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RADAR-
Graphing the Derivative Function REVIEW Math Topic: graphing the derivative
function Key Terms: slopes, tangents The Math behind the Observations: Many of your
observations in experimenting with the Radar applet are characteristic of the
basic relationships between the graph of�
a function and the graph of the derivative function. In the following
we explore these relationships. I.
A Static View of the Radar Applet:
This
screen capture shows the original graph in blue and the correct graph of the derivative in green. How was the graph of the derivative obtained? ����������������������������������������������������������������������������������� ����������� In the next
section we present some of the basic tools and observations needed to arrive
at this correct graph of the derivative function.  II. Background: Sketching the Graph of the Derivative Function
To begin, we
recall two basic facts about the derivative f'(x) of a function f(x): 1. The value f'(a) of f'(x) at x = a is
the slope of the tangent to the graph of the function f at the point where x
= a.
The graph of the
derivative function f'(x) gives us interesting information about the original
function f(x). The following example shows us how to sketch the graph of
f'(x) from a knowledge of the graph of f(x).
Let f(x) have the graph shown below.
Give a rough sketch of the graph of f'(x). Solution Remember that f'(x) is the slope of the tangent at the point (x, f(x)) on the graph of f. To sketch the graph of f', we make a table with several values of x (the corresponding points are shown on the graph) and rough estimates of the slope of the tangent f'(x).
(Note that rough estimates are the best we can do; it is difficult to measure the slope of the tangent accurately without using a grid and a ruler, so we couldn't reasonably expect two people's estimates to agree. However, all that is asked for is a rough sketch of the derivative.) Plotting these points suggests the curve shown below.
Note: 1.� The graph f'(x)
intersects the x-axis at points where the original
function has horizontal tangent lines. 2.� Where f is steep, the values of f'
are large; where f slopes gently, the values of f'
are�� small. ����������� We now return to our Radar applet
to apply these observations.  II.
Observations from the Radar Applet: We return to the screen capture at the
beginning of this review.
What observations were needed to arrive
at the correct graph of the derivative function, f' ? (
Recall the kinds of observations presented in Section II. ) 1.
We first note that the values of f'
must be positive where f is increasing and negative where f is
decreasing. 2.
We also note that where f is steep,
the value of f' is large; likewise, where f slopes
gently, the value of f'� is small. 3.
Finally, we observe that f' crosses the
x-axis at points where f has horizontal
tangents.  IV.� ** Another Site for Graphing f': The following site contains a short movie and description of graphing f': ( See the site-Montana University ) (The movie illustrates this idea. In the first frame we see the graph of a function f(x). As the movie plays we slide a tangent along the curve from left to right and as the tangent slides we draw the graph of the new function y = f'(x) (in blue). ) �����
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