smc-973-20-Find Curve Equation

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)

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`b = -3,\ \ a = -2`

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`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)`