91Ó°ÊÓ

Calculate a power series for \(1 /\left(1-x^{2}\right)\) by using long division.

Use Newton's method to solve the equation \(x^{2}-2=0\) to a result accurate to eight decimal places. How many steps does this take? Compare the efficacy of this method with that of the Chinese square root algorithm.

Find the curvature of the ellipse \(x^{2}+4 y^{2}=1\) by using Newton's procedure.

Construct Leibniz's harmonic triangle by beginning with the harmonic series \(1 / 1,1 / 2,1 / 3,1 / 4, \ldots\) and taking differences. Develop a formula for the elements in this triangle.

Prove the quotient rule \(d\left(\frac{x}{y}\right)=\frac{y d x-x d y}{y^{2}}\) by an argument using differentials.

Derive the power series for the logarithm by beginning with the differential equation \(d y=\frac{1}{x+1} d x\), assuming that \(y\) is a power series in \(x\) with undetermined coefficients, and solving simple equations to determine each coefficient in turn.

Compare and contrast the "calculuses" of Newton and Leibniz in terms of their notation, their ease of use, and their foundations.

Outline a series of lessons on power series using the ideas of Newton. Is it useful to introduce such series early in a calculus course? Why or why not?