Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 13 - Section 13.3 - Arc Length and Curvature - 13.3 Exercise - Page 868: 32



Work Step by Step

In order to find the equation of the parabola we need to find the curvature ,for this we will have to use formula 11, such that $\kappa(x)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}$ Consider $f(x)=ax^2+bx$ [ General equation of a parabola] $f'(x)=2ax+b$ and $f''(x)=2a$ $\kappa(x)=\dfrac{|2a|}{[1+(2ax+b)^2]^{3/2}}$ As we are given that the $ \kappa(0)$ is the curvature at origin. Therefore, $\kappa(0)=\dfrac{|2a|}{[1+b^2]^{3/2}}$ $\implies 4=\dfrac{|2a|}{[1+b^2]^{3/2}}$ or, $a=\pm 2 (1+b^2)^{3/2}$ Hence, the equation of parabola will be: $y=\pm 2 (1+b^2)^{3/2}x^2+bx$ If we take the parabola to have the vertex at the origin, then $b=0$ and $y=2x^2$
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