Question

Construct clamped cubic splines for the data from Problem 4 using the additional information below. (The parts are in bijection with the parts from the previous problem) A. f′ (0.8)=2.1691753 and f′ (1.0)=−0.625 B. f′ (0)=2 and f′(0.5)=5.43656 C. f′ (0.1)=−2.8004996 and f′ (0.3)=−2.9734038

Answer 1

To create clamped cubic splines for given data, ensure smoothness by specifying derivatives at endpoints. At x=0.8 and x=1.0, set f′ to 2.1691753 and −0.625, respectively. At x=0 and x=0.5, set f′ to 2 and 5.43656. At x=0.1 and x=0.3, set f′ to −2.8004996 and −2.9734038 for continuity.

Step-by-step explanation:

To construct clamped cubic splines for the given data, ensure the splines are smooth by considering the derivatives at the endpoints.

A. At x=0.8, set f′=2.1691753, and at x=1.0, set f′=−0.625 to ensure continuity.

B. At x=0, set f′=2, and at x=0.5, set f′=5.43656 for smoothness.

C. At x=0.1, set f′=−2.8004996, and at x=0.3, set f′=−2.9734038 to maintain spline continuity.

These steps ensure the clamped cubic splines interpolate the data points smoothly with specified derivatives at the endpoints.