Solution: Differentiation of Vector-Valued Functions In Exercises 16, find and for the

Chapter 12, Problem 5

(choose chapter or problem)

Differentiation of Vector-Valued Functions In Exercises 1-6, find r’(t), \(\mathrm{r}\left(t_{0}\right)\), and \(\mathrm{r’}\left(t_{0}\right)\) for the given value of \(t_{0}\). Then sketch the plane curve represented by the vector-valued function, and sketch the vectors \(\mathrm{r}\left(t_{0}\right)\), and \(\mathrm{r’}\left(t_{0}\right)\). Position the vectors such that the initial point of \(\mathrm{r}\left(t_{0}\right)\) is at the origin and the initial point of \(\mathrm{r’}\left(t_{0}\right)\) is at the terminal point of \(\mathrm{r}\left(t_{0}\right)\). What is the relationship between \(\mathrm{r’}\left(t_{0}\right)\) and the curve?

\(\mathbf{r}(t)=\left\langle e^{t}, e^{2 t}\right\rangle, \quad t_{0}=0\)

Text Transcription:

r(t_0)

r’(t_0)

r(t)=<e^t, e^2t>, t_0=0