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: 51


\begin{align} r(t)&= \left\langle 1, t, 8 t^2\right\rangle . \end{align}

Work Step by Step

By integration, we have \begin{align} r'(t)&= \left\langle c_1, c_2, 16 t +c_3 \right\rangle . \end{align} By the condition $r'(0)=\lt 0,1,0\gt$, we get $$0=c_1, \quad 1=c_2, \quad 0=c_3$$ Hence we have \begin{align} r'(t)&= \left\langle 0, 1, 16 t \right\rangle . \end{align} Again, by integration we have \begin{align} r(t)&= \left\langle a, t+b, 8 t^2+c \right\rangle . \end{align} By the condition $r(0)=\lt1,0,0\gt$, we get $$1=a, \quad 0=c_2, \quad 0=c$$ Hence we have \begin{align} r(t)&= \left\langle 1, t, 8 t^2\right\rangle . \end{align}
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