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$f(x)=x^{3}-3 x^{2}+5 x-4$
(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.
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(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}
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