Question

What is the equation of the line tangent to `f(x)=-x^2 + 7x - 1` at `x=-1`?

Answer 1

`y=9x`

Explanation:

The equation of the tangent in `color(blue)"point-slope form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y-y_1=m(x-x_1))color(white)(2/2)|)))`
where m represents the slope and `(x_1,y_1)" a point on the line"`

We require to find m and `(x_1,y_1)`

Given f(x) then `f'(a)=m_"tgt"` where a is the x-coordinate of a point on f(x).

`rArrf'(x)=-2x+7`

and `f'(-1)=-2(-1)+7=9=m_"tgt"`

Substitute x = - 1 into f(x) to obtain coordinates of point on tangent.

`f(-1)=-(-1)^2+7(-1)-1=-9`

Using `m=9" and " (x_1,y_1)=(-1,-9)`

`y+9=9(x+1)rArry+9=9x+9`

`rArry=9x" is the equation of the tangent"`