Problem

Fill in the missing numbers in the following derived table.

x

f(x)

1st derived

2nd derived

1
2
3
4
5
6
7
8
9

0
0.347
1.236
2.773
5.029
8.063
11.919
16.636
---

0.347
---
1.537
---
---
3.856
4.717
5.611

0.543
0.647
0.720
0.777
---
---
---

Solution

The values in any column depend on the values in the previous column and in the very first column. The third number in the 1st derived column is found by computing the difference in the third and fourth numbers in the previous column, divided by the difference in the third and fourth values of x:

Conveniently, the differences in the x values are all 1, so we need only be concerned with the numerators when doing our calculations. We will denote the entries in the 1st and 2nd derived columns as F_i and S_j, for i = 1 to 8 and j = 1 to 7.

Starting with the second column, we need to find f(9). We have the first derived difference for x = 8, so we can find f(9):

In the third column, we need the 1st derived difference for x = 2, 4, and 5.

In the fourth column, we need the 2nd derived differences for x = 5, 6, and 7.

Here is the completed table.

x

f(x)

1st derived

2nd derived

1
2
3
4
5
6
7
8
9

0
0.347
1.236
2.773
5.029
8.063
11.919
16.636
22.247

0.347
0.889
1.537
2.256
3.034
3.856
4.717
5.611

0.543
0.647
0.720
0.777
0.822
0.861
0.894