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 712: 45

Answer

Please see the figure attached.

Work Step by Step

We have ${\bf{r}}\left( t \right) = \left( {\left| t \right| + t,\left| t \right| - t} \right)$. So, $x = \left| t \right| + t$ and $y = \left| t \right| - t$. We evaluate several points for the interval $ - 5 \le t \le 5$ and list them in the following table: $\begin{array}{*{20}{c}} t&{\left( {x,y} \right)}\\ { - 5}&{\left( {0,10} \right)}\\ { - 4}&{\left( {0,8} \right)}\\ { - 3}&{\left( {0,6} \right)}\\ { - 2}&{\left( {0,4} \right)}\\ { - 1}&{\left( {0,2} \right)}\\ 0&{\left( {0,0} \right)}\\ 1&{\left( {2,0} \right)}\\ 2&{\left( {4,0} \right)}\\ 3&{\left( {6,0} \right)}\\ 4&{\left( {8,0} \right)}\\ 5&{\left( {10,0} \right)} \end{array}$ Then, we plot the points in rectangular coordinates and join them to get the curve.
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