91Ó°ÊÓ

Use Fermat's tangent method to determine the relation between the abscissa \(x\) of a point \(B\) and the subtangent \(t\) that gives the tangent line to \(y=x^{3}\).

Modify Fermat's tangent method to be able to apply it to curves given by equations of the form \(f(x, y)=c\). Begin by noting that if \((x+e, \bar{y})\) is a point on the tangent line near to \((x, y)\), then \(\bar{y}=\frac{t+\varepsilon}{t} y\). Then adequate \(f(x, y)\) to \(f(x+\) \(e, \frac{t+\varepsilon}{t} y\) ). Apply this method to determine the subtangent to the curve \(x^{3}+y^{3}=p x y\)

Show that in modern notation, Fermat's method of finding the subtangent \(t\) to \(y=f(x)\) determines \(t\) as \(t=\) \(f(x) / f^{\prime}(x)\). Show similarly that the modified method of Exercise 7 is equivalent in modern terms to determining \(t\) as \(t=-y(\partial f / \partial y) /(\partial f / \partial x)\)

Use Barrow's \(a\), e method to determine the slope of the tangent line to the curve \(x^{3}+y^{3}=c^{3}\).