Question

Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=3 x-2, \quad x=a $$

Short answer

The curvature is \(0\) and the radius of curvature is not defined for \(x=a\).

Step-by-step solution

Find the first derivative

The first derivative of a function \(f(x)\) is given by \(f'(x)\). For the function \(y = 3x - 2\), the first derivative \(y' = 3\).

Find the second derivative

The second derivative of a function \(f(x)\) is given by \(f''(x)\). For the first derivative \(y' = 3\), the second derivative \(y'' = 0\).

Calculate the curvature

Using the formula for curvature \(k = \left|y''\right| / (1 + (y')^2)^{3/2}\), where \(y' = 3\) and \(y'' = 0\), the curvature \(k = \left|0\right| / (1 + 3^2)^{3/2} = 0\).

Calculate the radius of curvature

The radius of curvature is the reciprocal of the curvature i.e. \(r = 1 / k\). Therefore, when \(k = 0\), the curvature is not defined.