Question

the figure shows the graph of a function f. suppose that newtons method is used to approximate the root s of the equation f(x) = 0 with initial approximation x₁ = 6.
(a) draw the tangent lines that are used to find x₂ and x₃, and estimate the numerical values of x₂ and x₃. (round your answers to one decimal place.)
x₂ = 8
x₃ = 6
(b) would x₁ = 8 be a better first approximation? explain.
x₁ = 8 would be a better first approximation because the tangent line at x = 8 intersects the x - axis closer to s than does the first approximation x₁ = 6.

Explanation:

Step1: Estimate \(x_2\)

By observing the graph, when using \(x_1 = 6\) as the initial approximation, the tangent line at \(x = 6\) intersects the \(x -\)axis at \(x_2\approx8\).

Step2: Estimate \(x_3\)

Using \(x_2 = 8\) as the new approximation, the tangent line at \(x = 8\) intersects the \(x -\)axis at \(x_3\approx6\).

Answer:

\(x_2 = 8\), \(x_3 = 6\)