function y = f(x) in terms of plotting points
(x, f(x)) for many different values of x and connecting the resulting points with straight
lines. This is a standard procedure when using puter and, if the function is well
behaved and sufficiently many points are plotted, will produce a reasonable picture of the
graph. However, as we noted at that time, this method assumes that the behavior of the
graph between any two essive points is approximated well by a straight line. With a
sufficient number of points and a differentiable function, this assumption will be reasonable.
Yet to understand a graph fully, it is important to have alternative techniques to verify
the picture at least qualitatively. We have already developed several important aids for
understanding the shape of a graph, including techniques for determining the location of
local extreme values and techniques for finding intervals where the function is increasing
and intervals where it is decreasing. In this section we will use this information, along
with additional information contained in the second derivative, to piece together a picture
of the graph of a given function.
2
To see√ the importance of the second derivative, consider the graphs of f(x