Compute the finite differences of y = x3, for x = 1, 1.1, ..., 1.5, naming the first, second, and third differences dy, d2y, and d3y.
|
x |
dx |
y= x3 |
dy |
d2y |
d3y |
|
|
|
|
|
|
|
|
1 |
|
1 |
|
|
|
|
|
0.1 |
|
0.331 |
|
|
|
1.1 |
|
1.331 |
|
0.066 |
|
|
|
0.1 |
|
0.397 |
|
0.006 |
|
1.2 |
|
1.728 |
|
0.072 |
|
|
|
0.1 |
|
0.469 |
|
0.006 |
|
1.3 |
|
2.197 |
|
0.078 |
|
|
|
0.1 |
|
0.547 |
|
0.006 |
|
1.4 |
|
2.744 |
|
0.084 |
|
|
|
0.1 |
|
0.631 |
|
|
|
1.5 |
|
3.375 |
|
|
|
Note: You can compute dy = 0.331 = 1.331-1, etc.
Remark.
You may also construct the table above as follows using lists on the TI-83/84:
|
{1, 1.1, 1.2, 1.3, 1.4, 1.5}3→L1 |
ENTER |
creates a list of y's; |
|
ΔList(L1)→L1 |
ENTER |
creates a list of dy's; |
|
ΔList(L1)→L1 |
ENTER |
creates a list of d2y's; |
|
ΔList(L1)→L1 |
ENTER |
creates a list of d3y's; |
Create a table of the approximate values of derivatives dy/dx, d2y/(dx)2, and d3y/(dx)3, and also the exact values of derivatives y' = 3x2, y'' = 6x, and y''' = 6. Compare the results.
|
x |
dy/dx |
d2y/(dx)2 |
d3y/(dx)3 |
y' |
y'' |
y''' |
|
|
|
|
|
|
|
|
|
1.05 |
.331/.1= 3.31 |
|
|
3.3075a |
|
|
|
1.1 |
|
.066/.01= 6.6 |
|
|
6.6b |
|
|
1.15 |
397/.1= 3.97 |
|
6 =.006/.001 |
3.9675 |
|
6c |
|
1.2 |
|
7.2 |
|
|
7.2 |
|
|
1.25 |
4.69 |
|
6 |
4.6875 |
|
6 |
|
1.3 |
|
7.8 |
|
|
7.8 |
|
|
1.35 |
5.47 |
|
6 |
5.4675 |
|
6 |
|
1.4 |
|
8.4 |
|
|
8.4 |
|
|
1.45 |
6.31 |
|
|
6.3075 |
|
|
Remark.
Compute y' in the same way that you computed y (use lists).
a To compute the y'
column beginning 3.3075, 3{1.05,1.15, 1.25, 1.35, 1.45}2 → L1
b To compute the y'' column beginning 6.6, 6{1.1, 1.2,1.3,1.4}→
L1
cTo
compute the y''' beginning 6, 6{1, 1, 1}→ L1
This example shows why dy/dx, d2y/(dx)2, ..., provide an alternative notation for the derivatives y', y'' ..., of variable y relative to variable x.