Question

Given the data in the following table, Newton's divided difference interpolation polynomial is
\[ y = b_0 + b_1(x-2) + b_2(x-2)(x-2.5) \]
If the values of \( b_0 \) and \( b_1 \) are 3 and 1, respectively, then the value of \( b_2 \) is

x	2.0	2.5	3.0
y	3.0	3.5	5.0

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

Recall that Newton's divided difference coefficients are the successive divided differences: b0, then b1, then b2.

Answer 1

The Correct Option is A

Step 1: Recall what b0, b1, b2 mean in Newton's divided difference form.
For the polynomial \( y = b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) \), the coefficients are the divided differences: \( b_0 = f[x_0] \), \( b_1 = f[x_0,x_1] \), and \( b_2 = f[x_0,x_1,x_2] \).
Here \( x_0 = 2.0 \), \( x_1 = 2.5 \), \( x_2 = 3.0 \), with y values 3.0, 3.5 and 5.0.

Step 2: Check b0 and b1 against the table.
\[ b_0 = f[x_0] = 3.0 \]
\[ b_1 = f[x_0,x_1] = \frac{f(x_1)-f(x_0)}{x_1-x_0} = \frac{3.5-3.0}{2.5-2.0} = \frac{0.5}{0.5} = 1.0 \]
Both match the values 3 and 1 given in the question, so the table is being read correctly.

Step 3: Find the second divided difference b2.
First get \( f[x_1,x_2] \):
\[ f[x_1,x_2] = \frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{5.0-3.5}{3.0-2.5} = \frac{1.5}{0.5} = 3.0 \]
Now combine it with \( f[x_0,x_1] \) over the full span \( x_2 - x_0 \):
\[ b_2 = f[x_0,x_1,x_2] = \frac{f[x_1,x_2]-f[x_0,x_1]}{x_2-x_0} = \frac{3.0-1.0}{3.0-2.0} = \frac{2.0}{1.0} = 2.0 \]
Option (B) 3 comes from mistakenly using \( f[x_1,x_2] \) alone without subtracting \( f[x_0,x_1] \). Option (C) 4 comes from dividing by \( x_1 - x_0 = 0.5 \) instead of the correct span \( x_2 - x_0 = 1.0 \).

Final Answer:
The value of b2 is 2. \[ \boxed{2} \]