Lagrange Numerical Analysis Questions

Problem. Consider the following data from How animals work [1], describing the amount of water stored in a liter of air at various air temperatures. Air temp Water vapor content T. (C) V. (mm H2O per L air) 0.0 4.865 5.0 7.026 15.0 10.021 20.0 18.256 25.0 23.773 40.0 53.301 1. Find the best quadratic Lagrange polynomial to approximate the water content of a liter of air with a temperature of T = 38°C (equal to the body temperature of a human). 2. Find a Lagrange polynomial to interpolate the entire dataset. Hint: To automate the calcu- lation, you should probably program this. 3. Use the Lagrange polynomial from the previous step to once again approximate V(38). 4. The measurement at T = 15.0 was known to be recorded with some error. The true value was actually V = 13.296, repeat steps (2) and (3) with this new data point. 5. The true value of V(38) = 46.0. Compare the estimates of V(38) before and after correcting the error in data. Plot both datasets and polynomials on the same graph and describe any differences. 6. Compute a natural cubic spline to interpolate the original data and corrected data and for each estimate V(38). 7. Compare the performance of the Lagrange polynomials with that of the cubic spline. In particular, how sensitive are the results of each approach to errors in data?