Question

What is the slope of the line normal to the tangent line of `f(x) = 2x-4sqrt(x-1)` at `x= 2`?

Answer 1

The slope will be undefined.

Explanation:

Start by finding the y-coordinate of the point of tangency.

`f(2) = 2(2) - 4sqrt(2 - 1)`

`f(2) = 4 - 4`

`f(2) = 0`

Find the derivative of `f(x)`.

`f'(x) = 2 - 4/(2sqrt(x - 1))`

`f'(x) = 2 - 2/sqrt(x - 1)`

Now find the slope of the tangent.

`f'(2) = 2 - 2/sqrt(2 - 1) = 2 - 2/1 = 0`

The normal line is perpendicular to the tangent line. The slope of `0` of the tangent line means the line will be `y = a`, where `a` is a constant. Then the line perpendicular to this will be of the form `x = b`, where the slope is undefined.

Hopefully this helps!