Problems:
Natural and clamped cubic splines
What is the difference between natural and clamped Cubic Splines?
Solve the following problems with a clear explanation.
[1] A natural cubic spline S on [0,2] is defined by
S(x) = { S0(x) = 1 + 2*x - x^3 , if 0 <= x <= 1
S(x) = { S1(x) = 2 + b*(x-1) + c*(x-1)^2 + d*(x-1)^3 , if 1 <= x <= 2
Find b, c and d.
[2] A clamped cubic spline S for a function f is defined by
S(x) = { S0(x) = 1 + B*x + 2*x^2 - 2*x^3 , if 0 <= x <= 1
S(x) = { S1(x) = 2 + b*(x-1) - 4*(x-1)^2 + 7*(x-1)^3 , if 1 <= x <= 2
Find f'(0) and f'(2).