correct answer of S(0.4) is approximately equal to 0.208.
To determine the value of S(0.4) using a clamped cubic spline, we need to find the cubic spline function that interpolates the given points (0,0), (1,1), and (2,-2), and then evaluate it at x = 0.4.
A clamped cubic spline consists of piece-wise cubic polynomials that are smooth and pass through each data point while maintaining continuous derivatives at the endpoints. In this case, the derivative values at the endpoints are not provided, so we'll assume the derivative values to be zero at both endpoints.
Let's denote the cubic spline function as S(x) = ax^3 + bx^2 + cx + d, where 'a', 'b', 'c', and 'd' are coefficients to be determined.
To find the coefficients, we'll set up a system of equations based on the conditions for the cubic spline:
S(0) = 0
S(1) = 1
S(2) = -2
S'(0) = 0 (Clamped endpoint)
S'(2) = 0 (Clamped endpoint)
Let's solve this system of equations to determine the coefficients 'a', 'b', 'c', and 'd':
Using the conditions:
S(0) = 0: Substitute x = 0: a(0)^3 + b(0)^2 + c(0) + d = 0
This simplifies to d = 0.
S(1) = 1: Substitute x = 1: a(1)^3 + b(1)^2 + c(1) + d = 1
This simplifies to a + b + c = 1.
S(2) = -2: Substitute x = 2: a(2)^3 + b(2)^2 + c(2) + d = -2
This simplifies to 8a + 4b + 2c - 2 = -2.
S'(0) = 0: Differentiate the cubic spline function with respect to x and substitute x = 0:
3a(0)^2 + 2b(0) + c = 0
This simplifies to c = 0.
S'(2) = 0: Differentiate the cubic spline function with respect to x and substitute x = 2:
3a(2)^2 + 2b(2) + c = 0
This simplifies to 12a + 4b = 0.
Using equations 2 and 4, we can solve for 'a' and 'b':
a + b + 0 = 1 (from equation 2)
12a + 4b = 0 (from equation 5)
From equation 2, we can rewrite it as b = 1 - a. Substituting this into equation 5:
12a + 4(1 - a) = 0
12a + 4 - 4a = 0
8a + 4 = 0
8a = -4
a = -4/8
a = -1/2
Substituting the value of 'a' back into b = 1 - a:
b = 1 - (-1/2)
b = 1 + 1/2
b = 3/2
We have determined the values of 'a' and 'b'. Since c = 0 and d = 0, the cubic spline function is:
S(x) = (-1/2)x^3 + (3/2)x^2 + 0 + 0
Now, let's evaluate S(0.4) using the derived cubic spline function:
S(0.4) = (-1/2)(0.4)^3 + (3/2)(0.4)^2
S(0.4) = (-1/2)(0.064) + (3/2)(0.16)
S(0.4) = -0.032 + 0.24
S(0.4) = 0.208
Therefore, S(0.4) is approximately equal to 0.208.