Final answer:
The question asks for derivation and plotting of interpolation functions for a given 1D problem using techniques from numerical methods in engineering. It involves mathematical analytical work and utilization of software like Excel or MATLAB to visualize the interpolation. Option C is correct.
Step-by-step explanation:
The question is related to constructing interpolation functions for a 1D problem, which is a topic covered in numerical methods within engineering and mathematics. Given the nature of the task, it involves creating functions n1(x), n2(x), n3(x), n4(x), and n5(x) that would theoretically pass through a set of given data points or function values.
The process typically includes the use of polynomial equations or other basis functions, which can be derived using techniques such as Lagrange interpolation, Newton's divided differences, or splines.
Once the interpolation functions are derived, they can be plotted within a specified range (in this case, x = {-1, 1}) using software such as Excel or MATLAB. Plotting these functions will demonstrate how each interpolation function estimates the curve passing through the data points.
It's important for the interpolation to be accurate, ensuring that the function passes through all the known data points. Furthermore, one can analyze these plots to understand the behavior of the interpolation within the interval and its accuracy.
It's not possible to provide an exact solution without having the specific data points or knowing the method of interpolation being used. However, the student needs to derive the interpolation functions and plot them, as stated in the task. This requires knowledge of numerical methods and access to software for creating the plots.