Exercise · ZW-SECANT-METHOD-010 · ★★★ Advanced

For \(f(x)=x^2-3\) with \(x_0=1\) and \(x_1=2\), compute \(x_2\) and \(x_3\) with the secant method, and check whether the result is closer to \(\sqrt{3}\) than \(x_1\).

Choose one option.

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D. \(x_2\approx1.667\) and \(x_3\approx1.727\), closer than \(x_1\)

Explanation

With f(1)=-2 and f(2)=1, x_2=2-1·1/3=5/3≈1.667; the next step moves toward \(\sqrt{3}\approx1.732\). Taking x_2=2.5 ignores the sign of f(1).

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Review the concept: Secant method

The secant method is an iterative procedure for approximating roots of a function \(f(x)=0\) using two previous approximations, without needing to compute the derivative.

\[x_{n+1}=x_n-f(x_n)\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}\]

For \(f(x)=x^2-3\) with \(x_0=1\) and \(x_1=2\), compute \(x_2\) and \(x_3\) with the secant method, and check whether the result is closer to \(\sqrt{3}\) than \(x_1\).

Explanation

Review the concept: Secant method

More practice problems: secant method