Question

\[
f(x)=x^{3}-3 x^{2}+5 x-4
\]
(a) Use differentiation to find \(f^{\prime}(x)\).

The equation \(\mathrm{f}(x)=0\) has a root \(\alpha\) in the interval \(1.4<x<1.5\)

(b) Taking \(1.4\) as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

Answer 1

(a) \(\mathrm{f}^{\prime}(x)=3 x^{2}-6 x+5\)

 

(b)
\[
\begin{aligned}
&\mathrm{f}(1.4)=-0.136 \\
&\mathrm{f}^{\prime}(1.4)=2.48 \\
&x_{0}=1.4, x_{1}=1.4-\frac{-0.136}{2.48} \\
&=1.455(3 \mathrm{dpl})
\end{aligned}
\]