Question

The average annual per capita consumption of fuel, in gallons, for a forty-year period is shown.

Year 0 5 10 15 20 25 30 35 40
Fuel Consumed (gallons) 654 711 712 675 619 563 527 527 584
What can be said about the finite differences of this data?

a. The third-order differences are almost non-zero and constant.
b. The second-order differences are almost non-zero and constant.
c. The first-order differences are almost non-zero and constant.
d. The fourth-order differences are almost non-zero and constant.

Answer 1

After computing the third-order finite differences for the given data, it is observed that these differences are close to being constant, leading to the conclusion that the third-order differences are almost non-zero and constant.

Step-by-step explanation:

Finite Differences:

To analyze the finite differences of the given data, we calculate the successive differences between fuel consumption values over the 40-year period. The first-order differences are the changes between each pair of consecutive years, the second-order are the differences between first-order differences, and so forth.

First-order differences: 57, 1, -37, -56, -56, -36, 0, 57

Second-order differences: -56, -38, -19, 0, 20, 36, 57

Third-order differences: 18, 19, 19, 20, 16, 21

Upon observing the third-order differences, we notice they are close to a constant value (almost constant), which can be interpreted as an approximately linear change in the second-order differences. Hence, the correct answer to the student's question is (a) The third-order differences are almost non-zero and constant.