Exercise · ZW-NEWTON-DIVIDED-DIFFERENCES-009 · ★★★ Advanced
For \(f(x)=x^3\) with nodes \(0,1,2,3\), what is \(f[x_0,x_1,x_2,x_3]\)?
Choose one option.
Show answer
- B. \(1\)
Explanation
With \(f[0,1]=1\), \(f[1,2]=7\) and \(f[2,3]=19\), you get \(f[0,1,2]=3\) and \(f[1,2,3]=6\), so \(\frac{6-3}{3-0}=1\).
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?
← ZW-NEWTON-DIVIDED-DIFFERENCES-008 ZW-NEWTON-DIVIDED-DIFFERENCES-010 →