Question

We have primarily discussed ways to construct polynomial interpolants but we can also consider the construction of interpolating functions with other forms. For the data (0,2),(0.5,5), and (1,4) :

(a) Find the quadratic polynomial interpolating this data.
(b) Find the function p(x)=a+bcos(πx)+csin(πx), which interpolates the data.

Answer 1

The question involves finding a quadratic polynomial and a trigonometric function that both interpolate a given set of data points. This requires setting up and solving systems of equations to find the coefficients for both forms.

Step-by-step explanation:

The student is asking about constructing interpolating functions for a given set of data points, both as a quadratic polynomial and in a form involving trigonometric functions. To find the quadratic polynomial interpolating the data (0,2), (0.5,5), and (1,4), we solve a system of equations derived from plugging the data points into the general quadratic form f(x) = ax^2 + bx + c. For the data points provided, this gives us a system of three equations which can be solved to find the coefficients a, b, and c.

To find the function p(x) = a + b cos(πx) + c sin(πx) that interpolates the data, we substitute the given data points into this equation and solve for the coefficients a, b, and c. This gives us another system of three equations to solve.