For the data points (1,,1/,1/,1/4), finde the natural cubic spline.

y = -1/24 x³ + 5/12 x² − 35/24 x + 25/12

Explanation:

A cubic has the form:

y = ax³ + bx² + cx + d

Given four points, we can write a system of equations:

1 = a + b + c + d

1/2 = 8a + 4b + 2c + d

1/3 = 27a + 9b + 3c + d

1/4 = 64a + 16b + 4c + d

Solving this algebraically would be time-consuming, but we can use matrices to make it easy.

First, we find the inverse of the coefficient matrix.  This is messy to do by hand, so let's use a calculator:

Now we multiply by the solution matrix (again using a calculator):

So the cubic is:

y = -1/24 x³ + 5/12 x² − 35/24 x + 25/12

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