Question

Using Newton's divided difference method, the second divided difference for the function values f(1)=2, f(2)=3, f(4)=7 is approximately equal to ____.

• A. 1.5
• B. 1
• C. 0.5
• D. 0.33

Hint:

Build the first divided differences f[x0,x1] and f[x1,x2], then combine them over (x2-x0) to get the second divided difference.

Answer 1

The Correct Option is D

Step 1: List the data points: \(x_0=1,\ f(x_0)=2\); \(x_1=2,\ f(x_1)=3\); \(x_2=4,\ f(x_2)=7\).

Step 2: Compute the first divided differences.

\[f[x_0,x_1] = \frac{f(x_1)-f(x_0)}{x_1-x_0} = \frac{3-2}{2-1} = 1\] \[f[x_1,x_2] = \frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{7-3}{4-2} = 2\]

Step 3: Compute the second divided difference using the first divided differences.

\[f[x_0,x_1,x_2] = \frac{f[x_1,x_2]-f[x_0,x_1]}{x_2-x_0} = \frac{2-1}{4-1} = \frac{1}{3}\]

Step 4: Express as a decimal.

\[\boxed{f[x_0,x_1,x_2] = \frac{1}{3} \approx 0.33}\]