Differentiation of Vector-Valued Functions In Exercises 16, find and for the given value

Chapter 12, Problem 1

(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?

\(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}, \quad t_{0}=2\)

Text Transcription:

r(t_0)

r’(t_0)

r(t)=t^2i+tj, t_0=2