Question

What is the equation of the line tangent to `f(x)= 12x+4` at `x=7`?

Answer 1

`y=12x+4`

Explanation:

`y=12x+4`.
This is clear from the fact that the function of `f` is linear, however, I will show the proof of it below.

The gradient of the tangent is represented by the derivative.

`f'(x)=12and f'(7)=12`.

So the gradient of the tangent to the curve is `12`.

But, `f(7)=12(7)+4=88`, so the point `(7,88)` lies on the curve and on the tangent.

We may hence substitute it into the general equation of the tangent, which is a straight line, of form `y=mx+c`.

`therefore 88=12(7)+c`

`therefore c=4`.

Hence the required tangent has equation `y=12x+4`.