Chapter 6.3, Problem 169E

The following exercises make use of the functions $S_5(x)=x+\frac{x^3}{6}+\frac{x^5}{120}$ and $C_4(x)=x-\frac{x^2}{2}+\frac{x^2}{24}$ on $[-\pi ,\pi ]$ .

169. [Taylor approximations and root finding.) Recall that Newton’s method $x_{n+1}=x_n-\frac{f(x_n)}{f\text{'}(x_n)}$ approximates solutions of f(x) = 0 near the input x₀.

a. If f and g are inverse functions, explain why a solution of g(x) = a is the value f(a) of f.

b. Let P_N (x) be the Nth degree Maclaurin polynomial of e^x. Use Newton’s method to approximate solutions of p_N(x) − 2 = 0 for N = 4, 5, 6.

c. Explain why the approximate roots of P_N(x) − 2 = 0 are approximate values of In(2).