history of mathematics - National STEM Centre

Calculus
including Descartes, understood it. Descartes used a similar method described in
Activity 9.5, without realising that it was essentially the same as Fermat's.
Activity 9.4 Fermat and constructing tangents
TO P
Figure 9.6
This activity is about drawing the tangent TM to the curve ax 2 = >> 3 at the point
M(x,y], shown in Figure 9.6. O is the origin.
Fermat's strategy was to find an expression for the length of the line NZ as Z moves
along the curve, and then to use the fact that, for this curve, the length NZ, which is
never negative, takes its minimum value of zero when Z coincides with the point M.
Call the distance PT = 5, and let the distance PQ be e.
1 a Use similar triangles to find an expression for NQ in terms of y, s and e.
b Write down the jc-coordinate of Q, and hence find the y-coordinate of Z.
2 You will find from your answers to question 1 that the expression for
ZN = (QN - QZ) is complicated. Find an expression for QN 3 - QZ 3 instead. Give
your answer in terms of a, s, e and x. Show that the coefficient of e in your
ax
expression is — (3x — Is).
s
3 Fermat now argues that this coefficient has to be zero. If it is not, he says, then
the value of QN 3 - QZ 3 will become negative close to e = 0.
a Find the value of s in terms of x.
b Choose a value of a and draw an example of this curve on your calculator, and
check Fermat's result.
c Use calculus to prove Fermat's result.
The connection between Activities 9.4 and 9.5 is that, in both cases, Fermat
considered a difference, which always had to be positive. Descartes used a similar
method, but directly on the construction of tangents. He also used a subtly different
definition of tangent: a tangent cuts a curve in two coincident points. He thus used a
slightly different diagram for his method, in which he also calculates the length TP.
Activity 9.5 Descartes's approach to tangents
1 a Write down the coordinates of N in terms of x, e and s.
b Write down the relationship between the coordinates of N, remembering that it
now lies on the curve.
c Multiply out the equation you have written down, and write it showing the
coefficients of e, e 1 and e 3
d Divide through by e.
Descartes then argued that, when TM is a tangent, N and M coincide, so e = 0 is a
solution of this equation.
2 Put e = 0 and find an expression for s.