Fill in the missing values in the table.
| x | y | first derived | second derived |
| 0 | 1 | 3.05 | |
| 0.2 | 1.494 | 3.08 | 1.625 |
| 0.4 | 2.11 | 3.405 | 0.925 |
| 0.6 | 2.791 | 3.59 | 0.575 |
| 0.8 | 3.509 | 3.705 | |
| 1 | 3.785 | ||
| 1.2 | 5.007 |
Using the definition for the average rate of change: (f(x+h) - f(x))/h, we can find the missing values in the table.
To find the first derived for when x = 0, we can say (f(x+0.2) - f(x))/0.2, where x = 0. Therefore (f(0.2) - f(0))/0.2 = (1.494 - 1)/0.2 = 2.47.
To find the value of y for when x = 1, we can say (f(x+0.2) - f(x))/0.2 = 3.785, where x = 1. Therefore (f(1.2) - f(1))/0.2 = 3.785 => f(1) = y = 4.25.
To find the second derived for when x = 0.8, we can say (f'(x+0.2) - f'(x))/0.2, where x = 0.8 and f' represents the first derived. Therefore (f'(1) - f'(0.8))/0.2 = (3.785-3.705)/0.2 = 0.4.
| x | y | first derived | second derived |
| 0 | 1 | 2.47 | 3.05 |
| 0.2 | 1.494 | 3.08 | 1.625 |
| 0.4 | 2.11 | 3.405 | 0.925 |
| 0.6 | 2.791 | 3.59 | 0.575 |
| 0.8 | 3.509 | 3.705 | 0.4 |
| 1 | 4.25 | 3.785 | |
| 1.2 | 5.007 |
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