Given that y=x^(5/2)- x^(-1/2) .

Question

a) Find the coordinates where the tangent line is parallel to the line y-4x-4=0.

b) Find the coordinates where the tangent line is perpendicular to the line 2y-3x-12=0.

Answer 1

First find the slope of the tangent line = y'

y' = 5/2 * x^(3/2) + 1/2 * x^(-3/2)

The line in question has slope m=4

So, you want y'=4

5/2 * x^(3/2) + 1/2 * x^(-3/2) = 4
let u = x^(3/2) We can rewrite that as

5u/2 + 1/2u = 4

Multiply by 2u

5u^2 + 1 = 8u
5u^2 - 8u + 1 = 0
u = 1.46332 or 0.13667
so, x = u^(2/3) = 1.2889 or .26532
Evaluate y for those values of x to get the actual points.

for the perpendicular line, the slope m = -2/3. Do the same steps.