By using the forward difference formula, find each missing entry in the following table (2 marks) 2. Suppose we choose integers n=4 and m=4, and partition x∈[2.1,2.5] and y∈[1.2,1.4] with the evenly spaced mesh points x0,x1,…,x4 and y0,y1,…,y4, respectively. Evaluate the following double integral using Composite Simpson's rule. ∫2.12.5∫1.21.4xy2dydx (6 marks) 3. An initial-value problem is defined as follows: y′=cos2t+sin3t,0≤t≤1,y(0)=1. Given step size h=0.25. Find the approximate solution of the above initial-value problem by using the Modified Euler method and the absolute error given that the exact solution is y(t)=2sin2t−cos2t+3. Please keep your calculation in 4 decimal places. (5 marks) 4. An initial-value problem is defined as follows: y′=te3t−2y,0≤t≤1,y(0)=0. Given step size h=0.5. Evaluate the approximate solution of the above initial-value problem by using the Runge-Kutta method of order four. Please keep your calculation in 4 decimal places.
