Chapter 5.1, Problem 58E

New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x₀and successively defines a sequence $z_{n+1}=x_n-\frac{f\left(x_n\right)}{f\text{'}\left(x_n\right)}$ . For the given choice of f and x₀. write out the formula for $x_{n+1}$ . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x_nagrees with x up to four decimal places.

58. [T] Suppose you start with one liter of vinegar and repeatedly remove 0.1 L. replace with water, mix, and

repeat.

a. Find a formula for the concentration after n steps.

b. After how many steps does the mixture contain less than 10% vinegar?