Chapter 5.1, Problem 59E

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.

59. [T] A lake initially contains 2000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 6% each month. However, factoring in all causes, 150 fish ate lost each month.

a. Explain why the fish population after ii months is modeled by P_n= 1 .06P _n— — 150 with P₀= 2000.

b. How many fish will be in the pond after one year?