Consider the following statements about the vector-valued function $$\mathbf{r}(t) = \langle x(t), y(t) \rangle$$.

I. The function can represent the position of an object moving in a plane.

II. Its derivative is found by differentiating each component, such that $$\mathbf{r}'(t) = \langle x'(t), y'(t) \rangle$$.

III. Its integral is found by integrating each component.

Which of the statements above are true?