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Natural and clamped cubic splines

Name: Solve the given natural and clamped cubic splines problems.
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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).

Solution shows every step methodically while determining the value of unknown constants and boundary values.

Natural and Clamped Cubic Spline are same in all the conditions except two -

Natural Cubic Spline requires that
S0''(x0) = SN_1''(xn) = 0 , N_1 represents the subscript value (n-1)

and Clamped Cubic Spline requires that
S0'(x0) = f'(x0)
SN_1'(xn) = f'(xn) , N_1 represents the subscript value (n-1)

[1] S0'(x) = 2 - 3x^2
S0''(x) = - 6x

S1'(x) = b + 2c(x-1) + 3d(x-1)^2
S1''(x) = 2c + 6d(x-1)

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