2) Let's make a similar estimation rule for - If we consider the function f(x) = - a then when f(x) = 0 we have that x =! a) Use Newton's method to give the formula for getting a better approximation given an initial approximation. b) What would be a quick approximation for the following: 1 1 1 1 1 1 52' 23' 98' 103'997' 1006 c) Using those initial approximations use 1 iteration of the rule to get a better approximation.

a and b

T Given f(x) = ax + In(cos(x)). Find the value(s) of a that make the statement f'(- 8 true: a =

19. The figure shows the graph of a function f. Suppose that Newton's method is used to approximate the root r of the equation f(x) = 0 with initial approximation x₁ = 1. (a) Draw the tangent lines that are used to find x₂ and x3, and estimate the numerical values of x2 and x3. (b) Would x₁ = 5 be a better first approximation? Explain. y -1 0 1 r S X