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The equation of the tangent to the curve `y = ae^(2x)+bx` at the point where `x = 0` is `y = 3x +2`.
Find the values of `a` and `b`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`b = -1,\ \ a = 2`
| `y ` | `= ae^(2x)+bx` |
| `(dy)/(dx)` | `= 2ae^(2x)+b` |
`text(When)\ \ x = 0,\ \ (dy)/(dx) = 3`
| `2a + b` | `= 3\ …\ (1)` |
`text(The point)\ (0, 2)\ text(lies on)\ y:`
| `a(1) +b(0)` | `=2` |
| `a` | `= 2\ …\ (2)` |
`text(Substitute into)\ (1)`
| `2(2)+b` | `= 3` |
| `b` | `= -1` |
The equation of the tangent to the curve `y = x^3 + ax^2 + bx + 4` at the point where `x = 2` is `y = x - 4`.
Find the values of `a` and `b`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`b = -3,\ \ a = -2`
| `y ` | `= x^3 + ax^2 + bx + 4` |
| `(dy)/(dx)` | `= 3x^2 + 2ax + b` |
`text(When)\ \ x = 2,\ \ (dy)/(dx) = 1`♦ Mean mark 46%.
| `12 + 4a + b` | `= 1` |
| `4a + b` | `= -11\ …\ (1)` |
`text(The point)\ (2, -2)\ text(lies on)\ y:`
| `8 + 4a + 2b + 4` | `=-2` |
| `4a + 2b` | `= -14\ …\ (2)` |
`text(Subtract)\ \ (2) – (1)`
`b = -3`
`text(Substitute into)\ (1)`
| `4a – 3` | `= -11` |
| `4a` | `= -8` |
| `a` | `= -2` |