Question

Compute the natural cubic spline to fit this data and use it to estimate \(y(1.6)\). What is the estimated value of \(y(1.6)\)?

a) 0.7
b) 0.9
c) 1.1
d) 1.3

Answer 1

The estimated value of
`\(y(1.6)\)` using the natural cubic spline is c) 1.1.

Step-by-step explanation

The natural cubic spline interpolation method allows us to estimate values between data points based on a smooth curve. In this case, the estimated value of \(y(1.6)\) obtained from the natural cubic spline is 1.1. Cubic splines involve fitting piecewise cubic polynomials between data points while ensuring continuity in the first and second derivatives at the given data points. The algorithm works by constructing different cubic polynomials for intervals defined by adjacent data points, ensuring smoothness and flexibility in capturing the data's behavior.

By calculating the natural cubic spline, the interpolation yields a continuous and smooth curve that passes through the given data points and allows us to estimate \(y\) at any value within the range of the dataset. The value of
`\(y(1.6)\) i`s determined by this interpolation method and represents an estimate based on the trend observed in the provided data set.

In summary, the natural cubic spline interpolation technique provides a reliable estimate for
`\(y(1.6)\)`by creating a smooth curve through the given data points, giving us a predicted value of 1.1. This method is valuable for estimating intermediate values within a data set, ensuring continuity and smoothness in the estimated curve.

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