Final answer:
To find the radius of curvature for the curve y^2 = x^3 + 8 at the point (-2, 0), we use the formula involving the first and second derivatives of y with respect to x and evaluate the derivatives at the given point.
Step-by-step explanation:
To find the radius of curvature for the curve y2 = x3 + 8 at the point (-2, 0), we will need to use the formula for radius of curvature, which is given by:
r = √((1 + (dy/dx)^2)^3) / |d^2y/dx^2|,
where dy/dx is the first derivative of y with respect to x, and d^2y/dx^2 is the second derivative of y with respect to x.
Steps to find the radius of curvature:
Differentiate y2 = x3 + 8 with respect to x to find dy/dx. This gives us 2y * (dy/dx) = 3x2.
Solve the first derivative equation for dy/dx when y = 0 at the point (-2, 0).
Differentiate the equation for dy/dx a second time to find d^2y/dx^2.
Solve the second derivative equation at the point (-2, 0).
Substitute the values of dy/dx and d^2y/dx^2 into the radius of curvature formula and calculate r.
Following the required steps will yield the specific radius of curvature for the given curve at the specified point.