Question

find the radius of curvature of the curve y2=x3+8at the point (-2,0)

Answer 1

Answer:

radius of curvature of the curve = 6

Step-by-step explanation:

find the radius of curvature of the curve y2=x3+8at the point (-2,0)

y² = x³  + 8

2y(dy/dx) = 3x²

=> dy/dx = 3x²/2y

d²y/dx² =  6x/2y  - (3x²/2y²)(dy/dx)

=> d²y/dx²  = 3x/y  - 9x⁴/4y³

R =  (  1  + (dy/dx)²)^(3/2) / (d²y/dx²)

=> R =  ( 1 + (3x²/2y)²)^(3/2) /( 3x/y  - 9x⁴/4y³)

=> R = ( (4y² + 9x⁴)^(3/2)/8y³) /((12xy² - 9x⁴)/4y³)

=> R =  (4y² + 9x⁴)^(3/2) / (2(12xy² - 9x⁴))

point (-2 , 0)  =>  x = -2  , y = 0

=> R = (0 + 9(-2)⁴)^(3/2) /(2 * ( 0 - 9(-2)⁴))

=> R =  (144)^(3/2) / (2 * (-144))

=> R = - (12²)^(3/2) /288

=> R = 12³/288

=> R = 6

radius of curvature of the curve = 6