Consider the following data: X f(x) -0.5 5 0 15 0.5 1 3 1.5 1 (4.1)...

 

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Consider the following data: X f(x) -0.5 5 0 15 0.5 1 3 1.5 1 (4.1) Set up a difference table through third differences. (4.2) What is the minimum degree that an interpolating polynomial, that fits all five data points exactly, can have? Explain. (4.3) Give the (forward) Newton-Gregory polynomial that fits the data points with X values 0.5, 1, and 1.5. Then compute f(1.25). (3) (4.4) Compute an approximate bound for the error in the approximation to f(1.25) in (4.3) using Newton's forward interpolating polynomial. (3) (4.5) Compute f(1.25) using the Lagrange interpolating polynomial through the data points with X values 0.5, 1, and 1.5. (3) (4.6) Construct the natural cubic spline for the last four data points. (3) Consider the following data: X f(x) -0.5 5 0 15 0.5 1 3 1.5 1 (4.1) Set up a difference table through third differences. (4.2) What is the minimum degree that an interpolating polynomial, that fits all five data points exactly, can have? Explain. (4.3) Give the (forward) Newton-Gregory polynomial that fits the data points with X values 0.5, 1, and 1.5. Then compute f(1.25). (3) (4.4) Compute an approximate bound for the error in the approximation to f(1.25) in (4.3) using Newton's forward interpolating polynomial. (3) (4.5) Compute f(1.25) using the Lagrange interpolating polynomial through the data points with X values 0.5, 1, and 1.5. (3) (4.6) Construct the natural cubic spline for the last four data points. (3)