Find the curvature of $y = x ^ { 7 }$ .
A) $\frac { | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$
B) $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 3 / 2 } }$
C) $\frac { 42 | x | ^ { 5 } } { \left( 1 - 49 x ^ { 12 } \right) ^ { 3 / 2 } }$
D) $\frac { | x | ^ { 5 } } { \left( 1 + x ^ { 12 } \right) ^ { 3 / 2 } }$
E) $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$

Question

Quizplus · Accepted Answer

The curvature of a curve is given by the formula $\kappa = \frac{|y''|}{(1 + y'^2)^{3/2}}$. For the given curve, $y = x^7$, we have $y' = 7x^6$ and $y'' = 42x^5$. Substituting these into the formula, we get $\kappa = \frac{|42x^5|}{(1 + 49x^{12})^{3/2}}$. Since $x$ can be positive or negative, we take the absolute value of $x$ to get the correct curvature.

Find the Curvature Of y=x7y = x ^ { 7 }y=x7

Find the Curvature Of $y = x ^ { 7 }$