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 - 14.2 Calculus of Vector-Valued Functions - Exercises - Page 721: 53


\begin{align} r(t)&= \left\langle 1, t^2-6t+10, t-3\right\rangle . \end{align}

Work Step by Step

By integration, we have \begin{align} r'(t)& = \left\langle c_1, 2t+ c_2, c_3 \right\rangle . \end{align} By the condition $r'(3)=\lt 0,0,1\gt$, we get $$0=c_1, \quad -6=c_2, \quad 1=c_3$$ Hence we have \begin{align} r'(t)& = \left\langle 0, 2t-6, 1 \right\rangle . \end{align} Again, by integration we have \begin{align} r(t)& = \left\langle a, t^2-6t+b, t+c \right\rangle . \end{align} By the condition $r(3)=\lt1,1,0\gt$, we get $$1=a, \quad 10=b, \quad -3=c$$ Hence we have \begin{align} r(t)&= \left\langle 1, t^2-6t+10, t-3\right\rangle . \end{align}
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