Let $y = 2 x ^ { 2 }$ . Then the curvature K at (1, 2) is

A) $\frac { \sqrt { 2 } } { 13 }$
B) $\frac { \sqrt { 2 } } { 11 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 4 } { 17 \sqrt { 17 } }$
E) $\frac{4}{\sqrt{17}}$

Question

Let $y = 2 x ^ { 2 }$ . Then the curvature K at (1, 2) is

A) $\frac { \sqrt { 2 } } { 13 }$
B) $\frac { \sqrt { 2 } } { 11 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 4 } { 17 \sqrt { 17 } }$
E) $\frac{4}{\sqrt{17}}$

Quizplus · Accepted Answer

The curvature $K$ of a curve defined by $y = f(x)$ at a point can be calculated using the formula $K = \frac{|y''|}{(1 + (y')^2)^{3/2}}$. For $y = 2x^2$, the first derivative $y' = 4x$ and the second derivative $y'' = 4$. At the point (1, 2), $y'(1) = 4$ and $y''(1) = 4$. Substituting these into the curvature formula gives $K = \frac{4}{(1 + (4)^2)^{3/2}} = \frac{4}{17\sqrt{17}}$.

Let y=2x2y = 2 x ^ { 2 }y=2x2 Then the Curvature K at (1, 2) Is

Let $y = 2 x ^ { 2 }$ Then the Curvature K at (1, 2) Is