Theoretical assignment Consider hyperbolic tangent function tanh(I) = exp(I) - exp(-I) exp(I) + exp(-I) where r E [-2, 2]. Use 5 nodes to uniformly cut the interval [-2, 2] into 4 sub-intervals. 1. Show these 5 nodes (r, tanh(x)) (5 pts) 2. Use these 5 nodes as given data to form the Van der Monde matrix (10 pts.) and find a polynomial to interpolate these 5 nodes (10 pts). The specific procedure of formulating the Van der Monde Matrix is required. 3. Write down the the cardinal functions for these 5 nodes (20 pts), and use the Lagrange polynomials to interpolate these 5 nodes, show the polyno- mial (15 pts). 4. Based on these 5 nodes, use linear spline function to approximate f(I) = tanh(x), r E [-2,2] (25 pts). The specific liner splines for each sub- intervals are required. 5. Based on these 5 nodes, use cubic spline function to approximate f(I) = tanh(r), r E [-2, 2] (25 pts). The specific cubic splines for each sub- intervals are required