Q9Advanced Engineering Mathematics

The Newton-Raphson algorithm is a rapid root-finding technique employing the iterative formula $x_1=x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$. For the given transcendental function $f(x)=x{e}^x-2$, the analytical derivative is $f^{\prime }(x)=e^x(1+x)$. Starting with an educated initial guess of $x_0=0.8$, we mathematically evaluate both functions and substitute them into the formula to obtain the improved first approximation $x_1\approx 0.855$.