2025-10-24 Quiz 5

1. Suppose you have \(n\) data points \(\left( x_i, y_i \right)\), with all \(x_i\) distinct. Which of the following statements about the degree \(n\) interpolating polynomial (so \(c_0 + c_1 x + \cdots + c_n x^n\)) is true?

  1. The interpolating polynomial is unique.

  2. There are infinitely many such interpolating polynomials.

  3. There is no polynomial of this degree that interpolates all \(n\) points.

Answerb

  1. Which of the following is not a valid reason for choosing piecewise cubic interpolation over piecewise linear interpolation.

  1. Piecewise cubics can yield an interpolating function that has continuous derivatives

  2. Piecewise cubics generally converge to the underlying function faster than linear interpolants when the number of data points is increased

  3. Piecewise cubics are cheaper to compute

  4. Piecewise cubics can predict derivative values better than linear interpolants

Answerc

3. Suppose we are fitting a global polynomial to a large number of noisy datapoints. Create a pair of labeled graphs of two possible polynomial fits, one of which is a good fit and one of which is overfitting to the noise in the data. Write a couple of sentences explaining your graphs. Bonus: Add and caption an example of underfitting.

AnswerSee the discussion on Wikipedia.