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Question 12.6: Given that x1 = 7.5 is an approximate root of e^x − 6x³ = 0,......

Given that   x_1  = 7.5 is an approximate root of   e^x-6 x^3=0,   use the Newton–Raphson technique to find an improved value.

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\begin{aligned}x_1 & =7.5 & & \\f(x) & =\mathrm{e}^x-6 x^3 & & f\left(x_1\right)=-723 \\f^{\prime}(x) & =\mathrm{e}^x-18 x^2 & & f^{\prime}\left(x_1\right)=796\end{aligned}

Using the Newton–Raphson technique the value of  x_2  is found:

x_2=x_1-\frac{f\left(x_1\right)}{f^{\prime}\left(x_1\right)}=7.5-\frac{(-723)}{796}=8.41

An improved estimate of the root of   e^x-6 x^3=0  is x = 8.41. To two decimal places the true answer is x = 8.05.

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