Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 709: 4


$ P $ does not lie on the path $ Q $ lies on the path

Work Step by Step

For the point $ P=(4,11,20)$, we have the system $$1+t=4, \quad 2+t^2=11, \quad t^4=20$$ which has no solutions since the first equation gives $ t=3$ which is not a solution for the third equation. Hence, $ P $ does not lie on the given path. For the point $ Q=(-1,6,16)$, we have the system $$1+t=-1, \quad 2+t^2=6, \quad t^4=16$$ which has the solution $ t=2$. Hence, $ Q $ lies on the given path.
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