Recall from Lecture the fourth order Runge-Kutta method:

x_{n+1} = x_n + frac{1}{6} left(k_1 + 2k_2 + 2k_3 + k_4 ight)xn+1=xn+61(k1+2k2+2k3+k4)

k_1 = h f(x_n, t_n)k1=hf(xn,tn)

k_2 = h fleft( x_n + frac{1}{2}k_1, t_n + frac{1}{2}h ight)k2=hf(xn+21k1,tn+21h)

k_3 = h fleft( x_n + frac{1}{2}k_2, t_n + frac{1}{2}h ight)k3=hf(xn+21k2,tn+21h)

k_4 = h fleft( x_n + k_3, t_n + h ight)k4=hf(xn+k3,tn+h)

Apply it, with h=1h=1, to the initial value problem of the previous Question to find a (better) approximation to sqrt{2}2(recall sqrt{2} approx 1.414212≈1.41421).

Provide a numeric answer rounded to five decimal places.

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