(a) Derive the Gaussian quadrature formula of the form f f(x) dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to X₁+k = x₁ + fik f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP (t) = f(t,x(t)), x(to) = xo. (c) Use the IVP x (t) = 10 x(t) + 11t-5t²-1,x(0) = 0 to test the validity of the formula on the interval [0, 2] with h=0.25. [Hint: the exact solution is x(t) = (1/2) -1.

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