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 Question1. (10 points) Use Taylor Expansion to compute the local truncation error for the implicit Trapezoidal method: Y^ k+1 = Y^ k + ∆t/2 (f ^k + f ^k+1)

Question2. (20 points) Suppose certain numerical method for ODE y ′ = f(t, y) has the form: Y^ k+2 = Y^ k+1 + ∆t/12 (−f^ k + 8f^k+1 + 5f^ k+2). (a) Compute the Local Tuncation Error for the scheme and judge if the scheme is consistent (b) Based on the eigenvalue calculation, judge if the scheme is zero-stable (c) Apply the scheme to the standard test problem and plot the region of absolute stability for the scheme. (explain in details how the plot is made and attach the related computer code) 1