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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