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

1 Answer
Feb 22, 2017

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!