Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 13 - Vector Functions - 13.3 Arc Length and Curvature - 13.3 Exercises - Page 908: 21



Work Step by Step

As we are given that $r(t)=t^3j+t^2k$ Write Theorem 10. $\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}$ Find $r'(t)=3t^2j+2tk$ and $r''(t)=6tj+2k$ it yields $|r'(t)|=\sqrt {(3t^2)^2+(2t)^2}=\sqrt {9t^4+4t^2}$ Thus, $\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}=\dfrac{| (3t^2j+2tk)\times(6tj+2k)|}{|\sqrt {9t^4+4t^2}|^3}=\dfrac{6t^2}{|\sqrt {9t^4+4t^2}|^3}$ After simplifications, we get $\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}=\dfrac{6t^2}{(9t^4+4t^2)^{3/2}}$
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