Graphics Reference
In-Depth Information
8
Derivatives
Tangents
Preliminary reading:
Bryant ch. 4, Courant and John ch. 2.
Concurrent reading:
Hart, Spivak ch. 9.
Further reading:
Tall (1982).
It is easy enough to say when a straight line is a tangent to a circle or
an ellipse. For these curves, a straight line
—
infinitely extended
—
meets
thecurvein 0, 1 or 2 points. Each linewith a uniquepoint of
intersection is a tangent. However if we try to use such a test to identify
tangents to other curves we are in for a disappointment, and on several
counts.
1 At how many points does the line
x
1 intersect the parabola
y
x
? Draw a sketch. Is this line a tangent to the curve?
2 At how many points does the line
y
2 intersect the cubic curve
y
x
3
x
? Draw a sketch. Is this line a tangent to the curve?
From qn 2 we learn that whether a line is a tangent to a curve or
not is a
local
question, which must be asked relative to the particular
point of intersection, that is, inside a suMciently small neighbourhood
of that point.
3 At how many points does the line
y
mx
intersect the cubic curve
y
x
? For how many values of
m
might thelinebea tangent to
thecurv?
Check that the line
y
h
x
is thechord joining thetwo points
(0, 0) and (
h
,
h
0.
If
h
0, to what does the slope (or gradient) of the chord tend?
) on the curve, provided
h
4 Writedown theequation of thelinewith slope
m
through thepoint
(
a
,
f
(
a
)).
