The radius of curvature of $y = x ^ { 3 } - 2 x$ at (2, 4) is

A)1
B) $\frac { 1 } { 2 \sqrt { 2 } }$
C) $\frac { 2 } { 5 \sqrt { 5 } }$
D) $\frac { 101 \sqrt { 101 } } { 12 }$
E) $\frac { 7 } { 125 }$

Question

The radius of curvature of $y = x ^ { 3 } - 2 x$ at (2, 4) is

A)1
B) $\frac { 1 } { 2 \sqrt { 2 } }$
C) $\frac { 2 } { 5 \sqrt { 5 } }$
D) $\frac { 101 \sqrt { 101 } } { 12 }$
E) $\frac { 7 } { 125 }$

Quizplus · Accepted Answer

The radius of curvature $ R $ is given by $ R = \frac{(1 + (y')^2)^{3/2}}{|y''|} $. For $ y = x^3 - 2x $, $ y' = 3x^2 - 2 $ and $ y'' = 6x $. At $ x = 2 $, $ y' = 10 $ and $ y'' = 12 $. Plugging these into the formula gives $ R = \frac{(1 + 10^2)^{3/2}}{12} = \frac{101\sqrt{101}}{12} $.

The Radius of Curvature Of y=x3−2xy = x ^ { 3 } - 2 xy=x3−2x

The Radius of Curvature Of $y = x ^ { 3 } - 2 x$