9 Craig is investigating the gradient of chords of the curve with equation \(\mathrm { f } ( x ) = x - x ^ { 2 }\)
Each chord joins the point \(( 3 , - 6 )\) to the point \(( 3 + h , \mathrm { f } ( 3 + h ) )\)
The table shows some of Craig's results.
| \(x\) | \(\mathrm { f } ( x )\) | \(h\) | \(x + h\) | \(\mathrm { f } ( x + h )\) | Gradient |
| 3 | -6 | 1 | 4 | -12 | -6 |
| 3 | -6 | 0.1 | 3.1 | -6.51 | -5.1 |
| 3 | -6 | 0.01 | | | |
| 3 | -6 | 0.001 | | | |
| 3 | -6 | 0.0001 | | | |
9
- Show how the value - 5.1 has been calculated.
9
- Complete the third row of the table above.
| 9 | State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to 0 |
| |
| [1 mark] |
| 9 | Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) |