Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 11 - Vectors and Vector-Valued Functions - 11.6 Calculus of Vector-Valued Functions - 11.6 Exercises: 30

Answer

$\langle \frac{\sqrt 7}{5},\frac{3}{5},\frac{3}{5}\rangle $

Work Step by Step

$\textbf{r}(t) = \langle f(t), g(t), h(t)\rangle$ $\textbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle$ $\textbf{r}(t) = \langle \sqrt 7e^t,3e^t,3e^t\rangle$ $\textbf{r}'(t) = \langle \sqrt 7e^t,3e^t,3e^t\rangle$ $|\textbf{r}'(t)| = \sqrt {(\sqrt 7e^t)^2 + (3e^t)^2 + (3e^t)^2} = \sqrt {25e^{2t}}$ $\textbf{T}(t) = \frac{\textbf{r}'(t)}{|\textbf{r}'(t)|} = \frac{\langle \sqrt 7e^t,3e^t,3e^t\rangle}{ \sqrt {25e^{2t}}} = 5e^t$ $\textbf{T}(ln(2)) = \frac{\langle \sqrt 7e^{ln(2)},3e^{ln(2)},3e^{ln(2)}\rangle}{ 5e^{ln(2)}} = \langle \frac{\sqrt 7}{5},\frac{3}{5},\frac{3}{5}\rangle $
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