The Slope of a Tangent Line
Graphical representation of a phenomenon provides
a very useful visual interpretation of that phenomenon. One very important
feature of any graph is whether it is rising (increasing) or falling (decreasing).
The rise and fall, peaks and troughs of a graph can be studied using the
tangent lines to the graph.
If a tangent has a positive slope the graph
is increasing around the point of tangency.
If a tangent has a negative slope the graph
is decreasing around the point of tangency.
If a tangent is horizontal the graph is often,
but not always, at a peak or trough
at the point of tangency.
We are interested in a graph that is determined by a function. But how
can one obtain a tangent line, and therefore its slope without drawing
a graph? We can calculate one approximately as follows:
Suppose there is a tangent line at a point P = (x, y)
on the graph of a function y = f(x). We take
another point near P, say Q = (x+h,
As we move Q closer to P (i.e. let h approach
zero), the secant line PQ gets closer
to the tangent line at P and the
slope of PQ gets closer to the slope of the tangent.
Now the slope of the secant line PQ is
This is known as a Newton’s quotient.
As h gets nearer to zero (we write ),
the above quotient gives the slope of the tangent. We write this as
and read this as the “limit of Newton’s quotient
as h tends to 0”.
Launch the tangent applet provided
below to investigate the connection between the slope of the secant
and the tangent for the curves
given, or enter equations of your own curves in the box provided.
Drag the red
point to see how the tangent changes along the curve. Drag the green
point to see how the secant line approaches the tangent as the green
dot (Q in our discussion above) approaches the red dot (P in our
discussion above). Not also how each of the slopes of the tangent
and secant lines change (shown in the box in the upper left hand
As the secant line approaches the tangent line, the slope of
the secant line gets closer to the slope of the tangent.
Can't see the above java applet? Click here
to see how to enable Java on your web browser. (This applet is based
on free Java applets from JavaMath
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