Chapter 15

Show that the largest parallelepiped that can be inscribed in a sphere is a cube. Determine the dimensions of the cube and its volume if the sphere has radius \(10 .\)

Show that the largest circular cylinder that can be inscribed in a sphere is one in which the ratio of diameter to altitude is \(\sqrt{2}: 1\) (Kepler).

Show that Fermat's two methods of determining a maximum or minimum of a polynomial \(p(x)\) are both equivalent to solving \(p^{\prime}(x)=0\)

Justify Fermat's first method of determining maxima and minima by showing that if \(M\) is a maximum of \(p(x)\), then the polynomial \(p(x)-M\) always has a factor \((x-a)^{2}\), where \(a\) is the value of \(x\) giving the maximum.

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 Fermat's method to determine the subtangent to the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). Compare your answer with that of Apollonius in Chapter 4 .

Use Descartes' circle method to determine the subnormal to \(y=x^{3 / 2}\).

Use Hudde's rule applied to Descartes' method to show that the slope of the tangent line to \(y=x^{n}\) at \(\left(x_{0}, x_{0}^{n}\right)\) is \(n x_{0}^{n-1}\).

Maximize \(3 a x^{3}-b x^{3}-\frac{2 b^{2} a}{3 c} x+a^{2} b\) using Hudde's rule. (This example is taken from Hudde's De maximis et minimis.)

Derive Sluse's rule for the special case \(f(x, y)=g(x)-y\) from Fermat's rule for determining the subtangent to \(y=\) \(g(x) .\) Derive Sluse's general rule from the modification of Fermat's rule discussed in Exercise \(7 .\)

Given that the volume of a cone is \((1 / 3) h A\), where \(h\) is the height and \(A\) the area of the base, use Kepler's method to divide a sphere of radius \(r\) into infinitely many infinitesimal cones of height \(r\), and then add up their volumes to get a formula for the volume of the sphere.

Show that Fermat's rule, $$ N\left(\begin{array}{c} N+k \\ k \end{array}\right)=(k+1)\left(\begin{array}{c} N+k \\ k+1 \end{array}\right) $$ is equivalent to $$ N \sum_{j=k-1}^{N+k-1}\left(\begin{array}{c} j \\ k-1 \end{array}\right)=(k+1) \frac{N(N+1) \cdots(N+k)}{(k+1) !} $$ and also to $$ \sum_{j=1}^{N} \frac{j(j+1) \cdots(j+k-1)}{k !}=\frac{N(N+1) \cdots(N+k)}{(k+1) !} $$

Fermat included the following result in a letter to Roberval dated August 23, 1636: If the parabola with vertex \(A\) and axis \(A D\) is rotated around the line \(B D\) perpendicular to its axis, the volume of this solid has the ratio \(8: 5\) to the volume of the cone of the same base and vertex (Fig. 15.20). Prove that Fermat is correct and show that this result is equivalent to the result on the volume of this same solid discovered by ibn al-Haytham, discussed in Chapter \(9 .\)

Determine the area under the curve \(y=p x^{k}\) from \(x=0\) to \(x=x_{0}\) by dividing the interval \(\left[0, x_{0}\right]\) into an infinite set, of subintervals, beginning from the right with the points \(a_{0}=x_{0}, a_{1}=\frac{n}{m} x_{0}, a_{2}=\left(\frac{n}{m}\right)^{2} x_{0}, \ldots\), where \(n

Using Wallis's method, interpolate the row \(p=3\) in his ratio table for \(n=1 / 2, n=3 / 2\), and \(n=5 / 2\).

Find the length of arc of the curve \(y^{4}=x^{5}\) from \(x=0\) to \(x=b\).

Show that to find the length of an arc of the parabola \(y=\) \(x^{2}\) one needs to determine the area under the hyperbola \(y^{2}-4 x^{2}=1\)

Gregory derived various formulas for calculating the subtangents of curves composed of other curves by addition, subtraction, and the use of proportionals. In particular, suppose that four functions are related by the proportion \(u: v=\) \(w: z\). Show that the subtangent \(t_{z}\) is given by the formula $$ t_{z}=\frac{t_{u} t_{v} t_{w}}{t_{u} t_{v}+t_{u t} t_{w}-t_{v} t_{w}} $$ Derive the product and quotient rules for derivatives from this formula, given that if a function \(u\) is a constant, then its subtangent \(t_{u}\) is infinite.

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

Compare the efficacy of the tangent method of Fermat and the circle method of Descartes to determine the slope of the tangent line to the curve \(y=x^{n}\). Note the kinds of calculations needed in each instance.

Outline a lesson introducing the concept of integration via the method of Fermat applied to curves whose equations are of the form \(y=x^{n}\) for \(n\) a positive integer.

Outline a lesson introducing the determination of arclength using the method of van Heuraet. How does this differ from the method normally presented in calculus texts?