Question

Compute the first four polynomial interpolations (linear, quadratic, etc.) for the function (x)=1−exp(−x) with evenly spaced data on the interval x∈(0,3). Make a plot of the interpolation and the function itself using MATLAB. Also, make a plot of the difference between the function and the interpolant. You can either work it out by hand or by writing a program. If you work it out by hand, include the values of all the coefficients for each interpolation formula.

Answer 1

The task is to perform the first four polynomial interpolations for the function ℓ(x) = 1 – exp(−x) on the interval (0, 3) using MATLAB to create visual comparisons.

Step-by-step explanation:

The question involves computing polynomial interpolations for the function ℓ(x) = 1 – exp(−x) and analyzing them over the interval x ∈ (0, 3). The task is to calculate the first four interpolations — linear, quadratic, cubic, and quartic — and compare these polynomial models with the given function. We use MATLAB to generate plots of the interpolations and function, as well as plots illustrating the discrepancies between them. Although the coefficients are not provided due to the complexity of calculating them by hand for each polynomial interpolation, one could leverage MATLAB or other numerical software to carry out this task.

Typically, linear interpolation involves fitting a line to two data points, while higher-order interpolations use more data points and fit a polynomial of the corresponding degree. To visualize the fit, one would draw a scatter plot of the original function and overlay the polynomial interpolations. In addition, the difference between the function and the interpolant can be shown in a separate plot, highlighting the approximation errors at different points.