Chapter 16

Calculate a power series for \(\sqrt{1+x}\) by applying the square root algorithm to \(1+x\).

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.

Solve \(y^{3}+y-2+x y-x^{3}=0\) for \(y\) as a power series in \(x\). Begin by finding the value of \(y\) when \(x=0\), that is, by solving \(y^{3}+y-2=0\). Since \(y=1\) is a solution, assume that \(y=1+p\) is a solution to the original equation. Substitute this value for \(y\) and get \(1+3 p+3 p^{2}+p^{3}+1+\) \(p-2+x+p x-x^{3}=0\). Removing all terms of degree higher than 1 in \(x\) and \(p\), solve \(4 p+x=0\) to get \(p=-\frac{1}{4} x\). Thus, \(1-(1 / 4) x\) are the first two terms of the desired power series for \(y\). To go further, substitute \(p=-(1 / 4) x+q\) in the equation for \(p\) and continue as before. Show that the next term in the series is \((1 / 64) x^{2}\)

Calculate, using the power series for \(\log (1+x)\), the values of the logarithm of \(1 \pm 0.1,1 \pm 0.2,1 \pm 0.01,1 \pm 0.02\) to eight decimal places. Using the identities presented in the text and others of your own devising, calculate a logarithm table of the integers from 1 to 10 accurate to eight decimal places.

Calculate the relationship of the fluxions in the equation \(x^{3}-a x^{2}+a x y-y^{3}=0\) using multiplication by the progression \(4,3,2,1\). What do you notice? What would happen if you used a different progression?

Find the relationship of the fluxions using Newton's rules for the equation \(y^{2}-a^{2}-x \sqrt{a^{2}-x^{2}}=0\). Put \(z=\) \(x \sqrt{a^{2}-x^{2}}\)

Solve the fluxional equation \(\dot{y} / \dot{x}=2 / x+3-x^{2}\) by first replacing \(x\) by \(x+1\) and then using power series techniques.

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

Check the third value in Newton's integral table (integral 16.3) by showing that the derivative of $$ z=\frac{2 a}{n c}\left(-\frac{2}{15} \frac{b}{c}+\frac{1}{5} x^{n}\right)\left(b+c x^{n}\right)^{3 / 2} $$ is \(y=a x^{2 n-1} \sqrt{b+c x^{n}}\)

Use modern techniques to integrate \(y=\frac{a x^{n-1}}{e+f x^{n}}\) and compare your answer with Newton's answer in integral \(16.5: z=\frac{1}{n} s\), where \(u=x^{n}\) and \(s\) is the area under the hyperbola \(v=\) \(\frac{a}{e+f u}\)

Find the derivative of \(z=\frac{8 a g s-4 a g x v-2 a f x}{4 n e g-n f^{2}}\), where \(u=x^{n}\) and \(s\) is the area under the curve \(v=\sqrt{e+f u+g u^{2}}\). This should equal Newton's value of \(y=\frac{a x^{n-1}}{\sqrt{e+f x^{n}+g x^{2 n}}}\), from integral \(16.6\).

Use a modern table of integrals to find the antiderivative of $$ y=\frac{a x^{2 n-1}}{\sqrt{e+f x^{n}+g x^{2 n}}} $$ Show that your answer is equivalent to Newton's answer in integral 16.7: \(z=\frac{-4 a f s+2 a f u v+4 a e v}{4 \text { neg }-n f^{2}}\), where \(u=x^{n}\) and \(s\) is the area under the curve \(v=\sqrt{e+f u+g u^{2}}\).

Find the ratio of the fluxion of \(x\) to the fluxion of \(1 / x\) using Newton's "synthetic" method of fluxions.

Suppose in a simplified solar system that all planets revolved uniformly in circles with the sun at the center. If the centripetal force is inversely as the square of the radius,| show that the squares of the periodic times of the planets are as the cubes of the radii. (This is a special case of Kepler's third law.)

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.

Show that the sum of the denominators in row \(n\) of the harmonic triangle is given by \(n 2^{n-1}\).

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 rule \(d\left(x^{3}\right)=3 x^{2} d x\) using differentials.

Derive the general power rule \(d\left(x^{n}\right)=n x^{n-1}\) for \(n\) a positive integer 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.

Use the method of Hayes and Ditton to calculate the fluxion of \(y=x^{x} .\) Compare with Exercise 28 .

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?