1.6E: Exercises for Section 1.7

Exercise \(\PageIndex{1}\)

Given data points \( (-1,10), (1,0), (3,6)\) find an interpolating polynomial \(p(x)\) of degree at most 2.

Answer

\(p\left(x\right)=2x^{2}-5x+3\)

Exercise \(\PageIndex{2}\)

Given data points \( (-2,25), (1,1), (2,1), (4,-23)\) find an interpolating polynomial \(p(x)\) of degree at most 3.

Answer

\(p\left(x\right)=-x^{3}+3x^{2}-2x+1\)

Exercise \(\PageIndex{3}\)

Given data points \( (-6,17), (2,1), (4,7)\) find an interpolating polynomial \(p(x)\) of degree at most 2.

Exercise \(\PageIndex{4}\)

Given data points \( (-1,-8), (0,-1), (2,-11), (4,-13)\) find an interpolating polynomial \(p(x)\) of degree at most 3.

Exercise \(\PageIndex{5}\)

Suppose you are given three distinct collinear points and tasked with finding an interpolating polynomial \(p(x)\) of degree at most 2. What must be true about your solution?

Answer

The coefficient of the quadratic term must be \(0\).

Exercise \(\PageIndex{6}\)

Suppose you are given \(n\) distinct points, each with \(y\) component of \(0\) and tasked with finding interpolating polynomial \(p(x)\) of degree at most \(n-1\). What must be true about your solution?

Answer

\(p(x)\) must be the \(0\) polynomial.

Exercise \(\PageIndex{7}\)

Suppose you are given \(n\) distinct points, each with \(y\) component of \(0\) and tasked with finding interpolating polynomial \(p(x)\) of degree at most \(n\). What must be true about your solution?

Exercise \(\PageIndex{8}\)

Suppose you are tasked with interpolating polynomial \(p(x)\) corresponding to a set of data points. The data points contain two distinct points with the same \(x\) components. What must be true about your solution?

Answer

There is no solution as the corresponding system will be an inconsistent system.