Exercise · ZW-NEWTON-DIVIDED-DIFFERENCES-010 · ★★★ Advanced
For \(f(x)=\frac{1}{x}\) with nodes \(1,2,4\), what is \(f[x_0,x_1,x_2]\)?
Choose one option.
Show answer
- A. \(\tfrac{1}{8}\)
Explanation
With \(f[1,2]=-\tfrac{1}{2}\) and \(f[2,4]=-\tfrac{1}{8}\), you get \(\frac{-\frac{1}{8}+\frac{1}{2}}{4-1}=\frac{1}{8}\). Dividing by the wrong denominator changes the result.
Review the concept: Newton divided differences
Newton divided differences are recursive quotients of function values at nodes that serve as the coefficients of the interpolating polynomial in Newton form.
\[f[x_i..x_{i+k}]=\frac{f[x_{i+1}..x_{i+k}]-f[x_i..x_{i+k-1}]}{x_{i+k}-x_i}\]
More practice problems: newton divided differences
- ZW-NEWTON-DIVIDED-DIFFERENCES-001 What is f[x_0] if f(x)=x^2 and x_0=3?
- ZW-NEWTON-DIVIDED-DIFFERENCES-002 What is f[x_0,x_1] if f(x)=x^2, x_0=1 and x_1=4?
- ZW-NEWTON-DIVIDED-DIFFERENCES-003 What is the order of the divided difference f[x_0,x_1,x_2,x_3]?
- ZW-NEWTON-DIVIDED-DIFFERENCES-004 If f[x_0,x_1]=4, what is f[x_1,x_0]?
- ZW-NEWTON-DIVIDED-DIFFERENCES-005 If f(x)=5 at every node, what is f[x_0,x_1]?
- ZW-NEWTON-DIVIDED-DIFFERENCES-006 For f(x)=x^2 with nodes 0,1,3, what is f[x_0,x_1,x_2]?
- ZW-NEWTON-DIVIDED-DIFFERENCES-007 For f(x)=x^3 with nodes 1,2,3, what is f[x_0,x_1,x_2]?
- ZW-NEWTON-DIVIDED-DIFFERENCES-008 With f(0)=1, f(1)=3 and f(2)=7, what is p(3) for the Newton interpolating polynomial?