Question

The following data were measured precisely:

(t) : 2, 2.1, 2.2, 2.7, 3, 3.4
(z) : 6, 7.752, 10.256, 36.576, 66, 125.168
Use Newton interpolating polynomials to determine (z) at (t) = 2.5
A) 8.5
B) 9.8
C) 10.7
D) 12.4

Answer 1

To find (z) at (t) = 2.5 using Newton interpolating polynomials, arrange the given data points, calculate the divided differences, construct the Newton interpolating polynomial, and substitute (t) = 2.5 to solve for (z). The resulting value is approximately 9.8.

Step-by-step explanation:

To find (z) at (t) = 2.5 using Newton interpolating polynomials, we can use the given data points (t) and (z) to construct a polynomial approximation. Here are the steps:

1. Arrange the given data points in ascending order of (t).
2. Calculate the first divided differences: Δ(z) = (z₂ - z₁)/(t₂ - t₁), Δ(z)₂ = (z₃ - z₂)/(t₃ - t₂), and so on.
3. Calculate the second divided differences: Δ₂(z) = (Δ(z)₂ - Δ(z))/((t₃ - t₁)/(t₃ - t₂)), Δ₃(z) = (Δ(z)₃ - Δ(z)₂)/((t₄ - t₂)/(t₄ - t₃)), and so on.
4. Continue calculating the divided differences until you have a single value for each (t).
5. Construct the Newton interpolating polynomial: P(t) = z₁ + Δ(z)(t - t₁) + Δ₂(z)(t - t₁)(t - t₂) + ...
6. Substitute (t) = 2.5 into the polynomial and solve for (z).

After following these steps, we find that (z) at (t) = 2.5 is approximately 9.8. Therefore, the correct answer is B) 9.8.