Question

What is the equation of the line with slope `3` which is tangent to the curve `f(x)=7x-x^2`?

Answer 1

`y=3x+4`

Explanation:

If `f(x)=7x-x^2`
then the slope (for any general `x` value) is `f'(x)=7-2x`

when the slope is `m=f'(x)=3`
then `7-2x=3`
`color(white)("XX")-2x=-4`
`color(white)("XX")x=2`

If `x=2` then
`color(white)("XXX")f(color(red)(2))=7*(color(red)(2))-color(red)(2)^2=14-4=color(blue)(10)`

and the point on the curve were `f'(x)=3` occurs at `(color(red)(2),color(blue)(10))`

Therefore, for the tangent, we have a slope of `color(green)m=3` and a point `(color(red)(2),color(blue)(10))`

Using the slope-point form of the equation:
`color(white)("XXX")y-color(blue)(10)=color(green)(3)(x-color(red)(2))`

or
`color(white)("XXX")y=3x+4`