1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
25
As we move from an average rate of change to an instantaneous one, we can think of
one point as “sliding towards” another. In particular, provided the function has a derivative
at (a, f (a)), the point (a + h, f (a + h)) will approach (a, f (a)) as h → 0. Because this
process of taking a limit is a dynamic one, it can be helpful to use computing technology
to visualize what the limit is accomplishing. While there are many different options 3 , one
of the best is a java applet in which the user is able to control the point that is moving.
See the examples referenced in the footnote here, or consider building your own, perhaps
using the fantastic free program Geogebra 4 .
In Figure 1.11, we provide a sequence of figures with several different lines through the
points (a, f (a)) and (a + h, f (a + h)) that are generated by different values of h. These
lines (shown in the first three figures in magenta), are often called secant lines to the curve
y = f (x). A secant line to a curve is simply a line that passes through two points that lie
on the curve. For each such line, the slope of the secant line is m =
f (a+h)− f (a)
h
, where the
value of h depends on the location of the point we choose. We can see in the diagram
how, as h → 0, the secant lines start to approach a single line that passes through the
point (a, f (a)). In the situation where the limit of the slopes of the secant lines exists, we
say that the resulting value is the slope of the tangent line to the curve. This tangent line
(shown in the right-most figure in green) to the graph of y = f (x) at the point (a, f (a)) is
the line through (a, f (a)) whose slope is m = f ′ (a).
x
y
f
a
x
y
f
a
x
y
f
a
x
y
f
a
Figure 1.11: A sequence of secant lines approaching the tangent line to f at (a, f (a)).
As we will see in subsequent study, the existence of the tangent line at x = a is
connected to whether or not the function f looks like a straight line when viewed up close
at (a, f (a)), which can also be seen in Figure 1.12, where we combine the four graphs in
Figure 1.11 into the single one on the left, and then we zoom in on the box centered at
(a, f (a)), with that view expanded on the right (with two of the secant lines omitted). Note
how the tangent line sits relative to the curve y = f (x) at (a, f (a)) and how closely it
3 For a helpful collection of java applets, consider the work of David Austin of Grand Valley State University
at http://gvsu.edu/s/5r, and the particularly relevant example at http://gvsu.edu/s/5s. For applets
that have been built in Geogebra, a nice example is the work of Marc Renault of Shippensburg University at
http://gvsu.edu/s/5p, with the example at http://gvsu.edu/s/5q being especially fitting for our work
in this section. There are scores of other examples posted by other authors on the internet.
4 Available for free download from http://geogebra.org.
25
As we move from an average rate of change to an instantaneous one, we can think of
one point as “sliding towards” another. In particular, provided the function has a derivative
at (a, f (a)), the point (a + h, f (a + h)) will approach (a, f (a)) as h → 0. Because this
process of taking a limit is a dynamic one, it can be helpful to use computing technology
to visualize what the limit is accomplishing. While there are many different options 3 , one
of the best is a java applet in which the user is able to control the point that is moving.
See the examples referenced in the footnote here, or consider building your own, perhaps
using the fantastic free program Geogebra 4 .
In Figure 1.11, we provide a sequence of figures with several different lines through the
points (a, f (a)) and (a + h, f (a + h)) that are generated by different values of h. These
lines (shown in the first three figures in magenta), are often called secant lines to the curve
y = f (x). A secant line to a curve is simply a line that passes through two points that lie
on the curve. For each such line, the slope of the secant line is m =
f (a+h)− f (a)
h
, where the
value of h depends on the location of the point we choose. We can see in the diagram
how, as h → 0, the secant lines start to approach a single line that passes through the
point (a, f (a)). In the situation where the limit of the slopes of the secant lines exists, we
say that the resulting value is the slope of the tangent line to the curve. This tangent line
(shown in the right-most figure in green) to the graph of y = f (x) at the point (a, f (a)) is
the line through (a, f (a)) whose slope is m = f ′ (a).
x
y
f
a
x
y
f
a
x
y
f
a
x
y
f
a
Figure 1.11: A sequence of secant lines approaching the tangent line to f at (a, f (a)).
As we will see in subsequent study, the existence of the tangent line at x = a is
connected to whether or not the function f looks like a straight line when viewed up close
at (a, f (a)), which can also be seen in Figure 1.12, where we combine the four graphs in
Figure 1.11 into the single one on the left, and then we zoom in on the box centered at
(a, f (a)), with that view expanded on the right (with two of the secant lines omitted). Note
how the tangent line sits relative to the curve y = f (x) at (a, f (a)) and how closely it
3 For a helpful collection of java applets, consider the work of David Austin of Grand Valley State University
at http://gvsu.edu/s/5r, and the particularly relevant example at http://gvsu.edu/s/5s. For applets
that have been built in Geogebra, a nice example is the work of Marc Renault of Shippensburg University at
http://gvsu.edu/s/5p, with the example at http://gvsu.edu/s/5q being especially fitting for our work
in this section. There are scores of other examples posted by other authors on the internet.
4 Available for free download from http://geogebra.org.